Numerical Control over Complex Analytic Singularities (Memoirs of the American Mathematical Society)

Numerical Control over Complex Analytic Singularities by David B. Massey

For my mother, Mary Alice Massey,

and in memory of my grandparents:

Leslie Ellsworth Porter Mary Frances Porter

William Walter Massey Bessie Ann Massey

PREFACE In 1983, I began work on my dissertation, “Families of Hypersurfaces with One-dimensional Singular Sets”, at Duke University. In that paper, I attempted to describe two numbers which one could effectively calculate from the defining equation of a hypersurface with a one-dimensional singular set – two numbers which control the topology and geometry in a similar fashion to how the Milnor number controls the topology and geometry for isolated hypersurface singularities. Since that time, my work has centered around finding numerical data which “control” various topological and geometric properties of complex analytic singularities. In 1987, while at The University of Notre Dame, I defined the Lˆe cycles and the Lˆe numbers for non-isolated hypersurface singularities. The Lˆe numbers are a generalization of the Milnor number, and they control the singularities; the constancy of the Lˆe numbers in a family implies the constancy of the Milnor fibres in the family, and also implies Thom’s af condition holds. My work on Lˆe numbers from 1987 to the present is contained in my recent monograph, Lˆ e Cycles and Hypersurface Singularities.

In 1988, I came to Northeastern University, and was immediately asked by Terry Gaffney how to generalize the Lˆe numbers of a hypersurface to the case of complete intersections. My answer to this was that I thought the generalization would have two distinct pieces: the first piece should be a method for associating numbers to an arbitrary constructible complex of sheaves on a complex analytic space – one should recover the Lˆe numbers of a hypersurface by applying this new method to the complex of vanishing cycles of the defining equation of the hypersurface. The second piece should be to decide what complex of sheaves should play the role of the sheaf of vanishing cycles in the case of a complete intersection. This first piece – finding a method for associating numbers to a constructible complex of sheaves – is described in my 1994 paper, “Numerical Invariants of Perverse Sheaves”. As the title indicates, a number of results for Lˆe numbers only generalize nicely in the case where the underlying complex is actually a perverse sheaf. After this first piece was completed, it became apparent that the second piece of the generalization was not to replace the sheaf of vanishing cycles by some other complex. Rather, one should continue to use the vanishing cycles a function, but the function could now have an arbitrarily singular domain. This changed the problem to one of finding a sufficiently algebraic characterization of the vanishing cycles – one that actually allows one to effectively produce the numbers that should control the singularities. The first part of such an algebraic description of the vanishing cycles appears in my paper “Hypercohomology of Milnor Fibres”, and the final piece appears in the paper “Critical Points of Functions on Singular Spaces”.

Having completed all of the above pieces, I thought it would be a relatively simple matter to merge them into one coherent whole; thus, I began writing a second book during the 1995-1996 academic year. Little did I suspect that the details, refinements, and corrections would take so long. Even now, I have only just realized that the correct topological treatment should use the v

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micro-local theory of Kashiwara and Schapira. Hence, I could be delayed longer by writing an appendix on micro-local theory, and by rewriting all of my Morse theory proofs in terms of the micro-local theory. As the micro-local theory would add another level of complexity to an already complicated work, I have decided to leave the topological part of this work for a later volume. Thus, what appears in this book is the general algebraic machinery (Vogel cycles and gap cycles), a mildly rewritten version of Lˆ e Cycles and Hypersurface Singularities in terms of this general set-up, the generalization of the Milnor number to functions on arbitrarily singular spaces, the generalization of the Lˆe numbers and cycles to functions on arbitrarily singular spaces, and new generalized Lˆe-Iomdine formulas and results on Thom’s af condition. Moreover, the correct treatment of these topics requires the inclusion of appendices on intersection theory and the derived category. One reason that I feel obligated to include the appendix on intersection theory is to correct the stupidest sentence that I have ever had published. In Lˆ e Cycles and Hypersurface Singularities, I attempted to give a quick summary of the needed intersection theory; I wrote “If we have two irreducible subschemes V and W in an open subset U of some affine space, V and W are said to intersect properly in U provided that codim V ∩ W = codim V + codim W ; when this is the case, the intersection product of [V ] and [W ] is defined by [V ] · [W ] = [V ∩ W ].” This statement is, in general, quite false. Moreover, I knew that it was false, and certainly never used it anywhere in the book; I have no idea how I wrote such a ridiculous thing. Hopefully, the included appendix will eliminate any confusion that I may have caused. Many people have contributed to the results which appear here. But, since I have thanked them in the individual works listed above, I will not give this extensive list here. However, it would be difficult to exaggerate the importance of Terry Gaffney’s and Lˆe D˜ ung Tr´ang’s contributions to this book. Not only have they helped me with or given me many results, but their continuing enthusiasm for my work is an incredible motivating force. I should also thank two former graduate students at Northeastern University, Mike Green and Robert Gassler; conversations with them contributed greatly to my own understanding. Finally, I want to thank some friends who have helped me greatly in ways not directly related to mathematics: General and Mrs. Hannon, Jenn Hannon, John Hannon, and Jed Hannon (the entire Hannon family), Mike Roberts, Tim Roberts, and especially Chad Brazee.

David B. Massey Boston, MA April 26, 2000

TABLE OF CONTENTS

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Part I. Algebraic Preliminaries: Gap Sheaves and Vogel Cycles Chapter 0. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Chapter 1. Gap Sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Chapter 2. Gap Cycles and Vogel Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Chapter 3. The Lˆe-Iomdine-Vogel Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 Chapter 4. Summary of Part I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 Part II. Lˆ e Cycles and Hypersurface Singularities Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 Chapter 1. Definitions and Basic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 Chapter 2. Elementary Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 Chapter 3. A Handle Decomposition of the Milnor Fibre . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 Chapter 4. Generalized Lˆe-Iomdine Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 Chapter 5. Lˆe Numbers and Hyperplane Arrangements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 Chapter 6. Thom’s af Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 Chapter 7. Aligned Singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 Chapter 8. Suspending Singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 Chapter 9. Constancy of the Milnor Fibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 vii

Chapter 10. Another Characterization of the Lˆe Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 Part III. Isolated Critical Points of Functions on Singular Spaces Chapter 0. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 Chapter 1. Critical Avatars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 Chapter 2. The Relative Polar Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 Chapter 3. The Link between the Algebraic and Topological Points of View . . . . . . . . . . . 128 Chapter 4. The Special Case of Perverse Sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 Chapter 5. Thom’s af Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 Chapter 6. Continuous Families of Constructible Complexes . . . . . . . . . . . . . . . . . . . . . . . . . . 147 Part IV. Non-Isolated Critical Points of Functions on Singular Spaces Chapter 0. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 Chapter 1. Lˆe-Vogel Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 Chapter 2. Lˆe-Iomdine Formulas and Thom’s Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 Chapter 3. Lˆe-Vogel Cycles and the Euler Characteristic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 Appendix A. Analytic Cycles and Intersections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 Appendix B. The Derived Category . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 Appendix C. Privileged Neighborhoods and Lifting Milnor Fibrations . . . . . . 221 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243

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ABSTRACT

The Milnor number is a powerful invariant of an isolated, complex, affine hypersurface singularity. It provides data about the local, ambient, topological-type of the hypersurface, and the constancy of the Milnor number throughout a family implies that Thom’s af condition holds and that the local, ambient, topological-type is constant in the family. Much of the usefulness of the Milnor number is due to the fact that it can be effectively calculated in an algebraic manner. The Lˆe cycles and numbers are a generalization of the Milnor number to the setting of complex, affine hypersurface singularities, where the singular set is allowed to be of arbitrary dimension. As with the Milnor number, the Lˆe numbers provide data about the local, ambient, topological-type of the hypersurface, and the constancy of the Lˆe numbers throughout a family implies that Thom’s af condition holds and that the Milnor fibrations are constant throughout the family. Again, much of the usefulness of the Lˆe numbers is due to the fact that they can be effectively calculated in an algebraic manner. In this work, we generalize the Lˆe cycles and numbers to the case of hypersurfaces inside arbitrary analytic spaces. We define the Lˆe-Vogel cycles and numbers, and prove that the Lˆe-Vogel numbers control Thom’s af condition. We also prove a relationship between the Euler characteristic of the Milnor fibre and the Lˆe-Vogel numbers. Moreover, we give examples which show that the Lˆe-Vogel numbers are effectively calculable. In order to define the Lˆe-Vogel cycles and numbers, we require, and include, a great deal of background material on Vogel cycles, analytic intersection theory, and the derived category. Also, to serve as a model case for the Lˆe-Vogel cycles, we recall our earlier work on the Lˆe cycles of an affine hypersurface singularity.

1991 Mathematics Subject Classification. 32B15, 32C35, 32C18, 32B10 Key words and phrases. Gap sheaf, Vogel cycle, Milnor fibre and number, Lˆe cycles and numbers, vanishing cycles, perverse sheaves, Thom’s af condition, Lˆe-Vogel cycles and numbers

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OVERVIEW

The Milnor number of an affine complex analytic hypersurface with an isolated singularity has been a ridiculously successful invariant: it can be effectively calculated, it determines the homotopy-type of the Milnor fibre, and its constancy in a family controls much of the geometry and topology of the family. It is no wonder that there have been myriad attempts to generalize the Milnor number to the cases where the singularity is non-isolated or where the underlying space is arbitrary. From the topological side, one might suspect that the Betti numbers or the Euler characteristic of the Milnor fibre might be reasonable substitutes for the Milnor number. From a differential geometry point-of-view, one can consider various notions of indices of vector fields. From the algebraic side, there are sheaf-theoretic generalizations of the Milnor number. A nice, but by no means complete, expository discussion of the Milnor number and its generalizations is contained in [Te1]. Our own work on generalizing Milnor numbers began with the Lˆe varieties, Lˆe cycles, and Lˆe numbers of a non-isolated affine hypersurface singularity; this work appeared in [Mas6], [Mas8], [Mas9], and [Mas14]. If U is an open subset of Cn+1 , f : U → C is an analytic function, and z := (z0 , . . . , zn ) is a linear choice of coordinates for Cn+1 , then the Lˆe numbers, λ0f,z , λ1f,z , . . . , λnf,z , have a number of very desirable properties. Let s denote the dimension of the critical locus of f at a point p ∈ f −1 (0). Then, the Lˆe numbers, i λf,z , are zero for i > s, and if s = 0, then λ0f,z is precisely the Milnor number. More generally, all of the Lˆe numbers are effectively calculable, and the Milnor fibre has a handle decomposition in which the number of handles attached of a given index is given by the corresponding Lˆe number. The constancy of the Lˆe numbers in a family implies that Thom’s af holds for the total space (the union of the members of the family) and that the Milnor fibrations are constant in the family. All of these properties, and more, are proved in Part II of this book.

There is only one question which is addressed by the results of this book: how does one generalize the Lˆ e numbers of an analytic function to the setting where the underlying space is no longer affine, but, rather, is an arbitrarily singular analytic space?

Obviously, to answer this, we need to consider how the Lˆe numbers are defined. The Lˆe numbers are intersection numbers of the Lˆe cycles with affine linear subspaces defined by the coordinate choice z. Hence, if we could generalize the Lˆe cycles, Λif,z , then we would know how to generalize the Lˆe numbers. The Lˆe cycles are defined by looking at the relative polar varieties ([L-T2], [Te3], and [Te4]) of f , with the correct cycle structure, and using them to give a “decomposition” of the Jacobian ideal of f into a collection of cycles. In a more general setting, this decomposition has been studied by Vogel [Vo] and van Gastel [Gas1], [Gas2]. The first problem, when the underlying space is arbitrary, is that there are three competing definitions for the polar cycles – all of which involve gap sheaves, a notion first studied in [Si-Tr]. Part I of this book discusses these three competing definitions in the general context of decomposing any ideal, not necessarily one related to the Jacobian of a function. In Part I, we prove that the Lˆe-Iomdine formulas, which are such an important result on Lˆe numbers, actually hold in a much more general setting. In Part II, we give all of our previously known results on Lˆe numbers, but give the proofs in 1

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DAVID B. MASSEY

terms of our work in Part I. In Part III, we deal with the second major problem when the ambient space, X, is singular: how does one define an analog of the Milnor number for an isolated critical point? For that matter, what does an “isolated critical point” even mean in this setting? As we shall see, the derived category and the vanishing cycles of the constant sheaf along f will be unavoidable tools for answering these questions. It turns out that if the constant sheaf C•X is perverse (up to a shift), then the whole theory becomes much easier; for instance, this would be the case if X was a local complete intersection. However, for arbitrary spaces, the fact that the complex links of strata can have non-trivial cohomology in more then one degree leads us to take perverse cohomology of the constant sheaf with all possible shifts. In a sense, perverse cohomology lets us decompose the relevant topological data about X into a collection of “positive” and “negative” pieces, and then when we work with these pieces, data from the various strata cannot cancel each other, because they all have the same sign. A result that appears at the end of Part III is a generalization of the result of Lˆe and Saito that the constancy of the Milnor number in a family implies Thom’s af condition [Lˆ e-Sa]; of course, in our theorem, the underlying space is arbitrary. With Parts I and III out of the way, and using Part II as a guide, it is relatively trivial to define our generalization of the Lˆe cycles and numbers in Part IV. We refer to these new gadgets as the Lˆe-Vogel cycles, or LˆeVo cycles for short. There are collections of LˆeVo cycles for various shifts – these shifts correspond to degrees in which the complex links of strata of X have non-trivial cohomology. The indexing on the shifts is set-up in such a way that local complete intersections have non-zero LˆeVo cycles only when the shift is zero. In Part IV, we prove an incredibly general Lˆe-Saito type result and prove a result relating the LˆeVo numbers to the Euler characteristic of the Milnor fibre.

Appendices A and B contain background information (without proofs) on intersection theory and the derived category. The intersection theory that we need is very simple – we need only proper intersections of analytic cycles in affine space; this is what is described in Appendix A. Appendix B contains more information than we really need; it is sort of a working mathematicians guide to the derived category, perverse sheaves, and vanishing cycles. Appendix C contains some extremely technical arguments which are needed in Part II, where we prove that constancy of the Lˆe numbers in a family implies the constancy of the Milnor fibrations. We believe that these arguments would only serve to obstruct the exposition in Part II; hence, we have relegated them to an appendix.

Finally, a word on what is not contained in this book. One will not find most of the extremely topological results on Lˆe numbers extended to the general setting of the LˆeVo numbers. This includes results on handle-decompositions, Morse inequalities, and the constancy of the LˆeVo numbers implying constancy of the Milnor fibrations. While we certainly believe that we can generalize most of these results, the correct treatment appears to require the micro-local of Kashiwara and Schapira [K-S1], [K-S2]. As this would add significant length and complexity to an already long and difficult work, we have elected to place such results in a future book.

Part I. ALGEBRAIC PRELIMINARIES: GAP SHEAVES AND VOGEL CYCLES

Chapter 0. INTRODUCTION Throughout this book, our primary algebraic tool consists of a method for taking a coherent sheaf of ideals and decomposing it into pure-dimensional “pieces”. Actually, we begin with an ordered set of generators for the ideal, and produce a collection of pure-dimensional analytic cycles, the Vogel cycles, which seem to contain a great deal of “geometric” data related to the original ideal. Part I of this book contains the construction of the Vogel cycles; it is, regrettably, very technical in nature. The Vogel cycles are defined using gap sheaves, together with the associated analytic cycles which they define, the gap cycles. A gap sheaf is a formal device which gives a scheme-theoretic meaning to the analytic closure of the difference of an initial scheme and an analytic set. If the underlying space is not Cohen-Macaulay, the main technical problem is that there are, at least, three different reasonable definitions of the gap sheaves and cycles; we select as “the” definition the one that works most nicely in inductive proofs. We show, however, that if one rechooses the functions defining the ideal in a suitably “generic” way, then all competing definitions for the gap cycles and Vogel cycles agree. In Chapter 3 of this part, we prove some extremely general Lˆe-Iomdine-Vogel formulas; as we shall see in later chapters, these formulas are an amazingly effective tool for transforming problems about a given singularity into problems involving a singularity of smaller dimension. The reader who wishes to bypass this technical portion of the book can jump to the Summary of Part I, which begins on page 31.

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Chapter 1. GAP SHEAVES Let W be analytic subset of an analytic space X and let α be a coherent sheaf of ideals in OX . At each point x of V (α), we wish to consider scheme-theoretically those components of V (α) which are not contained in |W |. This leads one to the notion of a gap sheaf. Our primary references for gap sheaves are [Si-Tr] and [Fi]. Let β be a second coherent sheaf of ideals in OX . We write αx for the stalk of α in OX,x .

S Definition 1.1. Let S be the multiplicatively closed set OX,x − p where the union is over all p ∈ Ass(OX,x /αx ) with |V (p)| * |W |. Then, we define αx ¬W to equal S −1 αx ∩ OX,x . Thus, αx ¬W is the ideal in OX,x consisting of the intersection of those (possibly embedded) primary ideals, q, associated to αx such that |V (q)| * |W |. Now, we have defined αx ¬W in each stalk. By [Si-Tr], if we perform this operation simultaneously at all points of V (α), then we obtain a coherent sheaf of ideals called a gap sheaf ; we write this sheaf as α¬W . If β is any coherent sheaf of ideals such that W = supp(OX /β), then α¬W =

∞ [

(α : β k ).

k=0

If V = V (α), we let V ¬W denote the scheme V (α¬W ). It is important to note that the scheme V ¬W does not depend on the structure of W as a scheme, but only as an analytic set. The scheme V ¬W is sometimes referred to as the analytic closure of V − W [Fi, p.41]; this is certainly the correct, intuitive way to think of V ¬W . We find it convenient to extend this gap sheaf notation to the case of analytic sets (reduced schemes) and analytic cycles. Hence, if Z and W are analytic sets, then P we let Z¬W denote the union of the components of Z which are not contained in W ; if C = mi [Vi ] is an analytic cycle in a complex manifold M and W is an analytic subset of M , then we define C¬W by X C¬W = mi [Vi ]. Vi 6⊆W

If α is a coherent sheaf of ideals in OM , C is a cycle in M , and W is an analytic subset of M , then clearly [V (α)¬W ] = [V (α)]¬W and |C¬W | = |C|¬W .

The following properties of gap sheaves are immediate from the definition. Proposition 1.2. i)

αx = βx for all x ∈ X − W if and only if α¬W = β¬W ;

ii)

if αi is a finite collection of coherent ideals in OX , then ∩(αi ¬W ) = (∩αi )¬W ;

iii)

if V (α) ∩ (X − W ) is reduced, then the sheaf of ideals of functions vanishing on the analytic set V (α) ∩ (X − W ) is α¬W .

PART I. ALGEBRAIC PRELIMINARIES

5

Later, the reader may wonder why we do not define something analogous to a gap sheaf, but where we keep those components which are contained in a given analytic set, W , instead of throwing them away. On the level of schemes, we can not make this approach work; the primary ideals in a primary decomposition (of a given ideal) which define varieties contained in W would not be independent of the decomposition. We could just take the isolated primary ideals which define varieties contained in W , but this disposes of too much algebraic structure. Similarly, we could consider V ¬(V ¬W ), which would not dispose of all embedded components, but would eliminate embedded components contained in both W and V ¬W . However, even this device would not aid us much later; as we shall see – beginning with Definition 2.14 – we need to deal more with the intersection product on analytic cycles, and not so much with primary decompositions. The following lemma is very useful for calculating V ¬W . Lemma 1.3. Let (X, OX ) be an analytic space, let α, β, and γ be coherent sheaves of ideals in OX , let f, g ∈ OX , and let W , Y , and Z be analytic subsets of X such that Z ⊆ W . Then, i)

α¬W = (α¬Z)¬W , and thus, as schemes, V (α)¬W = (V (α)¬Z)¬W ;

ii)

(α + β)¬W = (α¬Z + β)¬W , and thus, as schemes,
V (α) ∩ V (β) ¬W =

iii)


if V (α + γ) ⊆ W , then (α ∩ β) + γ ¬W = (β + γ)¬W , and thus, as schemes, 

iv)


V (α) ∪ V (β) ∩ V (γ) ¬W =


V (β) ∩ V (γ) ¬W ;

if V (α+ < g >) ⊆ W , then (α+ < f g >)¬W = (α+ < f >)¬W , and thus, as schemes,
V (α) ∩ V (f g) ¬W =

v)


V (α¬Z) ∩ V (β) ¬W ;


V (α) ∩ V (f ) ¬W.

α¬(W ∪ Y ) = (α¬W )¬Y , and thus, as schemes, V (α)¬(W ∪ Y ) = (V (α)¬W )¬Y.

The analog of ii) for sets and cycles is also trivial to verify; that is,  

|V (α)| ∩ |V (β)| ¬W = |V (α)|¬Z ∩ |V (β)| ¬W, and, if all intersections are proper,  

[V (α)] · [V (β)] ¬W = [V (α)]¬Z · [V (β)] ¬W.

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Proof. Statements i), ii), iii), and iv) are merely exercises in localization (see [Mas3]). Statement v) is trivial. 

Remark 1.4. While it is a trivial observation, it is frequently important and useful to note that, for any coherent sheaf of ideals, α, in OX and for any f ∈ OX , V (α)¬V (f ) and V (f ) intersect properly and V (f ) contains no embedded subvarieties of V (α)¬V (f ); thus, the intersection product cycle [V (α)¬V (f )] · [V (f )] in X is well-defined (without having to mention an ambient manifold) and is equal to [V (< α ¬ V (f ) > + < f >)]. If V (α) and V (f ) intersect properly, then [V (α)] = [V (α)¬V (f )] and, hence, [V (α)] · [V (f )] = [V (α)¬V (f )] · [V (f )] = [V (< α ¬ V (f ) > + < f >)].

Lemma 1.5. Let X be purely d-dimensional and Cohen-Macaulay. Let f1 , . . . , fk ∈ OX and let W be an analytic subset of X. If V (f1 , . . . , fk ) ¬ W is purely (d − k)-dimensional, then it contains no embedded subvarieties. Proof. By definition, V (f1 , . . . , fk ) ¬ W can not have any embedded subvarieties contained in W . At points, p, outside of W , f1 , . . . , fk determines a regular sequence in the Cohen-Macaulay ring OX,p ; hence, there are no embedded subvarieties outside of W . 

Example 1.6. For the remainder of this chapter, we wish to describe the blow-up of a space along an ideal; the description via gap sheaves is very nice. Let (X, OX ) be an analytic space, and let f := (f0 , . . . , fk ) be an ordered (k+1)-tuple of elements of OX . Then, the blow-up of X along f consists of an analytic subspace Blf X ⊆ X × Pk , together with the projection morphism π : Blf X → X, which is the restriction of the standard projection from X × Pk to X. If we use [w0 : · · · : wk ] for homogeneous coordinates on Pk , then the blow-up is given as a scheme by

Blf X := V {wi fj − wj fi } 06i,j6k ¬ V (f0 , . . . , fk ) × Pk . In order to describe the exceptional divisor as a cycle, we need to work on affine coordinate patches in Pk . We shall describe both the blow-up and the exceptional divisor on each affine patch {wj 6= 0}. On the patch {wj 6= 0}, we use coordinates w ei := wi /wj for all i 6= j. Then,
(1.7) {wj 6= 0} ∩ Blf X = V ({fi − w ei fj }i6=j ) ¬ V (fj ) × Pk , and the exceptional divisor, E, is the cycle defined on each affine patch in the following manner 

   (1.8) {wj 6= 0} ∩ E := V {fi − w ei fj }i6=j ¬ V (fj ) × Pk · V (fj ) × Pk . We have made these definitions with respect to a chosen (k + 1)-tuple f . In fact, the analytic isomorphism-type of the morphism π : Blf X → X only depends on the ideal, I, generated by the components f0 , . . . , fk ; this isomorphism-type is referred to as the blow-up of X along I. Of course, the isomorphism-type of the exceptional divisor also depends only on the ideal I, and this isomorphism-type is simply called the the exceptional divisor of the blow-up of X along I.

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Chapter 2. GAP CYCLES AND VOGEL CYCLES Let X be a d-dimensional analytic space and let f := (f0 , . . . , fk ) be an ordered (k + 1)-tuple of elements of OX . We will define a sequence of cycles, the Vogel cycles ([Vo], [Gas1], [Gas2]) of f ; these cycles provide effectively calculable data about the coherent sheaf of ideals < f0 , . . . , fk >. Before we can define the Vogel cycles, we must first define the gap varieties and gap cycles of f . It will prove useful (in Part IV) to define gap and Vogel P objects with respect to a given cycle. Hence, throughout Part I, we let M denote the cycle l ml [Vl ] in X; we assume that this is a minimal presentation of M – that is, we assume that the Vl are distinct, irreducible analytic subsets of X and that none of the ml equal zero. In addition, to avoid cancellation of contributions from various Vl , we assume that all of the ml have the same sign, i.e., that ±M > 0. If X is a union of irreducible components {Xi }, we will define the gap and Vogel cycles in X as sums of the gap and Vogel cycles from each Xi ; similarly, we will define gap and Vogel cycles with respect to M simply by taking weighted sums of the gap and Vogel cycles of the irreducible components. The case of an irreducible space X can be recovered from the cycle case by simply taking M = [X]. Thus, we find that we need to first define the gap varieties, gap cycles, and Vogel cycles in the case where X is irreducible. However, even if we assume that the underlying space is irreducible, there is a further complication in the general setting: OX may not be Cohen-Macaulay. This causes numerous problems, for we must worry about embedded subvarieties. To deal with this problem, we introduce three avatars of gap varieties and examine the relations them.  between  i e , and the We will define the (ordinary) gap varieties, Πif i , the modified gap varieties, Π f i  i b inductive gap varieties, Πf i . We shall use the inductive gap varieties to define the Vogel cycles, but need to make assumptions about the (ordinary) gap varieties in order for the definition to make sense; the modified gap varieties are merely a convenient tool for proving results about Πif bi . and Π f Definition 2.1. Assume that X is irreducible (though, not necessarily reduced). For all i, we define the gap varieties, the modified gap varieties, and the inductive gap varieties of f , which we e i , and Π b i , respectively. denote by Πi , Π f

f

f

ei = Π b i = ∅. First, if i < d − (k + 1) or i > d, we set Πif = Π f f ed = Π b d := X¬V (fk ). We define Πdf := X¬V (f ), and Π f f For d − (k + 1) 6 i < d, the i-th gap variety of f , Πif , is defined as Πif := V (fi+k+1−d , . . . , fk ) ¬ V (f ). d−(k+1)

Note that Πf

= ∅.

e i , is defined as For d − (k + 1) < i < d, the i-th modified gap variety of f , Π f e i := V (fi+k+1−d , . . . , fk ) ¬ V (fi+k−d ); Π f e d−(k+1) := ∅. we define Π f

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DAVID B. MASSEY

b i , is defined by downward induction For d − (k + 1) < i < d, the i-th inductive gap variety of f , Π f b d is defined above) (recall that Π f   b i := Π b i+1 ∩ V (fi+k+1−d ) ¬ V (fi+k−d ); Π f f b d−(k+1) := ∅. we define Π f Naturally, we define the i-th gap cycle, modified  i i-th  gap  cycle,  and  inductive i-th gap cycle of f i i e b to be the cycles defined by these schemes, i.e., Πf , Πf , and Πf , respectively. k+1

If X is a union of irreducible components {Xj } and f ∈ (OX ) , then we define the i-th gap     i  i  P  e := P Π e cycle of f by Πif := j Πif| , the i-th modified gap cycle of f by Π f f|X , and the j Xj j  i  i  P b := b i-th inductive gap cycle of f by Π Π . f f| j Xj

We the i-th inductive gap set of f to  define the i-th gap  set of f , the i-th modified gap set of f , and i i i i   e i  bi be Πf , Πf , and Πf , respectively. We will write simply Πf , e Πf , and b Πf , respectively. Finally, we need to define gap cycles and sets  to the cycle M . We define the i-th Pwith respect gap cycle of f with respect to M by Πif (M ) := l ml Πif| , the i-th modified gap cycle of f with Vl  i  P i e e respect to M by Πf (M ) := l ml Πf| , and the i-th inductive gap cycle of f with respect to M Vl  i  b i (M ) := P ml Π b by Π . f f| l Vl

Of course, we define the associated gap sets with respect to M to be the sets underlying the various gap cycles. Note that we have not defined gap varieties for f unless X is irreducible. The following proposition gives a number of basic results and interrelationships between the various gap varieties. Proposition 2.2. Let X be irreducible. Then, i i i i i) there is an inclusion of sets b Πf ⊆ e Πf ⊆ Πf , b Πf is purely i-dimensional, and all components i i e of Πf and Πf have dimension at least i;
ii) there is an equality of schemes Πif = Πi+1 ∩ V (fi+k+1−d ) ¬V (f ); f i i−1 i ⊆ Π and b Πf ⊆ b Πf ; iii) if i 6 d, then Πi−1 f f i+1 iv) the sets V (fi+k+1−d ) and b Πf intersect properly, and there is an equality of cycles  i   i+1    bf = Π b Π ¬ V (fi+k−d ); · V (f ) i+k+1−d f i i e i and Πi are equal up to embedded v) if there is an equality of sets e Πf = Πf , then the schemes Π f  i  i f e = Π ; subvariety, and so there is an equality of cycles Π f f j j  i  i b 6 Π ; vi) if there is an equality of sets b Πf = Πf for all j > i + 1, then Π f f

PART I. ALGEBRAIC PRELIMINARIES

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b i+1 , then there is an equality of schemes Π ei = Π bi . vii) if there is an equality of schemes Πi+1 =Π f f f f Proof. i) is obvious from the definitions. ii) follows immediately from Lemma 1.3.ii (using V (f ) for both Z and W ). v) is immediate. i i−1 i ⊆ Π . That b Proof of iii): ii) implies Πi−1 Πf ⊆ b Πf follows from the inductive definition. f f b i+1 has no components or embedded subvarieties contained in Proof of iv): By definition, Π f  i+1   i+1    b b V (fi+k+1−d ). Thus, Π ∩ V (fi+k+1−d ) = Π · V (fi+k+1−d ) . The desired conclusion f f follows.  d   b 6 Πd , since they are, in fact, Proof of vi): By downward induction on i. Note first that Π f jf j equal. Suppose now that i < d and that there is an equality of sets b Πf = Πf for all j > i + 1.  i+1   i+1  b From induction, we know that Π 6 Πf . Thus, f  i   i+1    Πf = Πf ∩ V (fi+k+1−d ) ¬ V (f ) > Πi+1 ∩ V (fi+k+1−d ) ¬ V (fi+k−d ). f intersects V (fi+k+1−d ) properly, we may apply [Fu, 8.2.a] to Since iv) tells us that Πi+1 f  i+1      conclude that Πf ∩ V (fi+k+1−d ) > Πi+1 · V (fi+k+1−d ) (the presence of embedded varieties f in Πi+1 can cause a strict inequality). Therefore, f  i   i+1    Πf > Πf · V (fi+k+1−d ) ¬ V (fi+k−d ).  i  i b 6 Π . Now, applying our inductive hypothesis and iv), we conclude that Π f f Proof of vii): We have

bi = Π b i+1 ∩ V (fi+k+1−d ) ¬ V (fi+k−d ) = Πi+1 ∩ V (fi+k+1−d ) ¬ V (fi+k−d ). Π f f f  
By 1.3.i, this equals Πi+1 ∩ V (f ) ¬ V (f ) ¬ V (fi+k−d ). By ii) of this proposition, this i+k+1−d f
i last expression equals Πf ¬ V (fi+k−d ) = V (fi+k+1−d , . . . , fk ) ¬ V (f ) ¬ V (fi+k−d ). Applying b i = V (fi+k+1−d , . . . , fk ) ¬ V (fi+k−d ) = Π ei .  1.3.i again, we find that Π f f

i Note that 2.2.i implies that e Πf and Πif are purely i-dimensional if and only if they are i-dimensional.

We wish to define the Vogel cycles now. However, before we can do this, we need to decide which of the different gap cycles to use to define the Vogel cycles. As a preliminary step, we first define the sets which will underlie the Vogel cycles. Definition 2.3. Assume that X is irreducible. If i 6= d, then we define the i-th Vogel set of f , Dfi , to be the union of the irreducible components of Πi+1 ∩ V (fi+k+1−d ) which are contained in f V (f ); by 2.2.ii, this is equivalent to ∩ V (fi+k+1−d ) − Πif . Dfi = Πi+1 f

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DAVID B. MASSEY

We set

Dfd

 =

∅,

if f 6≡ 0

X,

if f ≡ 0.

Note that, if i < d − (k + 1) or i > d, then Dfi = ∅. If X is a union of irreducible components {Xj }, we define Dfi :=

S

j

We define the i-th Vogel set of f with respect to M to be Dfi (M ) :=

Dfi| S

j

. Xj

Dfi| . Vl

Proposition 2.4. Every component of Dfi (M ) has dimension at least i and |M | ∩ V (f ) = S i i i Df (M ). If Πf (M ) is i-dimensional and C is an i-dimensional irreducible component of |M | ∩ |V (f )|, then C ⊆ Dfi (M ). If X is irreducible of dimension d, then, for all i 6 d − 1, i+1 [ k Π ∩ V (f ) = Df . f k6i

Proof. We may work on each irreducible set, Vl , separately; therefore, we assume that we are in the case where X is irreducible and M = [X]. That every component of Dfi has dimension at least i follows immediately from the fact that each component of Πi+1 has dimension at least i + 1 (by 2.2.i). f [ d Dfi By definition X = Π ∪Dd . Hence, V (f ) = Πd ∩V (f ) ∪Dd and so, the equation V (f ) = f

f

f

f

i

follows once we show the final claim of the proposition. Suppose that i 6 d − 1. Then, i+1
Π ∩ V (f ) = Πi+1 ∩ V (fi+k+1−d ) ∩ V (f ) = Πif ∪ Dfi ∩ V (f ) = Πif ∩ V (f ) ∪ Dfi . f f As Πif is eventually empty, the desired conclusion follows. Finally, suppose that C is an i-dimensional irreducible component of |V (f )| and Πif is i-dimensional. Then, C is contained in a component C 0 of |V (fi+k+2−d , . . . , fk )|; such a C 0 necessarily has dimension at least i + 1. Thus, C 0 cannot be contained in V (f ). It follows that C 0 is contained in Πi+1 . Therefore, f C ⊆ C 0 ∩ V (fi+k+1−d ) ⊆ Πi+1 ∩ V (fi+k+1−d ) = Πif ∪ Dfi . f If Πif is i-dimensional, then – since C ⊆ V (f ) and is i-dimensional – it follows that C 6⊆ Πif , and so C ⊆ Dfi .  Below, we prove the Dimensionality Lemma in which we state as hypotheses/conclusions that “ Πif (M ) is i-dimensional” and “Dfi (M ) is i-dimensional”. Since sets cannot be negative-dimensional, for i < 0, we mean that the respective set is empty. Note that 2.4 implies that Dfi (M ) is purely i-dimensional if and only if it is i-dimensional.

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Lemma 2.5 (Dimensionality Lemma). The following are equivalent: i) for all i, Πif (M ) is i-dimensional; i ii) for all i, Πif (M ) = e Πf (M ) ; i Πf (M ) . iii) for all i, Πif (M ) = b In addition, these equivalent conditions imply iv)

for all i, Dfi (M ) is i-dimensional;

and, for all p ∈ |M | ∩ V (f ), there exists a neighborhood of p in which iv) implies i), ii), and iii).

Proof. Again we may consider each component appearing M separately; hence, we may assume that X is irreducible and M = [X]. As all the statements are set-theoretic, to cut down on notation, we shall omit the vertical lines around the various gap sheaves. We will show that i) and iii) are each equivalent to ii), that i) implies iv), and that, near points of V (f ), iv) implies i). e i , what we need to show is: if C is a component of i)⇒ ii): Assume i). From the definition of Π f V (fi+k+1−d , . . . , fk ), then C is contained in V (f ) if and only if C is contained in V (fi+k−d ). As V (f ) ⊆ V (fi+k−d ), one implication is trivial, and so what we must show is that if C is a component of V (fi+k+1−d , . . . , fk ) and C ⊆ V (fi+k−d ), then C ⊆ V (f ). Suppose not. As C is a component of V (fi+k+1−d , . . . , fk ), the dimension of C is at least i. If C 6⊆ V (f ), then – by definition – C is a component of Πif . But C is also contained in V (fi+k−d ), and so C is a component of Πif ∩ V (fi+k−d ) = Πi−1 ∪ Dfi−1 . As C is not contained in V (f ), we f i−1 conclude that C is a component of Πf of dimension at least i. This contradicts i). ii)⇒ i): Assume ii). From Definition 2.1, Πif is purely i-dimensional for i > d. Suppose that i0 is the largest integer i (less than d) such that Πif is not purely i-dimensional. Then, Πif0 +1 is purely (i0 + 1)-dimensional and, by ii), the set Πif0 +1 is equal to V (fi0 +k+2−d , . . . , fk ) ¬ V (fi0 +k+1−d ). Hence, the intersection Πif0 +1 ∩ V (fi0 +k+1−d ) is proper, and so Πif0 +1 ∩ V (fi0 +k+1−d ) is purely i0 -dimensional. As there is an equality of sets Πif0 +1 ∩ V (fi0 +k+1−d ) = Πif0 ∪ Dfi0 , this contradicts the fact that Πif0 is not purely i0 -dimensional. iii)⇒ ii):

bi ⊆ Π e i ⊆ Πi (see 2.2.i). Assume iii). Then ii) follows immediately from the fact that Π f f f

ii)⇒ iii): Assume ii). The proof is by induction. iii) is certainly true by definition for i > d. b i for i > m, where m 6 d. We need to show that Πm−1 = Π b m−1 . We Now, suppose that Πif = Π f f f have

b m−1 = Π b m ∩ V (fm+k−d ) ¬ V (fm+k−1−d ) = Πm ∩ V (fm+k−d ) ¬ V (fm+k−1−d ). Π f f f

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By combining the definition of Πm f as V (fm+k+1−d , . . . , fk ) ¬ V (f ) with Lemma 1.3.ii, we conclude that
Πm f ∩ V (fm+k−d ) ¬ V (fm+k−1−d ) = V (fm+k−d , . . . , fk ) ¬ V (fm+k−1−d ) b m−1 = Π e m−1 . By ii), this implies that Π b m−1 = Πm−1 and we are finished. and so, Π f f f f i)⇒ iv): Assume i), and suppose that i0 is such that Dfi0 is not purely i0 -dimensional. Then, Πif0 +1 ∩ V (fi0 +k+1−d ) is not purely i0 -dimensional. As Πif0 +1 is purely (i0 + 1)-dimensional by assumption, it follows that V (fi0 +k+1−d ) contains a component, C, of Πif0 +1 . As C is a component of Πif0 +1 , C is not contained in V (f ). Thus, C is a component of Πif0 +1 ∩ V (fi0 +k+1−d ) = Πif0 ∪ Dfi0 which is not contained in V (f ), and so C is an (i0 + 1)-dimensional component of Πif0 – this contradicts our assumption. iv)⇒ i): Assume iv), and that we are interested in the germ of the situation at a point p ∈ V (f ). Let i0 be the smallest i such that Πif is not purely i-dimensional. By Proposition 2.2.i, Πif0 must have dimension at least i0 + 1. Thus, since p ∈ V (f ), Πif0 ∩ V (fi0 +k−d ) has dimension at least i0 . But, as sets, Πif0 ∩ V (fi0 +k−d ) = Πif0 −1 ∪ Dfi0 −1 , and by assumption Dfi0 −1 is purely (i0 − 1)dimensional. Therefore, we conclude that Πif0 −1 has dimension at least i0 – a contradiction of the choice of i0 . 

Remark 2.6. Our phrasing of Lemma 2.5 is the most elegant, and is in the form that we will usually need. However, it is occasionally helpful to note that our proof does not require that one knows i), ii), or iii) for all i. Specifically, what our proof actually shows is that: i Πf (M ) ; (M ) is (i − 1)-dimensional, then Πif (M ) = e • if Πi−1 f •

i Πf (M ) for all i > k, then Πkf (M ) is k-dimensional; if Πif (M ) = e

•

i i if Πif (M ) = b Πf (M ) , then Πif (M ) = e Πf (M ) ;

•

m−1 i Πf (M ) , (M ) = e Πf (M ) for all i > m, and Πm−1 if Πif (M ) = b f m−1 i b Πf (M ) ; in particular, if Πif (M ) = e Πf (M ) for all i > m − 1, then for all i > m − 1;

•

if Πif (M ) is i-dimensional and Πi+1 (M ) is (i + 1)-dimensional, then Dfi (M ) is f i-dimensional; and

•

if p ∈ |M | ∩ V (f ), Πi−1 (M ) is (i − 1)-dimensional at p, and Dfi−1 (M ) is (i − 1)f dimensional at p, then Πif (M ) is i-dimensional at p.

then Πm−1 (M ) = f i i Π (M ) = b Πf (M ) f

Definition 2.7. If the equivalent conditions i), ii), and iii) of Lemma 2.5 hold, we say that the gap sets of f with respect to M have the correct dimension. If the equivalent conditions i), ii), iii), and iv) of Lemma 2.5 hold at a point p ∈ |M | ∩ V (f ), we say that the Vogel sets of f with respect to M have the correct dimension at p. We say simply that

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the Vogel sets of f with respect to M have the correct dimension provided that they have correct dimension at all points of |M | ∩ V (f ). Remark 2.8. Note that, since every component of Dfi (M ) has dimension at least i (see 2.4), if the Vogel sets all have correct dimension at p, then all the Vogel sets have correct dimension at points near p. Note also that if the gap sets have correct dimension, then the Vogel sets have correct dimension. Moreover, since we are interested only in what happens near V (f ), the natural assumption for us to make seems like it should be that the Vogel cycles have correct dimension. However, our usual assumption will be that gap sets have correct dimension; for 2.5 tells us that, in a neighborhood of V (f ), these assumptions are equivalent, and requiring the gap sets to have the correct dimension saves us from having to state over and over again that we take a small neighborhood of a point of V (f ). It is important to remember that one implication of the Vogel and gap sets having correct e i (M ), and Π b i (M ) are all empty if i < 0, and Π0 (M ) = dimension is that Dfi (M ), Πif (M ), Π f f f e 0 (M ) = Π b 0 (M ) = ∅ at points of |M | ∩ V (f ). Π f f Finally, consider the special case where p is an isolated point of |M | ∩ V (f ). Then, 2.4 implies that, near p, Dfi (M ) = ∅ if i > 1, and Df0 (M ) = {p}. Thus, 2.5 implies that the gap sets and the Vogel sets have correct dimension at p.

Proposition 2.9. If X is irreducible and Cohen-Macaulay, and all of the gap sets of f have correct e i , and Π b i are equal. dimension, then, for all i, the schemes Πif , Π f f e i are equal Proof. By 2.2.v, if the gap sets have correct dimension, then the schemes Πif and Π f i i e have no embedded subvarieties; therefore, up to embedded subvariety. By Lemma 1.5, Πf and Π f they are equal as schemes. b i agrees with the other two, we must, of course, use To prove that the scheme structure of Π f ei = Π bi . induction. Let d denote the dimension of X. For i > d, we know that Πif = Π f f i+1 i+1 i+1 e b . Then, 2.2.vii tells us that Π ei = Π b i and, by the Suppose, inductively, that Πf = Π =Π f f f f first paragraph above, we know that this equals Πif .  While we have been selecting (k + 1)-tuples, f , our primary object of interest is, in fact, the ideal < f > generated by the f0 , . . . , fk . As far as the ideal < f > is concerned, the functions comprising f may not be suitably generic. However, as we shall see, to obtain a well-behaved ordered collection of generators, one only needs to replace (f0 , . . . , fk ) by generic linear combinations of the fi ’s themselves. However, the term “generic” here is used in a non-standard way; what we need is to replace f0 by a generic linear combination, then – fixing this new f0 – replace f1 by a generic linear combination, and so on. Since “generic” should always mean open and dense in some topology, we will define a new, convenient one. Definition 2.10. The pseudo-Zariski topology (pZ-topology) on a topological space (X, T ) is a new topological space (X, TpZ ) given by U ∈ TpZ if and only if U is empty or is an open, dense subset in (X, T ). (One verifies easily that this, in fact, yields a topology on X.) Given two topological spaces X and Y , let πX and πY denote the projections from X × Y onto

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X and Y , respectively. The inductive pseudo-Zariski topology (IPZ-topology) on X × Y is given by: W ⊆ X × Y is open in the IPZ-topology
if and only if πX (W) is open in the pZ-topology on X −1 and, for all x ∈ πX (W), πY W ∩ πX (x) is open in the pZ-topology on Y . (It is trivial to verify that this is a topology on X × Y , and that a non-empty open set in the IPZ-topology on X × Y is a dense set in the cross-product topology on X × Y .) Finally, given a finite number of topological spaces X1 , X2 , . . . , Xm , the IPZ-topology on X  1 ×X2 ×· · ·×Xm
is given inductively  by using the IPZ-topology on each product in the expression (X1 × X2 ) × X3 × · · · × Xm−1 × Xm . A generic linear reorganization of a (k + 1)-tuple f is a matrix product f A, where the matrix A is invertible and is an element of some given generic subset in the IPZ-topology on the (k + 1)-fold product Ck+1 × · · · × Ck+1 (where we consider each column of A to be contained in one copy of Ck+1 ). Note that, if X1 = X2 = · · · = Xm = CN (or PN ), then the IPZ-topology on the product is more fine than the Zariski topology, but sets which are open in the IPZ-topology need not be open in the classical topology on the product. Proposition/Definition 2.11. If X is irreducible, then, for all p ∈ X, for a generic linear reorganization, ˆf , of f , the gap sets of ˆf all have correct dimension at p and, for all i, there is an ei = Π b i in a neighborhood of p. equality of schemes Πˆif = Π ˆ ˆ f f Therefore, for p ∈ |M |, for a generic linear reorganization, ˆf , of f , the gap sets of ˆf with respect to |M | all have correct dimension at p and, for all i, there is an equality of cycles Πˆif (M ) = e i (M ) = Π b i (M ) at p. Π ˆ f

ˆ f

If we are working in the algebraic category, then we may produce such generic linear reorganizations globally. We refer to a reorganization ˆf such that the above equality of cycles holds as an agreeable reorganization of f (with respect to M at p) (for it makes the various cycle structures agree). Proof. Assume that X is irreducible. We fix a point p ∈ X. Our sole reason for stating the results “at p” is that, at several places in the proof, we will need to know that certain analytic sets have a finite number of analytic components. This is, of course, guaranteed near a given point or in the algebraic category. Hence, throughout the proof, we will make no further reference to working in a neighborhood of p, but will assume that all of the analytic sets that arise have a finite number of components. We first show: (†) for a generic linear reorganization, ˆf , of f , for all i, V (fˆi+k−d ) contains no component or embedded subvariety of Πˆif . We produce the (k + 1)-tuple ˆf one element at a time, by downward induction. If f is identically zero on X, then (†) is trivial. So, suppose that one of the fi does not vanish on X. Then, for a generic linear combination fˆk := a0 f0 + · · · + ak fk , fˆk does not vanish on X. Thus, V (fˆk ) contains no component or embedded subvariety of Πˆdf . Now, suppose that we have made generic linear reorganizations of f to produce ˆf , and that

PART I. ALGEBRAIC PRELIMINARIES

15

V (fˆi+k−d ) contains no component or embedded subvariety of Πˆif for all i > m. Then, for every component or embedded subvariety, W , of Πm−1 , W is contained in V (fˆm+k−d , . . . , fˆk ), but there ˆ f

exists some fˆj with j < m + k − d such that W 6⊆ V (fˆj ). Thus, a generic linear combination of the fˆ’s will not vanish on any component of embedded subvariety of Πˆm−1 . This proves (†). f
= Πˆif ∩ V (fˆi+k−d ) ¬V (ˆf ) by 2.2.ii, (†) implies that the Vogel sets of ˆf all have correct As Πˆi−1 f ei = Π b i by downward induction on i. dimension. We show that Πi = Π ˆ f

ˆ f

ˆ f

e i+1 = Π b i+1 . Then, When i = d, the statement is clear. Assume now that Πˆi+1 =Π ˆ ˆ f f f

b ˆi = Π b i+1 ∩ V (fˆi+k+1−d ) ¬V (fˆi+k−d ) = Πi+1 ∩ V (fˆi+k+1−d ) ¬V (fˆi+k−d ), Π ˆ ˆ f f f which, by 1.3.i, equals




ˆi+k+1−d ) ¬V (ˆf ) ¬V (fˆi+k−d ). Πˆi+1 ∩ V ( f f

b i = Πi ¬ V (fˆi+k−d ) = Πi . Therefore, applying 2.2.ii, followed by (†), we conclude that Π ˆ ˆ ˆ f f f b i = Πi for all i, by applying 2.2.vii, we conclude that Π ei = Π b i = Πi .  As Π ˆ ˆ ˆ ˆ ˆ f f f f f

We now wish to endow the Vogel sets a cycle structure. First, we need the following easy proposition. is (j − 1)-dimensional for all j > i, then Proposition 2.12. If X is irreducible, and Πj−1 f  j  j b Πf = Πf for all j > i, and there is an equality of cycles given by      i bf = b i+1 · V (fi+k+1−d ) ¬ V (f ). Π Π f

Therefore, on X−V (f ), all of V (fk ), V (fk−1 ), . . . , V (fi+k+1−d ) intersect properly and, on X−V (f ),  i b f = V (fk ) · V (fk−1 ) · . . . · V (fi+k+1−d ). Π

being (j − 1)-dimensional for all j > i implies that Proof. Using Remark 2.6, we see that Πj−1 f  j  j b for all j > i. Πf = Π f      i b i+1 · V (fi+k+1−d ) ¬ V (f ) is equivalent to the setb = Π Now, by 2.2.iv, the statement Π f f   i i+1 b b theoretic statement Πf = Πf ∩ V (fi+k+1−d ) ¬ V (f ). This set-theoretic statement follows easily from 2.5.iii; for it tells us that     b ∩ V (fi+k+1−d ) ¬ V (f ) = Πi+1 ∩ V (fi+k+1−d ) ¬ V (f ) Πi+1 f f and 2.2.ii tells us that this equals Πif . Applying 2.5.iii again yields the desired equality of cycles. 

16

DAVID B. MASSEY

 i
b = V (fk ) · V (fk−1 ) · . . . · V (fi+k+1−d ) ¬V (f ). We Remark 2.13. It is tempting to write that Π f could do this if we were willing to use intersection theory with non-proper intersections; this seems especially innocuous when the non-proper part of the intersection lies in a portion that we are going to throw away, as it does here. Nonetheless, we do not want wish to write any formulas involving intersection theory which are not discussed in Appendix A.  i b is the closure in X of this cycle on X − V (f ). This However, 2.12 does tell us that, on X, Π f cycle structure turns out to be the correct one to use in order to endow the Vogel sets with a cycle structure. However, in order to guarantee that the cycles we define actually have as their underlying sets the Vogel sets of f , we only define the Vogel cycles when the gap sets (or Vogel sets) have the correct dimensions and, even then, we must restrict ourselves to what happens in a neighborhood of V (f ). is (j − 1)-dimensional at each point in V (f ) for Definition 2.14. If X is irreducible, and Πj−1 f all j > i, then we define the i-th Vogel cycle of f , ∆if , to be the sum of the components of      i  b i+1 · V (fi+k+1−d ) − Π b Π f f which intersect V (f ). In other words, if  X       i  b i+1 · V (fi+k+1−d ) − Π bf = Π pj W j , f j

X

then ∆if =

  pj W j .

Wj ∩V (f )6=∅

is (j − 1)-dimensional at each point in V (f| ) for all j > i and for all components Vl If Πj−1 Vl f| Vl

of |M |, then Xwe say that the i-th Vogel cycle of f with respect to M is defined and its definition is ∆if (M ) := ml ∆if| . Vl

l

Note that the Dimensionality Lemma implies that there is no difference between saying that all the Vogel sets have correct dimension and that all the Vogel cycles are defined; we prefer to say that the Vogel cycles are defined, as the Vogel cycles are the objects in which we are most interested.

We have defined the Vogel cycles to consist of pieces which intersect V (f ); however, 2.12 yields immediately: Proposition 2.15. If ∆if (M ) is defined, then each ∆if| is non-negative and purely i-dimensional. Vl Moreover, ∆i (M ) = Di (M ) ⊆ |M | ∩ V (f ). f

f

Remark 2.16. If X is irreducible, Proposition 2.12 and Proposition 2.15, together with the Dimensionality Lemma, tell us how the Vogel cycles should be calculated; we will describe this now,

PART I. ALGEBRAIC PRELIMINARIES

17

omitting the square brackets for the cycles. b d = X¬V (fk ); thus, Π b d is either 0 or X. Next, one calculates the intersection One begins with Π f f b d · V (fk ). This intersection cycle has components contained in V (f ) and components which are Π f not contained in V (f ). By 2.12, the sum of the components which are not contained in V (f ) b d−1 and the sum of the components which are contained in V (f ) is ∆d−1 . Having is precisely Π f f b d ·V (fk ) = Π b d−1 +∆d−1 , we use our newly found Π b d−1 in the next step: the calculation calculated Π f f f f b d−1 · V (fk−1 ). One proceeds downward inductively. of Π f The subtle point in the above description is that, if one is working in a neighborhood of a point of V (f ), one may check while performing the calculation that the Vogel sets, ∆if , have correct dimension. For, by splitting the intersections into pieces which are contained in V (f ), and pieces which are not, we are actually obtaining a cycle ∆if whose underlying set is precisely Dfi (this follows from 2.2.ii). Thus, one proceeds with the inductive calculation described above, and then checks that the calculated ∆if have correct dimension, which then tells one that the calculation is actually correct. Consider the special case where p is an isolated point of |M | ∩ V (f ). As we saw in Remark 2.8, it is automatic that the Vogel cycles are defined at p, and only ∆0f can be non-zero.

Example 2.17. We continue to suppress the square brackets around cycles. Let X = C5 and let f = (f0 , f1 , f2 , f3 , f4 ) = (−2ux2 , −2vx2 , −2wx2 , −3x2 − 2x(u2 + v 2 + w2 ), 2y). (The reason for the strange, seemingly pointless, coefficients is that we will use this example later b 5 = C5 . in a different context. See Example II.2.4.) Then, V (f ) = V (x, y) and Π f b 5f · V (f4 ) = Π b 5f · V (−2y) = V (y). Π b 4 = V (y), and we continue. As V (y) is not contained in V (f ), Π f

b 4 · V (f3 ) = V (y) · V (−3x2 − 2x(u2 + v 2 + w2 )) = V (−3x − 2(u2 + v 2 + w2 ), y) + V (x, y) = Π b 3 + ∆3 . Π f f f

b 3 · V (f2 ) = V (−3x − 2(u2 + v 2 + w2 ), y) · V (−2wx2 ) = Π f b 2f + ∆2f . V (−3x − 2(u2 + v 2 ), w, y) + 2V (u2 + v 2 + w2 , x, y) = Π b 2f · V (f1 ) = V (−3x − 2(u2 + v 2 ), w, y) · V (−2vx2 ) = Π b 1 + ∆1 . V (−3x − 2u2 , v, w, y) + 2V (u2 + v 2 , w, x, y) = Π f f

b 1f · V (f0 ) = V (−3x − 2u2 , v, w, y) · V (−2ux2 ) = V (u, v, w, x, y) + 2V (u2 , v, w, x, y) = 5[0] = ∆0f . Π Hence, we find the Vogel sets all have correct dimension, and so the Vogel cycles are defined and ∆3f = V (x, y), ∆2f = 2V (u2 + v 2 + w2 , x, y), ∆1f = 2V (u2 + v 2 , w, x, y), and ∆0f = 5[0].

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DAVID B. MASSEY

Remark 2.18. Suppose that all the Vogel cycles of f are defined and k + 1 > d. Consider the truncated d-tuple ftr := (fk+1−d , . . . , fk ); we claim that, in a neighborhood of V (f ), |V (f )| = |V (ftr )| e i , and Π b i for all i (all of them will be empty and both f and ftr will produce the same Di , ∆i , Πi , Π for i < 0). ei = Π e i and Π bi = Π b i . We would know that, near It is immediate from the definitions that Π f ftr f ftr i i i i i i V (f ), Πf = Πftr and, hence, that Df = Dftr and ∆f = ∆ftr , if we could show that there is an equality of sets |V (f )| = |V (ftr )|. This is easy; by definition of Πif , |V (ftr )| = |V (f )| ∪ |Π0f |. As we are assuming that Π0f is 0dimensional (and, of course, has no components contained in V (f )), there is a neighborhood of V (f ) in which |V (f )| = |V (ftr )|. Suppose that all the Vogel cycles of f are defined and k + 1 < d. Consider the extended d-tuple fex := (f0 , . . . , f0 , . . . , fk ) (where there are d − k occurrences of f0 ); clearly, |V (f )| = |V (fex )|, and e i , and Π b i for all i (all of them will be empty for f and fex will produce the same Di , ∆i , Πi , Π i < d − (k + 1)). Looking at the two cases above, we see that, if all the Vogel cycles are defined, the whole theory remains unchanged if we assume that d = k + 1, i.e., if we assume that the dimension of the underlying space X is exactly equal to the number of functions in our tuple f .

Proposition 2.19. Suppose that X is irreducible of dimension k + 1. Let s := dimp V (f ) > 0. is (j −1)-dimensional for all j > s, then, in a neighborhood of p, all of V (fk ), V (fk−1 ), If Πj−1 f . . . , V (fs+1 ) intersect properly,  s+1  b Π = V (fk ) · V (fk−1 ) · . . . · V (fs+1 ), f

and ∆sf equals the sum of those components of V (fk ) · V (fk−1 ) · . . . · V (fs+1 ) · V (fs ) which are contained in V (f ). In particular, if p is an isolated point in V (f ), then ∆0f


p

  = V (fk ) · V (fk−1 ) · . . . · V (f1 ) · V (f0 ) . p

is (j − 1)-dimensional for all j > s, then 2.12 tells us that, on X − V (f ), Proof. If Πj−1 f V (fk ), V (fk−1 ), . . . , V (fs+1 ) intersect properly and  s+1  b Π = V (fk ) · V (fk−1 ) · . . . · V (fs+1 ). f b s+1 is purely (s+1)-dimensional, and every component of V (fk , . . . , fs+1 ) has dimension However, Π f at least s + 1. As s = dimp V (f ), it follows that there is a neighborhood of p in which the closure of |V (fk , . . . , fs+1 ) − V (f )| = |V (fk , . . . , fs+1 )|, and thus in which V (fk ), V (fk−1 ), . . . , V (fs+1 ) intersect properly and  s+1  b Π = V (fk ) · V (fk−1 ) · . . . · V (fs+1 ). f

PART I. ALGEBRAIC PRELIMINARIES

19

Therefore,  s+1  b Π · V (fs ) = V (fk ) · V (fk−1 ) · . . . · V (fs+1 ) · V (fs ) f and 2.12 tells us that the components of this that are contained in V (f ) are precisely ∆sf . Recalling Remark 2.8, if p is an isolated point in V (f ), then all of the gaps sets have correct dimension at p, which implies that Π0f is empty. Thus, ∆0f


p

  = V (fk ) · V (fk−1 ) · . . . · V (f1 ) · V (f0 )

p

follows at once from the above. 

We now prove a theorem which gives the basic relation between Vogel cycles and the blow-up. In fact, we show that the Vogel cycles are representatives of the Segre classes, as defined in [Fu, §4.2]. In the generic case, this is Theorem 3.3 of [Gas1], and is also proved in Lemma 2.2 of [G-G]. However, we are interested in cases which may not be quite so generic.

Theorem 2.20. Let X be an irreducible analytic subset of an analytic manifold U, let π : Blf X → X denote the blow-up of X along f (see Example 1.6), and let Ef denote the corresponding exceptional divisor. If Ef properly intersects U × Pm × {0} in U × Pk for all m, then i)

the Vogel cycles of f are defined;

ii)

there exists a neighborhood Ω of V (f ) such that, for all m, Blf X intersects Ω × Pm × {0} properly in Ω × Pk ; and

iii)

inside Ω, for all i, b i+1 = π∗ (Blf X · (U × Pi+k+1−d × {0})) Π f

and ∆if = π∗ (Ef · (U × Pi+k+1−d × {0})), where the intersection takes place in U × Pk and π∗ denotes the proper push-forward. Moreover, for all p ∈ X, there exists an open neighborhood W of p in U such that, for a generic linear reorganization, ˜f , of f , E˜f properly intersects W × Pm × {0} inside W × Pk for all m. In the algebraic category, we may produce such generic linear reorganizations globally, i.e., such that E˜f properly intersects U × Pm × {0} inside U × Pk for all m. Proof. We show the last two statements first. As in 2.11, the reason that we can only make local statements in the analytic case is because we must worry about analytic sets having an infinite number of irreducible components. For all p ∈ X, π −1 (p) is compact, and so, any analytic set can have only a finite number of irreducible components which meet π −1 (p). In the algebraic setting, we know that we have a finite number of irreducible components globally. For notational ease, we assume in the following paragraph, in the analytic case, that U is rechosen as small as necessary at

20

DAVID B. MASSEY

each stage so that U ×Pk contains a finite number of analytic components (of any specified analytic set) which intersect π −1 (p); this will mean that we will write U in place of the open neighborhood W which appears in the statement of the theorem. Now, as each point in each component of Ef cannot have all of its homogeneous coordinates equal to zero, for each component ν of Ef , there exists a homogeneous coordinate wk(ν) such that V (wk(ν) ) properly intersects ν. Therefore, for generic (a0,0 , . . . , a0,k ) ∈ Ck+1 , the linear form w ek := a0,0 w0 + · · · + a0,k wk is such that V (w ek ) contains no component of Ef . We continue in this manner; for generic (a1,0 , . . . , a1,k ) ∈ Ck+1 , the linear form w ek−1 := a1,0 w0 + · · · + a1,k wk is such that V (w ek−1 ) contains no component of Ef ∩ V (w ek ). Continuing, we produce a generic linear e of w such that, for all m, Ef properly intersects V (w reorganization, w, em+1 , . . . , w ek ) inside U × Pk . This proves the last two claims of the theorem. We now prove i), ii), and iii) of the theorem. We use [w0 : · · · : wk ] as homogeneous coordinates on Pk . Let η : Blf X → Pk denote the restriction of the projection. Until the end of the proof, we shall simply write fj in place of fj ◦ π; no confusion will arise, since it is clear that we must mean fj ◦ π when the domain is contained in Blf X. Certainly, π −1 induces an isomorphism from Πi+1 − V (f ) to f η −1 (Pi+k+1−d × {0}) − Ef = Blf X ∩ (U × Pi+k+1−d × {0}) − Ef . Hence, Πi+1 is purely (i + 1)-dimensional if and only if f Blf X ∩ (U × Pi+k+1−d × {0}) − Ef is purely (i + 1)-dimensional. But, every component of Blf X ∩ (U × Pi+k+1−d × {0}) has dimension at least i + 1, while – by hypothesis – Ef ∩ (U × Pi+k+1−d × {0}) is purely i-dimensional. Thus, Blf X ∩ (U × Pi+k+1−d × {0}) − Ef = Blf X ∩ (U × Pi+k+1−d × {0}), and every component has dimension at least i + 1. As Ef is locally defined in Blf X by a single equation and Ef ∩ (U × Pi+k+1−d × {0}) is purely i-dimensional, it follows that Blf X ∩ (U × Pi+k+1−d × {0}) is purely (i + 1)-dimensional, for all i, at all points which lie in Ef . This proves ii) from the statement of the theorem, and proves that Πi+1 is purely (i + 1)-dimensional, for all f i, at all points of V (f ), and so the Vogel cycles are defined. This proves i). Note that the Dimensionality Lemma and the above paragraphs imply that, in a neighborhood of any point p ∈ V (f ), (*)

Blf X ∩ V (wi+k+2−d , . . . , wk ) = Blf X ∩ V (wi+k+2−d , . . . , wk ) − V (fi+k+1−d ).

Let p be a point in V (f ). As the Vogel cycles are defined, there exists a neighborhood of p such that X − V (f ), V (fk ) − V (f ), . . . , V (fi+k+2−d ) − V (f ) all intersect properly and π induces an isomorphism       Blf X − E · V (fk ) − E · . . . · V (fi+k+2−d ) − E ∼ =       X − V (f ) · V (fk ) − V (f ) · . . . · V (fi+k+2−d ) − V (f ) .

PART I. ALGEBRAIC PRELIMINARIES

21

By the Dimensionality Lemma, no component of this intersection is contained in V (fi+k+1−d ), and b i+1 is equal to so we conclude that Π f π∗



     Blf X − V (fi+k+1−d ) · V (fk ) − V (fi+k+1−d ) · . . . · V (fi+k+2−d ) − V (fi+k+1−d ) .

We claim that this implies the first equality of the theorem:
b i+1 = π∗ Blf X · V (wi+k+2−d , . . . , wk ) , Π f

(†)

in a neighborhood of any point in V (f ). To see this, note that Blf X − V (fi+k+1−d ) ⊆ {wi+k+1−d 6= 0}. On the open set, W ⊆ U × Pk , where fi+k+1−d 6= 0 and wi+k+1−d 6= 0, there is an equality of schemes   wj fj − . Blf X = V fi+k+1−d wi+k+1−d j6=i+k+1−d  At points of W,

fj

wj

−



is easily seen to be a regular sequence. Therefi+k+1−d wi+k+1−d j6=i+k+1−d fore, on W, the cycle [Blf X] is equal to the intersection product of the cycles    fj wj V − . fi+k+1−d wi+k+1−d j6=i+k+1−d Moreover, on W, for j > i + k + 2 − d,         fj wj fj wj V − · V (fj ) = V − , fj = fi+k+1−d wi+k+1−d fi+k+1−d wi+k+1−d       V (fj , wj ) = V (fj ) · V (wj ) =   V

fj fi+k+1−d

−



wj wi+k+1−d

, wj

=

  V

fj fi+k+1−d

−

wj wi+k+1−d



  · V (wj ) .

Hence, on W,         [Blf X] · V (fk ) · . . . · V (fi+k+2−d ) = [Blf X] · V (wk ) · . . . · V (wi+k+2−d ) =   [Blf X] · V (wi+k+2−d , . . . , wk ) , and so (†) follows from our previous paragraphs and (∗). bi = Π b i+1 · V (fi+k+1−d ). Applying (†) and the push-forward forNow, by definition, ∆if + Π f f mula (see Appendix A.14) – which we may use since V (fi+k+1−d ◦ π) properly intersects Blf X ∩ V (wi+k+2−d , . . . , wk ) by (*) – we conclude that
b i = π∗ V (fi+k+1−d ◦ π) · Blf X · V (wi+k+2−d , . . . , wk ) . ∆if + Π f By the Dimensionality Lemma, ∆if consists of those components of the proper push-forward which are contained in V (f ). Hence, we will have proved the second equality of the theorem if we can

22

DAVID B. MASSEY

show that the components of V (fi+k+1−d ◦ π) · Blf X · V (wi+k+2−d , . . . , wk ) which are contained in Ef are equal to Ef · V (wi+k+2−d , . . . , wk ). On the open set where wi+k+1−d 6= 0, Ef is defined to be V (fi+k+1−d ◦ π) · Blf X. Thus, it is enough to show that V (fi+k+1−d ◦ π) · Blf X · V (wi+k+2−d , . . . , wk ) has no components contained in Ef which are also contained in V (wi+k+1−d ). However, by hypothesis, V (wi+k+1−d , . . . , wk ) properly intersects Ef , and so every component of Ef ∩ V (wi+k+1−d , wi+k+2−d , . . . , wk ) has dimension i − 1. As every component of V (fi+k+1−d ◦ π) · Blf X · V (wi+k+2−d , . . . , wk ) has dimension at least i, we are finished.  Remark 2.21. Note that the proof of 2.20 shows that, for each i, if Ef properly intersects U × Pi+k+1−d × {0} in U × Pk , then Πi+1 is (i + 1)-dimensional near V (f ) – the point being that we f do not need to assume that we have proper intersections for all i.

The following corollary follows immediately from Theorem 2.20.

Corollary 2.22 (The Segre-Vogel Relation). Let X be an analytic subset of an analytic manifold U, and let π : U × Pk → U denote the projection. Assume that M is purely d-dimensional. k k For each Vl appearing in M , consider and let Efl denote the corresponding P Blf Vl ⊆ Vl ×P ⊆ U ×P ,P exceptional divisor. Let Blf M := l ml [Blf Vl ] and Ef (M ) := l ml [Efl ]. If |Ef (M )| properly intersects U × Pm × {0} in U × Pk for all m, then i)

the Vogel cycles of f with respect to M are defined;

ii)

there exists a neighborhood Ω of |M |∩V (f ) such that, for all m, | Blf M | intersects Ω×Pm ×{0} properly in Ω × Pk ; and

iii)

inside Ω, for all i, b i+1 (M ) = π∗ (Blf M · (U × Pi+k+1−d × {0})) Π f

and ∆if (M ) = π∗ (Ef (M ) · (U × Pi+k+1−d × {0})), where the intersection takes place in U × Pk and π∗ denotes the proper push-forward. Moreover, for all p ∈ |M | ∩ X, there exists an open neighborhood W of p in U such that, for a generic linear reorganization, ˜f , of f , |E˜f (M )| properly intersects W × Pm × {0} inside W × Pk for all m. In the algebraic category, we may produce such generic linear reorganizations globally, i.e., such that |E˜f (M )| properly intersects U × Pm × {0} inside U × Pk for all m.

Corollary 2.23. Let X be an analytic subset of an analytic manifold U. Assume that M is purely (k + 1)-dimensional. k Then, continuing
with the notation from the previous corollary, the multiplicity of {p} × P in 0 Ef (M ) is ∆˜f (M ) p , where ˜f is a generic linear reorganization of f .

PART I. ALGEBRAIC PRELIMINARIES

23

Moreover, if p is an isolated point in |M | ∩ V (f ), then {p} × Pk
is the unique component of Ef (M ) over p and the multiplicity of {p} × Pk in Ef (M ) is ∆0f (M ) p ; if, in addition, there is a

regular sequence ˆf := (fˆ0 , . . . , fˆk ) in OU such that f := ˆf| , then ∆0 (M ) = M · V (ˆf ) . f

X

p

p

Proof. We use the notation from 2.22.
That the multiplicity of {p} × Pk in Ef (M ) is ∆˜0f (M ) p follows immediately from 2.22.iii, for we will be able to pick some copy of P0 ⊆ Pk so that, over p, |Ef (M )| ∩ (U × P0 ) = {p} × P0 . If p is an isolated point in |M | ∩ V (f ), then since Ef (M ) ⊆ U × Pk is purely k-dimensional, the only component of Ef (M ) over p has to be {p} × Pk , and so the proper intersection condition of 2.22 is automatically satisfied over a neighborhood of p. Thus, the multiplicity of {p} × Pk in
0 Ef (M ) is ∆f (M ) p . By 2.19, if we restrict to each Vl , then ∆0f 


p

  = V (fk ) · V (fk−1 ) · . . . · V (f1 ) · V (f0 ) = p

 Vl · V (fˆk ) · V (fˆk−1 ) · . . . · V (fˆ1 ) · V (fˆ0 ) = (Vl · V (ˆf ))p , p

where we used that ˆf is a regular sequence for the last equality (see Appendix A, section 4 for this equality and the one before it). 

Definition 2.24. We call a generic linear reorganization of f , such as appears in Corollary 2.22, a Vogel reorganization of f with respect to M . A generic linear reorganization of f which is both agreeable and Vogel is called unifying. Remark 2.25. Theorem 3.3 of [Gas1] actually shows that, by replacing f by a generic linear transformation applied to f , one obtains a unifying ˜f ; the point being that the linear transformation is actually generic, not just generic in the IPZ topology. However, as one can see in the proof of 2.20, proving that one can use an IPZ-generic transformation to obtain a suitable ˜f is quite trivial, and is actually what one uses in examples.

24

DAVID B. MASSEY

Chapter 3. THE LE-IOMDINE-VOGEL FORMULAS As in the previous chapter, X will denote an analytic space of dimension d contained in an analytic P manifold U, f := (f0 , . . . , fk ) will be an ordered (k + 1)-tuple of elements of OX , and M = l ml [Vl ] will be an analytic cycle in X such that ±M > 0. We wish to examine the effect on the Vogel cycles of adding scalar multiples of a large power of a new function g : X → C to f0 . The formulas that we derive are a powerful tool for inductive proofs. Throughout most of this chapter, we will be making the assumption that the Vogel cycles of f have correct dimension; as discussed in Remark 2.18, this means that we may as well assume that the number of elements of f is exactly d. Therefore, we will find it convenient to let n := d − 1, and then write that the dimension of X is n + 1 and that f = (f0 , . . . , fn ). Moreover, as all of our results will concern gap and Vogel cycles, the contributions from various irreducible components of M will simply add, and so – for simplicity – we will make the assumption that X is irreducible (though not necessarily reduced) and prove most results in the case where M = [X]. 0 Since we will be assuming that the gap sets
have correct dimension, ∆f will be purely 00 dimensional, and for any p ∈ X, we write ∆f p for the coefficient (possibly zero) of p appearing in the cycle ∆0f . The following lemma relates the Vogel cycles of (f0 , . . . , fn ) to the Vogel cycles of (f1 , . . . , fn , g), where g is a new function. We think of this as relating the Vogel cycles of f to the Vogel cycles of f restricted to V (g) – the elimination of f0 corresponds to the drop in dimension of the ambient space. As we shall see later, this “restriction” lemma is an essential step in proving the Lˆe-Iomdine-Vogel (LIV) formulas. Lemma 3.1 (The Restriction Lemma). Let X be an irreducible analytic space of dimension
n+1 n+1, and let f := (f0 , . . . , fn ) ∈ OX . Let g ∈ OX , let h := (f1 , . . . , fn , g), and let p ∈ V (f , g). i) Suppose that Π1f is 1-dimensional at p. Then, Π1f properly intersects V (g) at p if and only if V (h) = V (f , g) as germs of sets at p. ii) Suppose that the Vogel sets of f have correct dimension at p, that V (h) = V (f , g) as germs of sets at p, and that V (g) properly intersects Dfi at p for all i > 1.
bi Then, dimp V (h) = dimp V (f ) − 1 provided that dimp V (f ) > 1, V (g) properly intersects Π f at p for all i, the Vogel sets of h have correct dimension at p, and, for all i such that 1 6 i 6 n, there are equalities of germs of cycles at p given by b ih = Π b i+1 · V (g) and ∆ih = ∆i+1 · V (g). Π f f In addition, when i = 0, we have the following equality of germs of cycles

b 1f · V (g) + ∆1f · V (g) . ∆0h = Π

Proof. Proof of i):

As germs of sets at p,

V (f1 , . . . , fn , g) = Π1f ∪ V (f ) ∩ V (g) = Π1f ∩ V (g) ∪ V (f , g).

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Since Π1f is purely 1-dimensional at p, that Π1f properly intersects V (g) at p is equivalent to Π1f ∩ V (g) being empty or equal to {p}. As p ∈ V (f , g), we have proved i). S Proof of ii): By 2.4, V (f ) = Dfi . As the Vogel cycles have correct dimension and those of dimension
at least one are properly intersected by V (g) at p, we conclude that dimp V (h) equals dimp V (f ) − 1 provided that dimp V (f ) > 1. b i at p, we work solely with germs of sets at p. For i 6 0, To see that V (g) properly intersects Π f i Πf = ∅, and so there is nothing to prove. We now proceed with a proof by contradiction. Let m be the smallest i such that Πif does not properly intersect V (g) at p. Note that i) implies that m m > 2, and we have that dimp Πm f ∩ V (g) = m. Thus, dimp Πf ∩ V (fm−1 ) ∩ V (g) > m. However, m−1 m−1 m Πf ∩V (fm−1 ) = Πf ∪Df , and so we would have to have that either dimp Πm−1 ∩V (g) > m−1 f m−1 or dimp Df ∩ V (g) > m − 1; the first possibility is excluded by definition of m, and the second b i at p possibility is excluded by hypothesis. Thus, we have shown that V (g) properly intersects Π f for all i. To show that the Vogel sets of h have correct dimension at p, we once again work on the level of germs of sets. By definition, Πih = V (fi+1 , . . . , fn , g) ¬ V (h). One of our assumptions is that V (h) = V (f
, g); hence, Πih = V (fi+1 , . . . , fn , g) ¬ V (f , g). We apply 1.3.iii to obtain Πih = Πi+1 ∩ V (g) ¬ V (f , g). However, V (f , g) contains no components of Πi+1 ∩ V (g), for f
S f i+1 m Πf ∩ V (g) is purely i-dimensional, while – as sets – Πi+1 ∩ V (f ) ∩ V (g) = D ∩ V (g), m6i f f i+1 i which has dimension less than i. Therefore, as germs of sets at p, Πh = Πf ∩ V (g) and is purely i-dimensional, and so the Vogel sets of h have correct dimension at p. bi = Π b i+1 · V (g) at p. As we saw above, this equality holds We wish to see that, for 1 6 i 6 n, Π h f for the underlying sets and neither set has a component contained in V (f ). Therefore, it is enough b i and Π b i+1 · V (g) are equal on X − V (f ). Applying Remark 2.13, we find to show that the cycles Π h f that both of these cycles on X − V (f ) are given by V (fi+1 ) · . . . · V (fn ) · V (g). b i+1 · V (fi+1 ) − Π b i . Thus, for 1 6 i 6 n − 1, Finally, for 0 6 i 6 n − 1, ∆ih = Π h h b i+2 · V (g) · V (fi+1 ) − Π b i+1 · V (g) = ∆ih = Π f f


b i+1 + ∆i+1 · V (g) − Π b i+1 · V (g) = ∆i+1 · V (g). Π f f f f

When i = 0, we have
b 1h · V (f1 ) = Π b 2f · V (g) · V (f1 ) = Π b 1f + ∆1f · V (g). ∆0h = Π b n+1 · V (g) − Π bn = Π b n+1 · V (g) − Π b n+1 · V (g). We need to show that When i = n, ∆nh = Π h h h f n+1 n+1 n+1 b b Πh − Πf = ∆f . b n+1 = 0, ∆n+1 = [X], and – as V (g) properly intersects ∆n+1 – we conclude If f ≡ 0, then Π f f f b n+1 = [X]. Thus, if f ≡ 0, Π b n+1 − Π b n+1 = ∆n+1 . If f 6≡ 0, then Π b n+1 = [X], that h 6≡ 0 and so Π h h f f f n+1 n+1 b b n+1 = [X]. ∆f = 0, and – as V (g) properly intersects Π – we conclude that h 6≡ 0 and so Π f h b n+1 − Π b n+1 = ∆n+1 .  Thus, if f 6≡ 0, Π h f f

b 1 ∩ V (g) and that dimp Π b 1 = 1. Let η be an irreducible Definition 3.2. Suppose that p ∈ Π f f 1 b component (with its reduced structure) of Πf which passes through p. If η ∩ V (g) is zero-dimensional at p, then we define the gap
ratio of η at p (for f with respect η · V (fk+1−d ) p
to g) to be the ratio of intersection numbers . η · V (g) p

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If η ∩ V (g) is not zero-dimensional at p (i.e., if η ⊆ V (g)), then we define the gap ratio of η at p (for f with respect to g) to be 0. A gap ratio (at p for f with respect to g) is any one of the gap ratios of any component of b 1 ∩ V (g) through p. Π f b 1 , then we say that all the gap ratios are zero. If p ∈ V (g), but p 6∈ Π f Finally, a gap ratio at p for f with respect to g and the cycle M is a gap ratio (at p for f with respect to g) of f|Vl for some Vl appearing in M .

Lemma 3.3. Let X be an irreducible analytic space of dimension n + 1, let f := (f0 , . . . , fn ) ∈
n+1 OX , let g ∈ OX , and let p ∈ V (g). Let a be a non-zero complex number, and let j > 1 be an integer. If j is greater than or equal to the maximum gap ratio at p for f with respect to g, then, for all but (possibly) a finite number of complex a, b 1 properly intersects V (f0 + ag j ) at p, and ∆0 Π f f

i)


p


b 1 · V (f0 + ag j ) . = Π f p

Moreover, if we have the strict inequality that j is greater than the maximum gap ratio at p
for f with respect to g, then i) holds for all non-zero a; in particular, this is the case if j > 1 + ∆0f p . ii) Suppose that Π1f is 1-dimensional at p, and that p ∈ V (f , g). Then, Π1f properly intersects V (f0 + ag j ) at p if and only if there is an equality of germs of sets at p given by V (f1 , . . . , fn , f0 + ag j ) = V (f , g). iii) Suppose that p ∈ V (f , g) and that, at p, there is an equality of germs of sets given by V (f1 , . . . , fn , f0 + ag j ) = V (f , g), the Vogel sets of f all have correct dimension, and that, for all i > 1, V (g) properly intersects each Dfi . b i+1 properly intersects V (f0 + ag j ), the Vogel sets of the (n + 1)-tuple If 1 6 i 6 n, then, at p, Π f j (f1 , . . . , fn , f0 + ag ) have correct dimension, and there is an equality of germs of cycles given by bi b i+1 · V (f0 + ag j ). Π (f1 ,...,fn ,f0 +ag j ) = Πf

Proof.  1 P b 1 is purely one-dimensional at p. Thus, we may write the cycle Π b as mν [ν], i) Recall that Π f f where each ν is a reduced, irreducible curve at p. Let αν (t) denote a local parameterization of ν b 1 properly intersects V (f0 + ag j ) at p, we need to show such that αν (0) = p. Then, to show that Π f

j b 1 ·V (f0 +ag j ) , we need to show that that, for all ν, (f0 +ag )|αν (t) 6≡ 0. To show that ∆0f p = Π f p

b 1 · V (f0 ) = P mν [ν] · V (f0 + ag j ) ; calculating intersection numbers as in A.9 of Appendix Π f

p

p

A, we find that what we need to show is that, for all ν, multt f0 (αν (t)) = multt ((f0 + ag j ) ◦ αν )(t). Thus, we may prove both the proper intersection statement and the intersection formula at the same time by proving this multiplicity statement.

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27

 Clearly, multt ((f0 + ag j ) ◦ αν )(t) = min multt (f0 ◦ αν )(t), multt (g j ◦ αν )(t) , unless the lowest
degree terms of f0 (αν (t)) and −a(g j ◦αν )(t) are precisely equal. As multt (f0 ◦αν )(t) = [ν]·V (f0 ) p
and multt (g j ◦αν )(t) = j [ν]·V (g) p , we conclude that multt ((f0 +ag j )◦αν )(t) = multt (f0 ◦αν )(t) if j is greater than the maximum gap ratio, and that this equality holds when j equals the maximum gap ratio except for the finite number of values of a which would cause cancellation of the lowest degree terms. This proves i). ii) This follows immediately by applying Lemma 3.1.i with the g of the lemma replaced by f0 + ag j . iii) This follows immediately by applying Lemma 3.1.ii with the g of the lemma replaced by f0 + ag j . 

Theorem 3.4 (The Lˆ e-Iomdine-Vogel formulas). Suppose that each Vl appearing in M has
n+1 dimension n + 1. Let f := (f0 , . . . , fn ) ∈ OX , let g ∈ OX , and let p ∈ |M | ∩ V (f , g). Let a be a non-zero complex number, let j > 1 be an integer, and let h := (f1 , . . . , fn , f0 + ag j ). Suppose that the Vogel cycles of f with respect to M are defined at p, and that V (g) properly intersects each of the Vogel cycles, ∆if (M ), at p for all i > 1. If j is greater than or equal to the maximum gap ratio at p for f with respect to g and M , then for all but (possibly) a finite number of complex a, in a neighborhood of p: i) ii)

there is an equality of sets given by |M | ∩ V (h) = |M | ∩ V (f , g),

dimp (|M | ∩ V (h)) = dimp |M | ∩ V (f ) − 1 provided that dimp (|M | ∩ V (f )) > 1,

iii) the Vogel cycles of h with respect to M exist at p, and iv)



∆0h (M ) = ∆0f (M ) + j ∆1f (M ) · V (g) and, for 1 6 i 6 n − 1, ∆ih (M ) = j ∆i+1 (M ) · V (g) . f

Moreover, if we have the strict inequality that j is greater than the maximum gap ratio at p for f with respect to g and M ,
then these equalities hold for all non-zero a; in particular, this is the case if j > 1 + maxl { ∆0f| p }. Vl

Proof. The assumption that all ml have the same sign prevents cancellation of contributions from various Vl ; thus, the assumption that V (g) properly intersects each ∆if (M ) implies that V (g) properly intersects each ∆if| for all l. Therefore, we are reduced to considering the case of Lemma Vl

3.3, where X is irreducible and M equals [X]. Now, the equality of sets in i) is precisely 3.3.ii; the statementSconcerning dimp V (h) follows from this equality of sets, combined with the facts that V (f ) = Dfi (see 2.4) and that V (g) properly intersects the non-zero-dimensional Vogel cycles of V (f ). b i +∆i = Π b i+1 ·V (fi+1 ). By 3.3.iii, this equals Now, suppose that 0 6 i 6 n−1. By definition, Π h h h
b i+1 +∆i+1 ·V (f0 +ag j ). of the Vogel cycles, this equals Π f
f = ∆i+1 · V (ag j ) = j ∆i+1 · V (g) . Therefore, we have f f

b i+2 ·V (f0 +ag j )·V (fi+1 ). By definition Π f ⊆ V (f0 ), ∆i+1 · V (f0 + ag j ) As ∆i+1 f f

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shown that

b i + ∆i = Π b i+1 · V (f0 + ag j ) + j ∆i+1 · V (g) . Π h h f f

(†)

b i = 0, and the first equality of iv) of the theorem follows from (†) and 3.3.i. If i = 0, then Π h
b i ; cancelling Π b i from each b i+1 · V (f0 + ag j ) = Π If 1 6 i 6 n − 1, then 3.3.iii tells us that Π h h f side of (†) yields the second equality of the theorem. 

Remark 3.5. A principal use of the LIV formulas is in families; one requires something about the constancy of the Vogel cycles of f in the family, and the LIV formulas imply the constancy of the Vogel cycles of a tuple of function with a smaller zero locus. However, it is possible to use these formulas “in reverse” – to calculate the Vogel cycles of h(a,j) := (f1 , . . . , fn , f0 + ag j ) and have them tell us about the Vogel cycles of (f0 , . . . , fn ). The difficulty of applying the LIV formulas in this manner is that it is not so easy to know when j is greater than or equal to the maximum gap ratio. We discuss this problem below, using the notation from the theorem. Suppose that the Vogel cycles of f are defined at p, and that V (g) properly intersects each of the Vogel cycles, ∆if , at p for all i > 1. Assume that, in a neighborhood of p, there is an equality of sets given by V (h(a,j) ) = V (f , g) (we are still assuming that a 6= 0). By assuming that V (g) properly intersects ∆if for i > 1, we are assuming that we can calculate the Vogel sets of f in dimensions one and higher. While it would be nice to be able to proceed without this assumption, there seems to be no way to avoid it. Notice that, if we could calculate (∆0f )p , then we would know that the LIV formulas hold for j > (∆0f )p . However, (∆0f )p is typically more difficult to calculate than (∆1f · V (g))p . So, we will assume that we can also calculate the intersection number (∆1f · V (g))p , and then consider the problem of how can one tell when j is large enough for the LIV formulas to hold using data gathered from ∆0h(a,j) and (∆1f · V (g))p . Our best answer is that: if j >

∆0h(a,j)


p

− j ∆1f


p

, then the LIV formulas hold, and so ∆0f


p

= ∆0h(a,j)


p

− j ∆1f


p

.




b 1 · V (f0 + ag j ) . To see this, note that the proof of 3.4 shows that ∆0h(a,j) p − j ∆1f p = Π f p

b 1 · V (f0 ) and so the LIV formulas b 1 · V (f0 + ag j ) , then j > (∆0 )p = Π We claim that, if j > Π f f f p p hold. This is easy; calculating intersection numbers as in the proof of 3.3,
 1

b 1f · V (f0 + ag j ) b f · V (f0 ) , j Π b 1f · V (g) Π > min Π . p p p The desired conclusion follows.


One might hope that if ∆0h(a,j+1) p − ∆0h(a,j) p = ∆1f p (which would be true if the LIV formulas held), then one could, in fact, conclude that the LIV formulas do hold. Unfortunately, the situation is slightly more complicated than this. b 1 at p such that Let us call (a, j) an exceptional pair if there exists a component ν of Π f


j ν · V (f0 + ag ) p 6= min ν · V (f0 ) p , j ν · V (g) p .

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29

Looking at
the proofs of
3.3 and
3.4, it is easy to see that, if (a, j) is not an exceptional pair, then ∆0h(a,j+1) p − ∆0h(a,j) p > ∆1f p with equality if and only if j is greater than or equal to the maximum gap ratio. Hence, if it were not for the existence of exceptional pairs, one could simply
make a table of values of ∆0h(a,j) p for fixed a and increasing j, and when a difference between
successive entries is exactly ∆1f p , one would have identified the maximum gap ratio and would know that the LIV formulas hold beyond that value for j.
On the other hand, if (a, j) is an exceptional pair, it is quite possible that ∆0h(a,j+1) p −

∆0h(a,j) p = ∆1f p and still j is smaller than the maximum gap ratio. Of course, this can only happen once for each possible exceptional pair, and the number of exceptional pairs is certainly no b 1 through p. Thus, if we know the number of components more than the number of components of Π f
1 b through p, call this number c, and we make a table of values of ∆0 of Π f h(a,j) p , once we see a
difference between successive values equalling ∆1f p more than c times, we know that j is high enough for the LIV formulas to hold. Alternatively, and only pseudo-rigorously, if one selects then a will
the constant
a “randomly”,
not be part of an exceptional pair and so, ∆0h(a,j+1) p − ∆0h(a,j) p > ∆1f p with equality if and only if j is greater than or equal to the maximum gap ratio. This approach is particularly well-suited for computer calculation.

The following lemma is related to 3.3 and 3.4 and will be of use to us later. Lemma 3.6 . Suppose that each Vl appearing in M has dimension n + 1, let f := (f0 , . . . , fn ) ∈
n+1 OX , let p ∈ |M | ∩ V (f ), and suppose that the Vogel cycles of f with respect to M at p exist. Let a be a non-zero complex number, and let j > 1 be an integer. Let π denote the projection from C × X to X, and let w denote the projection from C × X to C. Then, the Vogel sets of h := (wj , f1 ◦ π, . . . , fn ◦ π, f0 ◦ π + awj ) with respect to C × M have b i (M ) properly intersects V (f0 ◦ π + awj ), and correct dimension at (0, p), for all i 6 n + 1, C × Π f there is an equality of germs of cycles at (0, p) given by b ih (C × M ) = Π


b if (M ) · V (f0 ◦ π + awj ). C×Π

Proof. As usual, we instantly reduce ourselves to the case where X is irreducible and M = [X]. b i properly intersects V (f0 ◦ π + awj ) is obvious. That C × Π f First, note that V (h) = {0} × V (f ). Suppose that 1 6 i 6 n + 1. Then,
Πih = V (fi ◦ π, . . . , fn ◦ π, f0 ◦ π + awj ) ¬ {0} × V (f ) .
We have V (fi ◦ π, . . . , fn ◦ π) = C × Πif ∪ V (f ◦ π), and V (f ◦ π) ∩ V (f0 ◦ π + awj ) ⊆ {0} × V (f ). Applying 1.3.iii, we find that

Πih = C × Πif ∩ V (f0 ◦ π + awj ) ¬ {0} × V (f ) .
Now, near p, C × Πif ∩ V (f0 ◦ π + awj ) is purely i-dimensional, and – not only does it have no components contained in {0} × V (f ) – in fact, it has no components contained in C × V (f ); for,

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by 2.4, the set Πif ∩ V (f ) equals the union of all of the
Vogel sets of dimension less than or equal to i − 1. Therefore, the set Πih equals the set C × Πif ∩ V (f0 ◦ π + awj ) and, hence, the Vogel sets of h have correct dimension i at (0, p). i b As we saw above, Πh = Πh has no components contained in C × V (f ). Thus, to prove that
j bi bi the cycles
Πh and C
× Πf · V (f0 ◦ π + aw ) are equal, it is enough to prove the equality on C × X − C × V (f ) . Once again, we apply Remark 2.13 and find that both cycles are equal to V (fi ◦ π) · . . . · V (fn ◦ π) · V (f0 ◦ π + awj )

on C × X − C × V (f ) . 

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Chapter 4. SUMMARY OF PART I Let W be analytic subset of an analytic space X and let α be a coherent sheaf of ideals in OX . Let V denote the scheme V (α). Then, the gap sheaf V ¬W is the analytic closure of V − W ; that is, V ¬W is the scheme obtained from V by removing any components or embedded subvarieties contained in W . Let X be a d-dimensional irreducible (though not necessarily reduced) analytic space and let k+1 f := (f0 , . . . , fk ) ∈ (OX ) . The i-th gap variety of f , Πif , is defined as Πif := V (fi+k+1−d , . . . , fk ) ¬ V (f ), e i , is defined as if d − (k + 1) < i < d. Similarly, the i-th modified gap variety of f , Π f e if := V (fi+k+1−d , . . . , fk ) ¬ V (fi+k−d ), Π b i , is defined by downward induction if d − (k + 1) < i < d. The i-th inductive gap variety of f , Π f  b df = X, if f 6≡ 0 Π ∅, if f ≡ 0 and

  b i+1 ∩ V (fi+k+1−d ) ¬ V (fi+k−d ), b if := Π Π f

if d − (k + 1) < i < d. If X is irreducible and Cohen-Macaulay, and each Πif is i-dimensional,then all three types of gap varieties are equal. If X is an arbitrary irreducible space, then, locally, we may replace each member of the tuple f by a “generic” linear combination of the elements of f to obtain a new tuple, a generic linear reorganization of f , for which the gap sheaves, modified gap sheaves, and inductive gap sheaves are all equal.  i b = If X is irreducible of dimension d and each Πif is i-dimensional, then, on X − V (f ), Π f  i b is the closure in X of this cycle on X − V (f ). V (fk ) · V (fk−1 ) · . . . · V (fi+k+1−d ); hence, on X, Π f S If X is a union of irreducible components, X = j Xj , then we do not define gap sheaves, but   only gap cycles. Writing V for the cycle defined by a scheme V , we define the i-th gap cycle  i      i P  e := P Π e of f by Πif := j Πif| , the i-th modified gap cycle of f by Π f f|X , and the i-th j Xj j  i  i  P b := b inductive gap cycle of f by Π f j Πf|X . j P More generally, if we have an analytic cycle M := l ml [Vl ] in X, where all of the ml have the same sign, then we define the various gap cycles relative to M by taking the sum of the appropriate gap cycles restricted to each of the Vl , weighted by the ml . The requirement that all of the ml have the same sign prevents the cancellation of contributions from the various Vl . The modified gap varieties and cycles are merely an intermediate tool. The inductive gap varieties are what we actually use to define (below) our primary objects of study: the Vogel cycles. However, the hypotheses that must be satisfied before we can define the Vogel cycles include, crucially, the hypothesis that each gap set Πif has dimension i. Thus, while one can safely forget the definition of the modified gap varieties, both the gap varieties and inductive gap varieties are important for our future results.

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DAVID B. MASSEY

If X is irreducible of dimension d and each Πif is i-dimensional, then the i-th Vogel cycle of f , is given by      i  b i+1 · V (fi+k+1−d ) − Π b . ∆i = Π f

f

f

If X is a union of irreducible components, then the i-th Vogel cycle is obtained by summing the i-th Vogel cycles of all of the irreducible components (as in the definition of the gap cycles). Similarly, one obtains Vogel cycles with respect to a given cycle M by taking the weighted sum of the Vogel cycles of f restricted to each subvariety appearing in M . If each ∆if is i-dimensional (which one can obtain locally by replacing f by a generic linear i reorganization), then each Vogel ∆f , is purely i-dimensional, non-negative, and is contained S cycle, i in V (f ). Moreover, V (f ) = i ∆f . Thus, we think of the Vogel cycles as decomposing V (f ) on the level of cycles. We proved the important Segre-Vogel Relation: Let X be an irreducible, d-dimensional, analytic k+1 subset of an analytic manifold U, let f = (f0 , . . . , fk ) ∈ (OX ) , let π : Blf X → X denote the blow-up of X along f , and let Ef denote the corresponding exceptional divisor. If Ef properly intersects U × Pm × {0} in U × Pk for all m, then Vogel cycles of f are defined and, in a neighborhood of V (f ), for all i, b i+1 = π∗ (Blf X · (U × Pi+k+1−d × {0})) Π f and ∆if = π∗ (Ef · (U × Pi+k+1−d × {0})), where the intersection takes place in U × Pk and π∗ denotes the proper push-forward. Moreover, for all p ∈ X, there exists an open neighborhood W of p in U such that, for a generic linear reorganization, ˜f , of f , E˜f properly intersects W × Pm × {0} inside W × Pk for all m. In the algebraic category, we may produce such generic linear reorganizations globally, i.e., such that E˜f properly intersects U × Pm × {0} inside U × Pk for all m. What we have just stated is the Segre-Vogel Relation for an irreducible space X, as it appears in Theorem 2.20. We give a more general version with respect to a pure-dimensional cycle in Corollary 2.22. Finally, we derived the Lˆe-Iomdine-Vogel (LIV) formulas: Let X be an irreducible analytic space
n+1 of dimension n + 1, let f := (f0 , . . . , fn ) ∈ OX , let g ∈ OX , and let p ∈ V (f , g). Let a be a non-zero complex number, let j > 1 be an integer, and let h := (f1 , . . . , fn , f0 + ag j ). Suppose that the Vogel cycles of f are defined at p, and that V (g) properly intersects each of the Vogel cycles, ∆if , at p for all i > 1. If j is sufficiently large, then there is an equality of sets given by V (h) = V (f , g), dimp V (h) =
dimp V (f ) − 1 provided that dimp V (f ) > 1, the Vogel cycles of h exist at p, and


∆0h = ∆0f + j ∆1f · V (g) and, for 1 6 i 6 n − 1,
∆ih = j ∆i+1 · V (g) . f

PART I. ALGEBRAIC PRELIMINARIES

33


In particular, if j > 1 + ∆0f p , then these conclusions hold. Once again, there is a more general version of this result with respect to the cycle M .

Fundamental Concepts from Part I: Gap sheaf, V ¬W . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 ei , Π b i ) . . . . . . . . . . . 2.1 Gap variety (resp. modified, inductive), Πif (resp. Π f f Vogel set, Dfi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Correct dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Pseudo-Zariski topology, (resp. inductive) . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.10 Generic linear reorganization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.10 Agreeable reorganization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.11 Vogel cycle, ∆if . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.14 Segre-Vogel relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.22 Vogel reorganization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.24 Unifying reorganization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.24 Gap ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Lˆe-Iomdine-Vogel formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4

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ˆ CYCLES AND HYPERSURFACE SINGULARITIES Part II. LE

Chapter 0. INTRODUCTION The Lˆe numbers and Lˆe cycles generalize the data given by the Milnor number of an isolated hypersurface singularity. In this introduction, we wish to quickly review why the Milnor number of an isolated hypersurface singularity is important. We will then give some previously-known general results on non-isolated hypersurface singularities, and indicate the types of results that can be obtained by the machinery contained in the remainder of Part II. We shall also describe how the results from Part I enter into the development of this Lˆe cycle machinery. Part II deals with the case of hypersurfaces in open subsets of affine space – this is the case described in [Mas6]. The case where the ambient space is itself allowed to be singular is much more difficult, and is the problem addressed in Parts III and IV. While we could, of course, conclude the affine results as a corollary of the more general case, we prefer to describe the affine situation first, and use it as a guide in developing the general case. Let U be an open neighborhood of the origin in Cn+1 and let f : (U, 0) → (C, 0) be an analytic function. Then, the Milnor fibration [Mi3], [Lˆ e7], [Ra] of f at the origin is an object of primary importance in the study of the local, ambient topology of the hypersurface, V (f ) := f −1 (0), defined by f at the origin. Milnor defined his fibration on a sphere of radius
; however, his Theorem 5.11 of [Mi3] leads one to consider a more convenient, equivalent, fibration which lives inside the open ball of radius
. Hence, throughout Part II, we will use the Milnor fibration as defined below.

f

Figure 0.1. The Milnor Fibration inside a ball ◦

For all
> 0, let B
denote the open ball of radius
centered at the origin in Cn+1 . For all η > 0, let Dη denote the closed disc centered at the origin in C, and let ∂Dη denote its boundary, which is a circle of radius η. Then, having fixed an analytic function, f , there exists
0 > 0 such that, for all
such that 0 <


0 , there exists η
> 0 such that, for all η such that 0 < η
η
, 35

36

DAVID B. MASSEY ◦

the restriction of f to a map B
∩ f −1 (∂Dη ) → ∂Dη is a smooth, locally trivial fibration whose diffeomorphism-type is independent of the choice of
and η. This fibration is called the Milnor fibration of f at the origin and the fibre is the Milnor fibre of f at the origin, which we denote by Ff,0 . Hence, the Milnor fibre is a smooth complex n-manifold (of real dimension 2n). The homotopy-type of the Milnor fibre is an invariant of the local, ambient topological-type of the hypersurface at the origin. The Results of Milnor We keep the notation from above; in particular, U is an open neighborhood of the origin in Cn+1 and f : (U, 0) → (C, 0) is an analytic function (actually, for Milnor, f was required to be a polynomial). We will use Σf to denote the critical locus of the map f . In [Mi3], Milnor proved the existence of the object that is now called the Milnor fibration. He also proved that the Milnor fibre, Ff,0 , has the homotopy-type of a finite n-dimensional CWcomplex ([Mi3], Theorem 5.1). This implies that all of the homology groups are finitely-generated, are zero above degree n, and that Hn (Ff,0 ) is free Abelian. In addition, Milnor proved that if f has an isolated critical point at the origin, i.e., dim0 Σf = 0, then Ff,0 is (n − 1)-connected ([Mi3], Lemma 6.4). Combining this with the previous result, it follows ([Mi3], Theorem 6.5) that, in the case of an isolated singularity, the Milnor fibre has the homotopy-type of a finite bouquet (one-point union) of n-spheres; the number of spheres in this bouquet is the Milnor number and is denoted by µ (or µ0 (f ), or some other such variant). In µ particular, the reduced homology is trivial except in degree n, and there the homology group is Z . The Milnor number can be calculated algebraically by taking the dimension as a complex vector space of the algebra O0n+1 /J(f ), where O0n+1 denotes the ring of analytic germs at the origin and ∂f ∂f J(f ) denotes the Jacobian ideal  ∂z , . . . , ∂z . 0 n So that we can do an example, there is one final result of Milnor’s that we wish to mention here. Suppose that f is a weighted homogeneous polynomial (i.e., there exist positive integers r0 , . . . , rn such that f (z0 r0 , . . . , zn rn ) is a homogeneous polynomial). Then, ([Mi3], Lemma 9.4) the Milnor fibre, Ff,0 , is diffeomorphic to f −1 (1). Example 0.2. As an example, consider f = xyz, which defines a hypersurface in C3 consisting of the three coordinate planes. Thus, V (f ) is a hypersurface with a one-dimensional singular set consisting of the three coordinate axes.

Figure 0.3. The coordinate hyperplanes By the above result on weighted homogeneous polynomials, the Milnor fibre is diffeomorphic to

ˆ CYCLES AND HYPERSURFACE SINGULARITIES PART II. LE

37

1 the set of points where xyz = 1; but, this is where x = 0, y = 0, and x = yz . ∗ ∗ ∗ Thus, Ff,0 is homeomorphic to C × C , where C = C − 0. In particular, Ff,0 is homotopyequivalent to the product of two circles, and so has non-zero homology in degrees 0, 1, and 2.

Further Results We wish to consider another classic example: the Whitney umbrella. Example 0.4. The Whitney umbrella is the hypersurface in C3 defined by the vanishing of f = y 2 − zx2 .

Figure 0.5. The Whitney umbrella Here, we have drawn the picture over the real numbers – this is the rarely-seen picture that explains the word “umbrella” in the name of this example. The “handle” of this umbrella is not usually drawn when one is in the complex setting, for the inclusion of this line gives the impression that the local dimension of the hypersurface is not constant; something which is not possible over the complex numbers. A second reason why one rarely sees the above picture is that one frequently encounters the Whitney umbrella as a family of nodes degenerating to a cusp; this representation is achieved by making the analytic change of coordinates z = x + t to obtain f = y 2 − x3 − tx2 (see Example 1.12). To determine the homotopy-type of the Milnor fibre of the Whitney umbrella at the origin, we need a new result. The result we need is that if we have an analytic function g(z0 , . . . , zn ) and a variable y, disjoint from the z’s, then the Milnor fibre of y 2 + g(z0 , . . . , zn ) is homotopy-equivalent to the suspension of the Milnor fibre of g. By an abuse of language, one frequently says that the singularity of y 2 + g(z0 , . . . , zn ) is the suspension of the singularity of g. So, in our example, Fy2 −zx2 ,0 is homotopy-equivalent to the suspension of Fzx2 ,0 . But, as zx2 is homogeneous,     
1 2 ∼  x = 0 ∼ , x Fzx2 ,0 = (z, x) | zx = 1 = = C∗ .  x2 Thus, Fy2 −zx2 ,0 is homotopy-equivalent to the suspension of a circle, i.e., the Milnor fibre of f at the origin is homotopy-equivalent to a 2-sphere.

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DAVID B. MASSEY

The suspension result used above is a special case of a much more general result proved in various forms: first by Sebastiani and Thom in [Se-Th], then by Oka in [Ok] and Sakamoto in [Sak]. This result states that: Theorem 0.6 (Sebastiani-Thom Result). If f : (U, 0) → (C, 0) and g : (U  , 0) → (C, 0) are analytic functions, then the Milnor fibre of the function h : (U × U  , 0) → (C, 0) defined by h(w, z) := f (w) + g(z) is homotopy-equivalent to the join, Ff,0 ∗ Fg,0 , of the Milnor fibres of f and g. This determines the homology of Fh,0 in a simple way, since the reduced homology of the join of two spaces X and Y is given by  j+1 (X ∗ Y ) = H



 k (X) ⊗ H  l (Y ) ⊕ H

k+l=j



  k (X), H  l (Y ) . Tor H

k+l=j−1

Returning now to Example 0.2 where f = xyz, we see that Ff,0 need not have the homotopytype of a bouquet of spheres when the singularity is non-isolated. However, there is the general result of Kato and Matsumoto [K-M]: Theorem 0.7. If s := dim0 Σf , then Ff,0 is (n − s − 1)-connected; in particular, when s = 0, we recover the result of Milnor. Moreover, this is the best possible general bound on the connectivity of the Milnor fibre, as is shown by: Example 0.8. Consider 2 g := z0 z1 . . . zs+1 + zs+2 + · · · + zn2 ;

we leave it as an exercise for the reader to verify, using our earlier methods, that this g has an s-dimensional critical locus at the origin and Fg,0 has non-trivial homology in dimension n − s. Lˆ e’s Attaching Result The result of Kato and Matsumoto can be obtained from a more general result of Lˆe; a result which is one of few which allows calculations concerning the homology of the Milnor fibre for an arbitrary hypersurface singularity. Let U be an open neighborhood of the origin in Cn+1 and let f : (U, 0) → (C, 0) be an analytic function. Let L : Cn+1 → C be a generic linear form. Then, it is easy to see that if dim0 Σf  1, then dim0 Σ(f|V (L) ) = (dim0 Σf ) − 1. Now, the main result of [Lˆ e1] is: Theorem 0.9. The Milnor fibre Ff,0 is obtained from the Milnor fibre Ff|V (L) ,0 by attaching a certain number of n-handles (n-cells on the homotopy level); this number of attached n-handles is

ˆ CYCLES AND HYPERSURFACE SINGULARITIES PART II. LE



given by the intersection number Γ1f,L with respect to L.

39

 · V (f ) , where Γ1f,L denotes the relative polar curve of f 0

We will define the polar curve and discuss how to calculate intersection numbers in Chapter 1, but we can already see that Kato and Matsumoto’s result follows inductively from Theorem 0.9 since we already know Milnor’s result for isolated singularities and because attaching handles of index k does not affect the connectivity in dimensions less than k − 1.

 Not only does Lˆe’s result imply Kato and Matsumoto’s, but – assuming that Γ1f,L · V (f ) is 0 effectively calculable – Lˆe’s result enables the calculation of the Euler characteristic of the Milnor fibre, together with some Morse-type inequalities on the Betti numbers of the Milnor  fibre; for 1 instance, the n-th Betti number, bn (Ff,0 ), must be less than or equal to Γf,L · V (f ) . 0

Unfortunately, the Morse inequalities above are usually far from being equalities. Of course, the real value of Lˆe’s result is that it allows one to calculate some important information even in the cases where the homotopy-type of the Milnor fibre cannot be determined by other means. The Result of Lˆ e and Ramanujam As the homotopy-type of the Milnor fibre is an invariant of the local, ambient topological-type of the hypersurface at the origin, if one has a family of hypersurfaces with isolated singularities in which the local, ambient, topological-type is constant, then the Milnor number must remain constant in the family. In 1976, Lˆe and Ramanujam [L-R] proved the converse of this; we describe their result now. ◦

Let D be an open disc about the origin in C, let U be an open neighborhood of the origin in ◦

◦

Cn+1 , and let f : (D × U, D × 0) → (C, 0) be an analytic function; we write ft for the function defined by ft (z) := f (t, z). Lˆe and Ramanujam proved: Theorem 0.10. Suppose that, for all small t, dim0 Σft = 0 and that the Milnor number of ft at the origin is independent of t. Then, for all small t, i)

the fibre-homotopy type of the Milnor fibrations of ft at the origin is independent of t;

and, if n = 2, ii) the diffeomorphism-type of the Milnor fibrations of ft and the local, ambient, topological-type of V (ft ) at the origin are independent of t. The Result of Lˆ e and Saito

◦

The result of Lˆe and Saito again deals with families of singularities, so we continue with f : ◦

(D × U, D × 0) → (C, 0) as above. The result of [Lˆ e-Sa] tells one how limiting tangent spaces to nearby level hypersurfaces of f approach the singularity. Theorem 0.11. Suppose that, for all small t, dim0 Σft = 0 and that the Milnor number of ft at

40

DAVID B. MASSEY ◦

the origin is independent of t. Then, D × 0 satisfies Thom’s af condition at the origin with respect ◦

to the ambient stratum, i.e., if pi is a sequence of points in D × U − Σf such that pi → 0 and such ◦

that Tpi V (f − f (pi )) converges to some T , then C × 0 = T0 (D × 0) ⊆ T . Generalizing the Milnor Number So, suppose we have a single analytic function, f : (U, 0) → (C, 0) with a critical locus of arbitrary dimension s := dim0 Σf . What properties would we want generalized Milnor numbers of f at 0 to have? First, associated to f , we want there to be s + 1 numbers which are effectively calculable; call the numbers λ0f , . . . , λsf . In the case of an isolated singularity, we want λ0f to be the Milnor number of f and all other λif to be zero. For arbitrary s, we would like to generalize Milnor’s result for isolated singularities and show that the Milnor fibre of f at the origin has a handle decomposition in which the number of attached handles of each index are given by the appropriate λif . Finally, we would like to have generalizations of the results of Lˆe and Ramanujam and Lˆe and Saito to families of hypersurface singularities of arbitrary dimension.

The λ0f , . . . , λsf that we define to achieve these goals are called the Lˆe numbers of f . In order to define the Lˆe numbers, we shall apply the machinery of Part I to the Jacobian ideal 

of f ; if we let ∂f ∂f z := (z0 , . . . , zn ) be coordinates on U, we obtain an ordered (n + 1)-tuple Jz (f ) := ∂z , , . . . ,
i 
0 i  ∂zn and the corresponding Vogel cycles, ∆Jz (f ) i , from Part I (I.2.14) are the Lˆe cycles, Λf,z i , of f with respect to z. We then obtain Lˆe numbers, λ0f,z , . . . , λsf,z , by intersecting these Lˆe cycles with affine linear subspaces defined by the coordinate functions (z0 , . . . , zn ). Thus, our generalization of the Milnor number depends on a choice of coordinates. Nonetheless, as we shall see, these Lˆe numbers have all of the properties that we expect to have in a generalization of the Milnor number to functions with non-isolated critical loci on affine spaces.

ˆ CYCLES AND HYPERSURFACE SINGULARITIES PART II. LE

41

Chapter 1. DEFINITIONS AND BASIC PROPERTIES In this chapter, we define and prove some elementary results about the fundamental objects of study in Part II – the Lˆe cycles and Lˆe numbers. The Lˆe cycles are analytic cycles which, in a sense, decompose the critical locus of an analytic function. The Lˆe numbers are intersection numbers of the Lˆe cycles with certain affine linear subspaces. To define the Lˆe cycles, we first need to define the relative polar cycles, which are the cycles associated to the relative polar varieties. The relative polar varieties were studied by Lˆe and Teissier in a number of places (see, for instance, [Te4], [Te5], and [Te7]). Lˆe and Teissier define the relative polar varieties of a function with respect to generic linear flags, and they usually assume that the flags have been chosen generically enough so that the relative polar varieties have many special properties. However, the whole theory seems to behave more nicely if one does not require the flags to be quite so generic, and then works with possibly non-reduced schemes and cycles. This means that we will define the relative polar varieties and cycles in terms of gap varieties and cycles, and then the Lˆe cycles will be obtained from the corresponding Vogel cycles. The key features of our definition of the relative polar varieties in terms of gap varieties are that the polar varieties are not necessarily reduced and that the dimension of the critical locus of the function is allowed to be arbitrary. The reader who is familiar with the works of Lˆe and Teissier ([Te4], [Te5], [Te7]) should note that we index by the generic dimension instead of the codimension. There is one further difference between our presentation of the relative polar varieties and that of Lˆe and Teissier; instead of fixing a complete flag inside the ambient affine space, we fix a linear choice of coordinates z := (z0 , . . . , zn ) for Cn+1 . We do this because we frequently find it useful to have the linear functions z0 , . . . , zn at our disposal. Let U be an open subset of Cn+1 , let z := (z0 , . . . , zn ) be a linear choice of coordinates for Cn+1 , and let h : (U, 0) → (C,  0) be an analytic function. We write Σh for the critical locus of h, i.e., ∂h ∂h Σh := V ∂z0 , . . . , ∂zn Definition 1.1. For 0
k
n, the k-th (relative) polar variety, Γkh,z , of h with respect to z is the

 ∂h ∂h scheme V ∂z ¬ Σh (see [Mas7], [Mas8], [Mas11]). If the choice of the coordinate , . . . , ∂zn k system is clear, we will often simply write Γkh . If f equals the Jacobian (n + 1)-tuple Jz (h) :=



∂h ∂h ∂z0 , . . . , ∂zn



, then Γkh,z agrees with the gap

variety, Πkf , of f (see Part I, Definition 2.1). Thus, on the level of ideals, Γkh,z consists of those components of the scheme V which are not contained in |Σh|. Note, in particular, that Γ0h,z is empty.



∂h ∂h ∂zk , . . . , ∂zn



Naturally, we define the k-th polar cycle of h with respect to z to be the analytic cycle Γkh,z . This agrees with our previous definition of the gap cycles (of the Jacobian tuple) from Part I, Definition 2.1.

42

DAVID B. MASSEY

Clearly, as sets, ∅ = Γ0h,z ⊆ Γ1h,z ⊆ . . . ⊆ Γn+1 h,z = U. In fact, by I.2.2.ii, we have that :



∂h Proposition 1.2. Γk+1 ¬ Σh = Γkh,z as schemes, and thus all the components of h,z ∩ V ∂zk 

∂h the cycle Γk+1 − Γkh,z are contained in the critical set of the map h. h,z ∩ V ∂zk ∂h ∂h As the ideal  ∂z , . . . , ∂z  is invariant under any linear change of coordinates which leaves n k V (z0 , . . . , zk−1 ) invariant, we see that the scheme Γkh,z depends only on h and the choice of the first k coordinates. At times, it will be convenient to subscript the k-th polar variety with only the first k coordinates instead of the whole coordinate system; for instance, we write Γ1h,z0 for the polar curve.

While it is from the number of defining equations that every component of the  immediate    analytic set Γkh,z  has dimension at least k, one usually requires that the coordinate system be suitably generic so that the dimension of Γkh,z equals k. In this case, we have the following: Proposition 1.3. If dimp Γkh,z = k, then Γkh,z has no embedded subvarieties through the point p. Proof. This is immediate from I.1.5.  Proposition 1.4. If dimp Σh < k, then Γkh,z and V



∂h ∂h ∂zk , . . . , ∂zn

 are equal up to embedded

subvariety and, hence, are equal as cycles at p. If dimp Σh < k and dimp Γkh,z = k, then Γkh,z and

 ∂h ∂h V ∂z are equal as schemes at p. , . . . , ∂zn k If f equals the Jacobian (n + 1)-tuple, Jz (h), and |Γkh,z | is purely k-dimensional for all k, then,  k , and Π  k are all equal as schemes. for all k, Γkh,z and the various gap schemes, Πkf , Π f f

Proof. As schemes, V := V

∂h ∂h ∂zk , . . . , ∂zn

 consists of the components not contained in Σh –

Γkh,z

these comprise – together with those contained in Σh. By the number of defining equations, every isolated component of V must have dimension at least k. Thus, if dimp Σh < k, the only  components of V which are contained in Σh must be embedded. Therefore, V

∂h ∂h ∂zk , . . . , ∂zn

equals Γkh,z up to embedded subvariety and, hence, they are equal as cycles. But this certainly implies that Γkh,z and V are equal as germs of sets at p. Thus, if dimp Γkh,z = k, then dimp V = k, i.e., V is a local complete intersection at p. Hence, V has no embedded subvarieties at p. The second statement follows. Since OU is Cohen-Macaulay, the final statement follows immediately from I.2.9 (since, by definition, Γkh,z = Πkf ).  Definition 1.5. If the intersection of Γkh,z and V (z0 −p0 , . . . , zk−1 −pk−1 ) is purely zero-dimensional

ˆ CYCLES AND HYPERSURFACE SINGULARITIES PART II. LE

43

at a point p = (p0 , . . . , pn ) (i.e., either p is an isolated point of the intersection or p is not in the k k intersection), then we say that the k-th polar number, γh,z (p), is defined and we set γh,z (p) equal to the intersection number 

 Γkh,z · V (z0 − p0 , . . . , zk−1 − pk−1 ) p .

(We use the term polar numbers, instead of polar multiplicities, since we are not assuming that the coordinates are so generic that this intersection number gives the multiplicities.) k Note that, if γh,z is defined at p, then it must be defined at all points near p. Note also that, k if γh,z (p) is defined, then Γkh,z must be purely k-dimensional at p and so – by 1.3 – Γkh,z has no embedded subvarieties at p.

Remark 1.6. As sets,  Σ(h|V (z0 −p0 ,...,zk−1 −pk−1 ) ) = V

z0 − p0 , . . . , zk−1 − pk−1 ,

∂h ∂h ,..., ∂zk ∂zn



  = V (z0 − p0 , . . . , zk−1 − pk−1 ) ∩ Σh ∪ Γkh,z . k Hence, if γh,z (p) is defined and p ∈ Σh, then

Σ(h|V (z0 −p0 ,...,zk−1 −pk−1 ) ) = V (z0 − p0 , . . . , zk−1 − pk−1 ) ∩ Σh at p.

We now wish to define the Lˆe cycles. Unlike the polar varieties and cycles, the Lˆe cycles are supported on the critical set of h itself. These cycles demonstrate a number of properties which generalize the data given by the Milnor number for an isolated singularity.

Definition 1.7. For 0
k
n, we define the k-th Lˆe cycle of h with respect to z, Λkh,z , to be      ∂h k+1 Γh,z ∩ V − Γkh,z . ∂zk   If the choice of coordinate system is clear, we will sometimes simply write Λkh . Also, as we have given the Lˆe cycles no structure as schemes, we will usually omit the brackets and write Λkh,z to denote the Lˆe cycle – unless we explicitly state that we are considering it as a set only. Note that, as every component of Γk+1 at least k + 1, every component of Λkh,z h,z has dimension  

  has dimension at least k. We say that the cycle Λkh,z or the set Λkh,z  has correct dimension at     a point p provided that Λkh,z  is purely k-dimensional at p. We define the k-th Lˆe number of h at p with respect to z, λkh,z (p), to equal the intersection num

 ber Λkh,z · V (z0 − p0 , . . . , zk−1 − pk−1 ) , provided this intersection is purely zero-dimensional at p

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DAVID B. MASSEY

p. If this intersection is not purely zero-dimensional at p, then we say that the k-th Lˆe number (of h at p with respect to z ) is undefined. Here, when k = 0, we mean that   λ0h,z (p) = Λ0h,z · U p =          ∂h ∂h Γ1h,z ∩ V = Γ1h,z · V . ∂z0 p ∂z0 p (This last equality holds whenever Γ1h,z is one-dimensional at p, for then Γ1h,z has no embedded

 ∂h subvarieties by 1.7 and Γ1h,z ∩ V ∂z is zero-dimensional. See Appendix A.4.) 0     Note that if λkh,z (p) is defined, then λkh,z is defined at all points near p and Λkh,z  must have correct dimension at p. Also note that, since Γk+1 and Γkh,z depend only on the choice of the h,z coordinates z0 through zk , the k-th Lˆe cycle, Λkh,z , depends only on the choice of (z0 , . . . , zk ). Finally, note that if h is a polynomial, then since we are taking linear coordinates, we remain inside the algebraic category. Remark 1.8. We have defined the Lˆe cycles as we did in all of our previous work  ∂h(see, for∂hinstance,  [Mas6]). We wish to see that this agrees with the Vogel cycles of Jz (f ) := V ∂z , . . . , ∂zn under 0 reasonable hypotheses (see 1.9, below). Note, however, that it follows from the definitions that the sets underlying the Lˆe cycles, |Λif,z |, are always equal to the Vogel sets DJi z (f ) . Proposition 1.9. If dim Γkf,z is purely k-dimensional for all k  0, then, for all k  0, the k-th Lˆe cycle of f with respect to z is equal to the k-th Vogel cycle of the Jacobian (n + 1)-tuple of f with respect to z, i.e., Λkf,z = ∆kJz (f ) . Proof. This follows immediately from the second paragraph of Proposition 1.4, Definition I.2.14, and Proposition I.2.12. 

While we shall defer most of our examples until later – when we will have more results to play with – it is instructive to include at least one at this early stage. Example 1.10. Let h = y 2 − x3 − tx2 ; this is the Whitney umbrella of Example 0.4, but written as a family of nodes degenerating to a cusp. We fix the coordinate system z = (t, x, y) and will suppress any further reference to it. We find Σh = V (−x2 , −3x2 − 2tx, 2y) = V (x, y). Thus, the critical locus of h is one-dimensional and consists of the t-axis.   Now the critical locus is one-dimensional, while the dimension of every component of V ∂h ∂y is

 ∂h at least two. Hence, V ∂y cannot possibly have any components contained in Σh and, therefore,

ˆ CYCLES AND HYPERSURFACE SINGULARITIES PART II. LE

we begin calculating polar varieties with Γ2h . We have simply  Γ2h

=V

∂h ∂y

 = V (2y) = V (y)

with no components to dispose of.

1

Γh, z 1

Λ h, z Γh,2 z

0

Λ h, z

Figure 1.11. Polar and Lê cycles

Next, we have  Γ2h ∩ V

∂h ∂x

 = V (y) ∩ V (−3x2 − 2tx) = V (y, −3x2 − 2tx).

Applying 1.2 and then I.1.3.iv, we find    ∂h Γ1h = Γ2h ∩ V ¬ Σh = V (y, −3x2 − 2tx) ¬ V (x, y) = V (y, −3x − 2t). ∂x From the definition of the Lˆe cycles (1.11), we obtain   Λ1h = V (y, −3x2 − 2tx) − [V (y, −3x − 2t)] =

 [V (y, x)] + [V (y, −3x − 2t)] − [V (y, −3x − 2t)] = [V (y, x)] .

Thus, Λ1h has as its underlying set the t-axis, and this axis occurs with multiplicity 1. Now we find      ∂h 0 1 Λ h = Γh ∩ V = V (y, −3x − 2t) ∩ V (−x2 ) = 2 [V (t, x, y)] = 2[0]. ∂t

45

46

DAVID B. MASSEY

Finally, we calculate the Lˆe numbers: λ1h (0) = (V (y, x) · V (t))0 = 1 and clearly λ0h (0) = 2.

Proposition 1.12. The  Lˆe cycles are all non-negative and are contained in the critical set of    h. Every component of Λkh,z  has dimension at least k. If s = dimp Σh then, for all k with         s < k < n + 1, p is not contained in Λkh,z , i.e., Λkh,z  is empty at p; thus, for s < k < n + 1, λkh,z (p) is defined and equal to 0. Proof. The first statement follows from 1.2. The second statement follows from the definition of the Lˆe cycles and the fact that every component of Γk+1 h,z has dimension at least k + 1. The third statement follows from the first two.  Due to the result of 1.12, we usually only consider λ0h,z (p), . . . , λsh,z (p). Proposition 1.13. As sets, for all k, Γk+1 ∩ Σh = h,z i dimp Σh, as germs of sets at p, Σh = i
s Λh,z .

 i
k

Λih,z . In particular, letting k = s :=

Proof. This follows immediately from I.2.4. 

1 Recall from Remark 1.6 that, if γh,z (p) is defined and p ∈ Σh, then

Σ(h|V (z0 −p0 ) ) = V (z0 − p0 ) ∩ Σh at p. This is especially useful for inductive proofs when combined with the easy: Proposition 1.14. If s := dimp Σh  1, Λih,z has correct dimension at p for all i
s − 1 (by this, we mean to allow that p may not be contained in some of the Λjh,z ’s), and λsh,z (p) is defined, then dimp (Σh ∩ V (z0 − p0 )) = s − 1. Proof. As we are assuming that Λih,z has correct dimension at p for all i
s − 1, it follows from 1.13 that we have only to show that the hyperplane slice V (z0 − p0 ) actually reduces the dimension of Λsh,z . But, this must be the case, since Λsh,z ∩ V (z0 − p0 , . . . , zs−1 − ps−1 ) is zero-dimensional at p.  Proposition 1.15. Fix an integer k  0. Suppose that p ∈ Σh. If, is purely j-dimensional at p, then, for all j such that 1
j
k + 1,

for all j with 0
j
k, Λjh,z  j  Γ  is purely j-dimensional h,z

ˆ CYCLES AND HYPERSURFACE SINGULARITIES PART II. LE

47

at p, and the cycles   

  ∂h  ∂h j j Γh,z ∩ V and Γh,z · V ∂zj−1 ∂zj−1 are equal at p. In addition, if for all j with 0
j
k, Λjh,z is purely j-dimensional at p, then, for 0
j
k+1,     every j-dimensional (isolated) component of the critical locus of h through p is contained in Λjh,z . Proof. Since Γ0h,z is empty, we may inductively apply the final statement in Remark I.2.6 to   conclude that Γjh,z  is purely j-dimensional for all j
k + 1; it follows from 1.3 that Γjh,z has no embedded subvarieties through p for j
k + 1.  

     = Γj−1  ∪ Now, for 1
j
k + 1, Γjh,z  is purely j-dimensional at p, and Γjh,z ∩ V ∂z∂h h,z j−1

  j−1  ∂h Λ  is purely (j − 1)-dimensional at p. Thus, V contains no components or embedded h,z ∂zj−1 subvarieties of Γjh,z , and therefore we may apply Appendix A.4 to conclude that   

  ∂h  ∂h Γjh,z ∩ V and Γjh,z · V ∂zj−1 ∂zj−1 are equal at p. The last statement follows immediately from Proposition I.2.4 

In practice, we use the first part of 1.15 to calculate the Lˆe cycles as follows: Assume for the moment that all the Lˆe cycles have the correct dimension, and let s denote the dimension of the the critical locus of h. Then, in a neighbor hood of the critical locus,   

∂h ∂h Γs+1 = V ; , . . . , h,z ∂zs+1 ∂zn

  ∂h      Γs+1 = Γsh,z + Λsh,z ; h,z · V ∂zs   



 s  ∂h s−1 Γh,z · V = Γs−1 + Λ h,z h,z ; ∂zs−1 .. .

    2      ∂h Γh,z · V = Γ1h,z + Λ1h,z ; ∂z1     1    ∂h Γh,z · V = Λ0h,z . ∂z0

In each line above, one obtains Γkh,z from the calculation in the previous line. Now, to write the above equalities, we have used that the Lˆe cycles have correct dimension – but, in any case, the equalities are true for sets (using intersection and union of sets, of course).

48

DAVID B. MASSEY

And so, after doing the above calculations, one verifies that the cycles that we have written above as the Lˆe cycles do, in fact, have the correct dimension and thus the equalities are correct. On the other hand, if the cycles that we have written above as the Lˆe cycles do not have the correct dimension, then the equalities above may be false. Remark 1.16. As we will see in Example 2.1, in the case of an isolated singularity, λ0h,z is nothing other than the Milnor number. In the general case, it is tempting to think of λ0h,z (p) as the local (generic) degree of the Jacobian map of h at p, i.e., the number of points in   ◦ ∂h ∂h B
∩ V − a0 , . . . , − an , ∂z0 ∂zn ◦

where B
is a small open ball centered at p and a is a generic point with length that is small compared to
; unfortunately, there is no such local degree. Consider the example h = z22 + (z0 − z12 )2 and let p be the origin.

Figure 1.17.

Then,

◦



B
∩ V

The hypersurface defined by h

∂h ∂h ∂h − a0 , − a1 , − a2 ∂z0 ∂z1 ∂z2

 =

◦

B
∩ V (2(z0 − z12 ) − a0 , 2(z0 − z12 )(−2z1 ) − a1 , 2z2 − a2 ). The solutions to these equations are z0 =

a0 a2 + 12 , 2 4a0

z1 = −

a1 , 2a0

z2 =

a2 . 2

The number of solutions of these equations inside any small ball does not just depend on picking small, generic a0 , a1 , and a2 , but also depends on the relative sizes of a0 and a1 . If a1 is small

ˆ CYCLES AND HYPERSURFACE SINGULARITIES PART II. LE

49

relative to a0 , then there will be one solution inside the ball; if a0 is small relative to a1 , then there will be no solutions inside the ball. Do either of these numbers actually agree with λ0h,z (0)? Yes; with these coordinates, λ0h,z (0) = 1. This can be seen from the above calculations together with the discussion below, which shows how “close” λ0h,z is to being the generic degree of the Jacobian map of h. We claim that, if dimp Γ1h,z = 1, then λ0h,z (p) exists and equals the number of points in 

◦

B
∩ V

 ∂h ∂h − a0 , . . . , − an , ∂z0 ∂zn

◦

where B
is a small open ball centered at p, a0 = 0 is small compared to
, and a1 , . . . , an are generic, with length that is small compared to that of a0 . To see this, note that this number of points equals the sum of the intersection numbers given by      ∂h ∂h ∂h V · V ,..., − a0 ∂z1 ∂zn ∂z0 q q where the sum is over all q in ◦



B
∩ V

∂h ∂h ∂h − a0 , ,..., ∂z0 ∂z1 ∂zn

 .

But, for a0 = 0, these points, q, do not occur on the critical locus of h, and so this sum equals    ∂h 1 Γh,z · V − a0 . ∂z0 q q This last sum is none other than

 λ0h,z (p) =

 Γ1h,z · V

∂h ∂z0

 . p

It is also possible to give a more intuitive characterization of λsh,z (p) where s = dimp Σh. Assuming that λsh,z (p) exists, by moving to a generic point, it is trivial to show that λsh,z (p) =



◦

nν µν ,

ν

where ν runs over all s-dimensional components of Σh at p, nν is the local degree of the map ◦ (z0 , . . . , zs−1 ) restricted to ν at p, and µν denotes the generic transverse Milnor number of h along the component ν in a neighborhood of p. In particular, if the coordinate system is generic enough so that nν is actually the multiplicity of ν at p for all ν, then λsh,z (p) is merely the multiplicity of the Jacobian scheme of h (the scheme defined by the vanishing of the Jacobian ideal) at p.

The next proposition tells us how the Lˆe numbers behave under the taking of hyperplane sections – a fundamental result; the statements concerning cycles could be derived from the Restriction

50

DAVID B. MASSEY

Lemma (I.3.1), but we want to use polar and Lˆe numbers both in the hypotheses and the conclusions. ˜ = Proposition 1.18. Suppose Σh ∩ V (z0 − p0 ) = Σ(h|V (z0 −p0 ) ), and use the coordinates z k (z1 , . . . , zn ) for V (z0 − p0 ). Let k  1 and suppose that γh,z (p) and λkh,z (p) are defined. Then, γhk−1 |

V (z0 −p0 )

,˜ z (p)

and λk−1 h|

V (z0 −p0 )

Γkh| Γk−1 h|

V (z0 −p0

,˜ z )

,˜ z (p)

V (z0 −p0 )

,˜ z

are defined, = Γk+1 h,z · V (z0 − p0 ),

+ Λk−1 h|

V (z0 −p0

,˜ z )

  = Γkh,z + Λkh,z · V (z0 − p0 ),

and γhk−1 |

V (z0 −p0 )

,˜ z (p)

+ λk−1 h|

V (z0 −p0 )

,˜ z (p)

k = γh,z (p) + λkh,z (p).

1 In the special case when k = 1, it follows that if γh,z (p) and λ1h,z (p) are defined, then so is λ0h| ,˜ z (p), and V (z0 −p0 )

λ0h|

V (z0 −p0 )

,˜ z (p)

1 = γh,z (p) + λ1h,z (p).

k+1 k Moreover, we conclude that if k  1 and γh,z (p), λkh,z (p), γh,z (p), and λk+1 h,z (p) are defined, then so are

γhk−1 |

V (z0 −p0 )

,˜ z (p),

λk−1 h|

V (z0 −p0 )

,˜ z (p),

γhk|

V (z0 −p0 )

,˜ z (p),

and λkh|

V (z0 −p0 )

,˜ z (p),

and Γkh| Λkh|

V (z0 −p0 )

V (z0 −p0 )

,˜ z

= Γk+1 h,z · V (z0 − p0 ),

,˜ z

= Λk+1 h,z · V (z0 − p0 ),

and so γhk|

V (z0 −p0 )

,˜ z (p)

k+1 = γh,z (p),

,˜ z (p)

= λk+1 h,z (p).

and λkh|

V (z0 −p0 )

k Proof. Clearly, it suffices to prove the assertions for p = 0. The assumption that γh,z (0) and k λh,z (0) are defined is equivalent to   ∂h dim0 Γk+1 ∩ V (z0 , . . . , zk−1 )
0. ∩ V h,z ∂zk

Hence, Γk+1 at the origin and thus has no embedded subvarieties h,z is purely (k + 1)-dimensional 

k+1 ∂h (Proposition 1.3). Also, Γh,z ∩ V ∂zk is purely k-dimensional at the origin and so, by Appendix A.4, we have an equality of cycles      ∂h ∂h k+1 k+1 Γh,z ∩ V = Γh,z · V = Γkh,z + Λkh,z . ∂zk ∂zk

ˆ CYCLES AND HYPERSURFACE SINGULARITIES PART II. LE

51





∂h In addition, we see that Γk+1 h,z ∩ V ∂zk ∩ V (z0 ) is purely (k − 1)-dimensional at the origin; we easily conclude that dim0 Γk+1 h,z ∩ Σh ∩ V (z0 )
k − 1.

Now, let us consider the cycle Γkh|

V (z0 )

,˜ z.

 Γkh|

V (z0 )

,˜ z

=V

z0 ,

By definition, ∂h ∂h ,..., ∂zk+1 ∂zn

 ¬ Σ(h|V (z0 ) ).

Using Lemma I.1.3.ii and our hypothesis that Σh ∩ V (z0 ) = Σ(h|V (z0 ) ), the equality above gives us  



k+1 Γkh| ¬ (Σh ∩ V (z0 )) = V (z0 ) ∩ Γk+1 ¬ Σh. ,˜ z = V (z0 ) ∩ Γh,z h,z V (z0 )

But, V (z0 ) ∩ is purely k-dimensional at the origin and, as we saw earlier, dim0 Γk+1 h,z ∩ Σh ∩ k+1 V (z0 )
k − 1; therefore, Σh contains no isolated components of V (z0 ) ∩ Γh,z and so, as cycles, Γk+1 h,z

Γkh|

V (z0 )

,˜ z

k+1 = Γk+1 h,z ∩ V (z0 ) = Γh,z · V (z0 ).



We find Γk−1 h|

V (z0 )

k−1 ,˜ z + Λh|

V (z0 )



Γk+1 h,z · V (z0 ) · V That γhk−1 |

V (z0 )

,˜ z (0)

,˜ z

∂h ∂zk

= Γkh| 

V (z0 )

,˜ z·V

,˜ z (0)

are defined and that

γhk−1 |

,˜ z (0)

+ λk−1 h|

V (z0 )

V (z0 )

 =

  = Γkh,z + Λkh,z · V (z0 ).

and λk−1 h|

V (z0 )

∂h ∂zk

,˜ z (0)

k = γh,z (0) + λkh,z (0)

follows by intersecting the cycle V (z1 , . . . , zn ) with each side of the above equality of cycles. The remaining equalities follow easily – we leave them as an exercise.  The following corollary is essentially a converse of the result stated in Remark 1.6. Corollary 1.19. Let k  0. Suppose Σh ∩ V (z0 − p0 , . . . , zk − pk ) = Σ(h|V (z0 −p0 ,...,zk −pk ) ), k+1 i and that γh,z (p) and λih,z (p) are defined for all i
k. Then, γh,z (p) is defined. In particular, if s := dimp Σh, λih,z (p) is defined for 0
i
s, and, for all k such that 0
k
n − 1, Σh ∩ V (z0 − p0 , . . . , zk − pk ) = Σ(h|V (z0 −p0 ,...,zk −pk ) ), i then γh,z (p) is defined for 0
i
n.

0 Proof. The last statement follows immediately from the first by induction, since γh,z (p) is always defined (and is zero).

52

DAVID B. MASSEY

It suffices to prove the first statement when p = 0. Case 1:

If 0 ∈ Σh, then near 0,  Γk+1 h,z

=V

∂h ∂h ,..., ∂zk+1 ∂zn



and so Γk+1 h,z ∩ V (z0 , . . . zk ) = Σ(h|V (z0 ,...,zk ) ) = Σh ∩ V (z0 , . . . , zk ) = ∅. k+1 Hence, γh,z (0) is defined and equal to zero.

Case 2:

0 ∈ Σh. The proof is by induction on k.

For k = 0, the claim is that if 0 ∈ Σh, λ0h,z (0) is defined, and Σh ∩ V (z0 ) = Σ(h|V (z0 ) ), then dim0 Γ1h,z ∩ V (z0 )
0. As 0 ∈ Σh and λ0h,z (0) is defined, we must have that dim0 Γ1h,z
1. So, if dim0 Γ1h,z ∩ V (z0 )  1, then V (z0 ) must contain a component of Γ1h,z through the origin. But, since Σh ∩ V (z0 ) = Σ(h|V (z0 ) ),  Γ1h,z

∩ V (z0 ) ⊆ V

∂h ∂h z0 , ,..., ∂z1 ∂zn



 =V

∂h ∂h z0 , ,..., ∂z0 ∂zn

 .

Hence, any component of Γ1h,z contained in V (z0 ) must also be contained in Σh; this contradicts the definition of Γ1h,z . Suppose now that the corollary is true up to k − 1, where k  1. Suppose 0 ∈ Σh, Σh ∩ i V (z0 , . . . , zk ) = Σ(h|V (z0 ,...,zk ) ), and that γh,z (0) and λih,z (0) are defined for all i
k. As 0 ∈ Σh i and γh,z (0) is defined for all i
k, Remark 1.6 implies that Σh ∩ V (z0 , . . . , zi ) = Σ(h|V (z0 ,...,zi ) ) for all i
k − 1. In particular, as k  1, Σh ∩ V (z0 ) = Σ(h|V (z0 ) ). Thus, we may apply Proposition 1.18 to i conclude that γhi | ,˜ z (0) and λh| ,˜ z (0) are defined for all i
k − 1 and, as sets, V (z0 )

V (z0 )

Γkh|

V (z0 )

,˜ z

= Γk+1 h,z ∩ V (z0 ).

Since Σ(h|V (z0 ) ) ∩ V (z1 , . . . , zk ) = Σ(h|V (z0 ,...,zk ) ), we are in a position to apply our inductive hypothesis to h|V (z0 ) . We conclude that γhk| ,˜ z (0) is defined, i.e., V (z0 )

dim0 Γkh| As Γkh|

V (z0 )

,˜ z

V (z0 )

,˜ z

∩ V (z1 , . . . , zk )
0.

= Γk+1 h,z ∩ V (z0 ), the proof is finished. 

We shall need the following relation between three intersection numbers. For isolated singularities, this formula appears in the proof of Proposition II.1.2 of [Te2] – our argument is essentially the same.

ˆ CYCLES AND HYPERSURFACE SINGULARITIES PART II. LE

53

Proposition 1.20. Let p ∈ Σh. Then, λ0h,z (p) is defined if and only if dimp Γ1h,z
1. 1 Moreover, if γh,z (p) is defined, then λ0h,z (p) is defined, the dimension of Γ1h,z ∩ V (h − h(p)) at p is at most zero, and  1  1 Γh,z · V (h − h(p)) p = λ0h,z (p) + γh,z (p).



∂h ∂h Proof. Γ1h,z consists of those components of V ∂z which are not contained in |Σh|. , . . . , ∂zn 1

 ∂h Thus, V ∂z contains no components of Γ1h,z . Therefore, Γ1h,z is purely one-dimensional at p if 0

 ∂h and only if Γ1h,z ∩ V ∂z is purely zero-dimensional at p, i.e., if and only if λ0h,z (p) is defined. 0 1 (p) is defined, then Γ1h,z must be purely one-dimensional at p and so λ0h,z (p) is defined, If γh,z by the above. The remainder of the proof is an argument which first appeared in Proposition II.1.2 of [Te2], and then appeared again in Proposition 1.3 of [Lˆ e1]; this argument shows that an easy application of the chain rule yields dimp Γ1h,z ∩ V (h − h(p))
0, and  1  1 Γh,z · V (h − h(p)) p = λ0h,z (p) + γh,z (p).

For convenience, weassume that p = 0 and that h(0) = 0. Suppose Γ1h,z = mW [W ] as cycles. We know that we can calculate the intersection number of a curve and a hypersurface by parameterizing the curve and looking at the multiplicity of the composition of the defining function of the hypersurface with the parameterization. So, for each component W , pick a local analytic parameterization α(t) of W such that α(0) = 0. We must show two things: that h(α(t)) is not identically zero, and that   ∂h multt h(α(t)) = multt + multt z0 (α(t)). ∂z0 |α(t) As we already know that the righthand side of the above equality is finite, we have only to prove that the equality holds in order to conclude that h(α(t)) is not identically zero. But this is easy:    ∂h d  multt h(α(t)) = 1 + multt {h(α(t))} = 1 + multt · α0 (t) , dt ∂z0 |α(t) where the remaining terms that come from the chain rule are zero since α(t) parameterizes a component of the polar curve. Thus,   ∂h multt h(α(t)) = 1 + multt + multt α0 (t) ∂z0 |α(t)  = multt

∂h ∂z0



 + multt α0 (t) = multt |α(t)

∂h ∂z0

 + multt z0 (α(t)) |α(t)

and we are finished. 

Of course, what we want to know is that, for a generic choice of coordinates, z, the polar numbers and the Lˆe numbers are actually defined. Our results in Part I – specifically, I.2.11 and

54

DAVID B. MASSEY

I.2.22 – tell us that, by replacing z by a generic linear reorganization, we can guarantee that the polar and Lˆe cycles exist and have the correct dimension. However, we must still do some work to know that the polar and Lˆe numbers are defined generically. To show this, we pick z generically with respect to a certain type of stratification of the hypersurface defined by h. Below, we define the type of stratification that we need, together with a proposition guaranteeing its existence. Definition 1.21. A good stratification for h at a point p ∈ V (h) is an analytic stratification, G, of the hypersurface V (h) in a neighborhood, U, of p such that the smooth part of V (h) is a stratum and so that the stratification satisfies Thom’s ah condition with respect to U − V (h). That is, if qi is a sequence of points in U − V (h) such that qi → q ∈ S ∈ G and Tqi V (h − h(qi )) converges to some hyperplane T , then Tq S ⊆ T . Proposition 1.22 (Hamm and Lˆe [H-L]). There exists a good stratification for all h : (U, 0) → (C, 0) at all p ∈ V (h).

The notion defined below, that of prepolar coordinates, is crucial throughout the remainder of Part II. It provides a generic condition on linear choices of coordinates which implies that all the Lˆe numbers and polar numbers are defined. Moreover, prepolarity seems to be the right condition to obtain many topological results. The importance of this definition cannot be overstated. Definition 1.23. Suppose that {Sα } is a good stratification for h in a neighborhood, U, of the origin. Let p ∈ V (h). Then, a hyperplane, H, in Cn+1 through p is a prepolar slice for h at p with respect to {Sα } provided that H transversely intersects all the strata of {Sα }– except perhaps the stratum {p} itself – in a neighborhood of p. If H is a prepolar slice for h at p with respect to {Sα }, then, as germs of sets at p, Σ(h|H ) = (Σh) ∩ H and dimp Σ(h|H ) = (dimp Σh) − 1 provided dimp Σh  1; moreover, {H ∩ Sα } is a good stratification for h|H at p (see [H-L]). By 2.1.3 of [H-L], for a fixed good stratification for h, prepolar slices are generic. We say simply that H is a prepolar slice for h at p provided that there exists a good stratification with respect to which H is a prepolar slice. Let (z0 , . . . , zn ) be a linear choice of coordinates for Cn+1 , let p ∈ V (h), and let {Sα } be a good stratification for h at p. For 0
i
n, (z0 , . . . , zi ) is a prepolar-tuple for h at p with respect to {Sα } if and only if V (z0 − p0 ) is a prepolar slice for h at p with respect to {Sα } and for all j such that 1
j
i, V (zj − pj ) is a prepolar slice for h|V (z0 −p0 ,...,zj−1 −pj−1 ) at p with respect to {Sα ∩ V (z0 − p0 , . . . , zj−1 − pj−1 )}. As prepolar slices are generic, a generic linear reorganization of z will produce a prepolar-tuple. Naturally, we say that (z0 , . . . , zi ) is a prepolar-tuple for h at p provided that there exists a good stratification for h at p with respect to which (z0 , . . . , zi ) is a prepolar-tuple. Finally, we say that the coordinates (z0 , . . . , zn ) are prepolar for h if and only if for all p ∈ V (h), if s denotes dimp Σh, then (z0 , . . . , zs−1 ) is a prepolar-tuple for h at p (if s = 0 or p ∈ Σh, we mean that there is no condition on the coordinates.) Note that, as prepolar for h is a condition at all points in Σh, it is not immediate that such coordinates exist (we shall, however, prove this in 10.2.)

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Remark 1.24. It will be helpful to interpret good stratifications and prepolar-tuples in terms of π conormal geometry and blowing-up the Jacobian tuple Jz (h). Let BlJz (h) U − → U denote the blowup; BlJz (h) U ⊆ U × Pn . Let E denote the corresponding exceptional divisor. We identify the projectivized cotangent space P(T ∗ U) with U × Pn . Then, a good stratification for h at the origin is a stratification {Sα } of V (h) in a neighborhood of 0 such that the smooth part of V (h) is a stratum and such that, for all Sα , π −1 (Sα ) ⊆ P(TS∗α U). The tuple (z0 , . . . , zk ) is prepolar at the origin with respect to {Sα } if and only if, for all i with 0
i
k, for all Sα with dim Sα  i + 1,   P(TS∗α U) ∩ V (z0 , . . . , zi ) × Pi × {0} = ∅.

We will show that by choosing coordinates which are prepolar, one guarantees the existence of the Lˆe and polar numbers. First, we need a lemma. We use the notation from Remark 1.24. Lemma 1.25. Suppose that (z0 , . . . , zk ) is a prepolar tuple for h at the origin. Then, over a neighborhood of the origin, for all i with 0
i
k,   E ∩ V (z0 , . . . , zi−1 ) × Pi × {0} ⊆ {0} × Pi × {0}.   (When i = 0, we mean that E ∩ U × P0 × {0} ⊆ {0} × P0 × {0}.) Proof. If (z0 , . . . , zk ) is a prepolar tuple, then (z0 , . . . , zi ) is a prepolar tuple for all i such that 0
i
k; thus, we only need to prove the claim when i = k. We use our characterizations from Remark 1.24. Let {Sα } be a good stratification for h at the origin. We first show that it suffices to prove the claim with zi−1 replaced by zi ; more precisely we show that: for all Sα , in a neighborhood of the origin, for all i such that 0
i
n,       P(TS∗α U)∩ V (z0 , . . . , zi−1 )×Pi ×{0} ⊆ V (z0 , . . . , zi )×Pi ×{0} ∪ V (z0 , . . . , zi−1 )×Pi−1 ×{0} .     (When i = 0, we mean that P(TS∗α U) ∩ U × Pi × {0} ⊆ V (z0 ) × P0 × {0} .) (†)

Suppose we have an analytic curve   β(t) := (r(t), [a0 (t)dz0 + · · · + ai (t)dzi ]) ∈ P(TS∗α U) ∩ V (z0 , . . . , zi−1 ) × Pi × {0} such that r(0) = 0 and such that, for all t = 0, r(t) ∈ Sα . Then, for t = 0, r (t) = (0, . . . , 0, ri (t), . . . , rn (t)) ∈ Tr(t) Sα and 0 ≡ (a0 (t)dz0 + · · · + ai (t)dzi )(r (t)) = ai (t)ri (t). Thus, either ai (t) ≡ 0 or ri (t) ≡ 0. Since r(0) = 0, if ri (t) ≡ 0, then ri (t) ≡ 0. This proves (†). Now, over a neighborhood of the origin,       (∗) E ∩ V (z0 , . . . , zi−1 ) × Pi × {0} ⊆ π −1 (Sα ) ∩ V (z0 , . . . , zi−1 ) × Pi × {0} ⊆ Sα ⊆Σh

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  Sα ⊆Σh

  P(TS∗α U) ∩ V (z0 , . . . , zi−1 ) × Pi × {0} .

We proceed by induction on k. When k = i = 0, (∗) and (†) combined yield that  

  E ∩ U × P0 × {0} ⊆

Sα ⊆Σh

  P(TS∗α U) ∩ V (z0 ) × P0 × {0}

over a neighborhood of the origin. However, the characterization of z0 being prepolar that we gave in Remark 1.24 tells us that this last quantity equals   P(T0∗ U) ∩ V (z0 ) × P0 × {0} = {0} × P0 × {0}. This proves the claim when k = 0. Now, suppose that the lemma is true for k; we wish to see that it is also true for k + 1 = i + 1. Combining (∗) and (†) again, over a neighborhood of the origin, we find   E ∩ V (z0 , . . . , zk ) × Pk+1 × {0} ⊆

  Sα ⊆Σh

  Sα ⊆Σh

  P(TS∗α U) ∩ V (z0 , . . . , zk ) × Pk+1 × {0} ⊆

  P(TS∗α U)∩ V (z0 , . . . , zk+1 )×Pk+1 ×{0} ∪

  Sα ⊆Σh

  P(TS∗α U)∩ V (z0 , . . . , zk )×Pk ×{0} .

Prepolarity, as described in Remark 1.24, implies that the image under π of this last quantity is contained in



   V (z0 , . . . , zk+1 ) ∩ Sα ∪ V (z0 , . . . , zk ) ∩ Sα . Sα ⊆Σh dim Sα
k+1

Sα ⊆Σh dim Sα
k

As (z0 , . . . , zk+1 ) is prepolar, near the origin, both of the above intersections are contained in {0}. Thus, we conclude that, over a neighborhood of the origin,   E ∩ V (z0 , . . . , zk ) × Pk+1 × {0} ⊆ {0} × Pk+1 × {0}. 

In the following theorem, we continue to use the notation from Remark 1.24. The characterization here of the Lˆe cycles in terms of blowing-up was first shown to us by T. Gaffney (without the description of how generic the coordinates must be). We generalize this result in IV.1.10. Theorem 1.26. Let p ∈ V (h) and let s = dimp Σh. If (z0 , . . . , zk ) is a prepolar tuple for h at p, then there exists a neighborhood Ω of p such that, for all i such that 0
i
k, the exceptional divisor E properly intersects Ω × Pi × {0} in Ω × Pn . 1 Hence, if z0 is prepolar for h at p, then γh,z (p) is defined. 0

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Moreover, if (z0 , . . . , zn ) is a prepolar tuple for h at p, then, for all i, the Lˆe numbers and polar i numbers λih,z (p) and γh,z (p) exist and, in a neighborhood of Σh, i Γi+1 h,z = π∗ (BlJz (h) U · (Ω × P × {0}))

and Λih,z = π∗ (E · (Ω × Pi × {0})), where the intersection takes place in Ω × Pn and π∗ denotes the proper push-forward. Proof. For convenience, we will assume that p = 0. Fix a good stratification {Sα } for h at 0. We show the first statement in the theorem by induction on k. We need to show that if (z0 , . . . , zk ) is a prepolar tuple, then there is a neighborhood of the origin over which E ∩ (U × Pi × {0}) is purely i-dimensional for all i
k. As (z0 , . . . , zk ) being a prepolar tuple implies that (z0 , . . . , zi ) is prepolar for all i
k, it suffices to show that if (z0 , . . . , zk ) is a prepolar tuple, then there is a neighborhood of the origin over which E ∩ (U × Pk × {0}) is purely k-dimensional. The lemma implies that, near 0, ∗ E ∩ (U × P0 × {0}) ⊆ P(T{0} U) ∩ (V (z0 ) × P0 × {0}) = {0} × P0 × {0}.

This proves the desired result when k = 0. Suppose now that E properly intersects U × Pk × {0} over the origin,  but does not properly  intersect U × Pk+1 × {0} over the origin. Let C be a component of E ∩ U × Pk+1 × {0} which has dimension at least k + 2 and such that 0 ∈ π(C). We will use w0 , . . . , wn as homogeneous coordinates on Pn . Our inductive hypothesis implies that 0 ∈ π(C ∩ V (wk+1 )). We shall derive a contradiction by using our other hypotheses to prove that 0 ∈ π(C ∩ V (wk+1 )).  Certainly, 0 ∈ π C ∩ (V (z0 , . . . , zk ) × Pk+1 × {0}) , and the lemma implies that C ∩ (V (z0 , . . . , zk ) × Pk+1 × {0}) ⊆ {0} × Pk+1 × {0}. Therefore, as each component of C ∩ (V (z0 , . . . , zk ) × Pk+1 × {0}) has dimension at least 1, it follows that each component must intersect V (wk+1 ). This is a contradiction, and establishes the first statement of the theorem. If z0 is prepolar for h at 0, then Σh ∩ V (z0 ) = Σ(h|V (z0 ) ), and we have just shown that E properly intersects U × P0 × {0} over a neighborhood of the origin. By Remark I.2.21, this implies 1 that Γ1h,z is purely 1-dimensional at 0, and now Corollary 1.19 allows us to conclude that γh,z (p) 0 is defined. If (z0 , . . . , zn ) is prepolar for h at the origin, then the Segre-Vogel Relation (Corollary I.2.22) allows us to conclude that there is a neighborhood, Ω, of the origin such that i Γi+1 h,z = π∗ (BlJz (h) U · (Ω × P × {0}))

and Λih,z = π∗ (E · (Ω × Pi × {0})). Lemma 1.25 tells us that, as germs of sets at the origin, Λih,z ∩ V (z0 , . . . , zi−1 ) = π(E ∩ (V (z0 , . . . , zi−1 ) × Pi × {0})) ⊆ {0}.

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Therefore, all of the Lˆe numbers are defined, and Corollary 1.19 implies that all of the polar numbers are also defined. 

Remark 1.27. While it is true that prepolar coordinates occur generically and guarantee the existence of the Lˆe numbers, it is not true that all sets of prepolar coordinates yield the same Lˆe numbers. If dimp Σh = 1, then {V (h) − Σh, Σh − p, p} is a good stratification for h in a neighborhood of p; hence, V (z0 − p0 ) is a prepolar slice if and only if dimp Σ(h|V (z0 −p0 ) ) = 0. This is the case if and 1 only if γh,z (p) and λ1h,z (p) are defined. Now, consider the example from Remark 1.16. The coordinates z = (z0 , z1 , z2 ) are prepolar for h = z22 + (z0 − z12 )2 at the origin, and λ0h,z (0) = 1 and λ1h,z (0) = 2. However, the coordinates z are really not very generic, as Σh is smooth at the origin, but V (z0 ) intersects Σh with multiplicity 2 at 0. The generic values of λ0h and λ1h (that is, the values with respect to generic coordinates) are 0 and 1, respectively. Note that the alternating sum of the Lˆe numbers is the same for the non-generic and generic coordinates. As we shall see in Chapter 3, this is a general fact: as long as the coordinates are prepolar, the alternating sum of the Lˆe numbers is independent of the coordinates and is, in fact, equal to the reduced Euler characteristic of the Milnor fibre. We know of no algebraic way to prove this independence. It is reasonable to ask why we do not strengthen our notion of prepolar in order to disallow examples such as the one above, where the Lˆe numbers do not have their generic values. The answer is that later (in Proposition 10.2) we shall show that, given h and a point p ∈ V (h), one may pick generic coordinates which are prepolar for h at every point in a neighborhood of p. This result allows us to give another characterization of the Lˆe cycles in Chapter 10. Example 2.4 shows that this result would be false if we were to strengthen the notion of prepolar to require the Lˆe numbers to obtain their generic values at each point in this open neighborhood of p. Finally, note that there are generic values for the Lˆe numbers; this follows easily from the characterization of the Lˆe cycles given in Theorem 1.26, combined with Kleiman’s Transversality Lemma [Kl].

We conclude this chapter with four results which do not seem to be of fundamental importance, but which are fairly surprising. Proposition 1.28. Suppose that dimp Σh = 1 and V (z0 − p0 ) is a prepolar slice for h at p. If V (z0 − p0 ) does not transversely intersect the set |Σh| at p (in particular, if |Σh| is not smooth at p), then λ0h,z (p) = 0. Proof. Despite the different appearance of the statement, this is precisely what Lˆe proves in [Lˆ e12].  Proposition 1.29. Let k  1. Suppose that Λ0h,z , . . . , Λk−1 h,z have correct dimension at p. Suppose, for all pairs of distinct irreducible germs, V and W , of Σh through p, that dimp (V ∩ W )
k − 1. Finally, suppose that λkh,z (p) = 0. Then, λjh,z (p) = 0 for all j
k.

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Proof. One applies 2.3 of [La] to the case where the variety is Cn+1 and the

irreducible normal  ∂h ∂h subvariety locally defined by n − k equations is V ∂zk+1 , . . . , ∂zn , which equals Γk+1 h,z ∪ Σh as a set. Let V be an irreducible component of Σh at p and let {Wi }i be the remaining irreducible components of Σh at p. Then, the lemma of Lazarsfeld says that if p∈



Γk+1 h,z

∩V



∪









Wi ∩ V

,

i

then

dimp

Γk+1 h,z

∩V



∪







Wi ∩ V

  k.

i

Now the proposition follows easily from 1.13.  j Proposition 1.30. Let s = dimp Σh and suppose that λjh,z (p) and γh,z (p) exist for all j
s. Suppose that the critical locus of h at p is itself singular and denote the singular set of the critical locus by ΣΣh. Then, every (s − 1)-dimensional component of ΣΣh through p is contained in the set |Λs−1 h,z |.

Proof. Let C be an (s − 1)-dimensional component of ΣΣh at p. As all the Lˆe and polar numbers exist, we may inductively apply Proposition 1.18, together with Remark 1.6 and Proposition 1.14, to conclude that p is a singular point of the one-dimensional critical locus of h|V (z0 −p0 ,...,zs−2 −ps−2 ) . Using (zs−1 , . . . , zn ) as coordinates for h|V (z0 −p0 ,...,zs−2 −ps−2 ) , we also conclude from Proposition 1.18 that, at p, λ1h| and γh1| V (z0 −p0 ,...,zs−2 −ps−2 )

V (z0 −p0 ,...,zs−2 −ps−2 )

exist – which, for a one-dimensional critical locus, is equivalent to V (zs−1 − ps−1 ) being prepolar for h|V (z0 −p0 ,...,zs−2 −ps−2 ) at p. Therefore, by Proposition 1.28, λ0h| (p) = 0. This is equivalent to saying that p ∈ Γ1h|

V (z0 −p0 ,...,zs−2 −ps−2 )

V (z0 −p0 ,...,zs−2 −ps−2 )

and now, by applying Proposition 1.18 once more, we find that p ∈ Γsh,z .

As we may apply this same argument at each point of C near p, we find that C ⊆ Γsh,z ∩ Σh. Finally, as the Lˆe numbers are defined, each of the Lˆe cycles has correct dimension at p and so the result follows from Proposition 1.13.  j Proposition 1.30. Let s = dimp Σh, suppose that λjh,z (p) and γh,z (p) exist for all j
s, and j s−1 suppose that λh,z (p) = 0. Then, λh,z (p) = 0 for all j
s − 1.

Proof. The result follows from Proposition 1.29, using i = s − 1, since the preceding proposition proves: if there exist two irreducible components, V and W , of Σh at p such that dimp (V ∩ W ) = s−1 s − 1, then p ∈ Λs−1 h,z and so λh,z (p) = 0. 

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Chapter 2. ELEMENTARY EXAMPLES Example 2.1. If 0 is an isolated singularity of h, then regardless of the coordinate system z, it follows from Proposition 1.11 that the only possibly non-zero Lˆe number is λ0h,z (0). Moreover,



 ∂h ∂h ∂h ∂h ∂h as V ∂z is zero-dimensional, V is one-dimensional with no compo, , . . . , , . . . , ∂zn ∂z1 ∂zn 0 ∂z1 nents contained in Σh and with no embedded subvarieties. Therefore,   ∂h ∂h Γ1h,z = V ,..., ∂z1 ∂zn 

and so λ0h,z (0)

=

 Γ1h,z

·V

∂h ∂z0

 0

     ∂h ∂h ∂h ·V = V ,..., = ∂z1 ∂zn ∂z0 0

   ∂h ∂h V ,..., = the Milnor number of h at 0. ∂z0 ∂zn 0

Example 2.2. Here, we generalize Example 1.12. Let h = y 2 − xa − txb , where a > b > 1. We fix the coordinate system z = (t, x, y) and will suppress any further reference to it.

Figure 2.3. Generalization of nodes degenerating to a cusp

We find: Σh = V (−xb , −axa−1 − btxb−1 , 2y) = V (x, y).   ∂h 2 Γh = V = V (2y) = V (y). ∂y   ∂h 2 Γh · V = V (y) · V (−axa−1 − btxb−1 ) = ∂x V (y) · (V (−axa−b − bt) + V (xb−1 )) = V (−axa−b − bt, y) + (b − 1)V (x, y)

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= Γ1h + Λ1h .  Γ1h

·V

∂h ∂t

 = V (−axa−b − bt, y) · V (−xb ) = bV (t, x, y) = b[0] = Λ0h .

Thus, λ0h (0) = b and λ1h (0) = b − 1. Notice that the exponent a does not appear; this is because h = y 2 − xa − txb = y 2 − xb (xa−b − t) which, after an analytic coordinate change at the origin, equals y 2 − xb u. Example 2.4 (The FM Cone). Let h = y 2 −x3 −(u2 +v 2 +w2 )x2 and fix the coordinates (u, v, w, x, y). Σh = V (−2ux2 , −2vx2 , −2wx2 , −3x2 − 2x(u2 + v 2 + w2 ), 2y) = V (x, y). As Σh is three-dimensional, we begin our calculation with Γ4h . Γ4h = V (−2y) = V (y).  Γ4h

·V

∂h ∂x

 = V (y) · V (−3x2 − 2x(u2 + v 2 + w2 )) =

V (−3x − 2(u2 + v 2 + w2 ), y) + V (x, y) = Γ3h + Λ3h .  Γ3h

·V

∂h ∂w

 = V (−3x − 2(u2 + v 2 + w2 ), y) · V (−2wx2 ) =

V (−3x − 2(u2 + v 2 ), w, y) + 2V (u2 + v 2 + w2 , x, y) = Γ2h + Λ2h .  Γ2h

·V

∂h ∂v

 = V (−3x − 2(u2 + v 2 ), w, y) · V (−2vx2 ) =

V (−3x − 2u2 , v, w, y) + 2V (u2 + v 2 , w, x, y) = Γ1h + Λ1h .  Γ1h

·V

∂h ∂u

 = V (−3x − 2u2 , v, w, y) · V (−2ux2 ) =

V (u, v, w, x, y) + 2V (u2 , v, w, x, y) = 5[0] = Λ0h . Hence, Λ3h = V (x, y), Λ2h = 2V (u2 + v 2 + w2 , x, y) = a cone (as a set), Λ1h = 2V (u2 + v 2 , w, x, y), and Λ0h = 5[0]. Thus, at the origin, λ3h = 1, λ2h = 4, λ1h = 4, and λ0h = 5.

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DAVID B. MASSEY

Σh 1

Λ h, z

2

Λ

h, z

Figure 2.5. The critical locus of h

Though λ1h is independent of a generic choice of coordinates, Λ1h depends on the coordinates – for, by symmetry, if we re-ordered u, v, and w, then Λ1h would change correspondingly. Moreover, one can check that this is a generic problem. Such “non-fixed” Lˆe cycles arise from the absolute polar varieties ([L-T2], [Te4], [Te5]) of the higher dimensional Lˆe cycles (this follows from two results in Part IV: Theorems 1.10 and 3.2). For instance, in the present case, Λ2h is a cone, and its one-dimensional polar variety varies with the choice of coordinates, but generically always consists of two lines; this is the case for Λ1h as well. Example 2.6. Let h = xyz, so that V (h) consists of the coordinate planes in C3 . (See Example 0.2.) Then, Σh = V (x, y) ∪ V (x, z) ∪ V (y, z) = union of the three coordinate axes. The coordinates (x, y, z) are extremely non-generic, so choose some other generic coordinates ˜ = (˜ z z0 , z˜1 , z˜2 ). Then, the set |Λ1h,˜z | = Σh. Hence,     λ1h,˜z (0) = Λ1h,˜z · V (˜ z0 ) 0 = z0 − ξ) p = λ1h,˜z (p), Λ1h,˜z · V (˜ p

p

◦

where the sum is over all p ∈ B
∩ Λ1h,˜z ∩ V (˜ z0 − ξ) for small
and 0 < ξ 
; this set consists of three points and, by symmetry, λ1 must be the same at each of these three points. We wish to use Proposition 1.27 to calculate λ1h,˜z (p). 1 0 As each p ∈ Σh, it follows from 1.10 that γh,˜ z is supported only at Λh,˜ z , which is generically 1 zero-dimensional. Thus, our points p are such that γh,˜z (p) = 0, and it follows from 1.27 that ˆ denotes the restriction of the coordinates z ˜ to V (˜ λ1h,˜z (p) = λ0h (p), where z z0 − p0 ). ,ˆ z |V (˜ z0 −p0 )

Now, h|V (˜z0 −p0 ) has an isolated singularity at each of our three points p, and so λ0h

,ˆ z |V (˜ z0 −p0 )

(p) =

the Milnor number of h|V (˜z0 −p0 ) at p, and this is easily seen to equal 1. It follows, finally, that λ1h,˜z (0) = 3. The generic value of λ0h,˜z (0) is somewhat messier to calculate, and is just as easy to treat in the more general case given in Example 2.8. (However, the answer is that λ0h,˜z (0) = 2.)

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Example 2.7. Let U be an open subset of Cn+1 , let z = (z0 , . . . , zn ) be the coordinates for Cn+1 , and h : U → C be any analytic function. The coordinates z may be non-generic for h. We wish to ˜. see how to calculate λ0h,˜z for a generic linear choice of z ˜ be a generic linear choice of coordinates for Cn+1 , and let aij denote ∂∂zz˜ji . So, let z Now,   ∂h ∂h 1 Γh,˜z = V ,..., ¬ Σh = ∂ z˜1 ∂ z˜n   ∂h ∂h ∂h ∂h V a01 ¬ Σh. + · · · + an1 , . . . , a0n + · · · + ann ∂z0 ∂zn ∂z0 ∂zn By performing elementary row operations, we find that the ideal   ∂h ∂h ∂h ∂h a01 + · · · + an1 , . . . , a0n + · · · + ann ∂z0 ∂zn ∂z0 ∂zn is generated by ∂h ∂h ∂h ∂h ∂h ∂h + b0 , + b1 , ..., + bn−1 , ∂z0 ∂zn ∂z1 ∂zn ∂zn−1 ∂zn where b0 , . . . , bn−1 are generic = 0. Thus,   ∂h ∂h ∂h ∂h ∂h ∂h 1 ¬ Σh, Γh,˜z = V + b0 , + b1 ,..., + bn−1 ∂z0 ∂zn ∂z1 ∂zn ∂zn−1 ∂zn ∂h ∂h and Λ0h,˜z is given by intersecting this with a00 ∂z + · · · + an0 ∂z . 0 n It is important to note that we are not claiming that the cycle Γ1h,˜z can be calculated by considering the cycle       ∂h ∂h ∂h ∂h ∂h ∂h V ·V · ... · V + b0 + b1 + bn−1 ∂z0 ∂zn ∂z1 ∂zn ∂zn−1 ∂zn

and then disposing of any portions of the cycle which are contained in Σh. There could easily be a problem with embedded subvarieties. Example 2.8. We can use the above example to calculate the generic value of λ0 in Example 2.6. Actually, we can just as easily do a more general calculation. n+1 and Σh =  Let h = z0 z1 . . . zn , so that V (h) is the union of the coordinate planes in C V (z , z ) = the union of intersections of pairs of the different coordinate planes. We wish to i j i
=j 0 ˜, that λh,˜z (0) = n. show, for a generic choice of coordinates, z By the above, we find that Γ1h,z equals  V z1 z2 . . . zn−1 (zn + b0 z0 ), z0 z2 . . . zn−1 (zn + b1 z1 ),  . . . , z0 z1 . . . zn−2 (zn + bn−1 zn−1 ) ¬ Σh. Applying 1.2.iii repeatedly, we conclude that Γ1h,z = V (zn + b0 z0 , zn + b1 z1 , . . . , zn + bn−1 zn−1 ). Finally, by intersecting this with   ∂h V = V (a00 z1 z2 . . . zn + · · · + an0 z0 z1 . . . zn−1 ) ∂ z˜0

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DAVID B. MASSEY

we obtain the desired result that λ0h,˜z (0) = n. We shall obtain this same result, but by inductive methods, in Example 5.2.  i Example 2.9.  Let h be an analytic map in the variables x and y, and suppose that h = P Qα i , i where P and Qα are relatively prime and α  2, i.e., h gives a non-reduced curve singularity. i i We wish to calculate the Lˆe numbers of h at the origin. Let z0 = ax + by, where a = 0, and let z1 = y. Then,         ∂h ∂h −b ∂h V = V + = ∂z1 ∂x a ∂y   ∂h  −b  ∂h !   + ∂y  ∂x a αi −1 V Qi = Λ1h,z + Γ1h,z . + V  αi −1 Qi Thus, whenever

  ∂h  −b  ∂x



V

a

+

∂h ∂y

!

i −1 Qα i

has no components contained in the critical locus of h (an easy argument shows that this is the case for a generic choice of (a, b) ), we have that λ1h,z = (αi − 1) (V (Qi ) · V (ax + by))0   ∂h  −b 

and λ0h,z =

where we have used that V Note that the formula

∂h ∂z0

V

 =V

λ1h,z =



∂x



 ∂h  ∂x

a

+

∂h ∂y



 ·V

i −1 Qα i

∂h ∂x

 , 0

.

(αi − 1) (V (Qi ) · V (ax + by))0

agrees with our earlier formula from the end of Remark 1.19, ◦ λ1h,z = nν µν , ν ◦

since we clearly have nν = (V (Qi ) · V (ax + by))0 and µν = αi − 1. Example 2.10. In this example, we show that – unlike the Milnor number – the Lˆe numbers in a family need not be upper-semicontinuous. While this may seem to be mildly disturbing at first, the example makes it clear what can happen; if a high-dimensional Lˆe number jumps up, then the lower-dimensional Lˆe numbers are free to jump up or down. Let ft (x, y, z) = z 2 − y 3 − txy 2 . The coordinates (x, y, z) are prepolar at the origin for ft for all t; we fix this set of coordinates and will suppress further reference to them. For t0 = 0, we are back in the situation of Example 2.2, with a = 3 and b = 2; therefore, λ0ft (0) = 2 and λ1ft (0) = 1. 0

0

ˆ CYCLES AND HYPERSURFACE SINGULARITIES PART II. LE

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On the other hand, the hypersurface defined by f0 is a cross-product singularity; hence, λ0f (0) = 0 0, and one trivially finds that λ1f (0) = 2. 0 Thus, at t = 0, λ1 jumps up to 2 from its generic value of 1; this allows the behavior of λ0ft (0) to be about as “bad” as possible; the generic value of λ0 is 2, while the special value is 0. The situation is not completely uncontrolled – as we shall see in Corollary 4.16, if we have a family ft , then the tuple of Lˆe numbers

 0 λsft ,z (0), λs−1 ft ,z (0), . . . , λft ,z (0)

is lexigraphically upper-semicontinuous in the t variable.

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Chapter 3. A HANDLE DECOMPOSITION OF THE MILNOR FIBRE

In this chapter, we give a handle decomposition of the Milnor fibre of an analytic function with a critical locus of arbitrary dimension. This decomposition is more refined than that obtained by iteratively applying Lˆe’s attaching result (Theorem 0.9). Throughout this chapter, h : U → C will be an analytic function on an open subset of Cn+1 . If p ∈ V (h), then we let Fh,p denote the Milnor fibre of h at p (more generally, for all p ∈ U, Fh,p denotes the Milnor fibre of f − f (p) at p). Our main tool is a proposition based on the argument of Lˆe and Perron [L-P], which is the same argument that is used in [Ti], [Va1], and [Va2]. In what follows, if we have a pair of topological spaces X ⊆ Y , then we say that Y is obtained from X by canceling k m-handles provided that X has a handle decomposition in which the handles of highest index are of index m and Y is obtained from X by attaching k (m + 1)-handles each of which cancels with an m-handle of X (in terms of Morse functions, this says that Y is obtained from X by passing through k critical points – all of index m + 1 – and these critical points cancel with k critical points in X each of index m. See [Mi1] and [Sm].) In particular, this implies that the cohomology groups of X and Y are identical except in degree m, where we have H m (X) ∼ = Zk ⊕ H m (Y ). Proposition 3.1. Let p ∈ V (h). If V (z0 − p0 ) is a prepolar slice for h at p, and n = 2, then the Milnor fibre of h at p is obtained – up to diffeomorphism – from the product of a disk with the 1 1 Milnor fibre of h|V (z0 −p0 ) at p by first attaching γh,z (p) n-handles, which cancel against γh,z (p) ◦

(n − 1)-handles of D × Fh|

V (z0 −p0 )

,p ,

and then attaching λ0h,z (p) more n-handles.

If n = 2, we have the same conclusion except that the canceling is only up to homotopy.

Proof. Essentially, this is Proposition 4.2 of [Mas7], except that here we have weakened the hypothesis on the genericity of the hyperplane slice. We use the coodinates (z0 , . . . , zn ) for our ambient space. Clearly, it suffices to prove the claim for p = 0. We will follow the argument of [Va2]. By Proposition C.6.iii, we may use neighborhoods of the form Dδ × B
, 0 < δ 
, to define the Milnor fibre of h at the origin up to homotopy. Choose
and δ such that B
is a Milnor ball for h|V (z0 ) and Γ1h,z0 ∩ (Dδ × ∂B
) = ∅ – we may accomplish this last equality since Theorem 1.26 implies that dim0 (Γ1h,z0 ∩V (z0 ))
0. Choose η such that (B
, Dη ) is a Milnor pair for h|V (z0 ) at the origin (see Appendix C.5). Let Ψ := (h, z0 ) and let ∆ denote Ψ(Γ1h,z0 ) in C2 (∆ is the Cerf diagram of h with respect to z0 ). ∆ is given its fitting ideal structure, which is possibly non-reduced (see [Loo]). Choose (α, β) ∈ C2 − ∆ sufficiently small and let hβ := h|V (z0 −β) . Let D be a small disc in Dη × {β} centered at (α, β), and let A be the region in Dη × {β} formed by joining to D small discs centered at each of the points of ∆ ∩ V (z0 − β), where the joining is via thickened paths which avoid (0, β) (see Figure 3.2). Note that, counted with multiplicity, there are (Γ1h,z0 · V (z0 ))0 points in ∆ ∩ V (z0 − β).

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β

D

(α,β)

∆

Dη × {β}

Figure 3.2. The Cerf diagram, and the sets A and C Then, the argument of Lˆe and Perron [L-P] and Vannier [Va1] shows  fibre of h

that the

Milnor −1 ∂h 0 1 is obtained from W := hβ (A) ∩ ({β} × B
) by attaching λh,z0 (0) = Γh,z0 · V ∂z0 n-handles 0

– this part of their argument does not depend on the dimension of Σh. (Though, as we are not assuming that the polar curve is reduced, the details of the generalization of the isotopy result used by Lˆe and Perron need to be checked – this is done in [Ti].) The problem is thus to show that W is obtained from the product of a disc with the Milnor 1 fibre of h|V (z0 ) by canceling γh,z (0) = (Γ1h,z0 · V (z0 ))0 (n − 1)-handles, even in the case where the 0 critical locus of h does not have dimension one. Let C ⊆ Dη × {β} be formed by taking a small disc around (0, β) and joining it to D with a thickened path (see Figure 3.2). Let U := h−1 β (C) ∩ ({β} × B
). Then, as A ∪ C is a strong deformation retract of Dη , U ∪ V is homotopy-equivalent to h−1 β (Dη ) ∩ ({β} × B
), which is in turn diffeomorphic to a real 2n-ball, B 2n . Moreover, U ∩ W is diffeomorphic to h−1 β (D) which is diffeomorphic to the product of a disc and the Milnor fibre of h|V (z0 ) . 1 Now, U ∪ W is obtained from U by attaching γh,z (0) n-handles (that is, one handle of index 0 n for each point of ∆ ∩ V (z0 − β) in A, counted with multiplicity). As U ∪ W is contractible, this 1 (0) (n − 1)-spheres. implies that U has the homotopy-type of a bouquet of γh,z 0

Consider the Mayer-Vietoris sequence of U and W . As U ∪ W is contractible, we have that Hk (U ∩ W ) ∼ = Hk (U ) ⊕ Hk (W ) for all k  1, where we know that U has the homotopy-type of a bouquet of (n − 1)-spheres, and U ∩ W is diffeomorphic to the product of a disc and the Milnor fibre of h|V (z0 ) , which has the homotopy-type of an (n − 1)-dimensional CW complex. Thus, we see that W has the homotopy-type of an (n − 1)-dimensional CW complex. But now, since h−1 β (D) is diffeomorphic to the product of a disc with the Milnor fibre of h|V (z0 ) , 1 and since W itself is obtained by attaching γh,z (0) n-handles to h−1 β (D), we see that up to 0 1 1 (0) (n − 1)-handles. But, if n  3, then homotopy these γh,z0 (0) n-handles must be canceling γh,z 0 the real dimension of W is greater than or equal to 6, and so this handle cancellation is up to diffeomorphism [Mi1], [Sm]. Finally, if n = 1, then – by the classification of surfaces – we have that the handle cancellation is up to diffeomorphism. 

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DAVID B. MASSEY

By an inductive application of Proposition 3.1 to each hyperplane slice, we arrive at Theorem 4.3 of [Mas7]. This theorem describes a handle decomposition of the Milnor fibre. Theorem 3.3. Let U be an open subset of Cn+1 , let h : U → C be an analytic map, let p ∈ V (h), let s denote dimp Σh, and let z = (z0 , . . . , zs−1 ) be prepolar for h at p. If s
n − 2, then Fh,p is obtained up to diffeomorphism from a real 2n-ball by successively attaching λn−k h,z (p) k-handles, where n − s
k
n; if s = n − 1, then Fh,p is obtained up to diffeomorphism from a real 2n-manifold with the n−k homotopy-type of a bouquet of λn−1 h,z (p) circles by successively attaching λh,z (p) k-handles, where 2
k
n Hence, the reduced Euler characteristic of the Milnor fibre of h at p is given by s χ (Fh,p ) = (−1)n−i λih,z (p) i=0

and the reduced Betti numbers, ˜bi (Fh,p ), satisfy Morse inequalities with respect to the Lˆe numbers, i.e., for all k with n − s
k
n, (−1)k

k

(−1)i˜bi (Fh,p )
(−1)k

i=n−s

and (−1)k

n

k

(−1)i λn−i h,z (p)

i=n−s

(−1)i˜bi (Fh,p )
(−1)k

i=k

n

(−1)i λn−i h,z (p)

i=k

Proof. By induction on s. When s = 0, the result follows from Milnor’s work. Now, assume the result for s − 1. As before, we consider only the case where p = 0. As V (z0 ) is prepolar, we may apply Proposition 3.1 to conclude that the Milnor fibre of h at 0 is obtained from the product of a disk with the Milnor fibre of h|V (z0 ) at 0 by first attaching ◦

1 1 γh,z (0) n-handles, which cancel against γh,z (0) (n−1)-handles of D×Fh|

V (z0 )

,0 ,

and then attaching

more n-handles – up to diffeomorphism if n = 2 and up to homotopy otherwise. ˜ = (z1 , . . . , zs−1 ) is prepolar for h|V (z0 ) at the origin and, by our inductive hypothesis, the But, z Milnor fibre of h|V (z0 ) at the origin is obtained by successively attaching λn−1−k h| ,˜ z (0) k-handles for λ0h,z (0)

V (z0 )

n−k (n − 1) − (s − 1)
k
n − 1. By Proposition 1.18, if n − 1 − k = 0, then λn−1−k h| ,˜ z (0) = λh,z (0),

and λ0h|

V (z0 )

V (z0

,˜ z (0) )

1 = γh,z (0) + λ1h,z (0). The conclusions concerning handles follow.

The Morse inequalities follow formally.  Siersma’s main result in [Si2] allows us to improve this result in a special case. Corollary 3.4 Let U be an open subset of Cn+1 , let h : U → C be an analytic map, let p ∈ V (h), and let s denote dimp Σh. Suppose that (z0 , . . . , zs−1 ) is prepolar for h at p, and suppose that λsh,z (p) = 1.

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69

Then, either λ0h,z (p) = λ1h,z (p) = · · · = λs−1 h,z (p) = 0, or the single (n − s)-handle in the handle decomposition of the previous theorem gets canceled – up to homotopy – by the attaching of one of the λs−1 h,z (p) (n − s + 1)-handles. Proof. The proof is exactly the inductive proof of 3.3, except in the first step one applies the result of [Si]. The function h|V (z0 −p0 ,...,zs−2 −ps−2 ) has a one-dimensional critical locus at p. Using (zs−1 , . . . , zn ) as coordinates for V (z0 − p0 , . . . , zs−2 − ps−2 ), we conclude from 1.18 that λ1h|

V (z0 −p0 ,...,zs−2 −ps−2 )

(p) = λsh,z (p) = 1

and λ0h|

V (z0 −p0 ,...,zs−2 −ps−2 )

s−1 (p) = γh,z (p) + λs−1 h,z (p).

Hence, h|V (z0 −p0 ,...,zs−2 −ps−2 ) is an isolated line singularity in the sense of Siersma [Si2], and his result is that either λ0h| (p) = 0 or that one only has homology in middle V (z0 −p0 ,...,zs−2 −ps−2 )

dimension, i.e., the one possible (n − s)-handle must get canceled up to homotopy. The equality s−1 λ0h| (p) = γh,z (p) + λs−1 h,z (p) = 0 V (z0 −p0 ,...,zs−2 −ps−2 )

corresponds to the case where λ0h,z (p) = λ1h,z (p) = · · · = λs−1 h,z (p) = 0, s−1 s−1 since λs−1 h,z (p) = 0 implies that p ∈ Γh,z and, by 1.13, Γh,z ∩ Σh =

p ∈

Γs−1 h,z ,

then it follows that all the lower Lˆe numbers are also zero.

 i
s−2

Λih,z . Hence, if



One might question whether the above result can possibly be correct. What about the case where λsh,z (p) = 1, λs−1 h,z (p) = 0, and one of the lower λ’s is not zero? In such a case, there would be no way to cancel the (n − s)-handle. Note, however, that 1.30 rules out the possibility of the existence of this case.

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DAVID B. MASSEY

ˆ Chapter 4. GENERALIZED LE-IOMDINE FORMULAS In this chapter, we generalize the formula of Lˆe and Iomdine (see [Lˆ e4], [Io], [Mas8], [Mas11], [M-S], and [Si3]) to functions with an arbitrary-dimensional critical locus; on the level of cycles, this will be a special case of the Lˆe-Iomdine-Vogel formulas from I.3.4. The Lˆe-Iomdine formulas that we present here tell us how the Lˆe numbers of a hypersurface singularity are related to the Lˆe numbers of a certain “sequence of hypersurface singularities” – a sequence which “approaches” the original singularity, but such that the critical loci of the terms in the sequence are of one dimension smaller than the original. These formulas have a large number of applications. The statement that we give here has an improvement in a certain bound over what we proved in [Mas14]; in the case of a one-dimensional critical locus, this is the form of the statement as it appears in [M-S] and [Si3]. To give this improved bound, we need a definition. Throughout this chapter, we concentrate our attention at the origin.

We are about to introduce the polar ratios. These quantities first appeared in Proposition 3.5.2 of [Te4], and the fact that they are invariants of the “equi-singularity type” appears in Theorem 6 of [Te 5]. In [Te 5], Teissier investigates a number of questions of “Iomdine/Sebastiani-Thom type” in the case of isolated hypersurface singularities.

Definition 4.1. Suppose that Γ1h,z0 is purely one-dimensional at the origin. Let η be an irreducible component of Γ1h,z0 (with its reduced structure) such that η∩V (z0 ) is zero-dimensional at the origin. Then, the polar ratio of η (for h at 0 with respect to z0 ) is

(η · V (h))0 = (η · V (z0 ))0

η·V



∂h ∂z0



+ (η · V (z0 ))0 0

(η · V (z0 ))0

=

(η · V



∂h ∂z0

 )0

(η · V (z0 ))0

+ 1.

(The equalities follow from our proof of Proposition 1.20.) If η ∩ V (z0 ) is not zero-dimensional at the origin (i.e., if η ⊆ V (z0 )), then we say that the polar ratio of η equals 1. A polar ratio (of h at 0 with respect to z0 ) is any one of the polar ratios of any component of the polar curve (if the polar curve is empty at 0, we say that the maximum polar ratio equals 1). If 0 ∈ Σ and all of the Lˆe cycles have the correct dimension at 0 (in particular, if the Lˆe numbers are defined at 0), this definition is (up to adding 1) a particular case of I.3.2.  ∂h ∂h Let f := ∂z , . . . , and g := z0 . Then, Γ1h,z0 = Π1f , and there is an equality of sets ∂z 0 n  1  1 Π  =  Π  by the Dimensionality Lemma (I.2.5). Let η be an irreducible component of Γ1 . f

f

h,z0

Looking at I.3.2 (and using d := n + 1 and k := n), we see that the polar ratio of η (for h) is precisely the gap ratio of η (for f ) plus one. Remark 4.2. The case where h is a homogeneous polynomial of degree d is particularly easy to analyze. Provided that Γ1h,z0 is one-dimensional at the origin, each component of the polar curve is a line, and so the polar ratios are all 1 or d.

ˆ CYCLES AND HYPERSURFACE SINGULARITIES PART II. LE

71

We are going to consider functions of the form h+az0j , where a is a non-zero complex number and j is suitably large. Clearly, however, the coordinate z0 is extremely non-generic for h + az0j . Hence, if we are using the coordinates (z0 , z1 , . . . , zn ) for h, we use the coordinates (z1 , z2 , . . . , zn , z0 ) for h + az0j . The purpose of this “rotation” of the coordinate system is merely to get the z0 coordinate out of the way. Normally, if h has an s-dimensional critical locus at the origin, then h + az0j will have an (s − 1)-dimensional critical locus at the origin; thus, it is only the choice of the coordinates z0 , . . . , zs−1 that we care about for h, and the coordinates z1 , . . . , zs−1 for h + az0j . Lemma 4.3. Let j  2. Let h : (U, 0) → (C, 0) be an analytic function, let s denote dim0 Σh, and assume that s  1. Let z = (z0 , . . . , zn ) be a linear choice of coordinates such that λih,z (0) is defined ˜ = (z1 , . . . , zn , z0 ) for for all i
s. Let a be a non-zero complex number, and use the coordinates z h + az0j . If j is greater than or equal to the maximum polar ratio for h then, for all but a finite number of complex a, 

j−1 ∂h i) dim0 Γ1h,z ∩ V ∂z = 0; + jaz 0 0 ii)





j−1 ∂h λ0h,z (0) = Γ1h,z · V ∂z + jaz ; 0 0 0

az0j )

is (s − 1)-dimensional at the origin and equal to Σh ∩ V (z0 ) as germs of sets at

iii) 0;

Σ(h +

iv)

if i  1, then we have an equality of cycles  Γih+azj ,˜z 0

=

Γi+1 h,z

·V

∂h + jaz0j−1 ∂z0



near the origin;

 v) Γ1h+awj ,w−z0 · V (h + awj ) = jλ0h,z (0), where w is a variable disjoint from those of h. 0

Moreover, if we have the strict inequality that j is greater than the maximum polar ratio for h, then the above equalities hold for all non-zero a; in particular, this is the case if j  2 + λ0h,z (0). Proof. Parts i), ii), iii), and iv) follow immediately Lemma I.3.3, where the X, f , g,  ∂h by applying  ∂h a, j, and p of Lemma I.3.3 are replaced by U, ∂z , . . . , , , z 0 j · a, j − 1, and 0, respectively. ∂zn 0 We will prove v) by applying Lemma I.3.6.  ∂h   j−1 ∂h  ∂h ∂h ∂h j−1 Let f := ∂z , . . . , , , . . . , , + jaw , and let g := w . Then, I.3.6 tells us ∂z ∂z ∂z ∂z 0 n 1 n 0 i that sets  ∂hthe Vogel  of g have correct dimension at 0, for all i
n + 1, C × Πf properly intersects j−1 V ∂z0 + jaw , and there is an equality of germs of cycles at 0 given by (†)

i = Π g



i C×Π f



· V

∂h ∂z0

 + jawj−1 .

As the Vogel sets of f and g have correct dimension at 0, we may use the Dimensionality Lemma and Proposition I.2.9 to conclude that we may replace each of the inductive gap varieties in (†) by the ordinary gap varieties.

72

DAVID B. MASSEY

Now, let L = w − z0 and use (L, z0 , . . . , zn ) as coordinates for C × U. Then, Γ1h+awj ,w−z0 = Γ1h+a(L+z0 )j ,L =  V

∂h ∂h ∂h + ja(L + z0 )j−1 , ,..., ∂z0 ∂z1 ∂zn

 ¬ Σ(h + a(L + z0 )j )

which, back in (w, z0 , . . . , zn ) coordinates, is equal to  V

∂h ∂h ∂h ,..., , + jawj−1 ∂z1 ∂zn ∂z0

 ¬ Σ(h + awj ).

Thus, we see that Γ1h+awj ,w−z0 = Π1g , and so (†) tells us that Γ1h+awj ,w−z0 =

(∗)



C × Γ1h,z



· V

∂h ∂z0

 + jawj−1 .

1 Now, let us assume for the moment that γh+aw j ,w−z (0) exists. Then, we may apply Proposition 0 I.1.20 to conclude that



Γ1h+awj ,w−z0 · V (h + awj )



 Γ1h+awj ,w−z0 · V

∂(h + awj ) ∂(w − z0 )



0

=

 + Γ1h+awj ,w−z0 · V (w − z0 ) = 0

0



 Γ1h+awj ,w−z0 · V (jawj−1 ) + Γ1h+awj ,w−z0 · V (w − z0 ) . 0

0

By (∗), this is equal to  (j − 1)λ0h,z (0) +





∂h V (w − z0 ) · C × Γ1h,z · V + jaz0j−1 , ∂z0 0

which, by i) and ii), is equal to (j − 1)λ0h,z (0) + λ0h,z (0) = jλ0h,z (0). 1 Thus, we would be finished if we could show that γh+aw j ,w−z (0) exists, but, as we saw above, 0 1 Γh+awj ,w−z0 ∩ V (w − z0 ) is zero-dimensional at the origin. 

Our next result will be to obtain the generalized Lˆe-Iomdine formulas; these formulas are a stunningly useful tool for reducing questions on general hypersurface singularities to the much easier case of isolated hypersurface singularities. The formulas tell how the Lˆe numbers of h change when a large power of one of the variables is added. By 4.3.iii, this modification of the function h will have a critical locus of dimension one smaller than that of h itself. Proceeding inductively, one arrives at the case of an isolated singularity.

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73

Figure 4.4. The effect of adding a large power of a variable

Theorem 4.5 (Lˆ e-Iomdine formulas). Let j  2, let h : (U, 0) → (C, 0) be an analytic function, let s denote dim0 Σh, and assume that s  1. Let z = (z0 , . . . , zn ) be a linear choice of coordinates such that λih,z (0) is defined for all i
s. Let a be a non-zero complex number, and use the ˜ = (z1 , . . . , zn , z0 ) for h + az0j . coordinates z If j is greater than or equal to the maximum polar ratio for h then, for all but a finite number of complex a, Σ(h + az0j ) = Σh ∩ V (z0 ) as germs of sets at 0, dim0 Σ(h + az0j ) = s − 1, λih+azj ,˜z (0) 0 exists for all i
s − 1, and λ0h+azj ,˜z (0) = λ0h,z (0) + (j − 1)λ1h,z (0), 0

and, for 1
i
s − 1, λih+azj ,˜z (0) = (j − 1)λi+1 h,z (0). 0

Moreover, if we have the strict inequality that j is greater than the maximum polar ratio for h, then the above equalities hold for all non-zero a; in particular, this is the case if j  2 + λ0h,z (0). Proof. This follows immediately from formulas (I.3.4) by letting X, f , g, a,  ∂hthe Lˆe-Iomdine-Vogel  ∂h j, and p of I.3.4 be replaced by U, ∂z , . . . , , j · a, j − 1, and 0, respectively.  , z 0 ∂zn 0

By applying the Lˆe-Iomdine formulas inductively, we immediately conclude Corollary 4.6. Let h : (U, 0) → (C, 0) be an analytic function, let s denote dim0 Σh, and let z = (z0 , . . . , zn ) be a linear choice of coordinates such that λih,z (0) is defined for all i
s. Then, for 0  j0  j1  · · ·  js−1 , j

s−1 h + z0j0 + z1j1 + · · · + zs−1

74

DAVID B. MASSEY

has an isolated singularity at the origin, and its Milnor number is given by µ(h +

z0j0

+

z1j1

+ ··· +

js−1 zs−1 )

=

s

 λih,z (0)

i=0

i−1 "

 (jk − 1)

=

k=0

λ0h,z (0) + (j0 − 1)λ1h,z (0) + (j1 − 1)(j0 − 1)λ2h,z (0) + . . . +(js−1 − 1) . . . (j1 − 1)(j0 − 1)λsh,z (0).

As another quick application of the Lˆe-Iomdine formulas, we have the following Pl¨ ucker formula. Corollary 4.7. Let h be a homogeneous polynomial of degree d in n + 1 variables, let s = dim0 Σh, and suppose that λih,z (0) exists for all i
s. Then, s

(d − 1)i λih,z (0) = (d − 1)n+1 .

i=0

Proof. By Remark 4.2, the maximum polar ratio is d. By an inductive application of the Lˆe-Iomdine formulas, we arrive at a function, d f := h + a0 z0d + a1 z1d + · · · + as−1 zs−1 ,

with an isolated singularity at the origin and such that the Milnor number of f = λ0f (0) = s i i n+1 .  i=0 (d − 1) λh,z (0). Now, by [M-O], this Milnor number is precisely (d − 1) In Chapter 9, we will see that the Lˆe cycles are actually Segre cycles. Knowing this, the above Pl¨ ucker formula is a special case of a much more general result of Van Gastel [Gas1, 1.2.c]. Remark 4.8. In general, Corollary 4.7 makes it slightly easier to calculate the Euler characteristic of the Milnor fibre of a homogeneous polynomial. In the case of a one-dimensional critical locus, 4.7 tells us that, if we know the degree and λ1 , then we know λ0 and, hence, the Euler characteristic (see also [M-S] and [Si3]). Being able to calculate the Euler characteristic of the Milnor fibre of a homogeneous singularity implies in many cases that we can also calculate the Euler characteristic of the Milnor fibre of a weighted-homogeneous polynomial. To see this, let f : Cn+1 → C be a weighted homogeneous polynomial. Then, there exist positive integers r0 , . . . , rn such that, if π : Cn+1 → Cn+1 is given by π(z0 , . . . , zn ) = (z0r0 , . . . , znrn ), then h := f ◦ π is homogeneous.

ˆ CYCLES AND HYPERSURFACE SINGULARITIES PART II. LE

75

We may define, up to diffeomorphism, the Milnor fibre of h at the origin by using the “weighted” ball ◦  
B 
:= (z0 , . . . , zn )  |z0 |2r0 + · · · + |zn |2rn <
2 ; ◦

that is, for 0  |ξ| 
 1, we define Fh,0 to be B 
∩ h−1 (ξ). That this yields the same diffeomorphism-type as the standard ball is well-known; see, for instance, [G-M2, II.2]. Clearly, the restriction of π induces a map from Fh,0 to the Milnor fibre (using standard balls), Ff,0 , of f at the origin; denote this map by π ˜ : Fh,0 → Ff,0 . Now, consider the stratification of Cn+1 derived from the hyperplane arrangement given by all the coordinate hyperplanes. That is, let I denote the indexing set {0, . . . , n}, and for each J ⊆ I, let wJ denote the intersection of hyperplanes (a.k.a. the flat) given by wJ := V (zj | j ∈ J), and let SJ denote the Whitney stratum 

SJ := wJ −

wK .

J
K

Near a given point, a representative of the Milnor fibre of a given function will transversely intersect all strata of any fixed Whitney stratification; this follows from the fact that, locally, the stratified critical values of an analytic function are isolated. Thus, the stratification {SJ } determines Whitney stratifications {SJ ∩ Fh,0 } and {SJ ∩ Ff,0 } of Fh,0 and Ff,0 , respectively, and with these stratifications, π ˜ becomes a stratified map. Moreover, the restriction of π ˜ to a map  from SJ ∩ Fh,0 to SJ ∩ Ff,0 is a topological covering map with fibre equal to i
∈J ri points. Hence,   "  ri  χ(SJ ∩ Ff,0 ). χ(Fh,0 ) = χ(SJ ∩ Fh,0 ) = J

i∈J

J

Some elementary combinatorics shows that this last quantity is equal to

 cJ 

J





χ(SK ∩ Ff,0 ) ,

J⊆K

where |J|

cJ := (−1)





"

|L|

(−1)

 ri  .

i∈L

L⊆J

The advantage of this last form is that

χ(SK ∩ Ff,0 ) = χ(Ff|

J⊆K

Therefore, we have that χ(Fh,0 ) =

J

where cJ is as above.

cJ χ(Ff|

,0

w

J

,0

w

J

),

).

76

DAVID B. MASSEY

It follows that χ(Fh,0 ) = (r0 . . . rn )χ(Ff,0 ) +

J=∅

cJ χ(Ff|

,0

w

),

J

and so, finally, we arrive at the formula χ(Fh,0 ) − χ(Ff,0 ) =

 J
=∅

cJ χ(Ff|

r0 . . . rn

,0

w

J

) .

This formula is inductively useful since, if J = ∅, then f|w is a weighted-homogeneous polynomial J in fewer variables (compare with [Di]). Note that, in this formula, we need not consider the term where J = {0, . . . , n}, for then wJ = 0 and hence χ(Ff| ,0 ) = 0. w

J

This is particularly useful in the case where f is a weighted-homogeneous polynomial with a one-dimensional critical locus and each restriction to a flat, f|w , also has a one-dimensional (or J zero-dimensional) critical locus.

Example 4.9. For instance, consider the case of a possibly non-reduced, weighted-homogeneous  plane curve singularity. Suppose that the irreducible factorization of f (z0 , z1 ) is z0a z1b fimi , where r0 r1 we allow for the case where a or b equals 0. Let π(z0 , z1 ) = (z0 , z1 ), and let h denote the homogeneous polynomial f ◦ π. Let hi denote the homogeneous polynomial fi ◦ π. Let d be the degree of h and let di be the degree of hi . Then, the formula of 4.7 becomes

χ(Ff,0 ) = Now, χ(Ff|

V (zk )

,0

χ(Fh,0 ) + (r0 − 1)r1 χ(Ff|

V (z0 )

,0

) + (r1 − 1)r0 χ(Ff|

V (z1 )

r 0 r1

,0

) .

) = 0 if f|V (zk ) ≡ 0 and simply equals the multiplicity of f|V (zk ) otherwise. In

addition, as h is homogeneous, we may calculate χ(Fh,0 ) from 4.6 by knowing only λ1h (0), which we may calculate as in 2.9. We find easily that   −d di ,     d(r −  d ), 0 i r0 r1 χ(Ff,0 ) =   d(r − d 1 i ),     d(r0 + r1 − di ),

if a = 0, b = 0 if a = 0, b = 0 if a = 0, b = 0 if a = b = 0.

Example 4.10. We can also apply the formula of 4.8 in harder cases. Consider the swallowtail singularity; this is given as the zero locus of f = 256z03 − 27z14 − 128z02 z22 + 144z0 z12 z2 + 16z0 z24 − 4z12 z23 (see, for instance, [Te3]).

ˆ CYCLES AND HYPERSURFACE SINGULARITIES PART II. LE

Figure 4.11. The swallowtail singularity

If π(z0 , z1 , z2 ) = (z04 , z13 , z22 ), then h = f ◦ π is homogeneous of degree 12. Using the notation of 4.8, we find c{0} = (−1)(4 · 3 · 2 − 3 · 2) = −18. f|w

{0}

= −27z14 − 4z12 z23 = z12 (−27z12 − 4z23 ).

Hence, from 4.9, we find that χ(Ff|

,0

w

) = −8.

{0}

Similarly, c{1} = (−1)(4 · 3 · 2 − 4 · 2) = −16. f|w

{1}

= 256z03 − 128z02 z22 + 16z0 z24 = 16z0 (16z02 − 8z0 z22 + z24 ) = 16z0 (4z0 − z22 )2 . χ(Ff|

,0

w

) = −3.

{1}

and c{2} = (−1)(4 · 3 · 2 − 4 · 3) = −12. = 256z03 − 27z14 .

f|w

{2}

χ(Ff|

,0

w

) = −5.

{2}

We also find f|w

{0,1}

≡ 0,

c{0,2} = 4 · 3 · 2 − 3 · 2 − 4 · 3 + 3 = 9, f|w

{0,2}

χ(Ff|

= −27z14 , ,0

w

{0,2}

) = 4,

77

78

DAVID B. MASSEY

and c{1,2} = 4 · 3 · 2 − 4 · 2 − 4 · 3 + 4 = 8, f|w

{1,2}

χ(Ff|

= 256z03 , ,0

w

) = 3.

{0,2}

Having made these calculations, it still remains for us to calculate χ(Fh,0 ). We can do this using 4.7, provided that we know that h has a one-dimensional critical locus and provided that we can calculate λ1h (0). While this calculation can be made by hand, it is rather tedious; a computer algebra program – such as Macaulay, a public domain program written by Michael Stillman and Dave Bayer – can tell us that not only does h have a one-dimensional critical locus, but that the multiplicity of the Jacobian scheme at the origin is 83, i.e., λ1h (0) = 83. Therefore, χ(Fh,0 ) = λ0h (0) − λ1h (0) + 1 = dλ1h (0) − (d − 1)n+1 + 1 = 12 · 83 − 113 + 1 = 336. Finally, χ(Ff,0 ) =

336 − (−18)(−8) − (−16)(−3) − (−12)(−5) − 9 · 4 − 8 · 3 = 1. 4·3·2

Remark 4.12. In the above example, we resorted to a computer calculation at one point. If we are willing to use a computer algebra program at each step, then there is a much easier way to calculate the Euler characteristic of the Milnor fibre in the case of a one-dimensional critical locus – whether the function, h, is a weighted-homogeneous polynomial or not. This is the method that we describe in [M-S]. Any computer program which can calculate the multiplicities of ideals in a polynomial ring, given a set of generators, can calculate the Lˆe numbers of a polynomial. (A number of programs have this capability, but by far the most efficient that we know of is Macaulay.) Given such a program and a polynomial, h, with a one-dimensional singular set, one proceeds as follows to calculate the Lˆe numbers, λ0 and λ1 , at the origin with respect to a generic set of coordinates. As we saw in 1.16, λ1 is nothing other than the multiplicity of the Jacobian scheme of h at the origin. So, one can have the program calculate it. Now, we need a hyperplane that is generic enough so that its intersection number (at the origin) with the (reduced) singular set is, in fact, equal to the multiplicity of the singular set. Usually, one knows the singular set (as a set) well enough to know such a hyperplane. (Alternatively, there are programs which can find the singular set for you – though how they present the answer is not always helpful.) We shall assume now, in addition to having λ1 , that we also have such a hyperplane, V (L), for some linear form, L . By the work of Iomdine [Io] and Lˆe [Lˆ e4] (or our generalization in 4.5 or [M-S]), we have that: for all k sufficiently large, h + Lk has an isolated singularity at the origin and the Milnor number µ(h + Lk ) equals λ0 + (k − 1)λ1 . But, the Milnor number is again nothing other than the multiplicity of the Jacobian scheme at the origin, and so we may use our program to calculate

ˆ CYCLES AND HYPERSURFACE SINGULARITIES PART II. LE

79

it. Thus, we can find λ0 – provided that we have an effective method for knowing when we have chosen k large enough so that the formula of Iomdine and Lˆe holds. However, we have such a method. If h + Lk has an isolated singularity, let µk denote its Milnor number. (Given a particular k, one must either check by hand whether h + Lk has an isolated singularity or have a program do it. Macaulay will tell you the dimension of the singular set in the course of calculating the multiplicity of the Jacobian scheme at the origin.) A quick look at the proof of the Iomdine-Lˆe formula in 4.5 shows that the formula holds provided that µk − (k − 1)λ1
k − 2. Therefore, to find λ0 , one starts with a relatively small k and checks whether µk
k − 2 + (k − 1)λ1 . If the inequality is false, pick a larger k. Eventually, the inequality will hold and then λ0 = µk − (k − 1)λ1 . This is the same argument that we used in Remark I.3.5 in the more general setting of the LˆeIomdine-Vogel formulas. As we also mentioned in I.3.5, there is an alternative method for calculating not only the Lˆe numbers but also the maximum polar ratio of h. Here, one needs to make sure that the linear form L has been chosen generically enough so that there are no exceptional pairs (see I.3.5). As we are assuming the use of a computer, all that one needs to do is select the linear form “randomly”. Then, to find the maximum polar ratio, one calculates µk for successive values of k – looking for a difference of λ1 . Once this occurs, k is at least the maximum polar ratio and, as before, we conclude that λ0 = µk − (k − 1)λ1 . Note, moreover, that this method works whether λ1 equals the multiplicity of the Jacobian scheme at the origin or not. As λ1 at least the multiplicity of the Jacobian scheme at the origin, with equality being the generic case, it follows that if µk+1 − µk equals the multiplicity of the Jacobian scheme at the origin, then λ1 equals the multiplicity of the Jacobian scheme at the origin and λ0 = µk − (k − 1)λ1 . While this method requires one to calculate at least two Milnor numbers, µk , it will still be a more efficient way of calculating λ0 – provided that the maximum polar ratio is significantly smaller than λ0 itself. This would be the case, for instance, if the polar curve had a large number of components. Consider again the swallowtail of Example 4.10 defined by h = 256z03 − 27z14 − 128z02 z22 + 144z0 z12 z2 + 16z0 z24 − 4z12 z23 . We use Macaulay to find that the multiplicity of the Jacobian scheme at the origin equals 5. (Alternatively, we know that the singularities of the swallowtail consist of a smooth curve of ordinary double points plus a multiplicity two curve of cusps; hence, the multiplicity of the Jacobian scheme at the origin = 1 + 2 · 2 = 5.) Now, using the notation above and letting L = z2 (this is “random” enough this time, but the reader is invited to check this by picking a “more random” linear form), we find k = µk =

2 6

3 12

4 18

5 24

6 30

7 35

80

DAVID B. MASSEY

From the table, we see that the maximum polar ratio is at most 6, λ1 is, in fact, equal to 5, and λ0 = 30 − (6 − 1)5 = 5. Hence, the Euler characteristic of the Milnor fibre of h equals λ0 − λ1 + 1 = 1, which agrees with our previous calculation. As the swallowtail is such an important singularity, one might wonder how close the Morse inequalities of Theorem 3.3 are to being equalities in this example. The answer is: not very. A. Suciu informs us that the degree 1 and 2 homology groups of the Milnor fibre of the swallowtail at the origin are both free Abelian of rank 2. Why are the Lˆe numbers off by so much from the Betti numbers? It is because the Lˆe numbers record information that the Betti numbers do not. In the case of the swallowtail, λ1 records the information that there is an entire cusp of cusp singularities coming into the origin plus a line of quadratic singularities. Thus, λ1 equals ( the multiplicity of the cusp )( the Milnor number of the cusp ) plus ( the Milnor number of the quadratic singularity ) = (2)(2) + 1 = 5. Now, as λ1 is forced to be 5, λ0 also has to be 5 – in order to make λ0 − λ1 + 1 come out to equal the Euler characteristic. Remark 4.13. For a projective hypersurface, X, defined by a homogeneous polynomial, h, in the projective coordinates (z0 : · · · : zn ), one may ask if there is some reasonable notion of global Lˆe numbers. It is easy to see that if one takes affine patches on X, calculates λ0 at points of each patch with respect to generic coordinates, and adds together the finite number of non-zero results that one gets, then the answer is precisely λ1h (0) (in the ordinary, affine sense) with respect to generic coordinates. It seems reasonable, then, to define the global λiX to be λi+1 h (0). If one makes this definition, it might initially look as though λ0h (0) should provide a new interesting invariant of X. However, Corollary 4.7 tells us that λ0h (0) can be calculated from the higher Lˆe numbers together with the degree of X.

We shall now prove a uniform version of the generalized Lˆe-Iomdine formulas for one-parameter ◦

families of germs of hypersurface singularities at the origin, i.e., D will be an open disc about the ◦

◦

origin in C, U will be an open neighborhood of the origin in Cn+1 and f : (D × U, D × 0) → (C, 0) will be an analytic function; naturally, we write ft for the function defined by ft (z) := f (t, z). First, we need a lemma Lemma 4.14. For all i and for all p = (t0 , z0 , . . . , zn ) near the origin such that t0 = 0, Γift0 ,z = Γi+1 ∩ V (t − t0 ) = Γi+1 · V (t − t0 ) f,(t,z) f,(t,z) as cycles at p, regardless of how generic (t, z0 , . . . , zn ) may be. Proof. Fix any good stratification, G, for f at the origin. The stratified critical values of the function t are isolated; hence, in a neighborhood of the origin, the map t restricted to each of the strata of G can have only 0 as a critical value (see [Mas11, Prop. 1.3]). Therefore, for all small t0 = 0, V (t − t0 ) is a prepolar slice of f at p. In particular, Σf ∩ V (t − t0 ) = Σ(f|V (t−t0 ) ). Thus,   ∂f ∂f i Γft0 ,z = V t − t0 , ,..., ¬ Σ(ft0 ) = ∂zi ∂zn

ˆ CYCLES AND HYPERSURFACE SINGULARITIES PART II. LE

 V  V

t − t0 ,

∂f ∂f ,..., ∂zi ∂zn

∂f ∂f t − t0 , ,..., ∂zi ∂zn



81

 ¬ Σf ∩ V (t − t0 ) =

 ¬ Σf = Γi+1 ∩ V (t − t ) ¬ Σf, 0 f,(t,z)

where the last equality uses 1.2.i. But, we claim that this equals Γi+1 ∩ V (t − t0 ) up to embedded subvariety for small t0 = 0. f,(t,z) ∩ V (t − t ) would have a component contained in Σf for an infinite number For otherwise, Γi+1 0 f,(t,z) i+1 of small t0 = 0, which would imply that Γf,(t,z) has a component contained in Σf – a contradiction . Therefore, Γift ,z = Γi+1 ∩ V (t − t0 ) up to embedded subvariety, and of the definition of Γi+1 f,(t,z) f,(t,z) 0 the conclusion follows easily.  As in Theorem 4.5, when we use the coordinates z = (z0 , . . . , zn ) for ft , we use the rotated ˜ = (z1 , z2 , . . . , zn , z0 ) for ft + z0j . coordinates z Theorem 4.15 (Uniform Lˆ e-Iomdine formulas). Let s := dim0 Σf0 , and suppose that s  1. Suppose that λift ,z (0) is defined for all i
s and for all small t. Then, there exist τ > 0 and j0 ◦

such that, for all j  j0 and for all t ∈ Dτ , dim0 Σ(f0 + z0j ) = s − 1, λif +zj ,˜z (0) is defined for all t 0 i
s − 1, and i)

λ0f

(0) = λ0ft ,z (0) + (j − 1)λ1ft ,z (0);

ii)

λif

(0) = (j − 1)λi+1 ft ,z (0), for 1
i
s − 1;

iii)

Σ(ft + z0j ) = Σft ∩ V (z0 ) near 0.

j z t +z0 ,˜

j z t +z0 ,˜

Proof. Given 4.5, all that we must show is that {λ0ft ,z (0)}t0 is bounded for small t0 . Clearly, it 0 suffices to show that {λ0ft ,z (0)}t0 is bounded for small t0 = 0. Of course, what we actually show 0 is that, for small t0 = 0, λ0ft ,z (0) is independent of t0 . 0 For small t0 = 0, we may apply the lemma to conclude  Λ0ft0 ,z  Γ2f,(t,z) · V (t − t0 ) · V

=

Γ1ft0 ,z

∂f ∂z0



·V

∂f ∂z0

 =

 = Γ1f,(t,z) + Λ1f,(t,z) · V (t − t0 ).

Thus, Γ1f,(t,z) + Λ1f,(t,z) has a one-dimensional component, nν [ν], which coincides with C × 0 near 0   (and so, must actually be a component of Λ1f,(t,z) ) and such that λ0ft ,z (0) = nν [ν]·V (t−t0 ) (t ,0) = 0 0 nν for all small non-zero t0 . The conclusion follows.  As we saw in Example 2.10, the Lˆe numbers in a family are not individually upper-semicontinuous. However, we do have the following.

82

DAVID B. MASSEY

Corollary 4.16. Using the notation of the theorem, the tuple of Lˆe numbers

 0 λsft ,z (0), λs−1 ft ,z (0), . . . , λft ,z (0)

is lexigraphically upper-semicontinuous in the t variable, i.e., for all t small, either λsf0 ,z (0) > λsft ,z (0) or s−1 λsf0 ,z (0) = λsft ,z (0) and λs−1 f0 ,z (0) > λft ,z (0)

or

.. .

or s−1 1 1 λsf0 ,z (0) = λsft ,z (0), λs−1 f0 ,z (0) = λft ,z (0), . . . , λf0 ,z (0) = λft ,z (0),

and λ0f0 ,z (0)  λ0ft ,z (0).

Proof. By applying 4.15 inductively, as in 4.6, we find that, if 0  j0  j1  · · ·  js−1 , then, js−1 for all small t, ft + z0j0 + z1j1 + · · · + zs−1 has an isolated singularity at the origin, and its Milnor number is given by js−1 )= µ(ft + z0j0 + z1j1 + · · · + zs−1 λ0ft ,z (0) + (j0 − 1)λ1ft ,z (0) + (j1 − 1)(j0 − 1)λ2ft ,z (0) + . . . +(js−1 − 1) . . . (j1 − 1)(j0 − 1)λsft ,z (0). Now, as the Milnor number is upper-semicontinuous, the conclusion is immediate. 

Before we leave this chapter, we want to see how adding a large power of z0 affects the prepolarity condition. Proposition 4.17. Let G be a good stratification for h at 0 and let V (z0 ) be a prepolar slice with respect to G at the origin. Suppose a = 0 and that j is such that Σ(h + az0j ) = Σh ∩ V (z0 ) as sets. Then, G = {V (h + az0j ) − Σh ∩ V (z0 )} ∪ {G ∩ V (z0 ) | G is a singular stratum of G} is a good stratification for h + az0j at 0. Proof. Suppose we have pi ∈ Σh ∩ V (z0 ) such that pi → p ∈ G ∩ V (z0 ), where G is a singular stratum, and such that Tpi V (h + az0j − (h + az0j )|pi ) → T . We wish to show that Tp (G ∩ V (z0 )) = Tp G ∩ Tp V (z0 ) ⊆ T .

ˆ CYCLES AND HYPERSURFACE SINGULARITIES PART II. LE

83

If T = Tp V (z0 ) of G = {0}, then we are finished. So suppose that T = Tp V (z0 )and G = {0}. Then, for all but a finite number of i, Tpi V (h+az0j −(h+az0j )|pi ) = Tpi V (z0 −z0 (pi )) and pi ∈ Σh. Hence, Tpi V (h − h(pi ), z0 − z0 (pi )) = Tpi V (h + az0j − (h + az0j )|pi , z0 − z0 (pi )) = Tpi V (h + az0j − (h + az0j )|pi ) ∩ Tpi V (z0 − z0 (pi )) → T ∩ Tp V (z0 ) where, by taking a subsequence, we may assume that Tpi V (h−h(pi )) approaches some hyperplane, T . As G is a good stratification for h, Tp G ⊆ T . Moreover, as V (z0 ) transversely intersects G, Tpi V (h − h(pi ), z0 − z0 (pi )) → T ∩ Tp V (z0 ) and thus T ∩ Tp V (z0 ) = T ∩ Tp V (z0 ). Therefore, Tp G ∩ Tp V (z0 ) ⊆ T ∩ Tp V (z0 ) = T ∩ Tp V (z0 ) ⊆ T . 

Corollary 4.18. Let k  0 and suppose (z0 , . . . , zk ) is prepolar for h at the origin. If j is such that   ∂h j−1 i+1 (*) dim0 Γh,z ∩ V ∩ V (z1 , . . . , zi )
0 + jaz0 ∂z0 for all i with 0
i
k, then (z1 , . . . , zk ) is prepolar for h + az0j at 0. Proof. When i = 0, (∗) yields dim0 Γi+1 h,z ∩ V  Σ(h + az0j ) = V



∂h ∂z0

 + jaz0j−1 = 0 and so, as sets,

∂h + jaz0j−1 ∂z0



 ∩V

∂h ∂h ,..., ∂z1 ∂zn

 =



   ∂h j−1 V ∩ Σh ∪ Γ1h,z = + jaz0 ∂z0      ∂h Σh ∩ V (z0 ) ∪ Γ1h,z ∩ V = Σh ∩ V (z0 ). + jaz0j−1 ∂z0 Thus, the hypothesis of 4.17 is satisfied and we apply it; this leaves us with only the problem of showing that each successive hyperplane slice transversely intersects the smooth part, i.e., as germs of sets at the origin, for all i with 0
i
k,     ∂h ∂h ∂h j j−1 Σ(h + az0|V (z ,...,z ) ) = V z1 , . . . , zi , + jaz0 ,..., ∩V = 1 i ∂z0 ∂zi+1 ∂zn 

   ∂h j−1 V z1 , . . . , zi , ∩ Σh ∪ Γi+1 + jaz0 h,z = ∂z0     ∂h j−1 i+1 (Σh ∩ V (z0 , . . . , zi )) ∪ Γh,z ∩ V ∩ V (z1 , . . . , zi ) + jaz0 ∂z0 which, by (∗), equals Σh ∩ V (z0 , . . . , zi ). 

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DAVID B. MASSEY

Proposition 4.19. Let k  0 and suppose (z0 , . . . , zk ) is prepolar for h at the origin. Then, for all large j,   ∂h j−1 i+1 dim0 Γh,z ∩ V ∩ V (z1 , . . . , zi )
0 + jaz0 ∂z0 for all i with 0
i
k and so (z1 , . . . , zk ) is prepolar for h + az0j at 0. i+1 Proof. As (z0 , . . . , zk ) is prepolar for h, we may apply 1.26 to conclude that γh,z (0) exists for all i with 0
i
k, i.e., dim0 Γi+1 h,z ∩ V (z0 , z1 , . . . , zi )
0.

It follows immediately that dim0 Γi+1 h,z ∩ V (z1 , . . . , zi )
1. Therefore,

 dim0 Γi+1 h,z ∩ V

if and only if V

∂h ∂z0

+ jaz0j−1



∂h + jaz0j−1 ∂z0

 ∩ V (z1 , . . . , zi ) 
0

contains a component of Γi+1 h,z ∩ V (z1 , . . . , zi ) through the ori-

gin. But, if a component W of Γi+1 , . . . , zi ) through the origin were contained in both h,z ∩ V (z



1 j −1 j −1 ∂h ∂h V ∂z and V ∂z for j1 = j2 , then z0 would have to equal 0 along that + j1 az01 + j2 az02 0 0 component – a contradiction, as dim0 W ∩ V (z0 )
0. The conclusion follows. 

ˆ CYCLES AND HYPERSURFACE SINGULARITIES PART II. LE

85

ˆ NUMBERS AND HYPERPLANE Chapter 5. LE ARRANGEMENTS The Pl¨ ucker formula of Corollary 4.7 states: Let h be a homogeneous polynomial of degree d in n + 1 variables, let s = dim0 Σh, and suppose that λih,z (0) exists for all i
s. Then, s i i n+1 . i=0 (d − 1) λh,z (0) = (d − 1) This formula allows us to calculate the Lˆe numbers for a central hyperplane arrangement in a purely combinatorial manner from the lattice of flats of the arrangement (see [O-T] and below). It was experimentally observed by D. Welsh and G. Ziegler that there was a fairly trivial relationship between the Lˆe numbers of the arrangement and the M¨ obius function (again, see [O-T] and below). This relationship generalizes to matroid-based polynomial identities (see [MSSVWZ]). In this chapter, we give the combinatorial characterization of the Lˆe numbers for central hyperplane arrangements and prove the relation between the Lˆe numbers and the M¨ obius function. A central hyperplane arrangement in Cn+1 is simply the zero-locus of an analytic function h : Cn+1 → C where h is a product of d linear forms on Cn+1 (here, we are not necessarily assuming that the forms are distinct). Though this may appear to be fairly trivial as a hypersurface singularity, this apparent simplicity is deceiving – the study of hyperplane arrangements is quite complex and touches on many areas of mathematics (see, for instance, [O-R], [O-S],[O-T]). Example 5.1. Suppose we have such an h. In this case, V (h) equals the union of hyperplanes,  {Hi }i∈I , where  I is the indexing set {1, . . . , d }, each Hi occurs with some multiplicity mi := mi = d (in particular, if h is reduced, then each mi = 1 and d = d). mult Hi , and There is an obvious good, Whitney stratification of V (h) obtained from the “flats” of the hyperplane arrangement; the collection of flats is given by {wJ }J⊆I , where +

wJ :=

Hi .

i∈J

If we now take the stratification {SJ }J⊆I , where S J = wJ −



wK ,

J
K

then clearly h is analytically trivial along the strata, and therefore one has trivially a Whitney stratification. In words, the strata are intersections of the hyperplanes minus smaller intersections of hyperplanes. We wish to calculate the Lˆe numbers of h at the origin with respect to generic coordinates z. As h is analytically trivial along the strata, it is easy to see that, as sets, the Lˆe cycles are given by the unions of the flats of correct dimension. Hence, as cycles, for all k, Λkh,z = aJ [wJ ] dim SJ =k

for some aJ . By 1.18, aJ may be calculated by taking any p ∈ SJ and a normal slice N to SJ in Cn+1 at p, and then aJ = λ0h| (p), where we use generic coordinates. After a translation to make N the point p the origin, we see that h|N at p is again (up to multiplication by units) a product of linear forms of degree eJ := i∈J mi .

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DAVID B. MASSEY

Therefore, we may use 4.7 to calculate the Lˆe numbers of h at the origin by a downward induction on the dimension of the flats. (In the following, it looks nicer if we suppress the subscripts.) We denote a hyperplane in the arrangement by H, a flat by w or v, and define e(w) := mult H. w⊆H

Next, we define the vanishing M¨ obius function, η, by downward induction on the dimension of the flats. For a hyperplane, H, in the arrangement, define η(H) := mult H − 1; for a smaller dimensional flat, w, 4.7 tells us that we need η(w) := (e(w) − 1)n+1−dim w − η(v) · (e(w) − 1)codimv w . vw

This equality is equivalent to

(e(w) − 1)dim v η(v) = (e(w) − 1)n+1 .

v⊇w

Finally, having calculated the vanishing M¨ obius function, one has that, for all i, λih,z (0) = η(w). dim w=i

By 3.3, knowing the Lˆe numbers of the hyperplane arrangement gives us the Euler characteristic of the Milnor fibre together with Morse inequalities on the Betti numbers. (Another method for computing the Euler characteristic of the Milnor fibre from the data provided by the containment relations among the flats, i.e., by knowing the intersection lattice, is given in [O-T].) Example 5.2. We wish to see what the above method gives us in the case of a generic central arrangement of d hyperplanes in Cn+1 (see [O-R]). Here, “generic” means as generic as possible considering that all the hyperplanes pass through the origin – that is, each hyperplane occurs with multiplicity 1, and if w is a flat of dimension k, and k = 0, then w is the intersection of precisely n + 1 − k hyperplanes of the arrangement; in terms of the above discussion, this says that if w = 0, then e(w) = n + 1 − k. We assume that d > n + 1 for, otherwise, after a change of coordinates, h = z0 z1 . . . zd−1 and the Milnor fibre is diffeomorphic to the (d − 1)-fold product of C∗ ’s. For a generic arrangement, it is easy to seethat for  all j-dimensional flats w = 0, the number of k-dimensional flats containing w is given by n+1−j k−j , provided that k  j. One also knows that,  d  if k  1, then the number of k-dimensional flats containing the origin is given by n+1−k . This is all the information that one needs to calculate the vanishing M¨ obius function, η, from the formula η(w) := (e(w) − 1)n+1−dim w − η(v) · (e(w) − 1)codimv w vw

together with the fact that for all hyperplanes, H, in the arrangement we have η(H) = 0. It is an amusing exercise to prove that this implies that, if dim w = j = 0, then η(w) = n − j. Alternatively, this also follows from Example 2.8. (The above is the inductive proof of the formula of 2.8 that is referred to in that example.)

ˆ CYCLES AND HYPERSURFACE SINGULARITIES PART II. LE

87

Therefore, for a generic central arrangement of d hyperplanes in Cn+1 , we have with respect to generic coordinates λnh (0) = 0,   d n−1 λh (0) = η(w) = (1), 2 dim w=n−1

.. . λih,z (0)

=





η(w) =

dim w=i

 d (n − i), n+1−i

.. . λ1h (0) =



η(w) =

dim w=1

So, finally,

n

λ0h (0) = (d − 1)n+1 −

  d (n − 1). n

(d − 1)i λih,z (0) =

i=1

(d − 1)

n+1

−

n

 (d − 1)

i=1

i

 d (n − i) = n+1−i

  d−1 (d − 1) , n where the last equality is an exercise in combinatorics. Now, by our earlier work, since we know the Lˆe numbers, we know the Euler characteristic of the Milnor fibre, Fh,0 , together with Morse inequalities on the Betti numbers, bi (Fh,0 ). But, in this special case, it is not difficult to obtain the Betti numbers precisely. By an observation of D. Cohen [Co1], if d > n+1, a generic central arrangement of d hyperplanes in Cn+1 is obtained by taking repeated hyperplane sections of a generic   hyperplane arrangement of d hyperplanes in Cd . It follows that for i
n − 1, bi (Fh,0 ) = d−1 i . Therefore, we have only to calculate bn (Fh,0 ); but, since we know the Euler characteristic, this is easy, and we find – after some more combinatorics- that   d−1 bn (Fh,0 ) = (d − n) , n which agrees with the results of [Co1] and [O-R]. Note that the Morse inequalities of 3.3 can be far from equalities; for instance, the two easiest inequalities are     d−1 d−1 (d − n) = bn (Fh,0 )
λ0h,z (0) = (d − 1) n n and d − 1 = b1 (Fh,0 )


λn−1 h,z (0)

  d d(d − 1) = = . 2 2

88

DAVID B. MASSEY

Now, we wish to describe the relation between the Lˆe numbers of a central arrangement and the M¨ obius function – this is the result which is generalized in [MSSVWZ]. Let h be the product of d distinct linear forms on Cn+1 , so that each hyperplane in the arrangement V (h) occurs with multiplicity 1. Let A denote the collection of hyperplanes which are components of V (h). We use the variable H to denote hyperplanes in A. We use the letters v and w to denote flats of arbitrary dimension. Finally, in agreement with our notation in 5.1, let eA (v) = the number of hyperplanes of A which contain the flat v. As we saw in 5.1 and 5.2, the Lˆe numbers of a central hyperplane arrangement can be described in terms of a function ηA defined inductively on the flats by: for all H ∈ A, ηA (H) = 0, and for all flats w, (eA (w) − 1)dim v ηA (v) = (eA (w) − 1)n+1 . w⊆v

The M¨ obius function, µA , on A (see [O-T]) is defined inductively on the flats by: µA (Cn+1 ) = 1 and for all flats v
w, µA (u) = 0. flats u v⊆u⊆w

Here, we subscript by η, e, and µ by A because our proof is by induction on the ambient dimension, and the inductive step requires slicing A by hyperplanes, N , not contained in A. This will produce new arrangements inside the ambient space N . So it is important that we indicate which arrangement is under consideration. More notation now, related to the slicing. We will be taking two kinds of hyperplane slices. N will denote a prepolar hyperplane slice through the origin in Cn+1 , i.e., a hyperplane slice which contains no flats of A other than the origin. We will also use normal slices to the one-flats; if v is a one-dimensional flat and pv ∈ v − 0, Nv will denote a normal slice to v at pv – that is, Nv is a hyperplane in Cn+1 which transversely intersects v at pv . We use A ∩ N to denote the obvious induced arrangement in N (which is identified with Cn ). The arrangement A ∩ Nv is considered as a central arrangement where pv becomes the origin and all hyperplanes not containing pv are ignored. Note that the number of hyperplanes in the arrangement A ∩ Nv is eA (v). An arrangement is essential provided that the origin is a flat of the arrangement (hence, the arrangement is not trivially a product). What we want to show is that, if A is a an essential, central hyperplane arrangement, then ηA (0) = (d − 1)(−1)n+1 µA (0) = (d − 1)|µA (0)|. To induct, we will first need the following three easy lemmas on η, µ, which describe the effects of slicing. We leave the first two as exercises using the inductive definitions of ηA andµA given above. However, we prove the third. Lemma 5.3. ηA∩N (0) =

ηA (0) + d−1

dim v=1

and, if v is a one-dimensional flat, ηA∩Nv (pv ) = ηA (v).

ηA (v)

ˆ CYCLES AND HYPERSURFACE SINGULARITIES PART II. LE

Lemma 5.4. µA∩N (0) = −



89

µA (v).

dim v≥2

and, if v is a one-dimensional flat, µA∩Nv (pv ) = µA (v).

Lemma 5.5. dµA (0) +



(d − eA (v))µA (v) = 0.

dim v=1

Proof. By one of Weisner’s formulas (see Lemma 2.40 of [O-T]), for all H ∈ A,

µA (v) = 0.

v∩H=0

Hence,

    0= µA (v) = dµA (0) + (d − e(v))µA (v).  µA (0) + H

dim v=1 v⊆H

dim v=1

Now, we can prove Theorem 5.6. If A is a an essential, central hyperplane arrangement consisting of d hyperplanes in Cn+1 , then ηA (0) = (d − 1)(−1)n+1 µA (0) = (d − 1)|µA (0)|.

Proof. The proof is by induction on the ambient dimension. The formula is stupidly true when n = 1. Now, suppose the formula is true for ambient dimension n. Then, ηA∩N (0) = (d − 1)(−1)n µA∩N (0). Hence, by Lemmas 5.3 and 5.4, ηA (0) + d−1 This gives us ηA (0) + d−1

dim v=1

dim v=1

 ηA (v) = (d − 1)(−1)n −



 µA (v) .

dim v2

ηA∩Nv (pv ) = (d − 1)(−1)n+1

dim v2

µA (v).

90

DAVID B. MASSEY

Using our inductive hypothesis again, we have ηA (0) + d−1



(eA (v) − 1)(−1)n µA∩Nv (pv ) = (d − 1)(−1)n+1

dim v=1



µA (v).

dim v2

Therefore, ηA (0) + d−1



(eA (v) − 1)(−1)n µA (v) = (d − 1)(−1)n+1

dim v=1



µA (v)

dim v≥2

and so  ηA (0) = (d − 1)(−1)n+1 (d − 1)



µA (v) +

dim v≥2

 (d − 1)(−1)

n+1



 (eA (v) − 1)µA (v) =

dim v=1

!   (d − 1) − µA (0) − µA (v) + (eA (v) − 1)µA (v) 

dim v=1

= (d − 1)(−1)n+1 µA (0) − dµA (0) −

dim v=1



!

(d − eA (v))µA (v) .

dim v=1

Now apply Lemma 5.5.  Our inductive proof given above is somewhat unsatisfactory, for it gives us no geometric insight as to why the theorem is true. Should there be a geometric explanation for the identity in 5.6? Probably so. The result of Orlik and Solomon [O-S] is that |µA (0)| is the (n + 1)-st Betti number of the complement of the arrangement A in Cn+1 . How this could be used to prove 5.6 still escapes us.

ˆ CYCLES AND HYPERSURFACE SINGULARITIES PART II. LE

91

Chapter 6. THOM’S af CONDITION In this chapter, we will use the Lˆe numbers to provide conditions under which a submanifold of affine space satisfies Thom’s af condition with respect to the ambient stratum. Let us recall the definition of the af condition (see [Mat]). Definition 6.1. Let U be an open subset of some affine space, let f : U → C be an analytic function, and let M ⊆ V (f ) be a submanifold of U. Thom’s af condition is satisfied between U − Σf and M (or along M , or by the pair (U − Σf, M )) if, whenever pi ∈ U − Σf , pi → p ∈ M , and Tpi V (f − f (pi )) → T , then Tp M ⊆ T .

axis

V(f ) level surfaces of f

limiting tangent plane

Figure 6.2. Failure of a f along an axis

We are about to prove five different results – all of the form: if the Lˆe numbers are constant, then Thom’s af condition holds. The order in which we must prove these results is interesting. First, we give a proof of Lˆe and Saito’s result that a constant Milnor number at the origin in a one-parameter family implies the Thom condition along the parameter axis. We then use Lˆe and Saito’s result, combined with the generalized Lˆe-Iomdine formulas, to prove that the constancy of the Lˆe numbers at the origin in a one-parameter family implies the Thom condition along the parameter axis [Mas14]. We use this parameterized version to prove a non-parameterized version: if the Lˆe numbers of a single function are constant along a submanifold, then Thom’s condition is satisfied along the submanifold. This non-parameterized version allows us to prove a multiparameter version of Lˆe and Saito’s result: if we have a family of isolated hypersurface singularities with constant Milnor number parameterized along a submanifold, then that submanifold satisfies Thom’s af condition with respect to the ambient stratum. Finally, we use this last result to prove our best result – the multi-parameter version of the Lˆe number result above: if we have a family of hypersurface singularities with constant Lˆe numbers parameterized along a submanifold, then that submanifold satisfies Thom’s af condition with respect to the ambient stratum.

92

DAVID B. MASSEY

In all of the results described above, it is extremely important that our assumptions on the genericity of the coordinate system will be solely that the Lˆ e numbers exist. This is a dimensional requirement which is very easy to check. This should be contrasted with the results of [H-M] and [HMS]. First, we need a well-known lemma. ◦

Lemma 6.3. Let D be an open disc about the origin in C, let U be an open neighborhood of the ◦

◦

origin in Cn+1 , and let f : (D × U, D × 0) → (C, 0) be an analytic function; we write ft for the function defined by ft (z) := f (t, z). Suppose that dim0 Σf0 = 0. Then, for all small t, the Milnor number of ft at the origin is independent of  if

t if and only ◦

there exists an open neighborhood, W, of the origin in D × U such that W ∩ V W ∩ (C × {0}).

∂f ∂f ∂z0 , . . . , ∂zn

=

Proof. There are many proofs of this fact. We shall use intersection numbers. of f0 at the origin, µ0 (f0 ), equals the multiplicity of the origin in the cycle   The

Milnor number ∂f0 ∂f0 ∂f ∂f is V ∂z0 , . . . , ∂zn . Because f0 has an isolated critical point at the origin, V t, ∂z , . . . , ∂zn 0 a local complete intersection, and so we have an equality of cycles          ∂f0 ∂f ∂f ∂f0 ∂f ∂f V = V t, = [V (t)] · V . ,..., ,..., ,..., ∂z0 ∂zn ∂z0 ∂zn ∂z0 ∂zn Therefore, 

 V (t) · V

∂f ∂f ,..., ∂z0 ∂zn

 0

µ0 (f0 ) =    ∂f ∂f V (t − η) · V = ,..., ∂z0 ∂zn p ◦ p∈B 

= µ0 (fη ) + R, ◦

where B
is a sufficiently small open ball around the origin in Cn+2 , η is chosen small with respect to
– that is, there exists δ
> 0 such that we may use any η satisfying 0 < |η| < δ
– and R denotes the sum of the remaining terms, i.e., the terms coming from points p which are not in C×{0}. Note 

◦ ∂f ∂f that the sum is actually finite since we are really summing over p ∈ B
∩V (t−η)∩V ∂z , . . . , ∂zn . 0 As all the intersection numbers are non-negative, R being zero is equivalent to there being no  ◦

∂f ∂f remaining terms, i.e., equivalent to (η, 0) being the only point in B
∩ V (t − η) ∩ V ∂z . , . . . , ∂z 0 n The desired conclusion follows immediately, where the set W in the statement can be taken to ◦ ◦  be B
∩ Dδ × U . 

Recall now the result of Lˆe and Saito [Lˆ e-Sa] as stated in the introduction. ◦

Theorem 6.4 (Lˆe-Saito [Lˆ e-Sa]). Let D be an open disc about the origin in C, let U be an open

ˆ CYCLES AND HYPERSURFACE SINGULARITIES PART II. LE ◦

93

◦

neighborhood of the origin in Cn+1 , and let f : (D × U, D × 0) → (C, 0) be an analytic function; we write ft for the function defined by ft (z) := f (t, z). Suppose that dim0 Σf0 = 0 and that, for all small t, the Milnor number of ft at the origin is ◦

independent of t. Then, D × {0} satisfies Thom’s af condition at the origin with respect to the ◦

ambient stratum, i.e., if pi is a sequence of points in D × U − Σf such that pi → 0 and such that ◦

Tpi V (f − f (pi )) converges to some T , then C × 0 = T0 (D × {0}) ⊆ T . Proof. We begin by noting that the existence of good stratifications as given in Proposition 1.22 ◦

implies that Thom’s af is satisfied, near the origin, by D × {0} − 0 with respect to the ambient stratum. ◦

Now, consider the blow-up of D × U by the Jacobian ideal of f :  ◦  ◦ BlJ(f ) D × U ⊆ D × U × Pn+1 π1 

 π2

◦

D×U

Pn+1

We first wish to show that the fibre π1−1 (0) has dimension at most n. The point q := [1 : 0 : · · · : 0] ∈ Pn+1 corresponds to the hyperplane V (t). As µ0 (ft ) is ◦

independent of t, the lemma implies that, in a neighborhood of the origin, π1 (π2−1 (q)) ⊆ D × {0}. ◦

However, as we noted above, the af condition holds generically on D × {0}. Therefore, near 0, either π1 (π2−1 (q)) is empty or consists only of the origin. But, the dimension of every component  ◦ of π2−1 (q) is at least dim BlJ(f ) D × U − dim Pn+1 = n + 2 − (n + 1) = 1. Thus, 0 ∈ π1 (π2−1 (q)), i.e., q ∈ π2 (π1−1 (0)). It follows that π1−1 (0) is a proper subset of Pn+1 and, hence, has dimension at most n. But, every component of the exceptional divisor E := π1−1 (Σf ) has dimension n + 1. Therefore, above an open neighborhood of the origin, E equals the topological closure of E − π1−1 (0), which is ◦

contained in (D × {0}) × ({0} × Pn ) since the af condition holds generically on the t-axis. It follows ◦

that π2 (π1−1 (0)) ⊆ {0} × Pn , i.e., that the af condition holds along D × {0} at the origin. 

Our first generalization of the result of Lˆe and Saito is: ◦

Theorem 6.5. Let D be an open disc about the origin in C, let U be an open neighborhood of the ◦

◦

origin in Cn+1 , and let f : (D × U, D × 0) → (C, 0) be an analytic function; we write ft for the function defined by ft (z) := f (t, z). Let s = dim0 Σf0 . Suppose that, for all small t, for all i with 0
i
s, λift ,z (0) is defined and ◦

is independent of t. Then, D × 0 satisfies Thom’s af condition at the origin with respect to the ◦

ambient stratum, i.e., if pi is a sequence of points in D × U − Σf such that pi → 0 and such that ◦

Tpi V (f − f (pi )) converges to some T , then C × 0 = T0 (D × 0) ⊆ T .

94

DAVID B. MASSEY

Proof. The proof is by induction on s. For s = 0, the theorem is exactly that of Lˆe and Saito in 6.4. Now, suppose that s  1 and that, for all small t, for all i with 0
i
s, λift ,z (0) is defined and ◦

is independent of t, but that there exists a sequence pi of points in D × U − Σf such that pi → 0, ◦

such that Tpi V (f − f (pi )) converges to some T , and T0 (D × 0) ⊆ T . As the collection of such limiting T is analytic, we may apply the curve selection lemma (see [Loo]) to conclude that there exists a real analytic curve ◦

α : [0,
) → {0} ∪ (D × U − Σf ) such that α(u) = 0 if and only if u = 0 and such that lim Tα(u) V (f − f (α(u))) = T .

u→0

As α is real analytic, it is trivial to show that, for all large j, lim

u→0

grad(f + z0j )|α(u) | grad(f +

z0j )|α(u) |

= lim

u→0

grad(f )|α(u) | grad(f )|α(u) |

. ◦

Therefore, for all large j, the family ft + z0j also has T as a limit to level hypersurfaces, i.e., D × 0 does not satisfy the af +zj condition at the origin with respect to the ambient stratum. 0 However, λ0ft ,z (0) is independent of t and, applying Theorem 4.5, if j  2 + λ0ft ,z (0), then the family ft + z0j has Lˆe numbers independent of t, and f0 + z0j has a critical locus of dimension s − 1. Thus, our inductive hypothesis contradicts the previous paragraph.  Corollary 6.6. Let h : U → C be an analytic function on an open subset of Cn+1 , let z = (z0 , . . . , zn ) be a linear choice of coordinates for Cn+1 , let M be an analytic submanifold of V (h), let q ∈ M , and let s denote dimq Σh. If, for each i such that 0
i
s, λih,z (p) is defined and is independent of p, for all p ∈ M near q, then M satisfies Thom’s ah condition at q with respect to the ambient stratum; that is, if qi is a sequence of points in U − Σh such that qi → q and such that Tqi V (h − h(qi )) converges to some T , then Tq M ⊆ T . Proof. This follows from 6.5 by a fairly standard trick. Let c(t) be a smooth analytic path in M such that c(0) = q. If we can show that any limiting tangent plane, T , contains the tangent to the image of c at q, then we will be finished. So, take such a c, and suppose that we have a sequence of points, qi , in U −Σh such that qi → q and such that Tqi V (h − h(qi )) → T . Define f (t, z) := h(z + c(t)), and consider the sequence of points (0, qi − q). If one now applies the theorem, the result is that c (0) ⊆ T ; we leave the details to the reader. 

Remark 6.7. It is important to note that, in 6.6, we only require that the coordinates are generic enough so that the Lˆe numbers are defined; we are not requiring that the coordinates are prepolar. On the other hand, Corollary 6.6 tells us how we can obtain good stratifications: if we have an analytic stratification of V (h) such that the Lˆe numbers are defined and constant along the strata,

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then the stratification is actually a good stratification. However, there is no guarantee that the coordinates used to define the Lˆe numbers are prepolar with respect to this good stratification. Now we can prove the multi-parameter version of the result of Lˆe and Saito. Theorem 6.8. Let M be an open neighborhood of the origin in Ck , let U be an open neighborhood of the origin in Cn+1 , and let f : (M × U, M × 0) → (C, 0) be an analytic function; we write ft for the function defined by ft (z) := f (t, z), where t ∈ M and z ∈ U. Suppose that dim0 Σf0 = 0 and that, for all t near the origin, the Milnor number of ft at the origin is independent of t. Then, M × 0 satisfies Thom’s af condition at the origin with respect to the ambient stratum, i.e., if pi is a sequence of points in M × U − Σf such that pi → 0 and such that Tpi V (f − f (pi )) converges to some T , then T0 (M × 0) ⊆ T . Proof. If the constant value of the Milnor number is 0, then there is nothing to prove. Note, though, that if the constant value of the Milnor number is non-zero, then it follows from Sard’s theorem that M × 0 ⊆ Σf , i.e., the critical points of the ft are not merely a result of critical points of the map t restricted to the smooth part of V (f ). Let a be an element of M near the origin. The Milnor number of fa at the origin satisfies the equality    ∂f ∂f µ0 (fa ) = V t0 − a0 , . . . , tk−1 − ak−1 , ,..., . ∂z0 ∂zn (a,0) In particular,

 dim(a,0) V

∂f ∂f t0 − a0 , . . . , tk−1 − ak−1 , ,..., ∂z0 ∂zn

 = 0.

This immediately implies that dim(a,0) Σf
k. Hence,    ∂f ∂f V = Γkf,(t,z) + Λkf,(t,z) , ,..., ∂z0 ∂zn k both γf,(t,z) (a, 0) and λkf,(t,z) (a, 0) exist, and k µ0 (fa ) = γf,(t,z) (a, 0) + λkf,(t,z) (a, 0). k As µ0 (fa ) is independent of a, and both γf,(t,z) (a, 0) and λkf,(t,z) (a, 0) are upper-semicontinuous k as functions of a, we conclude that both γf,(t,z) (a, 0) and λkf,(t,z) (a, 0) are independent of a. k (a, 0) is independent of a for (a, 0) in a k-dimensional component of This implies that γf,(t,z) k k Σf . But, Γf,(t,z) cannot contain a component of Σf . Therefore, the constant value of γf,(t,z) (a, 0) k for a ∈ M must be 0; that is, Γf,(t,z) does not intersect M × 0 near the origin. But, all the lower-dimensional relative polar cycles are contained in Γkf,(t,z) ; thus, none of them hit M × 0. This implies that λif,(t,z) (a, 0) = 0 for all a ∈ M and all i with 0
i
k − 1. As we already saw that λkf,(t,z) (a, 0) is independent of a ∈ M , we see that all the Lˆe numbers of f are constant along M × 0. Now, apply Corollary 6.6. 

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Finally, we have the multi-parameter generalization of the result of Lˆe and Saito, where the critical loci may have arbitrary dimension. Theorem 6.9. Let M be an open neighborhood of the origin in Ck , let U be an open neighborhood of the origin in Cn+1 , and let f : (M × U, M × 0) → (C, 0) be an analytic function; we write ft for the function defined by ft (z) := f (t, z), where t ∈ M and z ∈ U. Let s = dim0 Σf0 . Suppose that, for all small t, for all i with 0
i
s, λift ,z (0) is defined and is independent of t. Then, M × 0 satisfies Thom’s af condition at the origin with respect to the ambient stratum, i.e., if pi is a sequence of points in M × U − Σf such that pi → 0 and such that Tpi V (f − f (pi )) converges to some T , then T0 (M × 0) ⊆ T . Proof. The proof is by induction on s. To obtain 6.9 from 6.8, one follows word for word our derivation of 6.5 from 6.4. 

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Chapter 7. ALIGNED SINGULARITIES In this chapter, we once again consider analytic functions h : U → C. We wish to investigate those h for which the critical locus, Σh, is of a particularly nice form – a form which generalizes isolated singularities, smooth one-dimensional singularities (line singularities), and the singularities found in hyperplane arrangements (see Chapter 5). The obvious generalization of merely requiring the irreducible components of Σh to be smooth appears to be too general to yield nice results; what one would like is to put some restrictions on the subset of Σh where h fails to be “equisingular”. For instance, in the case where Σh is smooth and 2-dimensional, any reasonable notion of equisingularity could fail on a subset of dimension at most one; a reasonable condition to impose is that this one-dimensional subset itself be smooth. Essentially this is what we require of an aligned singularity.

For convenience, throughout this section, we concentrate our attention on hypersurface germs at the origin. Definition 7.1. If h : (U, 0) → (C, 0) is an analytic function, then an aligned good stratification for h at the origin is a good stratification for h at the origin in which the closure of each stratum of the singular set is smooth at the origin. If such an aligned good stratification exists, we say that h has an aligned singularity at the origin. If {Sα } is an aligned good stratification for h at the origin, then we say that a linear choice of coordinates, z, is an aligning set of coordinates for {Sα } provided that for each i, V (z0 , . . . , zi−1 ) transversely intersects the closure of each stratum of dimension at least i at the origin. Naturally, we say simply that a set of coordinates, z, is aligning for h at the origin provided that there exists an aligned good stratification for h at the origin with respect to which z is aligning. Note that, given an aligned singularity, aligning sets of coordinates are generic (in the IPZtopology) and prepolar. It is important that, in fact, aligning coordinates are prepolar at all points in an entire neighborhood of the origin; the importance of this fact stems from the following result (in which we are not assuming that h has an aligned singularity).

Theorem 7.2. If the coordinates z are prepolar at all points in a neighborhood Ω of a point p with respect to a good stratification {Sα } for h at p, then, inside Ω, for all i, |Λih,z | ⊆



Sα .

dim0 Sα
i

Proof. When i = 0, we must show that if p is in Λ0h,z , then p is also a stratum. Suppose not. Then, p is in some stratum S of dimension at least 1 and, as p ∈ Λ0h,z , we must have p ∈ Γ1h,z . As our coordinates are prepolar, V (z0 − p0 ) transversely intersects S at p. This, however, is a contradiction, since p ∈ Γ1h,z implies that there is a sequence of limiting tangent planes to level hypersurfaces which converges to {0} × Cn at p and, hence, Tp S should be contained in {0} × Cn since S is a good stratum.

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Now, suppose that we have a point q ∈ Ω such that q ∈ |Λih,z | and q ∈



Sα .

dim0 Sα
i

Then, q ∈ |Λih,z | ∩ Sβ for some good stratum Sβ of dimension at least i + 1. As z is prepolar at q, S  := {Sα ∩ V (z0 − q0 , . . . , zi−1 − qi−1 )}α is a good stratification for h|V (z0 −p0 ,...,zi−1 −pi−1 ) at q; in addition, V (z0 − p0 , . . . , zi−1 − pi−1 ) transversely intersects Sβ at q in a set of dimension at least 1. Thus, S  does not contain {q} as a stratum. ˆ := (zi , . . . , zn ). Then, z ˆ is prepolar with respect to S  and so, we conclude from the Now, let z i = 0 case (at the beginning of the proof) that q ∈ Λ0h

|V (z −p ,...,z 0 0 i−1 −pi−1 )

,ˆ z.

By repeated applications of Theorem 1.26 and Proposition 1.18, it follows that q ∈ Λih,z ; this is a contradiction. 

Closely related to the notion of aligning sets of coordinates is Definition 7.3. If h : (U, 0) → (C, 0) is an analytic function on an open subset of Cn+1 , then a linear choice of coordinates, z, for Cn+1 is pre-aligning for h at the origin provided that for each Lˆe cycle, Λih,z , and for each irreducible component, C, of Λih,z passing through the origin, the following conditions are satisfied: i)

dim0 C = i;

ii)

C is smooth at the origin;

iii)

V (z0 , z1 , . . . , zi−1 ) transversely intersects C at the origin.

Proposition 7.4. If h has an aligned singularity at the origin, then for a generic linear reorganization of the coordinates z, z is pre-aligning for h at the origin, and for all p near the origin, the reduced Euler characteristic of the Milnor fibre of h at p is given by χ (Fh,p ) =

s

(−1)n−i λih,p (0).

i=0

Proof. One may simply choose z to be aligning. Theorem 7.2 then implies that z is pre-aligning. The Euler characteristic statement follows at once from Theorem 3.3. 

Remark 7.5. It is tempting to think that if z is a set of pre-aligning coordinates for h at the origin, then we can produce an aligned good stratification by considering the components of  Λih,z − Λjh,z . j
i−1

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This might seem reasonable in light of Corollary 6.6 and Remark 6.7. However, we see no reason for the higher Lˆe numbers to be constant along these proposed “strata”. We could define a more restricted class of singularities – super aligned singularities – by requiring the existence of an aligned good stratification in which, for each i with 0
i
dim0 Σh =: s, there is at most one connected stratum, say S i , of dimension i and S 0 ⊆ S 1 ⊆ · · · ⊆ S s−1 ⊆ S s . It is easy to see that this is equivalent to the existence of a set of pre-aligning coordinates, z, for h at the origin such that each Λih,z has a single smooth component at the origin and, as germs of sets at the origin, Λ0h,z ⊆ Λ1h,z ⊆ · · · ⊆ Λsh,z . For such super aligned singularities, we obtain a good stratification for h by taking the stratification {Λj+1 − Λjh,z }. h,z

Our main interest in aligned singularities is due to Proposition 7.6. Suppose that h has an aligned s-dimensional singularity at the origin and that the coordinates z are aligning. Then, the Lˆe cycles and Lˆe numbers can be characterized topologically in the following inductive manner: As a set, Λsh,z equals the union of the s-dimensional components of the singular set of h. To determine the Lˆe cycle, to each s-dimensional component, C of Σh, we assign the multiplicity mC = (−1)n−s χ (Fh,p ) for generic p ∈ C, where Fh,p denotes the Milnorfibre of h at p and χ  is the reduced Euler characteristic. Moreover, for all p ∈ |Λsh,z |, λsh,z (p) = p∈C mC . Now, suppose that we have defined the Lˆe numbers, Λih,z (p) for all i  k + 1 and for all p near the origin. Then, as a set, Λkh,z equals the closure of the k-dimensional components of the set of points p ∈ V (h) where s χ (Fh,p ) = (−1)n−i λih,z (p). i=k+1

The Lˆe cycle is defined by assigning to each irreducible component C of this set the multiplicity   s n−k n−i i mC = (−1) χ (Fh,p ) − (−1) λh,z (p) , i=k+1

for generic p ∈ C. Finally, for all p ∈ |Λkh,z |, we have λkh,z (p) =

 p∈C

mC .

Proof. As the aligning coordinates are prepolar at each point near the origin, this essentially follows from Theorem 3.3. However, in writing that λkh,z (p) = mC , p∈C

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we are crucially using that the components of the Lˆe cycle are smooth and tranversely intersected by V (z0 − p0 , . . . , zk−1 − pk−1 ). If this were not the case, the intersection multiplicities of V (z0 − p0 , . . . , zk−1 − pk−1 ) with the components of the Lˆe cycles would enter the picture, and the characterization would no longer be purely topological.  The following two corollaries are immediate: Corollary 7.7. If h has an aligned singularity at the origin, then all aligning coordinates z determine the same Lˆe cycles and Lˆe numbers. Corollary 7.8. Let f and g be reduced, analytic germs with aligned singularities at the origin in ˜ be aligning sets of coordinates for f and g, respectively. If H is a local, ambient Cn+1 . Let z and z homeomorphism from the germ of V (f ) at the origin to the germ of V (g) at the origin, then as germs of sets at the origin, H(Λif,z ) = Λig,˜z , for all i, and for all p near the origin in Cn+1 , λif,z (p) = λig,˜z (H(p)), for all i.

Now, we will give an amusing application of the results of this section. In [Z], Zariski conjectures that the multiplicity of a hypersurface at a point is an invariant of the local, ambient topological type of the hypersurface. A number of people have concentrated on the case of a one-parameter family of isolated singularities, but even this case has not been settled (however, for families of quasi-homogeneous isolated singularities, the proof of the conjecture has been given by Greuel [Gr1] and O’Shea [O’S]). In our paper [Mas13], we prove a result which perhaps supplies a better place to look for counterexamples to the Zariski Multiplicity Conjecture; we prove, for families of hypersurfaces of dimension unequal to 2, that the Zariski Multiplicity Conjecture is true for families of hypersurfaces with isolated singularities if and only if it is true for families of hypersurfaces with smooth onedimensional critical loci. The results of this section allow us to generalize this. Theorem 7.9. The following are equivalent: i) for all n  3, the Zariski Multiplicity Conjecture is true for families of reduced analytic hypersurfaces ft : (U, 0) → (C, 0), where U is an open subset of Cn+1 and dim0 Σft = 0; ii) for all n  3, there exists a k such that the Zariski Multiplicity Conjecture is true for families of reduced analytic hypersurfaces ft : (U, 0) → (C, 0) with aligned singularities, where U is an open subset of Cn+1 and dim0 Σft = k; iii) for all n  3, for all k, the Zariski Multiplicity Conjecture is true for families of reduced analytic hypersurfaces ft : (U, 0) → (C, 0) with aligned singularities, where U is an open subset of Cn+1 and dim0 Σft = k.

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Proof. Certainly, iii) implies ii). We will show that ii) implies i) and that i) implies iii). Suppose that ii) is true for some k  1. Let ft : (U, 0) → (C, 0) be an analytic family, where U is an open subset of Cn+1 , n  3, dim0 Σft = 0, and such that the local ambient topological type of the hypersurfaces V (ft ) at the origin is independent of t. Then clearly, the family f˜t : (U × Ck , 0) → (C, 0) defined by f˜t (z, w) := ft (z) is a family of aligned singularities of dimension k with constant topological type. Hence, by ii), mult0 f˜t is independent of t. Now, as mult0 ft clearly equals mult0 f˜t , we are finished with the implication that ii) implies i). The interesting implication is, of course, that i) implies iii). Ideally, we would like to be able to select linear coordinates, z, for Cn+1 which are aligning for ft at the origin for all small t; however, a proof that this is possible seems problematic. We will avoid needing such a result by being somewhat devious and applying the Baire Category Theorem. Suppose that i) is true, and that we have a family of reduced analytic hypersurfaces ft : (U, 0) → (C, 0) with aligned singularities, where U is an open subset of Cn+1 , n  3, dim0 Σft = k, and such that the local ambient topological type of the hypersurfaces V (ft ) at the origin is independent of t. 1 Let tm be an infinite sequence in C which approaches 0, e.g., tm = m . For each tm , there exists n+1 ) representing aligned coordinates for ftm . We may apply the Baire a generic subset of P GL(C Category Theorem to conclude that there exists a choice of coordinates, z, which is aligning for f0 and for ftm for all m. Let us fix such a choice of coordinates. Then, by 7.8, the Lˆe numbers λif ,z (0) are equal to the Lˆe numbers λift ,z (0) for all large m. 0

m

j

s−1 By 4.6, if we take 0  j0  j1  · · ·  js−1 , then f0 + z0j0 + z1j1 + · · · + zs−1 has an isolated js−1 singularity at the origin; this implies that, for all small t, ft + z0j0 + z1j1 + · · · + zs−1 has, at worst, jk−1 j0 j1 an isolated singularity at the origin. In addition, 4.6 tells us that f0 + z0 + z1 + · · · + zk−1 has jk−1 the same Milnor number at the origin as ftm + z0j0 + z1j1 + · · · + zk−1 for all large m. As the Milnor jk−1 is upper-semicontinuous, it follows number at the origin in the family ft + z0j0 + z1j1 + · · · + zk−1 that, in fact, the Milnor number in this family is independent of t for all small t. Hence, by [L-R], the local, ambient topological type is independent of t in the family ft + z0j0 + jk−1 j1 z1 + · · · + zk−1 . Since we are assuming i), this implies that the multiplicity is independent of t in jk−1 the family ft + z0j0 + z1j1 + · · · + zk−1 . Finally, as the jm ’s are arbitrarily large, this implies that the multiplicity is independent of t in the family ft . 

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Chapter 8. SUSPENDING SINGULARITIES In [Ok], [Sak], and [Se-Th], the general question is addressed of how the structure of the Milnor fibre of f (z) + g(w) (where z and w are disjoint sets of variables) depends on the Milnor fibres of f and g. However, in each of these papers, f and g have isolated singularities or are quasi-homogeneous. Sakamoto remarks at the end of his paper that, by using Lˆe’s notion of a good stratification, he can prove his main lemmas without the isolated singularity assumptions. As we crucially need this result in the special case of f (z) + wj , we will use our results in the appendix to indicate how one needs to modify Sakamoto’s proof. After we describe the homotopy-type of the Milnor fibre of f (z)+wj , we will use this description to give a new generalization of the formula of Lˆe and Iomdine [Lˆ e4] – a different generalization than the formulas of Chapter 4. This new generalization appears in [Mas12]. ˜ Proposition 8.1. If j  2, then up to homotopy, the Milnor fibre of h(w, z) := h(z) + wj at the origin is the one-point union (wedge) of j − 1 copies of the suspension of the Milnor fibre of h at the origin. Proof. By Proposition C.14, we may use neighborhoods of the form Dω × B
, 0 < ω 
, to define ˜ at the origin. the Milnor fibre of h Now, for 0 < |ξ|  ω 
, consider the map 

 w Dω × B
∩ V (h + wj − ξ) −−−→ Dω .

This map is a proper, stratified submersion above all points of Dω − V (wj − ξ), i.e., except at the j roots of ξ. Thus, except above these j points, the fibre is the same as that above 0, which is clearly nothing more than the Milnor fibre of h at the origin. In addition, above each point of V (wj − ξ), the fibre is B
∩ V (h), which is contractible.

Figure 8.2. The fibres over the disc In fact, around each point α1 , . . . , αj in V (wj − ξ) ⊆ Dω , there is an arbitrarily small disc Dαi ⊆ Dω above which the total space is contractible. We choose the Dαi disjoint. Connect all of the Dαi to the origin by disjoint paths. Let P denote the subset of Dω consisting of the paths; so, P is a contractible set which has exactly one point in common with each of the Dαi , and the fibre above each point of P has the homotopy-type of the Milnor fibre of h at the origin.

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The result now follows easily. For the details, we refer the reader to Sakamoto [Sak] – the remainder of our proof now follows his exactly. 

We shall now use 8.1 to give our second generalization of the formula of Lˆe and Iomdine. Let U be an open neighborhood of the origin in Cn+1 , let h : (U, 0) → (C, 0) be an analytic function, and suppose that the linear from L : Cn+1 → C is prepolar with respect to h at the origin. The formula of Lˆe and Iomdine says that, if dim0 Σh = 1, then, for all large j, h + Lj has an isolated singularity at the origin and bn (h + Lj ) = µ(h + Lj ) = bn (h) − bn−1 (h) + j mν δν (h), ν

where bi () denotes the i-th Betti number of the Milnor fibre of a function at the origin, µ denotes the Milnor number of the isolated singularity at the origin, the summation is over all components, ν, of Σh, mν is the local degree of L restricted to ν at the origin, and δν (h) is the Milnor number of a generic hyperplane slice of h at a point p ∈ ν − 0 sufficiently close to the origin. This formula has, at least, two possible generalizations. One generalization is in terms of Lˆe numbers, as given in Chapter 4. But, while there are Morse inequalities between the Lˆe numbers and the Betti numbers of the Milnor fibre, the Lˆe numbers are not themselves (generally) Betti numbers of the Milnor fibre. So, one might ask for a generalization of the formula of Lˆe and Iomdine which generalizes the Betti number information. In remainder of this chapter, we prove that, if dim0 Σh = s  1, then, for all large j, dim0 Σ(h + Lj ) = s − 1 and   1 (0) . bn (h + Lj ) = bn (h) − bn−1 (h) + j bn−1 (h|V (L) ) − γh,L In the case where h has a one-dimensional critical locus at the origin, it is easy to show that this new formula reduces to that of Lˆe and Iomdine. We consider this new Lˆe-Iomdine formula interesting because it implies that 1 bn−1 (h|V (L) )  γh,L (0).

In terms of deformations, this says that the top possible non-zero Betti number of the Milnor fibre 1 of h|V (L) is greater than or equal to γh,L (0) for all h which have V (L) as a prepolar slice. Thus, if we define h to be a prepolar deformation of h|V (L) precisely when V (L) is a prepolar slice of h, we obtain a class of deformations of h|V (L) which give lower bounds on the top Betti number of the Milnor fibre. This also suggests that it might be useful to study prepolar deformations for which 1 γh,L (0) obtains its maximum value. We will need Proposition 8.3. If j  2 and S is a good stratification for h at the origin, then {V (h + wj ) − Σ(h + wj )} ∪ {0 × S | S is a singular stratum of S}

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is a good stratification for h + wj at the origin. Proof. As j  2, Σ(h + wj ) = {0} × Σh. Let p = (0, q) ∈ {0} × Σh, where S ∈ S, and let pi = (ui , qi ) be a sequence of points in C × U − {0} × Σh such that pi → p and Tpi V (h + wj − (h + wj )|pi ) → T . We wish to show that Tp ({0} × S) = {0} × Tq S ⊆ T . If T = Tp V (w) = {0} × Cn+1 , then we are finished. So, suppose otherwise. Then, by taking a subsequence, we may assume that qi ∈ Σh and that Tqi V (h − h(qi )) → η. As T transversely intersects Tp V (w), Tpi V (h+wj −(h+wj )|pi ) transversely intersects Tpi V (w− wi ) for all pi close to p. Thus, T ∩ ({0} × Cn+1 ) = lim Tpi V (h + wj − (h + wj )|pi ) ∩ Tpi V (w − wi ) = lim Tpi V (h + wj − (h + wj )|pi , w − wi ) = lim Tpi V (h − h(qi ), w − wi ) = {0} × η. Now, as S is a good stratum for h, Tq (S) ⊆ η and the proposition follows. 

Corollary 8.4. If V (z0 ) is a prepolar slice for h at 0 then, for all j  2 + λ0h,z (0), V (z0 − w) is a prepolar slice for h + wj at 0. Proof. In light of the proposition, all that we must show is that, for all j  2 + λ0h,z (0), Σ(h + wj ) ∩ V (z0 − w) = Σ(h + wj |V (z0 −w) ). Now, Σ(h + wj ) ∩ V (z0 − w) = ({0} × Σh) ∩ V (z0 − w) = {0} × (Σh ∩ V (z0 )). On the other hand, Σ(h + wj |V (z0 −w) ) = (C × Σ(h + z0j )) ∩ V (z0 − w). But, near the origin and for j  2 + λ0h,z (0), Σ(h + z0j ) = Σh ∩ V (z0 ) by 4.3.iii. The conclusion follows. 

Theorem 8.5. If V (z0 ) is a prepolar slice of h at 0 then, for all j  2 + λ0h,z (0),   1 bn (h + z0j ) = bn (h) − bn−1 (h) + j bn−1 (h|V (z0 ) ) − γh,z (0) , 0

where bi () denotes the i-th Betti number of the Milnor fibre at the origin. 1 In particular, bn−1 (h|V (z0 ) )  γh,z (0). 0

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Proof. After applying Proposition 3.1 to h + wj and the slice V (z0 − w), and considering the long exact sequence of the pair, we have

 bn+1 (h + wj ) − bn (h + wj ) + bn (h + z0j ) = Γ1h+wj ,z −w · V (h + wj ) 0

0

which, by 4.3.v, equals jλ0h,z (0). Now, as the Milnor fibre of h + wj has the homotopy-type of the one-point union of j − 1 copies of the suspension of the Milnor fibre of h, we obtain (j − 1)bn (h) − (j − 1)bn−1 (h) + bn (h + z0j ) = jλ0h,z (0). Using 3.1 on h itself and rearranging, we get

   bn (h + z0j ) = bn (h) − bn−1 (h) + j λ0h,z (0) − Γ1f,z · V (f ) 0 − bn−1 (h|V (z0 ) ) . 0

Finally, using the formula of Proposition 1.20 that

 1 Γ1h,z · V (h) = γh,z (0) + λ0h,z (0), 0

0

0

0

we obtain the desired result.  The result of Theorem 8.5 is best thought of in terms of prepolar deformations: every prepolar deformation, h, of a fixed h0 yields a lower bound on the top Betti number of the Milnor fibre of h0 . 1 One might hope that, by considering a prepolar deformation, h, for which γh,z (0) obtains its maximal value, one would actually obtain the top Betti number of the Milnor fibre of h0 . This seems unlikely however; certain singularities seem to be “rigid” with respect to prepolar deformations, in the weak sense that any prepolar deformation, h, has no polar curve at the origin. Nonetheless, the lower bounds provided by prepolar deformations give new data which helps describe the Milnor fibre of a completely general affine hypersurface singularity; these data do not appear to follow from our Morse inequalities between the Betti numbers of the Milnor fibre and the Lˆe numbers of the hypersurface, as given in Theorem 3.3. As part of these Morse inequalities, we showed that λ0h0 ,˜z (0), provides an upper-bound on the top Betti number of the Milnor fibre of h0 . Also, it follows from 1.18 that if h is a prepolar deformation of h0 , then 1 λ0h0 ,˜z (0) = λ1h,z (0) + γh,z (0).

Thus, given a prepolar deformation, h, of h0 , we have bounded the top Betti number of the Milnor fibre of h0 : 1 1 γh,z (0)
bn−1 (h0 )
λ1h,z (0) + γh,z (0). 1 As λ0h0 ,˜z (0) is fixed, a prepolar deformation, h, with maximal γh,z (0) will have minimal λ1h,z (0). We prefer to call such a deformation a minimal prepolar deformation, instead of a maximal one. Note that a minimal prepolar deformation will not only have the maximal possible lower bound on the top Betti number of the Milnor fibre, it also provides the smallest difference between our general upper and lower bounds. One might hope that it is always possible to find a prepolar 1 deformation, h, for which λ1h,z (0) = 0, for then we would have bn−1 (h0 ) = γh,z (0); unfortunately, 0 Proposition 1.29 implies that it is usually impossible to find such a deformation.

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Chapter 9. CONSTANCY OF THE MILNOR FIBRATIONS In this chapter, we prove what is perhaps our most important result, and certainly the result which requires the most machinery – we generalize the result of Lˆe and Ramanujam [L-R] as stated in Theorem 0.10 in the introduction. Basically, we prove that if the Lˆe numbers are constant in a one-parameter family, then the Milnor fibrations are constant in the family, regardless of the dimension of the critical loci. Unfortunately, we do not obtain the result that the local, ambient topological-type of the hypersurfaces remains constant in the family. It is an open question whether the constancy of the Lˆe numbers is strong enough to imply this topological constancy. While the idea behind our proof of this generalized Lˆe-Ramanujam is simple, the technical details are horrendous. It is this chapter alone which is responsible for the existence of Appendix C of this book; we have relegated most of the technical details to the appendix. Before we prove the main result, there remain only two lemmas which we need (besides the results which appear in the appendix). Also, we will restate one of the results from the appendix in a form which is comprehensible without reading the entire appendix. First, however, we wish to sketch the proof of the main theorem, so that the reader can see that the idea really is fairly easy. On the other hand, the proof is not straightforward – instead, it uses a trick which gives one very little insight as to why the result should be true. Throughout this chapter, U will denote an open neighborhood of the origin in Cn+1 and ft : (U, 0) → (C, 0) will be an analytic family in the variables z = (z0 , . . . , zn ). Let s = dim0 Σf0 .

A sketch of the proof is as follows: The result of Proposition 8.1 is that the Milnor fibre of ft + wj at the origin is homotopyequivalent to the one-point union of j − 1 copies of the suspension of the Milnor fibre of ft at the origin. So, it certainly seems reasonable to expect that the Milnor fibrations are independent of t in the family ft if and only if the Milnor fibrations are independent of t in the family ft + wj . But why should the family ft + wj be any easier to study than the family ft itself? It is easier because we have very nice hyperplanes defined by L = w −z0 such that, when we take the sections (ft + wj )|V (L) , we get the family ft + z0j which, for generic z0 and for large j, is a family of singularities of one less dimension (by the results of Chapter 4); that is, dim0 Σ(f0 + z0j ) = s − 1. Moreover, Lˆe’s attaching result (Theorem 0.9) tells us how the Milnor fibre of ft + wj is obtained from the Milnor fibre of a generic hyperplane section. The Milnor fibre of ft + wj is obtained from

 j 1 j the Milnor fibre of ft + z0 by attaching Γft +wj ,w−z0 · V (ft + w ) (n + 1)-handles. 0

By induction on s, we may require the Milnor fibrations of ft + z0j to be independent of t. If we also require the number of attached (n + 1)-handles to be independent of t, it seems reasonable to expect that the Milnor fibrations of the family ft + wj should be independent of t and, thus, that the Milnor fibrations of ft are independent of t. The Lˆ

e numbers enter the picture  because Lemma 4.3 says that, for large j, the intersection 1 j number Γft +wj ,w−z0 · V (ft + w ) = jλ0ft ,z (0). Combining this with the Lˆe-Iomdine formulas 0

of Theorem 4.5, we find that the inductive requirement that the Milnor fibrations of ft + z0j are independent of t amounts to requiring all the Lˆe numbers of ft to be independent of t.

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We first wish to prove a result which will tell us that the main theorem of this chapter is not vacuously true. Lemma 9.1. For all i with 0
i
n, for a generic linear reorganization of the coordinates (z0 , . . . , zi ), (z0 , . . . , zi ) is prepolar at the origin for ft for all small t. Proof. Fix a good stratification, G, for f in a neighborhood, V, of the origin. By refining if necessary, we may also assume that G satisfies Whitney’s condition a). We will also assume that S := V ∩ {t-axis} − {0} = V ∩ (C × {0}) − {0} and {0} are strata of G. As the function t has isolated stratified critical values, V (t−t0 ) transversely intersects all strata of G near (t0 , 0); hence, V (t − t0 ) ∩ G provides a good stratification for ft0 at 0 which still satisfies Whitney’s condition a). By induction on i, we will prove that: for all i with 0
i
n, for a generic linear reorganization of the coordinates (z0 , . . . , zi ), (z0 , . . . , zi ) is prepolar at the origin for ft0 with respect to the good stratification V (t − t0 ) ∩ G for all small, non-zero t0 . i = 0: Using the terminology of Goresky and MacPherson [G-M2], the set of degenerate conormal covectors to S is a complex analytic subvariety of codimension at least 1 inside the total space of the conormal bundle of S inside Cn+2 (see [G-M2, Prop. 1.8, p.44]). Projectivizing and dualizing, this says that the set Ω, defined by {(p, H) ∈ S × Gn (Cn+1 ) | Tp (C × H) contains a generalized tangent plane of G at p}, has dimension at most n. Hence, Ω ⊆ (V ∩ (C × 0)) × Gn (Cn+1 ) has dimension at most n and the fibre over 0, call it Ψ, must therefore have dimension at most n − 1. Thus, Gn (Cn+1 ) − Ψ is open and dense in Gn (Cn+1 ). We claim that, for all H in Gn (Cn+1 ) − Ψ, H is a prepolar slice for ft at 0 for all small, non-zero t. Certainly, if H ∈ Ψ, then, for all small, non-zero t0 , T(t0 ,0) (C × H) contains no generalized tangent plane from G at (t0 , 0). Also, Whitney’s condition a) guarantees that all limiting tangent planes from strata of G at (t0 , 0) actually contains T(t0 ,0) S = C × 0. Combining these two facts, it follows easily that T0 H contains no generalized tangent plane from V (t − t0 ) ∩ G at 0. Thus, H is prepolar for ft0 at 0 with respect to the good stratification V (t − t0 ) ∩ G. (Actually, this implies much more – it implies that H is polar, as defined in [Mas8].) i  1: Now, assume that we have already chosen (z0 , . . . , zi−1 ) generically (in the IPZ-topology) so that (z0 , . . . , zi−1 ) is prepolar at the origin for ft0 with respect to the good stratification V (t−t0 )∩G for all small, non-zero t0 . Then, there exists a good stratification, G , for f|V (z0 ,...,zi−1 ) at the origin which satisfies Whitney’s condition a). Though G may not necessarily be chosen to equal G ∩ V (z0 , . . . , zi−1 ), after refining G using Proposition C.2, we may certainly assume that each stratum of G is contained in a stratum of G and that, in some neighborhood of the origin, (C × 0) − 0 and 0 are strata of G . By the i = 0 case, for a generic choice of zi , V (zi ) is a prepolar slice for ft0 |V (z ,...,z ) at 0

i−1

the origin with respect to V (t − t0 ) ∩ G for all small non-zero t0 . But, as each stratum of G is

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contained in a stratum of G, this last statement is stronger than saying that V (zi ) is prepolar with respect to V (t − t0 ) ∩ G. This concludes the induction. Finally, to finish the proof, one simply chooses coordinates as generic as given above and generically enough so that the coordinates are prepolar for f0 at the origin.  We shall also need the following uniform version of Proposition 4.19: Proposition 9.2. If (z0 , . . . , zi ) is prepolar at the origin for ft for all small t, then, for all large j, (z1 , . . . , zi ) is prepolar for ft + z0j for all small t. Proof. In light of Corollary 4.18, what we need to show is that, for all large j, for all k with 0
k
i and for all small t0 ,   ∂f j−1 (∗) dim0 Γk+1 ∩ V (z1 , . . . , zk )
0. ∩ V t − t , + jz 0 0 ft0 ,z ∂z0 In fact, we only have to show that (∗) holds for small non-zero t0 , for then – by 4.19 – we may impose the extra largeness condition on j so that (∗) also holds for t0 = 0. k+2 By Lemma 4.14, for all small non-zero t0 , Γk+1 ft ,z = Γf,(t,z) ∩ V (t − t0 ) as sets, in a neighborhood 0

of the origin. In addition, by Theorem 1.26, γfk+1 (0) exists for all small t0 ; thus, dim0 Γk+1 ft0 ,z ∩ t0 ,z V (z0 , z1 , . . . , zk )
0. Putting these two facts together, we find that, for all small non-zero t0 , dim0 V (t − t0 ) ∩ Γk+2 f,(t,z) ∩ V (z0 , z1 , . . . , zk )
0. Let W denote the union of those irreducible analytic components, C, of Γk+2 f,(t,z) ∩ V (z1 , . . . , zk ), at the origin, which satisfy the property that the t-axis is contained in C ∩V (z0 ). By the above, W ∩ V (z0 ) is at most one-dimensional at the origin and, for all small non-zero t0 , Γk+1 ft0 ,z ∩V (z1 , . . . , zk ) = W ∩ V (t − t0 ) as germs of sets at (t0 , 0). It follows that each irreducible component of

W at the origin  is at most 2-dimensional. We j−1 ∂f is at most 1-dimensional at the would like to show that, for all large j, W ∩ V ∂z0 + jz0 origin, for then – by intersecting with V (t − t0 ) – we obtain (∗).

 ∂f But, this is easy. For if a 2-dimensional component C of W is contained in V ∂z + j0 z0j0 −1 0

 ∂f and V ∂z + j1 z0j1 −1 for j0 = j1 , then C ⊆ V (z0 ). However, we know that W ∩ V (z0 ) is at most 0 1-dimensional. Hence, there are only a finite number of “bad” values for j. 

Proposition 9.3 Suppose that V (z0 ) is a prepolar slice for ft at the origin for all small t, and that V (t) does not occur as the limit of tangent spaces to level hypersurfaces of f or f|V (z0 ) at the origin. Then, for all small non-zero t0 , there is a natural inclusion of pairs of Milnor fibres (Fft0 ,0 , Fft0 ,0 ∩ V (z0 )) → (Ff0 ,0 , Ff0 ,0 ∩ V (z0 )).

 If we also assume that the intersection number Γ1ft ,z0 · V (ft ) is independent of t for all small 0 t, then we have the following three results: i) if the inclusion Fft0 ,0 ∩ V (z0 ) → Ff0 ,0 ∩ V (z0 ) induces isomorphisms on integral homology groups, then so does the inclusion Fft0 ,0 → Ff0 ,0 ;

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ii) if s
n − 2 and the inclusion Fft0 ,0 ∩ V (z0 ) → Ff0 ,0 ∩ V (z0 ) is a homotopy-equivalence, then so is the inclusion Fft0 ,0 → Ff0 ,0 , and the fibre homotopy-type of the Milnor fibrations is independent of t for all small t; and iii) if s
n − 3 and the inclusion Fft0 ,0 ∩ V (z0 ) → Ff0 ,0 ∩ V (z0 ) is a homotopy-equivalence, then the inclusion Fft0 ,0 → Ff0 ,0 is a diffeomorphism and, moreover, the diffeomorphism-type of the Milnor fibrations is independent of t for all small t. Proof. This is primarily a restatement of Theorem C.13 from the appendix. The condition on V (t) is that V (t) is not in the Thom set of either f or f|V (z0 ) at the origin (see Definition C.8). Therefore, by Proposition C.9, the families ft and ft |V (z ) satisfy the universal 0 conormal condition at the origin (see Definition C.10) and this produces the inclusion of pairs of Milnor fibres. Now, i) and ii) follow from Theorem C.13, and iii) follows from ii) together with Proposition C.12. 

We are now able to prove the main result of this chapter: our generalization of part of the result of Lˆe and Ramanujam. Essentially, we prove that the constancy of the Lˆe numbers in a family implies the constancy of the Milnor fibrations in the family. Theorem 9.4. Let s := dim0 Σf0 . Suppose that, for all t small, (z0 , . . . , zs−1 ) is prepolar for ft at 0 and that the Lˆe numbers, λift ,z (0), are independent of t for each i with 0
i
s. Then, i)

the homology of the Milnor fibre of ft at the origin is independent of t for all t small;

if s
n − 2, ii) the fibre homotopy-type of the Milnor fibrations of ft at the origin is independent of t for all t small; and, if s
n − 3, iii) the diffeomorphism-type of the Milnor fibrations of ft at the origin is independent of t for all t small. Proof. By induction on s. For s = 0, this is the result of Lˆe and Ramanujam [L-R]. Now, assume that s  1 and that we know the result for families of hypersurfaces with critical loci of dimension
s − 1. Let j be so large that the uniform Lˆe-Iomdine formulas of Theorem 4.15 hold and so large that Proposition 9.2 holds. Finally, using that λ0ft ,z (0) is independent of t, let j be so large that j  2 + λ0ft ,z (0) for all small t, so that we may apply Lemma 4.3 and Corollary 8.4. Consider the family ft + wj , where w is a variable disjoint from the z’s. The dimension of the critical locus at t = 0 is still equal to s. As the Lˆe numbers of ft are independent of t, we may apply Theorem 6.5 to conclude that V (t) is not the limit of tangent spaces to level hypersurfaces of f

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at the origin. It follows trivially that V (t) is not the limit of tangent spaces to level hypersurfaces of f + wj at the origin. Moreover, using the uniform Lˆe-Iomdines formulas of Theorem 4.15, we find that the Lˆe numbers of (ft + wj )|V (z0 −w) = ft + z0j at 0 with respect to (z1 , . . . , zs−1 ) are

independent of t, and that dim0 Σ(f0 + z0j ) = s − 1. Therefore, by our inductive hypothesis, we already have the constancy results for the family (ft +wj )|V (z0 −w) . In addition, we may use Theorem 6.5 again to conclude that V (t) is not the limit of tangent spaces to level hypersurfaces of (f +wj )|V (z0 −w) at the origin. By 8.4, V (z0 −w) is a prepolar

 slice for ft + wj at the origin for all small t. Also, by Lemma 4.3.v, Γ1ft +wj ,w−z0 , ·V (ft + wj ) = 0

jλ0ft ,z (0), which is independent of t for all small t. Thus, we are in a position to apply Proposition 9.3 to the family ft + wj and we conclude the desired constancy results for this family. Finally, now that we know the results for ft + wj , we apply Proposition C.16 to conclude that the result actually holds for ft itself. 

Remark 9.5. While we are not very fond of discussing it, as we mentioned in Remark 1.27, for a fixed function h and a fixed point p, there exist generic Lˆe numbers of h at p – that is, as one varies the linear choice of coordinates, z, through coordinates for which the Lˆe numbers are defined, one finds a generic value for each of the Lˆe numbers, λih,z (p); let us denote this generic value simply by λih (p). We are not fond of discussing these generic Lˆe numbers because the Lˆe numbers are intended to be effectively calculable, and we know of no effective way of knowing when a coordinate choice is sufficiently generic to give λih (p). We do know, by Corollary 4.16, that the tuple of generic Lˆe numbers (λsh (p), . . . , λ0h (p)) is minimal with respect to the lexigraphic ordering. We mention all this here because the tuple (λsh (p), . . . , λ0h (p)) is an analytic invariant (this follows from the relationship between the Lˆe numbers and the polar multiplicities; see [Mas11] or Part IV, Theorems 1.10 and 3.2) , and so the reader may wonder whether Theorem 9.4 stills holds under the assumption that the generic Lˆe numbers of ft are independent of t at the origin. The answer to this question is easily seen to be: yes. This follows quickly from 9.4 itself. Suppose that the generic Lˆe numbers of ft at the origin are independent of t. Choose coordinates z that are so generic that z is prepolar for ft at the origin for all small t and so that z is generic enough to give the generic Lˆe numbers of f0 at the origin. Then, for all small t, we have (λsft (0), . . . , λ0ft (0))
(λsft ,z (0), . . . , λ0ft ,z (0))
(λsf0 ,z (0), . . . , λ0f0 ,z (0)) = (λsf0 (0), . . . , λ0f0 (0)), where we are using the lexigraphic ordering. Since (λsft (0), . . . , λ0ft (0)) = (λsf0 (0), . . . , λ0f0 (0)), it follows that (λsft ,z (0), . . . , λ0ft ,z (0)) equals the tuple at t = 0, and so we may apply Theorem 9.4.

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Chapter 10. ANOTHER CHARACTERIZATION OF THE ˆ CYCLES LE In this chapter, we give an alternative characterization of the Lˆe cycles and Lˆe numbers of a hypersurface singularity. This alternative characterization is generalized in Part IV, Chapter 3, where we see that the case of the Lˆe numbers of a function f is just the case where the underlying complex of sheaves is the sheaf of vanishing cycles along f . As a consequence of Theorem 3.3, the Lˆe cycles can be characterized formally by requiring that the alternating sum of the Lˆe numbers yields the reduced Euler characteristic of the Milnor fibre at each point – provided that we know that a fixed choice of coordinates is prepolar at every point. We will show this now.

Lemma 10.1. Let X be an analytic subset containing the origin in some CN and let {Sα } be an analytic stratification of X with connected strata. Let p : CN → Ck be a linear map such that p|Sα   is a submersion if dim Sα  k and dim0 p−1 (0) ∩ S α = 0 if dim Sα
k − 1. Then, for generic linear π : Ck → Ck−1 , there exists a refinement {Rβ } of {Sα } in some neighborhood of the origin which preserves the strata of dimension greater than or equal to k and  such that π ◦ p|Rβ is a submersion if dim Rβ  k − 1 and dim0 (π ◦ p)−1 (0) ∩ Rβ = 0 if dim Rβ
k − 2. Proof. Let X  denote the union of those strata Sα such that dim Sα is less than k. Then, as dim0 (p−1 (0) ∩ X  ) = 0, there exists an open neighborhood, U, of the origin in CN and an open neighborhood of the origin, V, in Ck such that the restriction of p to a map from U ∩ X  to V is finite. Therefore, as the conclusion of the lemma is purely local in nature, we may reduce ourselves to considering only the case where p|X  is a finite map. Thus, p(X  ) is an analytic subset of Ck of dimension at most k − 1 and so, for generic lines, L, through 0 in Ck , we have that L ∩ p(X  ) = {0} near the origin. Hence, for generic linear π : Ck → Ck−1 , we must have that π −1 (0) ∩ p(X  ) = {0}, and so   (π ◦ p)−1 (0) ∩ X  = p−1 π −1 (0) ∩ p(X  ) ∩ X  = p−1 (0) ∩ X  . But, by hypothesis, p−1 (0) ∩ X  = {0}, and thus – for such a generic π – any refinement of X  will give a refinement of {Sα } which preserves the strata of dimension greater than or equal to k and realizes the desired intersection condition. Now, we have the restriction π ◦ p : X  → Ck−1 . By the above, if we once again take sufficiently small neighborhoods around the origins, π ◦ p restricts to a finite map on X  . Assume then that (π ◦ p)|X  is finite. If dim X  = k − 1, then π ◦ p is a submersion on X  − W , where W is a closed subset of X  of dimension at most k − 2. Thus, we may refine the stratification so that X  ∩ (π ◦ p)−1 (π ◦ p)(W ) is a union of strata, where the strata have dimension at most k − 2 since π ◦ p restricted to X  is a finite map. This yields the desired refinement {Rβ }. 

Proposition 10.2. Let U be an open subset of Cn+1 containing the origin, and let h : (U, 0) → (C, 0) be an analytic map. Then, for a generic linear reorganization, z, of coordinate systems for

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Cn+1 , there exists an open neighborhood, U  , of the origin such that z is prepolar for h|U  , i.e., for all p ∈ U  ∩ V (h), if we let s denote dimp Σh, then (z0 , . . . , zs−1 ) is prepolar for h at p. Proof. Begin by applying the lemma to X = V (h), endowed with some good stratification, and p = id : Cn+1 → Cn+1 . By an inductive application of the lemma, for a generic linear reorganization of coordinates, z, for Cn+1 , we arrive at a stratification {Sα }, which is a refinement of the original good stratification (in a possibly smaller neighborhood of the origin), such that for each i, the map (z0 , . . . , zi ) is a submersion when restricted to a stratum of dimension greater than or equal to i + 1. This would say precisely that (z0 , . . . , zn ) is prepolar at each point of V (h) with respect to the stratification {Sα } – provided that {Sα } is actually a good stratification for h. But, any refinement of a good stratification is another good stratification – except for the fact that we may have refined the smooth part of V (h), which we required to be a stratum of any good stratification. However, {V (h) − Σh} ∪ {Sα | Sα ⊆ Σh} is certainly a good stratification for h, and what remains for us to show is that, for generically reorganized coordinates, z, for each p ∈ Σh, V (z0 − p0 , . . . , zs−1 − ps−1 ) transversely intersects the smooth part of V (h) near p, where again s = dimp Σh. Assume then that we have chosen the coordinates z generic as above and also generic enough s so that (z0 , . . . , zs−1 ) is prepolar for h at 0, where s = dim0 Σh. By Theorem 1.26, γh,z (0) exists, s and so γh,z (p) exists for all p near 0. But, by Remark 1.6, this implies that if p ∈ Σh, then Σ(h|V (z0 −p0 ,...,zk−1 −pk−1 ) ) = V (z0 − p0 , . . . , zk−1 − pk−1 ) ∩ Σh at p, i.e., V (z0 − p0 , . . . , zs−1 − ps−1 ) transversely intersects the smooth part of V (h) near p (and, of course, s = dim0 Σh  dimp Σh for all p near 0). 

In Part IV, Chapter 3, we will generalize the result below to the case where the underlying space is arbitrary.

Theorem 10.3. For a generic linear reorganization of coordinates, z, for Cn+1 , the Lˆe cycles are a collection of analytic cycle germs, Λih,z , in Σh at the origin such that each Λih,z is purely i-dimensional and properly intersects V (z0 , . . . , zi−1 ) at the origin, and χ (Fh,p ) =

s i=0

 (−1)n−i Λih,z · V (z0 − p0 , . . . , zi−1 − pi−1 ) , p

for all p ∈ Σh near 0, where χ (Fh,p ) is the reduced Euler characteristic of the Milnor fibre of h at p; specifically, this is the case if z is prepolar for h in a neighborhood of the origin. Moreover, if z is any linear coordinate system such that such cycles exist, then they are unique. Proof. The first statement follows immediately from the previous proposition and Theorem 3.3. As for the uniqueness assertion, this is a fairly standard argument for constructible functions. Suppose that we had two such collections, Λih,z and Ωih,z . Let s denote dim0 Σh. We will show that Λih,z and Ωih,z agree by downward induction on i.

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For a generic point, p, in an s-dimensional component, ν, of the support of Λsh,z , p will be a smooth point of ν, V (z0 − p0 , . . . , zs−1 − ps−1 ) will transversely intersect ν at p, and p will not be in the support of any of the lower-dimensional Λih,z or Ωih,z . Thus, at such a p,



 Ωsh,z · V (z0 − p0 , . . . , zs−1 − pi−1 ) = Λsh,z · V (z0 − p0 , . . . , zs−1 − pi−1 ) . p

p

Of course, the same conclusion would have followed it we had chosen a generic point, p, in an s-dimensional component, ν, of the support of Ωsh,z , It follows that Λsh,z = Ωsh,z . Now, suppose that we have shown that Λih,z = Ωih,z for all i greater than some k. Then, k

 (−1)n−i Λih,z · V (z0 − p0 , . . . , zi−1 − pi−1 ) =

i=0 k i=0

p

 (−1)n−i Ωih,z · V (z0 − p0 , . . . , zi−1 − pi−1 ) , p

and we repeat the argument above with k in place of s. The conclusion follows. 

Theorem 10.3 leaves us with a very strange set of affairs; it tells us that, for generic z, we could have defined the Lˆe cycles by their characterization in the theorem. This means that the Lˆe cycles and hence, the Lˆe numbers, are determined by the choice of z and the data of the Euler characteristic of the Milnor fibre at each point. But, had we defined the Lˆe cycles and numbers this way, then we would have produced the Morse inequalities on the Betti numbers of the Milnor fibres in Theorem 3.3 from seemingly much less data. This phenomenon occurs in a much more general setting; the explanation appears in [Mas11].

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Part III. ISOLATED CRITICAL POINTS OF FUNCTIONS ON SINGULAR SPACES

Chapter 0. INTRODUCTION In the introduction to Part II, we discussed known results for functions, or families of functions, with isolated critical points. The remainder of Part II was devoted to the development of the Lˆe cycles and Lˆe numbers – a generalization to functions with non-isolated critical loci of the data provided by the Milnor number of an isolated critical point. All of the functions considered in Part II had open subsets of affine space as their domains. Here, in Part III, we will begin our generalization of the Lˆe numbers to the case where the functions considered have arbitrary analytic spaces as their domains. However, since the Lˆe numbers generalize the Milnor number of an isolated critical point (on affine space), if we want to have Lˆe numbers for functions with arbitrary domains, then we must first develop some sort of “Milnor number” theory for functions with “isolated critical points” with arbitrary domains. This immediately leads us to two fundamental questions for functions with arbitrary domains: •

What is the proper notion of the “critical locus” of such a function?

•

What is the proper notion of the “Milnor number” of such a function when its “critical locus” consists of an isolated point?

In the remainder of this introduction, we will discuss possible definitions of the “critical locus” for functions; we will first discuss the non-controversial case of functions on affine space, and then move on to the much more complicated general case. There is a slight bit of overlap here with some of the material presented in the introduction to Part II, but we feel that this minor repetition aids the exposition.

Let U be an open subset of Cn+1 , let z := (z0 , z1 , . . . , zn ) be coordinates for Cn+1 , and suppose that f˜ : U → C is an analytic function. Then, all conceivable definitions of the critical locus, Σf˜, of f˜ agree: one can consider the points, x, where the derivative vanishes, i.e., dx f˜ = 0, or one can consider the points, x, where the Taylor series of f˜ at x has no linear term, i.e., f˜ − f˜(x) ∈ m2U ,x (where mU ,x is the maximal ideal in the coordinate ring of U at x), or one can consider the points, x, where the Milnor fibre of f˜ at x, Ff˜,x , is not trivial (where, here, “trivial” could mean even up to analytic isomorphism). Now, suppose that X is an analytic subset of U, and let f := f˜|X . Then, what should be meant by “the critical locus of f ”? It is not clear what the relationship is between points, x, where f − f (x) ∈ m2X,x and points where the Milnor fibre, Ff,x , is not trivial (with any definition of trivial); moreover, the derivative dx f does not even exist. We are guided by the successes of Morse Theory and stratified Morse Theory to choosing the Milnor fibre definition as our primary notion of critical locus, for we believe that critical points should coincide with changes in the topology of the level hypersurfaces of f . Therefore, we make the following definition: 115

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Definition 0.1. The C-critical locus of f , ΣC f , is given by ΣC f := {x ∈ X | H ∗ (Ff,x ; C) 6= H ∗ (point; C)}. (The reasons for using field coefficients, rather than Z, are technical: we want Lemma 4.1 to be true.) In Chapter 1, we will compare and contrast the C-critical locus with other possible notions of critical locus, including the ones mentioned above and the stratified critical locus. After Chapter 1, the remainder of Part III is dedicated to showing that Definition 0.1 really yields a useful, calculable definition of the critical locus. We show this by looking at the case of a generalized isolated singularity, i.e., an isolated point of ΣC f , and showing that, at such a point, there is a workable definition of the Milnor number(s) of f ; we show that the Betti numbers of the Milnor fibre can be calculated (3.7.ii), and we give a generalization of the result of Lˆe and Saito [L-S] that constant Milnor number throughout a family implies Thom’s af condition holds. Specifically, in Corollary 6.14, we prove (with slightly weaker hypotheses) that: Theorem 0.2. Let W be a (not necessarily purely) d-dimensional analytic subset of an open subset ◦

of Cn . Let Z be a d-dimensional irreducible component of W . Let X := D × W be the product of ◦

an open disk about the origin with W , and let Y := D × Z. ◦

Let f : (X, D × {0}) → (C, 0) be an analytic function, and let ft (z) := f (t, z). Suppose that f0 is in the square of the maximal ideal of Z at 0. Suppose that 0 is an isolated point of ΣC (f0 ), and that the reduced Betti number ˜bd−1 (Ffa ,(a,0) ) is independent of a for all small a. ◦ ◦ Then, ˜bd−1 (Ff ,(a,0) ) 6= 0 and, near 0, Σ(f| ) ⊆ D×{0} and the pair (Yreg −Σ(f| ), D×{0}) a

Yreg

Yreg

satisfies Thom’s af condition at 0. Thom’s af is important for several reasons, but perhaps the best reason is because it is an hypothesis of Thom’s Second Isotopy Lemma. General results on the af condition have been proved by many researchers: Hironaka, Lˆe, Saito, Henry, Merle, Sabbah, Brian¸con, Maisonobe, Parusi´ nski, etc., and the above theorem is closely related to the recent results contained in [BMM] and [P2]. However, the reader should contrast the hypotheses of Theorem 0.2 with those of the main theorem of [BMM] (Theorem 5.2.1); our main hypothesis is that a single number is constant throughout the family, while the main hypothesis of Theorem 4.2.1 of [BMM] is a condition which requires one to check an infinite amount of data: the property of local stratified triviality. Moreover, the Betti numbers that we require to be constant are actually calculable. While much of this part is fairly technical in nature, there are three new, key ideas that guide us throughout. The first of these fundamental precepts is: controlling the vanishing cycles in a family of functions is enough to control Thom’s af condition and, perhaps, the topology throughout the family. While this may seem like an obvious principle – given the results of Lˆe and Saito in [L-S] and of Lˆe and Ramanujam in [L-R] – in fact, in the general setting, most of the known results seem to require the constancy of much stronger data, e.g., the constancy of the polar multiplicities [Te6]

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or that one has the local stratified triviality property [BMM]. In a very precise sense, controlling the polar multiplicities corresponds to controlling the nearby cycles of the family of functions, instead of merely controlling the vanishing cycles. As we show in Corollary 5.4, controlling the characteristic cycle of the vanishing cycles is sufficient for obtaining the af condition. Our second fundamental idea is: the correct setting for all of our cohomological results is where perverse sheaves are used as coefficients. While papers on intersection cohomology abound, and while perverse sheaves are occasionally used as a tool (e.g., [BMM, 4.2.1]), we are not aware of any other work on general singularities in which arbitrary perverse sheaves of coefficients are used in an integral fashion throughout. The importance of perverse sheaves in Part III begins with Theorem 4.2, where we give a description of the critical locus of a function with respect to a perverse sheaf. The third new feature of Part III is the recurrent use of the perverse cohomology of a complex of sheaves. This device allows us to take our general results about perverse sheaves and translate them into statements about the constant sheaf. The reason that we use perverse cohomology, instead of intersection cohomology, is because perverse cohomology has such nice functorial properties: it commutes with Verdier dualizing, and with taking nearby and vanishing cycles (shifted by [−1]). If we were only interested in proving results for local complete intersections (l.c.i.’s), we would never need the perverse cohomology; however, we want to prove completely general results. The perverse cohomology seems to be a hitherto unused tool for accomplishing this goal. Part III is organized as follows: In Chapter 1, we discuss seven different notions of the “critical locus” of a function. We give examples to show that, in general, all of these notions are different. In Chapter 2, we discuss the relative polar curve of Lˆe and Teissier. We need to relate the intrinsic definition to a conormal characterization in terms of gap sheaves. Chapter 3 is devoted to proving an “index theorem”, Theorem 3.10, which provides the main link between the topological data of the Milnor fibre and the algebraic data obtained by blowingup the image of df˜ inside the appropriate space. This theorem is presented with coefficients in a bounded, constructible complex of sheaves; this level of generality is absolutely necessary in order to obtain the results in the remainder of Part III. Chapter 4 uses the index theorem of Chapter 3 to show that ΣC f and the Betti numbers of the Milnor fibre really are fairly well-behaved. This is accomplished by applying Theorem 3.10 in the case where the complex of sheaves is taken to be the perverse cohomology of the shifted constant sheaf. Perverse cohomology essentially gives us the “closest” perverse sheaf to the constant sheaf. Many of the results of Chapter 4 are stated for arbitrary perverse sheaves, for this seems to be the most natural setting. Chapter 5 contains the necessary results from conormal geometry that we will need in order to conclude that topological data implies that Thom’s af condition holds. The primary result of this chapter is Corollary 5.4, which once again relies on the index theorem from Chapter 3. Chapter 6 begins with a discussion of “continuous families of constructible complexes of sheaves”. We then prove in Theorem 6.7 that additivity of Milnor numbers occurs in continuous families of perverse sheaves, and we use this to conclude additivity of the Betti numbers of the Milnor fibres, by once again resorting to the perverse cohomology of the shifted constant sheaf. Finally, in Corollaries 6.11 and 6.12, we prove that the constancy of the Milnor/Betti number(s) throughout a family implies that the af condition holds – we prove this first in the setting of arbitrary perverse sheaves, and then for perverse cohomology of the shifted constant sheaf. By translating our hypotheses from the language of the derived category back into more down-to-Earth terms, we obtain Corollary 6.12, which leads to Theorem 0.2 above.

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Chapter 1. CRITICAL AVATARS. We continue with U, z, f˜, X, and f as in the introduction. In this chapter, we will investigate seven possible notions of the “critical locus” of a function on a singular space, one of which is the C-critical locus already defined in 0.1. Definition 1.1. The algebraic critical locus of f , Σalg f , is defined by Σalg f := {x ∈ X | f − f (x) ∈ m2X,x }.

Remark 1.2. It is a trivial exercise to verify that Σalg f = {x ∈ X | there exists a local extension, fˆ, of f to U such that dx fˆ = 0}. Note that x being in Σalg f does not imply that every local extension of f has zero for its derivative at x. One might expect that Σalg f is always a closed set; in fact, it need not be. Consider the example where X := V (xy) ⊆ C2 , and f = y|X . We leave it as an exercise for the reader to verify that Σalg f = V (y) − {0}. There are five more variants of the critical locus of f that we will consider. We let Xreg denote ∗ U denote the regular (or smooth) part of X and, if M is an analytic submanifold of U, we let TM the conormal space to M in U (that is, the elements (x, η) of the cotangent space to U such that x ∈ M and η annihilates the tangent space to M at x). We let N (X) denote the Nash modification of X, so that the fibre Nx (X) at x consists of limits of tangent planes from the regular part of X. We also remind the reader that complex analytic spaces possess canonical Whitney stratifications (see [Te6]). Definition 1.3. We define the regular critical
locus of f , Σreg f , to be the critical locus of the restriction of f to Xreg , i.e., Σreg f = Σ f|Xreg . We define the Nash critical locus of f , ΣNash f , to be 

x ∈ X | there exists a local extension, fˆ, of f to U such that dx fˆ(T ) ≡ 0, for all T ∈ Nx (X) .

We define the conormal-regular critical locus of f , Σcnr f , to be 

x ∈ X | there exists a local extension, fˆ, of f to U such that (x, dx fˆ) ∈ TX∗reg U



;

it is trivial to see that this set is equal to 

x ∈ X | there exists a local extension, fˆ, of f to U such that dx fˆ(T ) ≡ 0, for some T ∈ Nx (X) .

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Let S = {Sα } be a (complexSanalytic)
Whitney stratification of X. We define the S-stratified critical locus of f , ΣS f , to be α Σ f|Sα . If S is clear, we simply call ΣS f the stratified critical locus. If S is, in fact, the canonical Whitney stratification of X, then we write Σcan f in place of ΣS f , and call it the canonical stratified critical locus.

We define the relative differential critical locus of f , Σrdf f , to be the union of the singular set of X and Σreg f . If x ∈ X and h1 , . . . , hj are equations whose zero-locus defines X near x, then x ∈ Σrdf f if and only if the rank of the Jacobian map of (f˜, h1 , . . . , hj ) at x is not maximal among all points of X near x. By using this Jacobian, we could (but will not) endow Σrdf f with a scheme structure (the critical space) which is independent of the choice of the extension f˜ and the defining functions h1 , . . . , hn (see [Loo, 4.A]). The proof of the independence uses relative differentials; this is the reason for our terminology. n o S Remark 1.4. In terms of conormal geometry, ΣS f = x ∈ X | (x, dx f˜) ∈ α TS∗α U or, using o n S Whitney’s condition a) again, ΣS f = x ∈ X | (x, dx f˜) ∈ α TS∗α U . Clearly, Σrdf f is closed, and it is an easy exercise to show that Whitney’s condition a) implies that ΣS f is closed. On the other hand, Σreg f is, in general, not closed and, in order to have any information at singular points of X, we will normally look at its closure Σreg f . Looking at the definition of Σcnr f , one might expect that Σreg f = Σcnr f . In fact, we shall see in Example 1.8 that this is false. That Σcnr f is, itself, closed is part of the following proposition. (Recall that f˜ is our fixed extension of f to all of U.)

In the following proposition, we show that, in the definitions of the Nash and conormal-regular critical loci, we could have used “for all” in place of “there exists” for the local extensions; in particular, this implies that we can use the fixed extension f˜. Finally, we show that the conormalregular critical locus is closed.

Proposition 1.5. The Nash critical locus of f is equal to 

x ∈ X | for all local extensions, fˆ, of f to U, dx fˆ(T ) ≡ 0, for all T ∈ Nx (X) = 

x ∈ X | dx f˜(T ) ≡ 0, for all T ∈ Nx (X) .

The conormal-regular critical locus of f is equal to 

x ∈ X | for all local extensions, fˆ, of f to U, (x, dx fˆ) ∈ TX∗reg U  = x ∈ X | (x, dx f˜) ∈ TX∗reg U .

In addition, Σcnr f is closed.



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 Proof. Let Z := x ∈ X | for all local extensions, fˆ, of f to U, dx fˆ(T ) ≡ 0, for all T ∈ Nx (X) . Clearly, we have Z ⊆ ΣNash f . Suppose now that x ∈ ΣNash f . Then, there exists a local extension, fˆ, of f to U such that dx fˆ(T ) ≡ 0, for all T ∈ Nx (X). Let fˇ be another local extension of f to U and let T∞ ∈ Nx (X); to show that x ∈ Z, what we must show is that dx fˇ(T∞ ) ≡ 0. Suppose not. Then, there exists v ∈ T∞ such that dx fˇ(v) 6= 0, but dx fˆ(v) = 0. Therefore, there exist xi ∈ Xreg and vi ∈ Txi Xreg such that xi → x, Txi Xreg → T∞ , and vi → v. Let V be an open neighborhood of x in U which in fˆ and fˇ are both defined. Let Φ : V ∩T Xreg → C be defined by Φ(p, w) = dp (fˆ − fˇ)(w). Then, Φ is continuous, and so Φ−1 (0) is closed. As (fˆ − fˇ)|X∩V ≡ 0, (xi , vi ) ∈ Φ−1 (0), and thus (x, v) ∈ Φ−1 (0) – a contradiction. Therefore, Z = ΣNash f .  It follows immediately that ΣNash f = x ∈ X | dx f˜(T ) ≡ 0, for all T ∈ Nx (X) .  Now, let W := x ∈ X | for all local extensions, fˆ, of f to U, (x, dx fˆ) ∈ TX∗reg U . Clearly, we have W ⊆ Σcnr f . Suppose now that x ∈ Σcnr f . Then, there exists a local extension, fˆ, of f to U such that (x, dx fˆ) ∈ TX∗reg U. Let (xi , ηi ) ∈ TX∗reg U be such that (xi , ηi ) → (x, dx fˆ). Let fˇ be another local extension of f to U; to show that x ∈ W , what we must show is that (x, dx fˇ) ∈ TX∗reg U.

Since (fˇ− fˆ)|X∩V ≡ 0, for all q ∈ Xreg , q, dq (fˇ− fˆ) ∈ TX∗reg U; in particular, xi , dxi (fˇ− fˆ) ∈

T ∗ U. Thus, xi , ηi + dx (fˇ − fˆ) ∈ T ∗ U, and xi , ηi + dx (fˇ − fˆ) → (x, dx fˇ). Therefore, Xreg

i

Xreg

i

(x, dx fˇ) ∈ TX∗reg U, and W = Σcnr f .  It follows immediately that Σcnr f = x ∈ X | (x, dx f˜) ∈ TX∗reg U . Finally, we need to show that Σcnr f is closed. Let Ψ : X → T ∗ U be given by Ψ(x) = (x, dx f˜). Then, Ψ is a continuous map and, by the above, Σcnr f = Ψ−1 (TX∗reg U).  Proposition 1.6. There are inclusions Σreg f ⊆ Σalg f ⊆ ΣNash f ⊆ Σcnr f ⊆ ΣC f ⊆ Σcan f ⊆ Σrdf f. In addition, if S is a Whitney stratification of X, then Σcan f ⊆ ΣS f . Proof. Clearly, Σreg f ⊆ Σalg f ⊆ ΣNash f ⊆ Σcnr f , and so the containments for their closures follows (recall, also, that Σcnr f is closed). It is also obvious that Σcan f ⊆ Σrdf f and Σcan f ⊆ ΣS f . That ΣZ f ⊆ Σcan f follows from Stratified Morse Theory [Go-Mac1], and so, since Σcan f is closed, ΣC f ⊆ Σcan f . It remains for us to show that Σcnr f ⊆ ΣC f . Unfortunately, to reach this conclusion, we must refer ahead to Theorem 4.6, from which it follows immediately. (However, that Σalg f ⊆ ΣC f follows from A’Campo’s Theorem [A’C].)  Remark 1.7. For a fixed stratification S, for all x ∈ X, there exists a neighborhood W of x in X such that W ∩ ΣS f ⊆ f −1 f (x). This is easy to show: the level hypersurfaces of f close to

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V (f − f (x)) will be transverse to all of the strata of S near x. All of our other critical loci which are contained in ΣS f (i.e., all of them except Σrdf f ) also satisfy this local isolated critical value property.

Example 1.8. In this example, we wish to look at the containments given in Proposition 1.6, and investigate whether the containments are proper, and also investigate what would happen if we did not take closures in the four cases where we do. The same example that we used in Remark 1.2 shows that none of Σreg f , Σalg f , ΣNash f , or ΣC f are necessarily closed; if X := V (xy) ⊆ C2 , and f = y|X , then all four critical sets are precisely V (y) − {0}. Additionally, since Σcnr f = V (y), this example also shows that, in general, Σcnr f 6⊆ ΣC f . If we continue with X = V (xy) and let g := (x + y)2|X , then Σalg g = {0} and Σreg g = ∅; thus, in general, Σreg f 6= Σalg f . While it is easy to produce examples where ΣNash f is not equal to Σalg f and examples where ΣNash f is not equal to Σcnr f , it is not quite so easy to come up with examples where all three of these sets are distinct. We give such an example here. Let Z := V ((y − zx)(y 2 − x3 )) ⊆ C3 and L := y|Z . Then, one easily verifies that Σalg L = ∅, ΣNash L = {0}, and Σcnr L = C × {0}. If X = V (xy) and h := (x+y)|X , then ΣC h = {0} and Σcnr h = ∅; thus, in general, Σcnr f 6= ΣC f . Let W := V (z 5 + ty 6 z + y 7 x + x15 ) ⊆ C4 ; this is the example of Brian¸con and Speder [B-S] in which the topology along the t-axis is constant, despite the fact that the origin is a point-stratum in the canonical Whitney stratification of W . Hence, if we let r denote the restriction of t to W , then, for values of r close to 0, 0 is the only point in Σcan r and 0 6∈ ΣC r. Therefore, 0 ∈ Σcan r − ΣC r, and so, in general, ΣC f 6= Σcan f . Using the coordinates (x, y, z) on C3 , consider the cross-product Y := V (y 2 − x3 ) ⊆ C3 . The canonical Whitney stratification of Y is given by {Y − {0} × C, {0} × C}. Let π := z|Y . Then, Σcan π = ∅, while Σrdf π = {0} × C. Thus, in general, Σcan f 6= Σrdf f . It is, of course, easy to throw extra, non-canonical, Whitney strata into almost any example in order to see that, in general, Σcan f 6= ΣS f . To summarize the contents of this example and Proposition 1.6: we have seven seemingly reasonable definitions of “critical locus” for complex analytic functions on singular spaces (we are not counting ΣS f , since it is not intrinsically defined). All of our critical locus avatars agree for manifolds. The sets Σreg f , Σalg f , ΣNash f , and ΣC f need not be closed. There is a chain of containments among the closures of these critical loci, but – in general – none of the sets are equal.

However, we consider the sets Σreg f , Σalg f , ΣNash f , and Σcnr f to be too small; these “critical loci” do not detect the change in topology at the level hypersurface h = 0 in the simple example X = V (xy) and h = (x + y)|X (from Example 1.8). Despite the fact that the Stratified Morse Theory of [Go-Mac1] yields nice results and requires one to consider the stratified critical locus, we also will not use Σcan f (or any other ΣS f ) as our primary notion of critical locus; Σcan f is often too big. As we saw in the Brian¸con-Speder example

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in Example 1.8, the stratified critical locus sometimes forces one to consider “critical points” which do not correspond to changes in topology. Certainly, Σrdf f is far too large, if we want critical points to have any relation to changes in the topology of level hypersurfaces: if X has a singular set ΣX, then the critical space of the projection π : X × C → C would consist of ΣX × C, despite the obvious triviality of the family of level hypersurfaces defined by π. Therefore, we choose to concentrate our attention on the C-critical locus, and we will justify this choice with the results in the remainder of Part III. Note that we consider ΣC f , not its closure, to be the correct notion of critical locus; we think that this is the more natural definition, and we consider the question of when ΣC f is closed to be an interesting one. It is true, however, that all of our results refer to ΣC f . We should mention here that, while ΣC f need not be closed, the existence of Thom stratifications [Hi] implies that ΣC f is at least analytically constructible; hence, ΣC f is an analytic subset of X. Before we leave this chapter, in which we have already looked at seven definitions of “critical locus”, we need to look at one last variant. As we mentioned at the end of the introduction, even though we wish to investigate the Milnor fibre with coefficients in C, the fact that the shifted constant sheaf on a non-l.c.i. need not be perverse requires us to take the perverse cohomology of the constant sheaf. This means that we need to consider the hypercohomology of Milnor fibres with coefficients in an arbitrary bounded, constructible complex of sheaves of modules. As we wish to discuss Euler characteristics, we need for the rank of a finitely-generated module to be defined and additive over exact sequences; thus, we must choose our base ring to be a p.i.d. However, since the rank of a module over a p.i.d. equals the dimension of the associated vector space over the quotient field, we may as well restrict ourselves to the case where the base ring is, in fact, a field. The C−critical locus is nicely described in terms of vanishing cycles (see [K-S] for general properties of vanishing cycles, but be aware that we use the more traditional shift): ΣC f = {x ∈ X | H ∗ (φf −f (x) C•X )x 6= 0}. This definition generalizes easily to yield a definition of the critical loci of f with respect to arbitrary bounded, constructible complexes of sheaves on X. Let R be a p.i.d. Let S := {Sα } be a Whitney stratification of X, and let F• be a bounded complex of sheaves of R-modules which is constructible with respect to S. Definition 1.9. The F• -critical locus of f , ΣF• f , is defined by ΣF• f := {x ∈ X | H ∗ (φf −f (x) F• )x 6= 0}. Remark 1.10. Stratified Morse Theory (see [Go-Mac1]) implies that ΣF• f ⊆ ΣS f (alternatively, this follows from 8.4.1 and 8.6.12 of [K-S], combined with the facts that complex analytic Whitney stratifications are w-stratifications, and w-stratifications are µ-stratifications.) We could discuss three more notions of the critical locus of a function – two of which are obtained by picking specific complexes for F• in Definition 1.9. However, we will defer the introduction of these new critical loci until Chapter 4; at that point, we will have developed the tools necessary to say something interesting about these three new definitions.

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Chapter 2. THE RELATIVE POLAR CURVE In this chapter, we discuss the relative polar curve of Lˆe and Teissier ([L-T2],[Te5], [Te6], [Te7], etc.). This object is now a standard part of singularity theory, and the reader is most likely familiar with some of the results that appear here. However, we shall use our early results on gap varieties and cycles to present the theory in the conormal form in which we will use it in the next chapter. Our treatment of the higher-dimensional relative polar varieties, when the underlying space is singular, does not appear here; it is a major portion of Part IV. We let U be an open subset of Cn+1 , and let X be a reduced analytic subspace of U with analytic components {Xj }. Let dj denote the dimension of Xj , let cj := n + 1 − dj denote its codimension, and let d denote the global dimension of X. Let f˜ : U → C be an analytic function, and let f := f˜|X . For g1 , . . . , gj ∈ OU , let Jac(g1 , . . . , gj ) denote the Jacobian matrix of (g1 , . . . , gj ), which has the partial derivatives of gi in its i-th row. For any matrix A of functions, we let Mini (A) denote the sheaf of ideals generated by the determinants of the i × i minors of A. Let Ji (g1 , . . . , gj ) denote Mini (Jac(g1 , . . . , gj )). We use z := (z0 , . . . , zn ) as coordinates on U. We let η : T ∗ U → U denote the cotangent bundle, and we identify the cotangent space T ∗ U with U × Cn+1 by using dz0 , . . . , dzn as a basis. We use w := (w0 , . . . , wn ) as coordinates for the cotangent vectors, i.e., a cotangent vector is ˆ is a linear change of coordinates applied to z, then we let w b denote w0 dz0 + · · · + wn dzn . If z cotangent coordinates with respect to the new basis dˆ z0 , . . . , dˆ zn , i.e., if A is in Gln+1 (C) and b ˆ := Az, then w = At w. z Definition 2.1. The relative polar curve of X with respect to f and z0 , Γ1f,z0 (X), is defined as a set by
Γ1f,z0 := Σ (f, z0 )|Xreg −Σ(f . ) |X

reg

Let V be an open subset of U, and suppose that h := (h1 , . . . , hl ) defines X in V. Then, Γ1f,z0 is defined as a scheme on V by [


 Xj ∩ V Jcj +2 (h, f˜, z0 ) ¬ (ΣX ∪ Σ(f|Xreg )).

j

(We remind the reader that the union is given a scheme structure by using the intersection of the underlying ideal sheaves.) These definitions are independent of the defining equations h and the choice of the extension, f˜, of f . Lˆe and Teissier prove: Proposition 2.2. Let p ∈ X. Then, for a generic choice of the linear form, z0 , in a neighborhood of p, Γ1f,z0 is reduced, is purely 1-dimensional, and the restriction of either of the maps f or z0 to this scheme is finite.

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Moreover, if X is irreducible at p, and Y is a proper analytic subset of X, then for a generic choice of z0 , Γ1f,z0 has no component contained in Y at p. Proof. The first paragraph is well-known and is proved in many places; see, for instance, [L-T2], 4.2.1 and [Te6], 4.1.3.2. However, for lack of a convenient reference, we will prove the second statement. Suppose that X is irreducible. If f is constant on X, then Γ1f,z0 is empty, and we are finished; so, assume that f is not constant. It clearly suffices to prove the result when Y is irreducible; so, we assume that it is. Also, assume that Y 6⊆ ΣX ∪ Σ(f|Xreg ), for otherwise there is nothing to prove. Finally, it suffices to prove that Γ1f,z0 has no component contained in Yreg at p; for if Γ1f,z0 has a component contained in ΣY , we replace Y by ΣY and induct on the dimension of Y . ◦
Let X := Xreg − Σ f|Xreg , and consider the relative conormal variety ◦ ◦
Tf∗| U := {(x, η) ∈ T ∗ U | x ∈ X, η Tx X ∩ ker dx f˜ = 0}. ◦ X

◦

The dimension of the fibre over a point x ∈ X is precisely n + 2 − dim X, and (x, dx z0 ) ∈ Tf∗| U ◦ X

if and only if x ∈

Γ1f,z0 .

It follows that

η −1 (Yreg )

∩

Tf∗| ◦

U is irreducible of dimension at most

X

n + 2 − dim X + dim Y 6 n + 1. Therefore, the fibre of this space over p is conic and of dimension at most n, and so does not contain dp z0 for generic z0 . The result follows. 

In general, we do not care is z0 is chosen so that Γ1f,z0 is reduced, but we do want that the restrictions of f and z0 are finite. We now wish to characterize the relative polar curve in terms of conormal spaces and gap sheaves. We remind the reader that TX∗reg U is purely (n + 1)-dimensional. Also, if X = V (h) in an open subset V of U, then it is trivial to see that, over V, the reduced space TX∗reg U is given by   [ w n+1 (Xj × C ) ∩ Mincj +1 ¬(ΣX × Cn+1 ). Jac(h) j

Note that a generic linear reorganization of z (and the corresponding reorganization of w) produces   ∂ f˜ ∂ f˜ a generic linear reorganization of w0 − ∂z0 , . . . , wn − ∂zn .

Definition 2.3. The relative conormal polar curve of X with respect to f˜ and z, Γ1f˜,z (TX∗reg U), is   ˜ ∂ f˜ ∂ f˜ defined to be the 1-th gap variety of w − ∂∂zf := w0 − ∂z , . . . , w − restricted to TX∗reg U, n ∂zn 0 i.e., !   ˜ ˜ ∂ f ∂ f ∂ f˜ Γ1f˜,z (TX∗reg U) := TX∗reg U ∩ V w1 − , . . . , wn − ¬ V w− . ∂z1 ∂zn ∂z

PART III. ISOLATED CRITICAL POINTS

125

Note that we are not claiming that the relative conormal polar curve is independent of the extension f˜. We need to investigate the relation between the relative conormal polar curve and the ordinary polar curve. The following lemma tells us that they agree over the regular part of X, regardless of whether the two varieties are even really curves. Lemma 2.4. Let p ∈ X be a regular point and suppose that dp z0 6∈ (TX∗reg U)p . Then, in a neighborhood of p, the projection map η restricted to Γ1f˜,z (TX∗reg U) induces an isomorphism onto Γ1f,z0 . Proof. Because X is smooth at p and dp z0 6∈ (TX∗reg U)p , we may use an analytic change of coordinates at p to reduce ourselves to the case where X = V, where V is an open subset of Cd × {0} and TX∗reg U = V × ({0} × Cn+1−d ).      ∂f ∂f Now, one sees easily that Γ1f,z0 = V ∂z , . . . , ∂z∂f ¬V ∂z × {0} and that 1 0 d−1 Γ1f˜,z (TX∗reg U)

=

V × ({0} × C

V × ({0} × C

n+1−d

n+1−d


∂ f˜ ∂ f˜  ) ∩ V w1 − , . . . , wn − ∂z1 ∂zn

!

 ∂ f˜  ¬ V w0 − = ∂z0

 ∂ f˜
∂ f˜ ∂ f˜ ∂ f˜  ) ∩V ,..., , wd − , . . . , wn − ∂z1 ∂zd−1 ∂zd ∂zn

! ¬V

 ∂ f˜  ∂z0

=


∂ f˜ ∂ f˜  Γ1f,z0 × Cn+1 ∩ V wd − , . . . , wn − . ∂zd ∂zn Thus, η restricted to Γ1f˜,z (TX∗reg U) (and to its image) has as its inverse the map τ : Γ1f,z0 →
∂ f˜ ∂ f˜ , . . . , .  Γ1f˜,z (TX∗reg U) given by τ (x) = x, 0, ∂z ∂z n d

We now prove the fundamental result for polar curves on general spaces. Theorem 2.5. Let p ∈ X. For a generic linear reorganization, z,


0)

dp z0 6∈ TX∗reg U

i)

Γ1f,z0 is purely 1-dimensional at p;

ii)

f|Γ1

p

;

is finite at p; and

f,z0

iii)


Γ1f˜,z TX∗reg U has no components contained in η −1 (ΣX) at (p, dp f˜).

In addition, whenever 0)-iii) hold, then we also have

126

iv)

DAVID B. MASSEY

z0|Γ1

is finite at p;

f,z0


Γ1f˜,z TX∗reg U is purely 1-dimensional at (p, dp f˜);

v)

(f ◦ η)|Γ1

vi)

f˜,z

vii)

(z0 ◦ η)|Γ1

(T ∗ U) Xreg

f˜,z

viii)



is finite at (p, dp f˜);

(T ∗ U) Xreg

is finite at (p, dp f˜); and


Γ1f˜,z TX∗reg U · V w0 −


∂ f˜ ∂z0

 (p,dp f˜)

=



Γ1f,z0 · V (f − f (p)) p − Γ1f,z0 · V (z0 − z0 (p)) p .

Proof. For notational convenience, we shall assume that f (p) = z0 (p) = 0. That 0) holds generically is trivial. That i) and ii) hold generically follows from 2.2. As η −1 (ΣX) ∩ TX∗reg U has dimension at most n, I.2.11 implies that for a generic linear reorganization !   ˜ ˜ ∂ f ∂ f ∂ f˜ −1 η (ΣX) ∩ TX∗reg U ∩ V w1 − , . . . , wn − ¬ V w− ∂z1 ∂zn ∂z
is purely 0-dimensional at (p, dp f˜). As every component of Γ1f˜,z TX∗reg U has dimension at least 1 (by I.2.2), iii) holds for generic z. Now suppose that 0)-iii) hold.
By the lemma, η yields a local isomorphism between the schemes Γ1f˜,z TX∗reg U − η −1 (ΣX) and Γ1f,z0 − ΣX; combining this with iii), we conclude that i) implies v), ii) implies vi), and iv) implies vii). It remains for us to show that iv) and viii) hold. Let C be an irreducible component of Γ1f,z0 and suppose that γ(t) is an analytic parameterization of C near p such that γ(0) = p. For t 6= 0, γ(t) ∈ Xreg , dγ(t) f˜ 6∈ TX∗reg U, and there must exist complex numbers a(t) and b(t), not both zero, such that a(t)dγ(t) f˜ − b(t)dγ(t) z0 ∈ T ∗ U. If a(t) Xreg

were zero for an infinite number of t, then, since TX∗reg U is conic, dγ(t) z0 would be in TX∗reg U for an infinite number of t; this would contradict 0). Thus, for small t 6= 0, there exists c(t) such that dγ(t) f˜ − c(t)dγ(t) z0 ∈ TX∗reg U. By evaluating this form on the tangent vector γ 0 (t), we conclude
0
0 that f (γ(t)) − c(t) z0 (γ(t)) ≡ 0. If iv) were false, then z0 (γ(t)) would be zero (for some component C), and, hence, we would
0 have that f (γ(t)) ≡ 0; but, this would imply that f (γ(t)) ≡ 0, in contradiction of ii). Therefore, we have shown iv). Note that c(t) is uniquely determined by
0 f (γ(t)) c(t) =
0 , z0 (γ(t)) 1 where z0 (γ(t)) 6≡ 0 by iv). If |c(t)| → ∞ as t → 0, then since c(t) dγ(t) f˜−dγ(t) z0 ∈ TX∗reg U, we would
once again conclude that dp z0 ∈ TX∗reg U p , in contradiction to 0). It follows that c(t) is analytic

PART III. ISOLATED CRITICAL POINTS

127

at 0, and thus that c(t) approaches some finite value qC as t → 0. Therefore, the component C
b is b of Γ1 T ∗ U through (p, dp f˜) if and only if q = 0; for C corresponds to a component, C, C Xreg f˜,z parameterized by γˆ (t) = (γ(t), dγ(t) f˜ − c(t)dγ(t) z0 ).
Now suppose that, at (p, dp f˜), the cycle Γ1f˜,z TX∗reg U equals ΣnV [V ]. As the restriction of η
to Γ1f˜,z TX∗reg U is a local isomorphism onto Γ1f,z0 over smooth points of X, we conclude that the cycle Γ1f,z0 can be written as Γ1f,z0 =

X V

nV [η(V )] +

X

nC [C].

qC 6=0

Therefore, to demonstrate viii), we need to show two things: a) b)

if qC 6= 0, then (C · V (f ))p − (C · V (z0 ))p = 0, and

b · V w0 − ∂ f˜ if qC = 0, then C = (C · V (f ))p − (C · V (z0 ))p . ∂z0 (p,dp f˜)

We show a) and b) by calculating intersection numbers via parameterizations (see Appendix A.9).
0
0 If qC 6= 0, then f (γ(t)) and z0 (γ(t)) must have the same t-multiplicity. Thus, f (γ(t)) and z0 (γ(t)) have the same t-multiplicity, and so (C · V (f ))p − (C · V (z0 ))p = multt f (γ(t)) − multt z0 (γ(t)) = 0. If qC = 0, then 

multt

  ˜  ∂ f˜  b · V w0 − ∂ f C = multt w0 (ˆ γ (t)) − = ∂z0 (p,dp f˜) ∂z0 γˆ (t)

 ∂ f˜


0
0 ∂ f˜  − c(t) − = multt (c(t)) = multt f (γ(t)) − multt z0 (γ(t)) = ∂z0 γˆ (t) ∂z0 γˆ (t)

multt f (γ(t)) − multt z0 (γ(t)) = (C · V (f ))p − (C · V (z0 ))p . 

Remark 2.6. The point of 2.5.viii is that, for generic z, the quantity 


∂ f˜  Γ1f˜,z TX∗reg U · V w0 − ∂z0 (p,dp f˜)

is, in fact, the multiplicity of the 0-th Vogel cycle of (w −

∂ f˜ ∂z )

at (p, dp f˜) (recall I.2.14). T∗ U Xreg

This enables us to apply results from Part I. Note, also, in the affine case where X = U, the equality of 2.5.viii reduces to the well-known formula from Proposition II.1.20: 

Γ1f,z0 · V

 ∂f  ∂z0

= p



Γ1f,z0 · V (f − f (p)) p − Γ1f,z0 · V (z0 − z0 (p)) p .

128

DAVID B. MASSEY

Chapter 3. THE LINK BETWEEN THE ALGEBRAIC AND TOPOLOGICAL POINTS OF VIEW. We continue with our previous notation: X is a d-dimensional complex analytic space contained in some open subset U of some Cn+1 , f˜ : U → C is a complex analytic function, f = f˜|X , S = {Sα } is a Whitney stratification of X with connected strata, the base ring R is a p.i.d., and F• is a bounded complex of sheaves of R-modules which is constructible with respect to S. In addition, Nα and Lα are, respectively, the normal slice and complex link of the dα -dimensional stratum Sα (see [Go-Mac1]). In this chapter, we are going to prove a general result which describes the characteristic cycle of φf F• in terms of blowing-up the image of df˜ inside the conormal spaces to strata. We will have to wait until the next chapter (on results for perverse sheaves) to actually show how this provides a relationship between ΣF• f and ΣS f in the case where F• is perverse. Because d is the global dimension of X, and we are not assuming that X is pure-dimensional, or that f is not constant on a d-dimensional component of X, if v ∈ C, then the dimension of V (f − v) could be anything between 0 and d. Hence, we let dˆv := 1 + dim V (f − v), and will usually ˆ Of course, if we work locally, or assume that X is pure-dimensional, and denote dˆ0 by simply d. require f not to vanish on a component of X, then dˆ will have attain its “expected” value of d.

Definition 3.1. hRecallithat the characteristic cycle, Ch(F• ), of F• in T ∗ U is the linear combinaP tion α mα (F• ) TS∗α U , where the mα (F• ) are integers given by mα (F• ) := (−1)d χ(φL|X [−1]F• )x = (−1)d χ(φL|N [−1]F• |Nα [−dα ])x = α


(−1)d−dα χ H∗ (Nα , Lα ; F• ) for any point x in Sα , with normal slice Nα at x, and any L : (U, x) → (C, 0) such that dx L is a non-degenerate covector at x (with respect to our fixed stratification; see [Go-Mac1]) and L|Sα has a Morse singularity at x. This cycle is independent of all the choices made (see, for instance, [K-S, Chapter IX]).

We need a number of preliminary results before we can prove the main theorem (Theorem 3.10) of this section. Definition 3.2. Recall that, if M is an analytic submanifold of U and M ⊆ X, then the relative conormal space (of M with respect to f in U), Tf∗| U, is given by M


Tf∗| U := {(x, η) ∈ T ∗ U | x ∈ M, η ker dx (f|M ) = 0} = M


{(x, η) ∈ T ∗ U | x ∈ M, η Tx M ∩ ker dx f˜ = 0}. ∗ ∗ We define the total relative conormal cycle, Tf,F U, by Tf,F U := • •

X Sα 6⊆f −1 (0)

h mα (F• ) Tf∗|

i U . Sα

PART III. ISOLATED CRITICAL POINTS

129

From this point, through Lemma 3.9, it will be convenient to assume that we have refined our stratification S = {Sα } so that V (f ) is a union of strata. By Remark 1.7, this implies that, in a neighborhood of V (f ), if Sα 6⊆ V (f ), then Σ(f|Sα ) = ∅.

We first stated Theorems 3.3 and 3.4 below in our earlier works [Mas2] and [Mas5]. In those papers, we were mainly concerned with local questions, and we also tacitly assumed that f was not constant on any irreducible component of X. We did not correctly adjust the sign for degenerate cases. We correct this error in the statements below – the proofs remain the same. We shall need the following important result from [BMM, 3.4.2]. Theorem 3.3. ([BMM]) The shifted characteristic cycle of the sheaf of nearby cycles of F•

ˆ d−d ∗ U · V (f ) × Cn+1 in along f , (−1) Ch ψf F• , is isomorphic to the intersection product Tf,F • U × Cn+1 . We should note here that the context of [BMM] is that of D-modules and, hence, in that work, the complex of sheaves was a complex of C-vector spaces. However, Theorem 3.3 can easily be recovered by combining the first formula of Theorem 3.4 with Lemma 3.5 (see below), keeping in mind that Lemma 3.5 relies on the result of Theorem 3.3 with C-complexes only. Let Γ1f,L (Sα ) denote the relative polar curve of f|S with respect to a generic linear form L (see α Chapter 2 and [M1] and [M3]). It is important to note that the second part of 2.2 implies that Γ1f,L (Sα ) has no components contained in any strata Sβ ⊆ Sα such that Sβ 6= Sα . It is convenient to have a specific point in X at which to work. Below, we concentrate our attention at the origin; of course, if the origin is not in X (or, if the origin is not in V (f )), then we obtain zeroes for all the terms below. Forh any bounded, constructible complex A• on a subspace i • ∗ of U, let m0 (A ) equal the coefficient of T{0} U in the characteristic cycle of A• . We need to state one further result without proof – this result can be obtained from [BMM], ˆ but we give the result as stated in [Mas5, 4.6], with the added corrections of (−1)d−d in various places. Theorem 3.4. For generic linear forms L, we have the following formulas: ˆ d−d

(−1)

X

m0 (ψf F• ) =


mα (F• ) Γ1f,L (Sα ) · V (f ) 0 ;

Sα 6⊆V (f ) ˆ d−d

m0 (F• ) + (−1)

m0 (F•|V (f ) ) =

X


mα (F• ) Γ1f,L (Sα ) · V (L) 0 ; and

Sα 6⊆V (f )

ˆ d−d

(−1)

m0 (φf F• ) = m0 (F• ) +

X Sα 6⊆V (f )

mα (F• )





 Γ1f,L (Sα ) · V (f ) 0 − Γ1f,L (Sα ) · V (L) 0 .

130

DAVID B. MASSEY

h i

∗ Lemma 3.5. If Sα 6⊆ f −1 (0), then the coefficient of P(T{0} U) in P Tf∗ U · V (f ) × Pn is |S α
given by Γ1f,L (Sα ) · V (f ) 0 . h i Proof. Take a complex of sheaves, F• , which has a characteristic cycle consisting only of TS∗α U (see, for instance, [M1]). Now, apply the formula for m0 (ψf F• ) from Theorem 3.4 together with Theorem 3.3. 

We need to establish some notation that we shall use throughout the remainder of this section. Using the isomorphism, T ∗ U ∼ = U × Cn+1 , we consider Ch(F• ) as a cycle in X × Cn+1 ; we use z := (z0 , . . . , zn ) as coordinates on U and w := (w0 , . . . , wn ) as the cotangent coordinates.

˜ ˜ Let I denote the sheaf of ideals on U given by the image of df˜, i.e., I = w0 − ∂ f , . . . , wn − ∂ f . ∂z0

∂zn

For all α, let Bα = Blim df˜ TS∗α U denote the blow-up of TS∗α U along the image of I in TS∗α U, and let Eα denote the corresponding exceptional divisor. For all α, we have Eα ⊆ Bα ⊆ X × Cn+1 × Pn . ˜ Let π : X ×Cn+1 ×Pn → X ×Pn denote the projection. Note that, if (x, w, [η]) ∈ EP α , then w = dx f and so, for all α, π induces an isomorphism from Eα to π(Eα ). We refer to E := hα mα Eiα as the P total exceptional divisor inside the total blow-up Blim df˜ Ch(F• ) := α mα Blim df˜ TS∗α U .

  Lemma 3.6. For all Sα , there is an inclusion π Blim df˜ TS∗α U ⊆ P Tf∗

|S


U . α

Proof. This is entirely straightforward. Suppose that (x, w, [η]) ∈ Blim df˜ TS∗α U = Blim df˜ TS∗α U. Then, we have a sequence (xi , wi , [ηi ]) ∈ Blim df˜ TS∗α U such that (xi , wi , [ηi ]) → (x, w, [η]). By definition of the blow-up, for each (xi , wi , [ηi ]), there exists a sequence (xji , wij ) ∈ TS∗α U −
im df˜ such that (xji , wij , [wij −dxj f˜]) → (xi , wi , [ηi ]). Now, (xji , [wij −dxj f˜]) is clearly in P Tf∗ U , i i |S α

and so each (xi , [ηi ]) is in P Tf∗ U . Therefore, (x, [η]) ∈ P Tf∗ U .  |S

|S

α

α


 ∗ Lemma 3.7. If Sα 6⊆ f −1 (0), then the coefficient of P T{0} U = {0} × Pn in π∗ (Eα ) equals

Γ1f,L (Sα ) · V (f ) 0 − Γ1f,L (Sα ) · V (L) 0 . Proof. By I.2.23, the multiplicity of {0}×Pn in π∗ (Eα ) equals ∆0


∂ f˜ w− ∂z


|

T∗

U

, (0,d0 f˜)

for a generic

Sα

choice of z. By 2.5.viii and 2.6, this is equal to Γf,L (Sα ) · V (f ) 0 − Γ1f,L (Sα ) · V (L) 0 for a generic linear choice of L. 

1

PART III. ISOLATED CRITICAL POINTS

131

Lemma 3.8. For all α such that Sα ⊆ V (f ), there is an inclusion of the exceptional divisor Eα ∼ = π(Eα ) ⊆ P Tf∗

|S



U ∩ V (f ) × Pn . α

Proof. That π is an isomorphism when restricted to the exceptional divisor is trivial: (x, w, [η]) ∈  
∗ ˜ Eα implies that w = dx f . From Lemma 3.6, π(Eα ) ⊆ π Blim df˜ TSα U ⊆ P Tf∗ U . The result |S

α

follows. 

Lemma 3.9. If Sα ⊆ f −1 (0), then Eα ∼ = π(Eα ) = P(TS∗α U). Proof. If Sα ⊆ f −1 (0), then P Tf∗

|S

α



U = P TS∗α U , and so, by 3.8, π(Eα ) ⊆ P(TS∗α U). We will

demonstrate the reverse inclusion. Suppose that we have (x, [η]) ∈ P(TS∗α U). Then, there exists a sequence (xi , ηi ) ∈ TS∗α U such
that (xi , ηi ) → (x, η). Hence, xi , 1i ηi + dxi f˜ ∈ TS∗α U − im df˜ and  h 1  i 1 xi , ηi + dxi f˜, ηi + dxi f˜ − dxi f˜ → (x, dx f˜, [η]) ∈ Eα .  i i

We come now to the main theorem of this section. This theorem relates the topological data provided by the vanishing cycles of a function f to the algebraic data given by blowing-up the image of the differential of an extension of f . Theorem 3.10. The projection π induces an isomorphism between the total exceptional divisor E ⊆ Blim df˜ Ch(F• ) and the sum over all v ∈ C of the projectivized characteristic cycles of the sheaves of vanishing cycles of F• along f − v, i.e., X ˆv d−d E∼ (−1) P(Ch(φf −v F• )). = π∗ (E) = v∈C

Proof. Remarks 1.7 and 1.10 imply that, locally, supp φf −v F• ⊆ f −1 (v). As the P(Ch(φf −v F• )) are disjoint for different values of v, we may immediately reduce ourselves to the case where we are working near 0 ∈ X and where f (0) = 0. We refine our stratification so that, for all α, Σ(f|Sα ) = ∅ unless Sα ⊆ V (f ). As any newly introduced stratum will appear with a coefficient of zero in the characteristic cycle, the total exceptional divisor will not change. We need to show that E ∼ = π(E) = P(Ch(φf F• )). Now, we will first show that π(E) is Lagrangian. If Sα ⊆ f −1 (0), then π(Eα ) = P(TS∗α U) by 3.9. If Sα 6⊆ f −1 (0), then, by Theorem 3.3,

P Tf∗ U ∩ V (f ) × Pn is Lagrangian and, in particular, is purely n-dimensional. By Lemma |S

α

132

DAVID B. MASSEY



U ∩ V (f ) × Pn . We need

3.8, π(Eα ) is a purely n-dimensional analytic set contained in P Tf∗

|S

α

to show that π(Eα ) is closed. Suppose we have a sequence (xi , [ηi ]) ∈ π(Eα ) and (xi , [ηi ]) → (x, [η]) in U × Pn . Then, there exists a sequence wi so that (xi , wi , [ηi ]) ∈ Eα ; by definition of the exceptional divisor, this implies wi = dxi f˜. Therefore, (xi , wi , [ηi ]) → (x, dx f˜, [η]), which is contained in Eα since Eα is closed in U × Cn+1 × Pn . Thus, (x, [η]) ∈ π(Eα ), and so π(Eα ) is closed and, hence, Lagrangian. Now, π(E) and P(Ch(φf F• )) are both supported over ΣS f and, by taking normal slices to strata, we are reduced to the point-stratum case. Thus, what we need to show is: the coefficient 
 
 ˆ d−d ∗ ∗ of P T{0} U in E equals the coefficient of P T{0} U in (−1) P(Ch(φf F• )). Using 3.4, this is 
 ∗ equivalent to showing that the coefficient of P T{0} U in E equals X

m0 (F• ) +



mα



 Γ1f,L (Sα ) · V (f ) 0 − Γ1f,L (Sα ) · V (L) 0

Sα 6⊆V (f )

for a generic linear form L. But, by 3.9, E =

X

X

mα Eα =

α


 ∗ and the coefficient of P T{0} U in

  mα P(TS∗α U) +

Sα ⊆V (f )

X

X

mα E α

Sα 6⊆V (f )

  mα P(TS∗α U) is precisely m0 (F• ).

Sα ⊆V (f )


 ∗ Therefore, we will be finished if we can show that the coefficient of P T{0} U in Eα equals

Γ1f,L (Sα ) · V (f ) 0 − Γ1f,L (Sα ) · V (L) 0 if Sα 6⊆ V (f ). However, this is exactly the content of Lemma 3.7. 

Remark 3.11. In special cases, Theorem 3.10 was already known. Consider the case where X = U and F• is the constant sheaf. Then, Ch(F• ) = U × {0}, and the image of df˜ in U ×{0} is simply defined by the Jacobian ideal of f . Hence, our result reduces to the result obtained from the work of Kashiwara in [K] and Lˆe-Mebkhout in [L-M] – namely, that the projectivized characteristic cycle of the sheaf of vanishing cycles is isomorphic to the exceptional divisor of the blow-up of the Jacobian ideal in affine space. As a second special case, suppose that X and F• are completely general, but that x is an isolated point in the image of Ch(φf F• ) in X (for instance, x might be an isolated point in supp φf F• ). Then, for every stratum for which mα 6= 0, (x, dx f˜) is an isolated point of im df˜ ∩ TS∗α U or is not contained in the intersection at all. Therefore, the last part of I.2.23 implies that the exceptional divisor of the blow-up of im df˜ in ∗ TSα U has one component over (x, dx f˜) and that that component occurs with multiplicity precisely   equal to the intersection multiplicity im df˜ · T ∗ U in T ∗ U. Thus, we recover the results of Sα

(x,dx f˜)

three independent works appearing in [Gi], [Lˆ e3], and [Sab2] – that the coefficient of {x} × Cn+1

PART III. ISOLATED CRITICAL POINTS ˆ d−d

in (−1)

  Ch(φf F• ) is given by im df˜ · Ch(F• )

(x,dx f˜)

133

. This result is usually stated in terms of

the Euler characteristic: if x is an isolated point in supp φf F• , then   χ(φf [−1]F• )x = (−1)d im df˜ · Ch(F• )

(x,dx f˜)

.

In addition to generalizing the above results, Theorem 3.10 fits in well with Theorem 3.4.2 of [BMM]; that theorem contains a nice description of the characteristic cycles of the nearby cycles and of the restriction of a complex to a hypersurface. However, [BMM] does not contain a nice description of the vanishing cycles, nor does our Theorem 3.10 seem to follow easily from the results of [BMM]; in fact, Example 3.4.3 of [BMM] makes it clear that the general result contained in our Theorem 3.10 was unknown – for Brian¸con, Maisonobe, and Merle only derive the vanishing cycle result from their nearby cycle result in the easy, known case where the vanishing cycles are supported on an isolated point and, even then, they must make half a page of argument.

Corollary 3.12. For each extension f˜ of f , let Ef˜ denote the exceptional divisor in Blim df˜ TX∗reg U.
Then, π Ef˜ is independent of f˜. Proof. We apply Theorem 3.10 to a complex of sheaves F• such that mα = 1 for each smooth component of Xreg and mα = 0 for every other stratum in some Whitney stratification of X (it is easy to produce such an F• – see, for instance, Lemma 3.1 of [M1]). The corollary follows from the fact that P(Ch(φf F• )) does not depend on the extension. 

134

DAVID B. MASSEY

Chapter 4. THE SPECIAL CASE OF PERVERSE SHEAVES. We continue with our previous notation, except that in this chapter we must assume that our base ring is a field. For the purposes of Part III, perverse sheaves are important because the vanishing cycles functor (shifted by −1) applied to a perverse sheaf once again yields a perverse sheaf and because of the following lemma. Lemma 4.1. If P• is a perverse sheaf on X, then Ch(P• ) =

h i ∗ U , where m T α α Sα

P

mα = (−1)d dim H 0 (Nα , Lα ; P•|Nα [−dα ]); in particular, (−1)d Ch(P• ) is a non-negative cycle. If P• is perverse on X (or, even, perverse up to a shift), then supp P• equals the image in X of the characteristic cycle of P• . Proof. The first statement follows from the definition of the characteristic cycle, together with the fact that a perverse sheaf supported on a point has non-zero cohomology only in degree zero. The second statement follows at once from the fact that if P• is perverse up to a shift, then so is the restriction of P• to its support. Hence, by the support condition on perverse sheaves, there is an open dense set of the support, Ω, such that, for all x ∈ Ω, H ∗ (P• )x is non-zero in a single degree. The conclusion follows.  The fact that the above lemma refers to the support of P• , which is the closure of the set of points with non-zero stalk cohomology, means that we can use it to conclude something about the closure of the P• -critical locus (recall Definition 1.9). • Theorem 4.2. Let P• be a perverse i on X, and suppose that the characteristic cycle of P h sheaf P • in U is given by Ch(P ) = α mα TS∗α U . Then, the closure of the P• -critical locus of f is given by

ΣP• f =

n o x ∈ X (x, dx f˜) ∈ | Ch(P• )| =

[

Σcnr f|S




.

α

mα 6=0

Proof. Let q ∈ X, and let v = f (q). Let W be an open neighborhood of q in X such that W ∩ ΣP• f ⊆ V (f −v) (see the end of Remark 1.7). Then, W∩ΣP• f = W∩supp φf −v P• . As φf −v P• [−1] is perverse, Lemma 4.1 tells us that supp φf −v P• equals the image in X of Ch(φf −v P• ). Now, Theorem 3.10 tells us that this image is precisely o [ n x ∈ Sα | (x, dx f˜) ∈ TS∗α U , mα 6=0

since there can be no cancellation as all the non-zero mα have the same sign. Therefore, we have the desired equality of sets in an open neighborhood of every point; the theorem follows. 

PART III. ISOLATED CRITICAL POINTS

135

We will use the perverse cohomology of the shifted constant sheaf, C•X [k], in order to deal with non-l.c.i.’s; this perverse cohomology is denoted by µH 0 (C•X [k]) = µH k (C•X ) (see [BBD], [K-S], or Appendix B). Like the intersection cohomology complex, this sheaf has the property that it is the shifted constant sheaf on the smooth part of any component of X with dimension equal to d = dim X. We now list some properties of the perverse cohomology and of vanishing cycles that we will need later. For further properties, see Appendix B. The perverse cohomology functor on X, µH 0 , is a functor from the derived category of bounded, constructible complexes on X to the Abelian category of perverse sheaves on X. If F• is constructible with respect to S, then µH 0 (F• ) is also constructible with respect to S,
µ 0 • and H (F ) | [−dα ] is naturally isomorphic to µH 0 (F•|N [−dα ]). α

Nα

The functor µH 0 , applied to a perverse sheaf P• is canonically isomorphic to P• . In addition, a bounded, constructible complex of sheaves F• is perverse if and only µH 0 (F• [k]) = 0 for all k 6= 0. In particular, if X is an l.c.i., then µH 0 (C•X [d]) ∼ = C•X [d] and µH 0 (C•X [k]) = 0 if k 6= d. µ 0 The functor H commutes with vanishing cycles with a shift of −1, nearby cycles with a shift of −1, and Verdier dualizing. That is, there are natural isomorphisms H 0 ◦ φf [−1] ∼ = φf [−1] ◦ µH 0 ,

µ

H 0 ◦ ψf [−1] ∼ = ψf [−1] ◦ µH 0 , and D ◦ µH 0 ∼ = µH 0 ◦ D.

µ

Let F• be a bounded complex of sheaves on X which is constructible with respect to a connected Whitney stratification {Sα } of X. Let Smax
be a maximal stratum contained in the support of F• , and let m = dim Smax . Then, µH 0 (F• ) | is isomorphic (in the derived category) to the Smax −m • in degree −m and zero in all other degrees. complex which has (H (F ))|S S max In particular, supp F• = i supp µH 0 (F• [i]), and if F• is supported on an isolated point, q, then H 0 (µH 0 (F• ))q ∼ = H 0 (F• )q .

Throughout the remainder of Part III, we let kP• denote the perverse sheaf µH 0 (C•X [k + 1]); it will be useful later to have a nice characterization of the characteristic cycle of kP• .

Proposition 4.3. The complex kP• is a perverse sheaf on X which is constructible with respect to S and the characteristic cycle Ch(kP• ) is equal to (−1)d

X

h i bk+1−dα (Nα , Lα ) TS∗α U ,

α

where bj denotes the j-th (relative) Betti number.
In particular, H ∗ (Lα ; C) ∼ = H ∗ (point; C) if and only if mα kP• = 0 for all k. Proof. The constructibility claim follows from the fact that the constant sheaf itself is clearly constructible with respect to any Whitney stratification. The remainder follows trivially from the definition of the characteristic cycle, combined with two properties of µH 0 ; namely, µH 0 commutes with φf [−1], and µH 0 applied to a complex which is supported at a point simply gives ordinary cohomology in degree zero and zeroes in all other degrees. See [K-S, 10.3]. 

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DAVID B. MASSEY

Remark 4.4. As Nα is contractible, it is possible to give a characterization of bk+1−dα (Nα , Lα ) without referring to Nα ; the statement gets a little complicated, however, since we have to worry about what happens near degree zero and because the link of a maximal stratum is empty. However, if we slightly modify the usual definitions of reduced cohomology and the corresponding reduced Betti numbers, then the statement becomes quite easy. e k (A; C) to be the relative cohomology vector What we want is for the “reduced” cohomology H k+1 space H (B, A; C), where B is a contractible set containing A, and we want ˜b∗ () to be the Betti numbers of this “reduced” cohomology. Therefore, letting bk () denote the usual k-th Betti number, we define ˜b∗ () by  bk (A), if k 6= 0 and A 6= ∅     b (A) − 1, if k = 0 and A 6= ∅ 0 ˜bk (A) =  0, if k 6= −1 and A = ∅    1, if k = −1 and A = ∅. Thus, ˜bk (A) is the k-th Betti number of the reduced cohomology, provided that A is not the empty set. The special definition of ˜bk () for the empty set implies that if Sα is maximal, so that Nα = point and Lα = ∅, then  0, if k + 1 6= dα bk+1−dα (Nα , Lα ) = = ˜bk−dα (Lα ). 1, if k + 1 = dα Thus, with this new notation, Ch(kP• ) = (−1)d

X

i h ˜bk−d (Lα ) T ∗ U . α Sα

α

By combining 4.2 with 4.3 and 4.4, we can now give a result about ΣC f . First, though, it will be useful to adopt the following terminology. Definition 4.5. We say that the stratum Sα is visible (or, C-visible) if H ∗ (Lα ; C) 6∼ = H ∗ (point; C) ∗ (or, equivalently, if H (Nα , Lα ; C) 6= 0). Otherwise, the stratum is invisible. The final line of hProposition 4.3 tells us that a stratum is visible if and only if there exists an i
∗ integer k such that TSα U appears with a non-zero coefficient in Ch kP• . Note that if Sα has an empty complex link (i.e., the stratum is maximal), then Sα is visible. Theorem 4.6. Then, ΣC f =

d−1 [ k=−1

ΣkP• f =

[ visible Sα

n o x ∈ Sα | (x, dx f˜) ∈ TS∗α U =

[

Σcnr f|S




.

α

visible Sα

In particular, since all maximal strata are visible, Σcnr f ⊆ ΣC f (as stated in Proposition 1.6). Moreover, if x is an isolated point of ΣC f ,then, for all Whitney stratifications, {Rβ }, of X, the only possibly visible stratum which can be contained in f −1 f (x) is {x}.

PART III. ISOLATED CRITICAL POINTS

137

S Proof. Recall that, for any complex F• , supp F• = k supp µH 0 (F• [k]). In addition, we claim that k • P = 0 unless −1 6 k 6 d − 1. By Lemma 4.1, kP• = 0 is equivalent to Ch(kP• ) = 0; if k is not between −1 and d − 1, then, using Proposition 4.3, Ch(kP• ) = 0 follows from the fact that the complex link of a stratum has the homotopy-type of a finite CW complex of dimension no more than the complex dimension of the link (see [Go-Mac1]). Now, in an open neighborhood of any point q with v := f (q), we have ΣC f = supp φf −v C• =

[

supp µH 0 (φf −v C•X [k]) =

k

[

[
supp φf −v [−1] µH 0 (C•X [k + 1]) = ΣkP• f .

k

k

Now, applying Theorem 4.2, we have ΣC f =

[

[

n o x ∈ Sα | (x, dx f˜) ∈ TS∗α U .

k mα (kP• )6=0

The desired conclusion follows.  Remark 4.7. Those familiar with stratified Morse theory should find the result of Theorem 4.6 very un-surprising – it looks like it results from some break-down of the C-critical locus into normal and tangential data, and naturally one gets no contributions from strata with trivial normal data. This is the approach that we took in Theorem 3.2 of [Ma1]. There is a slightly subtle, technical point which prevents us from taking this approach in our current setting: by taking normal slices at points in an open, dense subset of supp φf −v C•X , we could reduce ourselves to the case where ΣC f consists of a single point, but we would not know that the point was a stratified isolated critical point. In particular, the case where supp φf −v C•X consists of a single point, but where f has a non-isolated (stratified) critical locus coming from an invisible stratum causes difficulties with the obvious Morse Theory approach. Remark 4.8. At this point, we wish to add to our hierarchy of critical loci from Proposition 1.6. Theorem 4.6 tells us that ΣkP• f ⊆ ΣC f for all k. If X is purely (m + 1)-dimensional, then 4.2 implies that Σcnr f ⊆ ΣmP• f . Now, suppose that X is irreducible of dimension m + 1. Let IC• be the intersection cohomology sheaf (with constant coefficients) on X (see [Go-Mac2]); IC• is a simple object in the category of perverse sheaves. As the category of perverse sheaves on X is (locally) Artinian, and since mP• is a perverse sheaf which is the shifted constant sheaf on the smooth part of X, it follows that IC• appears as a simple subquotient in any composition series for mP• . Consequently, | Ch(IC• )| ⊆ | Ch(mP• )|, and so 4.2 implies that ΣIC• f ⊆ ΣmP• f . Moreover, 4.2 also implies that Σcnr f ⊆ ΣIC• f . Therefore, we can extend our sequence of inclusions from Proposition 1.6 to: Σreg f ⊆ Σalg f ⊆ ΣNash f ⊆ Σcnr f ⊆ ΣIC• f ⊆ ΣmP• f ⊆ ΣC f ⊆ Σcan f ⊆ Σrdf f. Why not use one of these new critical loci as our most fundamental notion of the critical locus of f ? Both ΣIC• f and ΣmP• f are topological in nature, and easy examples show that they can be distinct from ΣC f . However, 4.6 tells us that ΣmP• f is merely one piece that goes into making up ΣC f – we should include the other shifted perverse cohomologies. On the other hand, given

138

DAVID B. MASSEY

the importance of intersection cohomology throughout mathematics, one should wonder why we do not use ΣIC• f as our most basic notion. Consider the node X := V (y 2 − x3 − x2 ) ⊆ C2 and the function f := y|X . The node has a small resolution of singularities (see [Go-Mac2]) given by simply pulling the branches apart. As a result, the intersection cohomology sheaf on X is the constant sheaf shifted by one on X − {0}, and the stalk cohomology at 0 is a copy of C2 concentrated in degree −1. Therefore, one can easily show that 0 6∈ ΣIC• f . As ΣIC• f fails to detect the simple change in topology of the level hypersurfaces of f as they go from being two points to being a single point, we do not wish to use ΣIC• f as our basic type of critical locus. That is not to say that ΣIC• f is not interesting in its own right; it is integrally tied to resolutions of singularities. For instance, it is easy to show (using the Decomposition Theorem π e− [BBD]) that if X → X is a resolution of singularities, then ΣIC• f ⊆ π(Σ(f ◦ π)).

Now that we can “calculate” ΣC f using Theorem 4.6, we are ready to generalize the Milnor number of a function with an isolated critical point. Definition 4.9. If P• is a perverse sheaf on X, and x is an isolated point in ΣP• f (or, if x 6∈ ΣP• f ), then we call dimC H 0 (φf −f (x) [−1]P• )x the Milnor number of f at x with coefficients in P• and we denote it by µx (f ; P• ). This definition is reasonable for, in this case, φf −f (x) [−1]P• is a perverse sheaf supported at the isolated point x. Hence, the stalk cohomology of φf −f (x) [−1]P• at x is possibly non-zero only in degree zero. Normally, we summarize that x is an isolated point in ΣP• f or that x 6∈ ΣP• f by writing dimx ΣP• f 6 0 (we consider the dimension of the empty set to be −∞).

Before we state the next proposition, note that it is always the case that 

∗ im df˜ · T{0} U

 (0,d0 f˜)

= 1.

Proposition 4.10. For notational convenience, we assume that 0 ∈ X and that f (0) = 0. Then, dim0 ΣC f 6 0 if and only if, for all k, dim0 ΣkP• f 6 0. Moreover, if dim0 ΣC f 6 0, then, i) for all visible strata, Sα , such that dim Sα > 1, the intersection of im df˜ and TS∗α U is, at most, 0-dimensional at (0, d0 f˜), and  

im df˜ · TS∗α U = Γ1f,L (Sα ) · V (f ) 0 − Γ1f,L (Sα ) · V (L) 0 , (0,d0 f˜)

where L is a generic linear form, and ii) for all k,
µ0 (f ; kP• ) = ˜bk (Ff,0 ) = (−1)dim X im df˜ · Ch(kP• ) (0,d0 f˜) =

PART III. ISOLATED CRITICAL POINTS

X

  ˜bk−d (Lα ) im df˜ · T ∗ U α Sα

visible Sα

X

  ˜bk−d (Lα ) im df˜ · T ∗ U α Sα

visible Sα Sα not maximal

(0,d0 f˜)

+

(0,d0 f˜)

X

139

=



im df˜ · TS∗α U

Sα maximal dim Sα =k+1

 (0,d0 f˜)

.

Proof. It follows immediately from 4.6 that dim0 ΣC f 6 0 if and only if, for all k, dim0 ΣkP• f 6 0. i) follows immediately from Lemma 3.7 (combined with Remark 3.11). It remains for us to prove ii). As in the proof of 4.6, we have
µ 0 H (φf C•X [k]) = φf [−1] µH 0 (C•X [k + 1]) = φf [−1]kP• . It follows that

µ0 (f ; kP• ) = dimC H 0 (φf [−1]kP• )0 = dimC H 0 µH 0 (φf C•X [k]) 0 = dimC H 0 φf C•X [k] 0 , where the last equality is a result of the fact that 0 is an isolated point in the support of φf C•X [k]. Therefore,

e k (Ff,0 ; C). µ0 (f ; kP• ) = dimC H 0 φf C•X [k] 0 = dimC H k φf C•X 0 = dim H That we also have the equality
µ0 (f ; kP• ) = (−1)dim X im df˜ · Ch(kP• ) (0,d0 f˜) is precisely the content of Theorem 3.10, interpreted as in the last paragraph of Remark 3.11. The remaining equalities in ii) follow from the description of Ch(kP• ) given in Proposition 4.3 and Remark 4.4. 

Remark 4.11. The formulas from 4.10 provide a topological/algebraic method for “calculating” the Betti numbers of the Milnor fibre for isolated critical points on arbitrary spaces. It should not be surprising that the data that one needs is not just the algebraic data – coming from the polar curves and intersection numbers – but also includes topological data about the underlying space: one has to know the Betti numbers of the complex links of strata. Example 4.12. The most trivial, non-trivial case where one can apply 4.10 is the case where X is an irreducible local, complete intersection with an isolated singularity (that is, X is an irreducible i.c.i.s). Let us assume that 0 ∈ X is the only singular point of X and that f has an isolated C-critical point at 0. Let d denote the dimension of X. Let us write LX,0 for the complex link of X at 0. By [Lˆ e1], LX,0 has the homotopy-type of a finite bouquet of (d − 1)-spheres. Applying 4.10.ii, we see, then, that the reduced cohomology of Ff,0 is concentrated in degree (d − 1), and the (d − 1)-th Betti number of Ff,0 is equal to     ∗ ˜bd−1 (LX,0 ) im df˜ · T ∗ U + im df˜ · TX U = 0 reg (0,d0 f˜)

(0,d0 f˜)

140

DAVID B. MASSEY

˜bd−1 (LX,0 ) +



Γ1f,L (Xreg ) · V (f ) 0 − Γ1f,L (Xreg ) · V (L) 0 ,

for generic linear L. Now, the polar curve and the intersection numbers are quite calculable in practice; see Remark 1.8 and Example 1.9 of [Ma1]. However, there remains the question of how one can compute ˜bd−1 (LX,0 ). Corollary 5.6 and Example 5.4 of [Ma1] provide an inductive method for computing the Euler characteristic of LX,0 (the induction is on the codimension of X in U) and, since we know that LX,0 has the homotopy-type of a bouquet of spheres, knowing the Euler characteristic is equivalent to knowing ˜bd−1 (LX,0 ). The obstruction to using 4.10 to calculate Betti numbers in the general case is that, if X is not an l.c.i., then a formula for the Euler characteristic of the link of a stratum does not tell us the Betti numbers of the link.

Example 4.13. In this example, X will be a hypersurface with a non-isolated singularity. Use (x, y, z) as coordinates for U := C3 , and let X := V (xy). Let f˜ := xα + y β + z γ , where α, β, γ > 2. The strata are S0 := V (x, y), S1 := V (x) − V (y), and S2 := V (y) − V (x), withh corresponding i P links L0 = two points, L1 = ∅, and L2 = ∅. As Ch(kP• ) = (−1)d α ˜bk−dα (Lα ) TS∗α U , we see that Ch(kP• ) = 0 unless k = 1, and h i h i h i Ch(1P• ) = TS∗0 U + TS∗1 U + TS∗2 U = [V (x, y, w2 )] + [V (x, w1 , w2 )] + [V (y, w0 , w2 )] . Now, we have that im df˜ = V (w0 − αxα−1 , w1 − βy β−1 , w2 − γz γ−1 ) and im df˜∩ | Ch(1P• )| = (0, 0). Therefore, dim0 ΣC f = 0, the only non-zero reduced Betti number of the Milnor fibre of f at 0 is ˜b1 (Ff,0 ) =


im df˜ · Ch(1P• ) (0,0) =

(c − 1) + (b − 1)(c − 1) + (a − 1)(c − 1) = (c − 1)(a + b − 1). One can actually verify this computation. The Milnor fibre Ff,0 is easily seen to be the union of the Milnor fibre, F1 , of y β + z γ restricted to V (x) and the Milnor fibre, F2 , of xα + z γ restricted to V (y); these two fibres intersect in c distinct points. The classical calculation of the Milnor numbers tells us that F1 is homotopy-equivalent to a bouquet of (β − 1)(γ − 1) 1-spheres, while F2 is homotopy-equivalent to a bouquet of (α − 1)(γ − 1) 1-spheres. Applying the Mayer-Vietoris exact sequence, we recover the equality above.

Example 4.14. In this example, X will be the simplest non-l.c.i. Use (u, x, y, z) as coordinates for U := C4 , and let X := V (u, x) ∪ V (y, z). Let f˜ := uα + xβ + y γ + z δ , where α, β, γ, δ > 2. The strata are S0 := {0}, S1 := V (u, x) − {0}, and S2 := V (y, z) − {0}, with corresponding links L0 = two complex disks (sets of complex dimension one), L1 = ∅, and L2 = ∅. We see that Ch(kP• ) = 0 unless k = 0 or 1, and h i h i Ch(1P• ) = TS∗1 U + TS∗2 U = [V (u, x, w2 , w3 )] + [V (y, z, w0 , w1 )] ,

PART III. ISOLATED CRITICAL POINTS

while

141

h i Ch(0P• ) = TS∗0 U = [V (u, x, y, z)] .

Now, we find im df˜ = V (w0 − αuα−1 , w1 − βxβ−1 , w2 − γy γ−1 , w3 − δz δ−1 ), im df˜ ∩ | Ch(1P• )| = {0}, and im df˜ ∩ | Ch(0P• )| = {0}. Therefore, dim0 ΣC f = 0, the only non-zero reduced Betti numbers of the Milnor fibre of f at 0 are ˜b1 and ˜b0 , and ˜b1 (Ff,0 ) =


im df˜ · Ch(1P• ) (0,0) = (γ − 1)(δ − 1) + (α − 1)(β − 1)

and ˜b0 (Ff,0 ) =


im df˜ · Ch(0P• ) (0,0) = 1.

Again, one can actually verify this computation. The Milnor fibre Ff,0 is easily seen to be the disjoint union of the Milnor fibre, F1 , of y γ + z δ restricted to V (u, x) and the Milnor fibre, F2 , of uα + xβ restricted to V (y, z). The classical calculation of the Milnor numbers tells us that F1 is homotopy-equivalent to a bouquet of (γ − 1)(δ − 1) 1-spheres, while F2 is homotopy-equivalent to a bouquet of (α − 1)(β − 1) 1-spheres. Thus, we recover the equalities above.

142

DAVID B. MASSEY

Chapter 5. THOM’S af CONDITION. We continue with the notation from Chapter 3. In this section, we explain the fundamental relationship between Thom’s af condition and the vanishing cycles of f . Definition 5.1. Let M and N be analytic submanifolds of X such that f has constant rank on N . Then, the pair af condition at a point x ∈ N if and only if we have  (M, N) satisfies  Thom’s  the containment Tf∗| U M

⊆

x

Tf∗| U N

of fibres over x.

x

In particular, if f is, in fact, constant on N , then the pair Thom’s af condition  (M, N ) satisfies   ∗ ∗ at a point x ∈ N if and only if we have the containment Tf| U ⊆ TN U of fibres over x. M

x

x

We have been slightly more general in the above definition than is sometimes the case; we have not required that the rank of f be constant on M . Thus, if X is an analytic space, we may write that (X
reg , N ) satisfies the af condition, instead of writing the much more cumbersome (Xreg − Σ f|Xreg , N ) satisfies the af condition. If f is not constant on any irreducible component of X, it is easy to see that these statements are equivalent: ◦
Let X := Xreg − Σ f|Xreg , which is dense in Xreg (as f is not constant on any irreducible U; clearly, this is equivalent to showing that components of X). We claim that Tf∗| U = Tf∗| Xreg ◦ X
Tf∗| U ⊆ Tf∗| U. This is simple, for if x ∈ Σ f|Xreg , then (x, η) ∈ Tf∗| U if and only if Xreg

◦ X

Xreg

(x, η) ∈ TX∗reg U, and TX∗reg U ⊆ T ∗◦ U ⊆ Tf∗| U. X

◦ X

The link between Theorem 3.10 and the af condition is provided by the following theorem, which describes the fibre in the relative conormal in terms of the exceptional divisor in the blow-up of im df˜. Originally, we needed to assume Whitney’s condition a) as an extra hypothesis; however, T. Gaffney showed us how to remove this assumption by using a re-parameterization trick. Theorem 5.2. Let π : U × Cn+1 × Pn → U × Pn denote the projection. Suppose that f is not constant on any irreducible component of X. Let E denote the exceptional divisor in Blim df˜ TX∗reg U ⊆ U × Cn+1 × Pn . 

 Then, for all x ∈ X, there is an inclusion of fibres over x given by π(E) x ⊆ P Tf∗| U . Xreg

x

Moreover, if x ∈ ΣNash f , then this inclusion is actually an equality. Proof. By 3.12, it does not matter what extension of f we use. 

 That π(E) x ⊆ P Tf∗| U is easy. Suppose that (x, [η]) ∈ π(E), that is (x, dx f˜, [η]) ∈ E. Xreg

x

Then, there exists a sequence (xi , ωi ) ∈ TX∗reg U −im df˜ such that (xi , ωi , [ωi −dxi f˜]) → (x, dx f˜, [η]). Hence, there exist scalars ai such that ai (ωi − dxi f˜) → η, and these ai (ωi − dxi f˜) are relative

 conormal covectors whose projective class approaches that of η. Thus, π(E) x ⊆ P Tf∗| U . Xreg

x

PART III. ISOLATED CRITICAL POINTS

 We must now show that P Tf∗|

U


 x

Xreg

143


⊆ π(E) x , provided that x ∈ ΣNash f .

◦

Let X := Xreg − Σ f|Xreg . Suppose that (x, [η]) ∈ P Tf∗| U . Then, there exists a complex ◦ X

analytic path α(t) = (x(t), ηt ) ∈ Tf∗| U such that α(0) = (x, η) and α(t) ∈ Tf∗| U for t 6= 0. ◦

◦ X

◦ X

As f has no critical points on X, each ηt can be written uniquely as ηt = ωt + λ(x(t))dx(t) f˜, ◦

where ωt (Tx(t) X) = 0 and λ(x(t)) is a scalar. By evaluating each side on x0 (t), we find that λ(x(t)) =

ηt (x0 (t)) d dt f (x(t))

.

Thus, as λ(x(t)) is a quotient of two analytic functions, there are only two possibilities for what happens to λ(x(t)) as t → 0. Case 1: |λ(x(t))| → ∞ as t → 0. ηt ωt → 0 and, hence, − → dx f˜. Therefore, λ(x(t)) λ(x(t))      ωt ωt ωt ˜ , − − dx(t) f = x(t), − , [ηt (x(t))] → (x, dx f˜, [η]), x(t), − λ(x(t)) λ(x(t)) λ(x(t))

In this case, since ηt → η, it follows that

and so (x, [η]) ∈ π(E).

Case 2: λ(x(t)) → λ0 as t → 0. In this case, ωt must possess a limit as t → 0. For t small
and unequal to zero,
let projt denote the complex orthogonal projection from the fibre Tf∗| U x(t) to the fibre T ∗◦ U x(t) . Let ◦ X

X

γt := projt (ηt ) = ωt + λ(x(t)) projt (dx(t) f˜). Since x ∈ ΣNash f , we have that projt (dx(t) f˜) → dx f˜ and, thus, γt → η. As η is not zero (since it represents a projective class), we may define the (real, non-negative) scalar s || projt (dx(t) f˜) − dx(t) f˜|| at := . ||γt || One now verifies easily that (x(t), at γt + projt (dx(t) f˜), [at γt + projt (dx(t) f˜) − dx(t) f˜]) −→ (x, dx f˜, [η]), and, hence, that (x, [η]) ∈ π(E). 

Remark 5.3. In a number of results throughout the remainder of Part III, the reader will find the hypotheses that x ∈ ΣNash f or that x ∈ Σalg f . While Theorem 5.2 explains why the hypothesis x ∈ ΣNash f is important, it may not be so clear why the hypothesis x ∈ Σalg f is of interest. If Y is an analytic subset of X, then one shows easily that Y ∩ Σalg f ⊆ Σalg (f|Y ). The Nash critical locus does not possess such an inheritance property. Thus, the easiest hypothesis to make

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DAVID B. MASSEY

in order to guarantee that a point, x, is in the Nash critical locus of any analytic subset containing x is the hypothesis that x ∈ Σalg f , for then if x ∈ Y , we conclude that x ∈ Σalg (f|Y ) ⊆ ΣNash (f|Y ).
A further remark is that the fibre π(E) x being non-empty is trivially seen to be equiva

lent to x ∈ Σcnr f . As the fibre P Tf∗| U is always non-empty, the equality π(E) x = Xreg x 
 P Tf∗| U implies that x ∈ Σcnr f . This is slightly short of being a converse to the statement Xreg

x

in the theorem, unless we are in a situation where we know that Σcnr f = ΣNash f .

We come now to the result which tells one how the topological information provided by the sheaf of vanishing cycles controls the af condition. Corollary 5.4. Let N be a submanifold of X such that N ⊆ V (f ), and let x ∈ N   P ∗ U , where {M } is a collection of connected analytic submanifolds of Let Ch(F• ) = α mα TM α α   P X such that either mα > 0 for all α, or mα 6 0 for all α. Let Ch(φf F• ) = β kβ TR∗ U , where β {Rβ } is a collection of connected analytic submanifolds.   Finally, suppose that, for all β, there is an inclusion of fibres over x given by TR∗ U ⊆ β x
∗ TN U x .  
Then, the pair Mα reg , N satisfies Thom’s af condition at x for every Mα for which f|Mα 6≡ 0, mα 6= 0, and x ∈ ΣNash (f|M ). α

 Proof. Let Sγ
be a Whitney stratification for X such that each Mα is a union of strata and such that Σ f|Sγ = ∅ unless Sγ ⊆ V (f ). Hence, for each α, there exists a unique Sγ such that   P Mα = Sγ ; denote this stratum by Sα . It follows at once that Ch(F• ) = α mα TS∗α U .
P From Theorem 3.10, E = α mα Eα ∼ the = P Ch(φf F• ) . Thus, since all non-zero mα have
• same sign, if mα is not zero, then Eα appears with a non-zero coefficient in P Ch(φf F ) . The result now follows immediately by applying Theorem 5.2 to each Mα in place of X.  Theorem 5.2 also allows us to prove an interesting relationship between the characteristic varieties of the vanishing and nearby cycles – provided that the complex of sheaves under consideration is perverse. Corollary 5.5. Let P• be a perverse sheaf on X. If x ∈ Σalg f and (x, η) ∈ | Ch(ψf P• )|, then (x, η) ∈ | Ch(φf P• )|. Proof. Let S := {Sα } be a Whitney stratification with connected strata such that P• is constructible with respect to S and such that V (f ) is a union of strata. For the remainder of the proof, we will work in a neighborhood of V (f ) – a neighborhood in which, if Sα 6⊆ V (f ), then Σ(f|Sα ) = ∅.

PART III. ISOLATED CRITICAL POINTS

145

h i P Let Ch(P• ) = mα TS∗α U . As P• is perverse, all non-zero mα have the same sign. Thus, 3.10 tells us – using the notation from 3.10 – that |P(Ch(φf P• ))| =

(†)

[

π(Eα ),

mα 6=0

where Eα denotes the exceptional divisor in the blow-up of TS∗α U along im df˜ (in a neighborhood of V (f )). In addition, 3.3 tells us that | Ch(ψf P• )| = V (f ) × Cn+1




[

∩

Tf∗|

U. Sα

mα 6=0 Sα 6⊆V (f )

Assume (x, η) ∈ | Ch(ψf P• )|. Then, there exists Sα 6⊆ V (f ) such that mα 6= 0 and (x, η) ∈ U. Clearly, then, (x, η) ∈ Tf∗| U. Now, if x ∈ Σalg f and η 6= 0, then x ∈ Σalg (f|S ) and

Tf∗| S

α

α

(S α )reg

so Theorem 5.2 implies that (x, [η]) ∈ π(Eα ), where [η] denotes the projective class of η and Eα ∗ denotes the exceptional divisor of the blow-up of T(S U = TS∗α U along im df˜. Thus, by (†), ) α reg

(x, η) ∈ | Ch(φf P• )|. We are left with the trivial case of when (x, 0) ∈ | Ch(ψf P• )|. Note that, if (x, 0) ∈ | Ch(ψf P• )|, then there must exist some non-zero η such that (x, η) ∈ | Ch(ψf P• )|. For, otherwise, the stratum (in some Whitney stratification) of supp ψf P• containing x must be all of U. However, ψf P• is supported on V (f ), and so f would have to be zero on all of U; but, this implies that | Ch(ψf P• )| = ∅. Now, if we have some non-zero η such that (x, η) ∈ | Ch(ψf P• )|, then by the above argument, (x, η) ∈ | Ch(φf P• )| and, thus, certainly (x, 0) ∈ | Ch(φf P• )|.  The following result helps to illuminate the connection between the Lˆe-Iomdine (-Vogel) cycles and Thom’s af condition (see II.6 and IV.2). The result tells us that adding a large power of a second function, g, to f reduces the critical locus, but expands the fibre of the relative conormal. For a generic choice of g, we can obtain effective lower bounds on the power to which g must be raised (see II.4.3.iii and IV.2.1.ii); however, the g below is completely general. Corollary 5.6 (Thom reduction). Suppose that x ∈ ΣNash f . Assume that f (x) = 0, and that
 ∗ U , we have a second function g : X → C such that g(x) = 0. Suppose that [η] ∈ P Tf| Xreg x  
∗ Then, for all j  2, [η] ∈ P T(f U , and there exists a neighborhood W of x in X +g j )| Xreg

x

such that, in W, Σreg (f + g j ) ⊆ V (g) ∩ Σreg f . Proof. Let g˜ denote a local extension of g to U. Let E denote the exceptional divisor in Blim df˜ TX∗reg U ⊆ U ×Cn+1 ×Pn , and let Ej denote the exceptional divisor in Blim d(f˜+˜gj ) TX∗reg U ⊆ U × Cn+1 × Pn . Let π : U × Cn+1 × Pn → U × Pn denote the projection.


It is trivial to show that if [η] ∈ π(E) x , then, for all j  2, [η] ∈ π(Ej ) x . For [η] ∈ π(E) x if and only if (x, dx f˜, [η]) ∈ E, which means that there is an analytic path α(t) = (p(t), ω(t)) in TX∗reg U such that α(0) = (x, dx f˜), α(t) ∈ TX∗reg U −im df˜ for t 6= 0, and [ω(t)−dp(t) f˜] → [η]. Clearly, since g(x) = 0, we may now choose j large enough so that [ω(t) − dp(t) f˜− j˜ g j−1 (p(t))dp(t) g˜] → [η].

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DAVID B. MASSEY

Moreover, if α(t) ∈ im d(f˜ + g˜j ) for two different j’s, then g˜(p(t)) ≡ 0 and we are finished with the proof of the first statement; otherwise, α(t) 6∈ im d(f˜ + g˜j ) for large j, and we are once again finished.

Therefore, if [η] ∈ π(E) x , then, for all j  2, [η] ∈ π(Ej ) x . Now, one shows easily that x ∈ ΣNash f implies that x ∈ ΣNash (f + g j ) for all j > 2. One now applies Theorem 5.2 twice to conclude the first part of the corollary. It is somewhat lengthier to prove that there exists a neighborhood W of x in X such that, in W, Σreg (f + g j ) ⊆ V (g) ∩ Σreg f , but the idea is simple: we prove it first when X is smooth at x (using an inequality of Lojasiewicz), and then we resolve the singularity in the general case. So, assume that X is smooth at x. Perform an analytic change of coordinates to place ourselves in an open subset of affine space. By an inequality of Lojasiewicz ([Loj], p. 238), there exists a neighborhood W of x and a real θ, with 0 < θ < 1, such that, for p ∈ W, |f (p)|θ 6 | grad f (p)|. We will show how to pick j large depending on the size of θ. Suppose that Σ(f + g j ) 6⊆ V (g) ∩ Σf ; we wish to derive a contradiction. Then, there would exist an analytic path α(t) ∈ X such that α(0) = x and, for t 6= 0, α(t) ∈ Σ(f + g j ) − V (g) ∩ Σf . By Remark 1.7, we know that, near x, Σ(f +g j ) ⊆ V (f +g j ). Thus, along α(t), grad f = −jg j−1 grad g and f = −g j . Hence, along α(t), |g|jθ 6 j|g|j−1 | grad g| and so, as g(α(t)) 6≡ 0, we conclude that |g|jθ−j+1 6 j| grad g|. As g(α(0)) = 0, we would have a contradiction if jθ − j + 1 < 0. Therefore, if j > 1/(1 − θ), we obtain the desired conclusion. Actually, in the smooth case, we have shown the stronger result that there exists a single neighborhood W which can be used for all large j. π e− Now, allow X to be singular at x. Let X → X be a local analytic resolution of the singularities e such that π is an isomorphism over Xreg . As π of X, i.e., a proper map from the smooth space X is proper, π −1 (x) is compact. Applying the smooth case to f ◦ π and g ◦ π at each point of π −1 (x), f of π −1 (x) such that, in W, f and using compactness, we conclude that there is a neighborhood, W, j for all j  2, Σ((f + g ) ◦ π) ⊆ V (g ◦ π); fix a j this large. As in the smooth case, suppose that Σ(f + g j ) 6⊆ V (g) ∩ Σf ; we wish to derive a contradiction. Then, there would exist an analytic path α(t) ∈ X such that α(0) = x and, for t 6= 0, α(t) ∈ e of Γ is a curve Σreg (f + g j ) − V (g) ∩ Σreg f . Let Γ denote the image of α. The proper transform, Γ, −1 which intersects π (x) in a unique point; the existence of such a curve contradicts the choice of f  j and W.

PART III. ISOLATED CRITICAL POINTS

147

Chapter 6. CONTINUOUS FAMILIES OF CONSTRUCTIBLE COMPLEXES. We wish to prove statements of the form: the constancy of certain data in a family implies that some nice geometric facts hold. As the reader should have gathered from the last section, it is very advantageous to use complexes of sheaves for cohomology coefficients; in particular, being able to use perverse coefficients is very desirable. The question arises: what should a family of complexes mean? Let X be a d-dimensional analytic space, let t : X → C be an analytic function, and let F• be a bounded, constructible complex of C-vector spaces. We could say that F• and t form a “nice” family of complexes, since, for all a ∈ C, we can consider the complex F•| −1 on the space X|t−1 (a) . t (a) However, this does yield a satisfactory theory, because there may be absolutely no relation between F•| −1 and F•| −1 for a close to 0. What we need is a notion of continuous families of complexes t (0) t (a) – we want F•| −1 to equal the “limit” of F•| −1 as a approaches 0. Fortunately, such a notion t (0) t (a) already exists; it just is not normally thought of as continuity.

Definition 6.1. Let X, t, and F• be as above. We define the limit of F•a := F•| −1 approaches b, lim

a→b

F•a ,

•

t

[−1] as a (a)

to be the nearby cycles ψt−b F [−1].

We say that the family F•a is continuous at the value b if the comparison map from F•b to ψt−b F• [−1] is an isomorphism, i.e., if the vanishing cycles φt−b F• [−1] = 0. We say that the family F•a is continuous if it is continuous for all values b. We say that the family F•a is continuous at the point x ∈ X if there is an open neighborhood W of x such that the family defined by restricting F• to W is continuous at the value t(x). If P• is a perverse sheaf on X and P•a := P•| −1 [−1] is a continuous family of complexes, then t (a) we say that P•a is a continuous family of perverse sheaves. Remark 6.2. The reason for the shifts by −1 in the families is so that if P• is perverse, and P•a is a continuous family, then each P•a is, in fact, a perverse sheaf (since P•a ∼ = ψt−a P• [−1]). It is not difficult to show that: if the family F•a is continuous at the value b, and, for all a 6= b, each F•a is perverse, then, near the value b, the family F•a is a continuous family of perverse sheaves. For the remainder of this section, we will be using the following additional notation. Let t˜ be an analytic function on U, and let t denote its restriction to X. Let P• be a perverse sheaf on X. Consider the families of spaces, functions, and sheaves given by Xa := X ∩ V (t − a), fa := f|Xa , and P•a := P•|X [−1] (normally, if we are not looking at a specific value for t, we write Xt , ft , a and P•t for these families). Note that, if we have as an hypothesis that P•t is continuous, then the family P•t is actually a family of perverse sheaves.

We will now prove three fundamental lemmas; all of them have trivial proofs, but they are nonetheless extremely useful. The first lemma uses Theorem 4.2 to characterize continuity at a point for families of perverse sheaves.

148

DAVID B. MASSEY

Lemma 6.3. Let x ∈ X. The following are equivalent: i)

The family P•t is continuous at x;

ii)

x 6∈ ΣP• t;

iii) (x, dx t˜) 6∈ | Ch(P• )| for some local extension, t˜, of t to U in a neighborhood of x; and iv)

(x, dx t˜) 6∈ | Ch(P• )| for every local extension, t˜, of t to U in a neighborhood of x.

Proof. The equivalence of i) and ii) follows from their definitions, together with Remark 1.7. The equivalence between ii), iii), and iv) follows immediately from Theorem 4.2.  The next lemma is a necessary step in several proofs. • Lemma 6.4. Suppose that h the ifamily Pt is continuous at t = b, and that the characteristic cycle P of P• is given by α mα TS∗α U . Then, Sα 6⊆ V (t − b) if mα 6= 0.

Proof. This follows immediately from 6.3.



The last of our three lemmas is the stability of continuity result. Lemma 6.5 (Stability of Continuity). Suppose that the family P•t is continuous at x ∈ X. ◦

Then, P•t is continuous at all points near x. In addition, if D is an open disk around the origin ◦

in C, h : D × X → C is an analytic function, hc (z) := h(c, z), and h0 = t, then the family P•hc is continuous at x for all c sufficiently close to 0. Proof. Let t˜ be an extension of t to a neighborhood of x in U, and let Π1 : T ∗ U → U be the cotangent bundle. As T ∗ U is isomorphic to U × Cn+1 , there is a second projection Π2 : T ∗ U → Cn+1 . −1 • • ˜ Now, Π−1 1 (x) ∩ | Ch(P )| and Π2 (dx t) ∩ | Ch(P )| are closed sets. Therefore, the lemma follows immediately from 6.3. 

The following lemma allows us to use intersection-theoretic arguments for families of generalized isolated critical points. Lemma 6.6. Suppose that the family P•t is continuous at x ∈ X. Let b := t(x). Let {Sα } be a Whitney stratification of X withhconnected strata with respect to which P• is constructible. Suppose i P • ∗ that Ch(P ) is given by α mα TSα U . If dimx ΣP• fb 6 0, then there exists an open neighborhood b W of x in U such that:

PART III. ISOLATED CRITICAL POINTS

i)

im df˜ properly intersects

X

h mα Tt∗|

i U in W;

Sα

α

ii)

149

for all y ∈ X ∩ W, V (t − t(y)) properly intersects X

im df˜ ·

h mα Tt∗|

U

i

Sα

α

at (y, dy f˜) in (at most) an isolated point; and iii) for all y ∈ X ∩ W, if a := t(y), then dimy ΣP• fa 6 0 and a

µy (fa ; P•a ) = (−1)d

h

X

im df˜ ·

h mα Tt∗|

U

i

i · V (t − a)

Sα

α

(y,dy f˜)

.

Proof. First, note that we may assume that X = supp P• ; for, otherwise, we would immediately replace X by supp P• . We may refine our stratification so that V (t − b) is a union of strata; for by Lemma 6.4, if Sα ⊆ V (t − b), then mα = 0. This also explains why we may index over all strata in the formulas. Finally, 6.4 implies that V (t − b) does not contain an entire irreducible component of X; thus, dim X0 = d − 1. We use f˜ as a common extension of ft to U, for all t. Proposition 4.10 tells us that µx (fb ; P•b ) =
(−1)d−1 im df˜ · Ch(P•b ) (b,d f˜) . Then, continuity, implies that Ch(P•b ) = Ch(ψt−b [−1]P• ), and b

(∗)

Ch(ψt−b [−1]P• ) = − Ch(ψt−b P• ) = − V (t − b) × Cn+1




h mα Tt∗|

X

·

i U ,

Sα

Sα 6⊆V (t−b)

by Theorem 3.3. Therefore, (†)

 µx (fb ; P•b ) = (−1)d im df˜ ·

V (t − b) × Cn+1




·

X

h mα Tt∗|



im df˜ ·

X α

h mα Tt∗|

U

i

·

Sα

i (x,dx f˜)

Sα

α

(−1)d

U

V (t − b) × Cn+1


 (x,dx f˜)

=

.

Thus,  h X C := (−1)d im df˜ · mα Tt∗| α

U

i

Sα

is a non-negative cycle such that (x, dx f˜) is an isolated point in (or, is not in) C · V (t − b). Statements i) and ii) of the lemma follow immediately. Now, Lemma 6.5 tells us that the family P•t is continuous at all points near x; therefore, if y is close to x and a := t(y), then, by repeating the argument for (∗), we find that Ch(P•a ) = − Ch(ψt−a P• ) = − V (t − a) × Cn+1




·

X α

h mα Tt∗|

Sα

U

i

150

DAVID B. MASSEY

and we know that the intersection of this cycle with im df˜ is (at most) zero-dimensional at (y, dy f˜) (since C ∩ V (t − b) is (at most) zero-dimensional at x). By considering f˜ an extension of fa and applying Theorem 4.2, we conclude that dimy ΣP• fa 6 0. a Finally, now that we know that P•t is continuous at y and that dimy ΣP• fa 6 0, we may argue a as we did at x to conclude that (†) holds with x replaced by y and b replaced by a. This proves iii). 

We can now prove an additivity/upper-semicontinuity result. We prove this result for a more general type of family of perverse sheaves; instead of parametrizing by the values of a function, we parametrize implicitly. We will need this more general perspective in Theorem 6.10. Theorem 6.7. Suppose that the family P•t is continuous at x ∈ X. Let b := t(x), and suppose that dimx ΣP• fb 6 0. b

◦

◦

Let D be an open disk around the origin in C, let h : D × X → C be an analytic function, for ◦

all c ∈ D, let hc (z) := h(c, z), let c P• := P•|V (h −b) [−1] and c f := f|V (hc −b) . Suppose that h0 = t. c Then, there exists an open neighborhood W of x in U such that, for all small c, for all y ∈ V (hc − b) ∩ W, dimy Σc P• c f 6 0. Moreover, for fixed c close to 0, there are a finite number of points y ∈ V (hc − b) ∩ W such that µy (c f ; c P• ) 6= 0 and X µx (b f ; b P• ) = µy (c f ; c P• ). y∈V (hc −b)∩W

In particular, for all small c, for all y ∈ V (hc − b) ∩ W, µy (c f ; c P• ) 6 µx (b f ; b P• ). Proof. We continue to let P•c = P•|V (t−c) [−1] and fc = f|V (t−c) . Note that, if we let h(w, z) := t(z) − w, then the statement of the theorem would reduce to a statement about the ordinary families P•c and fc . Moreover, this statement about the families P•c and fc follows immediately from Lemma 6.6. We wish to see that this apparently weak form of the theorem actually implies the stronger form. ◦

◦

◦

˜ : D × U → C denote a local extension of h to D × U. We use Shrinking D and U if necessary, let h ◦

w as our coordinate on D. Note that replacing h(w, z) by h(w2 , z) does not change the statement ˜ vanishes on C × {0}. of the theorem. Therefore, we can, and will, assume that d(0,x) h ◦

Let p˜ : D × U → U denote the projection, and let p := p˜|◦ ◦

D×X

. Let Q• := p∗ P• [1]; as P• is ◦

perverse, so is Q• . Let Y := (D × X) ∩ V (h − b), and let w b : Y → D denote the projection. Let R• := Q•|Y [−1]. Let fˆ : Y → C be given by fˆ(w, z) := f (z). As we already know that the theorem is true for ordinary families of functions, we wish to apply it to the family of functions fˆwˆ and the family of sheaves R•wˆ ; this would clearly prove the desired result. Thus, we need to prove two things: that R• is perverse near (0, x), and that the family R•wˆ is continuous at (0, x).

PART III. ISOLATED CRITICAL POINTS

151

Let {Sα } be a Whitney stratification, with connected strata, of X with respect to which P• is constructible. Refininghthe stratification if necessary, assume that V (t − b) is a union of i P • ∗ strata. Let Ch(P ) = mα TSα U . Clearly, Q• is constructible with respect to the Whitney ◦

◦

stratification {D × Sα }, and the characteristic cycle of Q• in T ∗ (D × U) is given by Ch(Q• ) = h i ◦ P − mα T ◦∗ (D × U) . D×Sα

Note that, for all (z, η) ∈ T ∗ U, (z, η) ∈ TS∗α U if and only if (0, z, η ◦ d(0,z) p) ∈ T ◦∗

◦

(D × U).

D×Sα

˜ vanishes on C × {0} and that h0 = t, we know that d(0,x) h ˜ = As we are assuming that d(0,x) h ◦

˜ ∈ T ◦∗ dx t˜ ◦ d(0,z) p˜. Thus, (x, dx t˜) ∈ TS∗α U if and only if (0, x, d(0,x) h)

(D × U). Therefore,

D×Sα

˜ ∈ | Ch(Q• )|. As we are assuming that the family (x, dx t˜) ∈ | Ch(P• )| if and only if (0, x, d(0,x) h) • Pt is continuous at x, we may apply Lemma 6.3 to conclude that (x, dx t˜) 6∈ | Ch(P• )| and, hence, ˜ 6∈ | Ch(Q• )|. It follows that, for all (w, z) near (0, x), (w, z, d(w,z) h) ˜ 6∈ | Ch(Q• )| and (0, x, d(0,x) h) • that the family Qh is continuous at (0, x); that is, there exists an open neighborhood, Ω × W, of ◦

(0, x) in D × U, in which φh−b [−1]Q• = 0 and such that, if (w, z) ∈ Ω × W and mα 6= 0, then ◦ ◦
˜ 6∈ T ◦∗ D × U . For the remainder of the proof, we assume that D and U have (w, z, d(w,z) h) D×Sα

been rechosen to be small enough to use for Ω and W. As φh−b [−1]Q• = 0, R• ∼ = ψh−b [−1]Q• is a perverse sheaf on Y . It remains for us to show that • the family Rwˆ is continuous at (0, x). Of course, we appeal to Lemma 6.3 again – we need to show that (0, x, d(0,x) w) 6∈ | Ch(R• )|. Now, | Ch(R• )| = | Ch(ψh−b [−1]Q• )|, and we wish to use Theorem 3.3 to describe this character◦
˜ 6∈ T ◦∗ istic variety. If (w, z) ∈ Ω × W and mα 6= 0, then (w, z, d(w,z) h) D × U ; thus, if mα 6= 0, D×Sα

◦

then h has no critical points when restricted to D × Sα , and, using the notation of 3.2 and 3.3, " # ◦ ◦
X
∗ ∗ Th−b,Q• D × U = mα Th| D×U . α

◦ D×Sα

Now, using Theorem 3.3, we find that [
| Ch(R• )| = V (h − b) × Cn+2 ∩ Th∗| mα 6=0

◦ D×Sα

◦
D×U .

We will be finished if we can show that, if mα 6= 0, then (0, x, d(0,x) w) 6∈ Th∗| Fix an Sα for which mα 6= 0. Suppose that (0, x, η) ∈ Th∗|

◦ ◦ D×Sα

◦ D×Sα

◦
D×U .




D × U . Then, there exists a

◦
sequence (wi , zi , ηi ) ∈ Th∗| D × U such that (wi , zi , ηi ) → (0, x, η). Thus, ηi (C × Tzi Sα ) ∩ ◦ D×Sα
˜ = 0. By taking a subsequence, if necessary, we may assume that Tz Sα converges ker d(wi ,zi ) h i ˜ → ker d(0,x) h ˜ = to some T in the appropriate Grassmanian. Now, we know that ker d(wi ,zi ) h ∗ C × ker dx t˜. As (x, dx t˜) 6∈ TSα U, C × ker dx t˜ transversely intersects C × T . Therefore, (C × Tzi Sα ) ∩ ˜ → (C × T ) ∩ (C × ker dx t˜) , and so C × {0} ⊆ ker η. However, ker d(0,x) w = {0} × Cn+1 , ker d(wi ,zi ) h and we are finished. 

152

DAVID B. MASSEY

We would like to translate Theorem 6.7 into a statement about Milnor fibres and the constant sheaf. First, though, it will be convenient to prove a lemma.
Lemma 6.8. Let x ∈ X, and let b := t(x). Suppose that dimx V (t − b) ∩ ΣC t 6 0. Fix an integer e k (Ft,x ; C) = 0, then the family kP•t is continuous at x. In addition, if H e k (Ft,x ; C) = 0 k. If H e k−1 (Ft,x ; C) = 0, then kP• ∼ and H = µH 0 (C• [k]) near x. b

Xb


Proof. By Remark 1.7, the assumption that dimx V (t − b) ∩ ΣC t 6 0 is equivalent to dimx ΣC t 6 0 and, by Theorem 4.6, this is equivalent to dimx ΣjP• t 6 0 for all j. Thus, supp φt−b [−1]kP• ⊆ e k (Ft,x ; C) = 0 implies that, in fact, {x} near x. We claim that the added assumption that H φt−b [−1]kP• = 0 near x. For, near x, supp φt−b [−1]C•X [k + 1] ⊆ {x}, and so φt−b [−1]kP• = φt−b [−1]µH 0 (C•X [k + 1]) ∼ = µH 0 (φt−b [−1]C•X [k + 1]) ∼ = H0 (φt−b [−1]C•X [k + 1]). Near x, φt−b [−1]C•X [k + 1] is supported at, at most, the point x and, hence, φt−b [−1]kP• = 0 e k (Ft,x ; C) = 0. This proves the provided that H0 (φt−b [−1]C•X [k + 1])x = 0, i.e., provided that H first claim in the lemma. Now, if the family kP•t is continuous at x, then, near x, P•b = kP•|V (t−b) [−1] ∼ = ψt−b [−1]µH 0 (C•X [k + 1]) ∼ = µH 0 (ψt−b [−1]C•X [k + 1]),

k

e k (Ft,x ; C) = 0 and H e k−1 (Ft,x ; C) = 0, then there is an isomorphism (in and we claim that, if H µ 0 • the derived category) H (ψt−b [−1]CX [k + 1]) ∼ = µH 0 (C•Xb [k]). To see this, consider the canonical distinguished triangle [1]

C•Xb [k] → ψt−b [−1]C•X [k + 1] → φt−b [−1]C•X [k + 1] −→ C•Xb [k]. A portion of the long exact sequence (in the category of perverse sheaves) resulting from applying perverse cohomology is given by H −1 (φt−b [−1]C•X [k + 1]) → pH 0 (C•Xb [k]) → µH 0 (ψt−b [−1]C•X [k + 1]) → µH 0 (φt−b [−1]C•X [k + 1]).

p

We would be finished if we knew that the terms on both ends of the above were zero. However, since φt−b [−1]C•X [k + 1] has no support other than x (near x), we proceed as we did above to show e k−1 (Ft,x ; C) that pH −1 (φt−b [−1]C•X [k + 1]) and pH 0 (φt−b [−1]C•X [k + 1]) are zero precisely when H k e and H (Ft,x ; C) are zero. 

Theorem 6.9. Let x ∈ X and let b := t(x). Suppose that x 6∈ ΣC t, and that dimx ΣC (fb ) 6 0. Then, there exists a neighborhood, W, of x in X such that, for all a near b, there are a finite e ∗ (Ff ,y ; C) 6= 0; moreover, for all integers, k, number of points y ∈ W ∩ V (t − a) for which H a ˜bk−1 (Ff ,y ) = µy (fa ; kP• ), and a a ˜bk−1 (Ff ,x ) = b

X y∈W∩V (t−a)

˜bk−1 (Ff ,y ), a

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153

e ∗ () and ˜b∗ () are as in Remark 4.4. where H Proof. Let v := fb (x). Fix an integer k. By the lemma, the family kP•t is continuous at x and kP•b ∼ = µH 0 (C•Xb [k]) near x. Thus, φfb −v [−1]kP•b ∼ = µH 0 (φfb −v [−1]C•Xb [k]). = φfb −v [−1]µH 0 (C•Xb [k]) ∼ We are assuming that dimx ΣC (fb ) 6 0; this is equivalent to: supp φfb −v [−1]C•Xb [k] ⊆ {x} near x, it follows from the above line and Theorem 4.6 that dimx ΣkP• fb 6 0 and that b

(‡)

µx (fb ; kP•b ) = dim H 0 (φfb −v [−1]C•Xb [k])x = ˜bk−1 (Ffb ,x ).

Applying Theorem 6.7, we find that there exists an open neighborhood W 0 of x in U such that, for all y ∈ W 0 , if a := t(y), then (∗) dimy ΣkP• fa 6 0, and, for fixed a close to b, there are a finite a

number of points y ∈ W 0 ∩ V (t − a) such that µy (fa ; kP•a ) 6= 0 and X (†) µx (fb ; kP•b ) = µy (fa ; kP•a ). y∈W∩V (t−a)

Now, using the above argument for all k with 0 6 k 6 d − 1 and intersecting the resulting W 0 -neighborhoods, we obtain an open neighborhood W of x such that (∗) and (†) hold for all such k. We claim that, if a is close to b, then W ∩ ΣC fa consists of isolated points, i.e., the points e ∗ (Ff ,y ; C) 6= 0 are isolated. y ∈ W ∩ V (t − a) for which H a If a = b, then there is nothing to show. So, assume that a 6= b, and assume that we are working in W throughout. By Remark 1.7, t satisfies the hypotheses of Lemma 6.8 at t = a; hence, for all k, not only is kP•t continuous at t = a, but we also know that kP•a ∼ = µH 0 (C•Xa [k]). By Theorem S 4.6, ΣC fa = ΣkP•a fa , where the union is over k where 0 6 k 6 dim Xa . As dim Xa 6 d − 1, the claim follows from (∗) and the definition of W. Now that we know that kP•t is continuous at t = a and that W ∩ ΣC fa consists of isolated points, we may use the argument that produced (‡) to conclude that µy (fa ; kP•a ) = ˜bk−1 (Ffa ,y ). The theorem follows from this, (‡), (∗), and (†). 

We want to prove a result which generalizes that of Lˆe and Saito [L-S]. We need to make the assumption that the Milnor number is constant along a curve that is embedded in X. Hence, it will be convenient to use a local section of t : X → C at a point x ∈ X; that is, an analytic function r from an open neighborhood, V, of t(x) in C into X such that r(t(x)) = x and t ◦ r equals the inclusion morphism of V into C. Note that existence of such a local section implies that x 6∈ Σalg t; in particular, V (t˜ − t˜(x)) is smooth at x. Theorem 6.10. Suppose that the family P•t is continuous at x ∈ X. Let b := t(x), and let v := fb (x). Let r : V → X be a local section of t at x, and let C := im r. Assume that C ⊆ V (f −v), that dimx ΣP• fb 6 0, and that, for all a close to b, the Milnor number µr(a) (fa ; P•a ) is non-zero b and is independent of a; denote this common value by µ. Then, C is smooth at x, V (t˜ − b) transversely intersects C in U at
x , and there exists
• a ∼ neighborhood, W, of x in X such that W ∩ ΣP• f ⊆ C and φf −v [−1]P• | CµW∩C [1] . In = W∩C

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particular, if we let tˆ denote the restriction of t to V (f − v), then the family φf −v [−1]P• continuous at x.




tˆ

is

If, in addition to the other hypotheses, we assume that x ∈ Σalg f , then the two families

ψf −v [−1]P• tˆ and P•|V (f −v) [−1] tˆ are continuous at x. (Though P•|V (f −v) [−1] need not be perverse.) Proof. Let us first prove that the last statement of the theorem follows easily from the first portion of the theorem. So, assume that φtˆ
−b [−1]φf −v [−1]P• = 0 near x. Therefore, working near x, we have that φtˆ−b [−1] P•|V (f −v) [−1] ∼ = φtˆ−b [−1]ψf −v [−1]P• , and we need to show that this is the zero-sheaf. By Lemma 6.3, what we need to show is that (x, dx t˜) 6∈ | Ch(ψf −v [−1]P• )| = | Ch(ψf −v P• )|. As we are assuming that x ∈ Σalg f , we may apply Corollary 5.5 to find that it suffices to show that (x, dx t˜) 6∈ | Ch(φf −v P• )| = | Ch(φf −v [−1]P• )|. By 6.3, this is equivalent to φtˆ−b [−1]φf −v [−1]P• = 0 near x, which we already know to be true. This proves the last statement of the theorem. Before proceeding with the remainder of the proof, we wish to make some simplifying assumptions. As x 6∈ Σalg t, we may certainly perform an analytic change of coordinates in U to reduce ourselves to the case where t is simply the restriction to X of a linear form t˜. Moreover, it is notational convenient to assume, without loss of generality, that x = 0 and that b and v are both zero. Let {Sα } be a Whitney stratification of X with connected strata with respect to which P• is • constructible h andisuch that V (t) and V (f ) are each unions of strata. Suppose that Ch(P ) is given P by α mα TS∗α U . e := {(r(a), dr(a) f˜) | a ∈ V}; the projection, ρ, onto the first component induces an Let C e to C. By Lemma 6.6, the assumption that the Milnor number, µr(a) (fa ; kP•a ), isomorphism from C is independent of a is equivalent to: f of (0, d0 f˜) in T ∗ U in which C e equals (†) there exists an open neighborhood W im df˜ ∩

[ mα 6=0

Tt∗|

U

Sα

e is a smooth curve at (0, d0 f˜) such that (0, d0 f˜) 6∈ Σ(t ◦ ρ| ). and C e C It follows immediately that C is smooth at 0 and 0 6∈ Σ(t|C ). We need to show that (†) implies

• µ ∼ f that W ∩ ΣP• f ⊆ C and φf [−1]P• | = CW∩C [1] , where W := ρ(W). W∩C S e As TS∗α U ⊆ Tt∗| U, we have that | Ch(P• )| ⊆ mα 6=0 Tt∗| U and, thus, im df˜ ∩ | Ch(P• )| ⊆ C Sα Sα f It follows from Theorem 4.2 that W ∩ ΣP• f ⊆ C. inside W.

• µ ∼ It remains for us to show that φf [−1]P• | = CW∩C [1] . As φf [−1]P• is perverse and W∩C we have just shown that the support of φf [−1]P• , near 0, is a smooth curve, it follows from the work of MacPherson and Vilonen in [M-V] that what we need to show is that, for a generic linear form L, Q• := φL [−1]φf [−1]P• = 0 near 0. By definition of the characteristic cycle (and since 0 ∗ is an isolated point in the support of Q• ), this is the same as showing that the coefficient of T{0} U

PART III. ISOLATED CRITICAL POINTS

155

in Ch(φf [−1]P• ) equals zero. To show this, we will appeal to Theorem 3.4 and use the notation from there. We need to show that m0 (φf [−1]P• ) = 0. By 3.4, if suffices to show that m0 (P• ) = 0 and Γ1f,L (Sα ) = ∅ near 0, for all Sα which are not contained in V (f ) and for which mα 6= 0 (where L still denotes a generic linear form). As P•t is continuous at 0, Lemma 6.3 tells us that m0 (P• ) = 0. e Now, near 0, if y ∈ Γ1f,L (Sα ) − {0}, then (y, dy f˜) ∈ TL∗| U. If we knew that, near (0, d0 f˜), C Sα S equals im df˜ ∩ T ∗ U, then we would be finished – for C is contained in V (f ) while Sα mα 6=0

L|S

α

is not; hence, Γ1f,L (Sα ) would have to be empty near 0. e equals im df˜ ∩ Looking back at (†), we see that what we still need to show is that if C ∗ ˜ mα 6=0 Tt|Sα U near (0, d0 f ), then the same statement holds with t replaced by a generic linear form L. We accomplish this by perturbing t until it is generic, and by then showing that this perturbed t satisfies the hypotheses of the theorem. As C is smooth and transversely intersected by V (t˜) at 0, by performing an analytic change of coordinates, we may assume that t˜ = z0 , that C is the z0 -axis, and that r(a) = (a, 0). Since the

S

◦

set of linear forms for which 3.4 holds is generic, there exists an open disk, D, around the origin ◦ ◦ ˜ : (D × U, D × {0}) → (C, 0) such that h ˜ 0 (z) := h(0, ˜ z) = t˜(z) and in C and an analytic family h ˜ c (z) := h(c, ˜ z) is a linear form for which Theorem 3.4 holds. such that, for all small non-zero c, h ˜| Let h := h . ◦ D×X

As the family P•t is continuous at 0, Lemma 6.5 tells us that P•hc is continuous at 0 for all small c. As we are now considering these two different families with the same underlying sheaf, the expression P•a for a fixed value of a is ambiguous, and we need to adopt some new notation. We continue to let P•a := P•|V (t−a) [−1] and fa := f|V (t−a) , and let c P•a := P•|V (h −a) [−1] and c c fa := f|V (hc −a) . ˜ 0 ) = V (z0 ) transversely intersects C at 0 in U, for all small c, V (hc ) transversely Since V (h intersects C at 0 in U. Hence, for all small c, there exists a local section rc (a) for hc at 0 such that im rc ⊆ C. We claim that, for all small c: i)

dim0 Σc P•0 (c f0 ) 6 0 and µ0 (c f0 ; c P•0 ) 6 µ0 (0 f0 ; 0 P•0 ) = µ0 (f0 ; P•0 );

ii)

for all small a, dimrc (a) Σc P•a (c fa ) 6 0 and µrc (a) (c fa ; c P•a ) 6 µ0 (0 f0 ; 0 P•0 ); and

iii) for all small a 6= 0, µrc (a) (c fa ; c P•a ) = µrc (a) (fz0 (rc (a)) ; P•z0 (rc (a)) ). Note that proving i), ii), and iii) would complete the proof of the theorem, for they imply that the hypotheses of the theorem hold with t replaced by hc for all small c. To be precise, we would know that P•hc is continuous at 0, dim0 Σc P•0 (c f0 ) 6 0, and, for all small a, µrc (a) (c fa ; c P•a ) = µ0 (c f0 ; c P•0 ); this last equality follows from i), ii), and iii), since, for all small a 6= 0, we would have µ = µrc (a) (fz0 (rc (a)) ; P•z0 (rc (a)) ) = µrc (a) (c fa ; c P•a ) 6 µ0 (c f0 ; c P•0 ) 6 µ0 (f0 ; P•0 ) = µ. However, i), ii) and iii) are easy to prove. i) and ii) follow immediately from Theorem 6.7, and ˜c −h ˜ c (rc (a))) are iii) follows simply from the fact that, for all small a 6= 0, V (z0 −z0 (rc (a))) and V (h smooth and transversely intersect all strata of any analytic stratification of X in a neighborhood of (0, 0). This concludes the proof. 

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Corollary 6.11. Suppose that the family P•t is continuous at x ∈ X. Let b := t(x), and let v := fb (x). Let r : V → X be a local section of t at x, and let C := im r. Assume that C ⊆ V (f −v), that dimx ΣP• fb 6 0, and that, for all a close to b, the Milnor number µr(a) (fa ; P•a ) is non-zero b h i P and is independent of a. Let Ch(P• ) = α mα TS∗α U , where {Sα } is a collection of connected analytic submanifolds of U. Then, C is smooth at x, and there exists a neighborhood, W, of x in X such that, for all Sα for which Sα 6⊆ V (f − v) and mα 6= 0:  

W ∩ Σ f|(S ) ⊆ C and, if x ∈ ΣNash (f|S ), then the pair Sα reg , C satisfies Thom’s af α reg α condition at x. Proof. One applies Theorem 6.10. The fact that W ∩ Σ f|(S


α )reg

⊆ C, for all Sα for which mα 6= 0

follows from Theorem 4.2, since W ∩ΣP• f ⊆ C. The remainder of the corollary follows by applying Corollary 5.4, where one uses C for the submanifold N . 

Just as we used perverse cohomology to translate Theorem 6.7 into a statement about the constant sheaf in Theorem 6.9, we can use perverse cohomology to translate Corollary 6.11. We will use the notation and results from Proposition 4.3 and Remark 4.4. Corollary 6.12. Let b := t(x), and let v := fb (x). Suppose that x 6∈ ΣC t. Suppose, further, that, dimx ΣC (fb ) 6 0. Let r : V → X be a local section of t at x, and let C := im r. Assume that C ⊆ V (f − v). Let Sα be a visible stratum of X of dimension dα , not contained in V (f − v), and let j be an integer such that ˜bj−1 (Lα ) 6= 0. Let Y := Sα and let k := dα + j − 1. In particular, Y could be any irreducible component of X, j could be zero, and k would be (dim Y ) − 1. Suppose that the reduced Betti number ˜bk−1 (Ffa ,r(a) ) is independent of a for all small a, and that either a)

x ∈ ΣNash (f|Y ); or that

b)

x 6∈ Σcnr (f|Y ), C is smooth at x, and (Yreg , C) satisfies Whitney’s condition a) at x.

Then, C is smooth at x, and the pair (Yreg , C) satisfies the af condition at x. Moreover, in case a), ˜bk−1 (Ffa ,r(a) ) 6= 0, C is transversely intersected by V (t˜ − b) at x, and Σ(f|Yreg ) ⊆ C near x. In addition, if x ∈ Σalg f and, for all small a and for all i, ˜bi (Ffa ,r(a) ) is independent of a, then x 6∈ ΣC (t|V (f −v) ). Proof. We will dispose of case b) first. Suppose that x 6∈ Σcnr (f|Y ), C is smooth at x, and (Yreg , C) ◦

satisfies Whitney’s condition a) at x. Let Y := Yreg − Σ(f|Yreg ).

PART III. ISOLATED CRITICAL POINTS

157

Suppose that we have an analytic path (x(t), ηt ) ∈ Tf∗| U, where (x(0), η0 ) = (x, η) and, for ◦ Y

t 6= 0, (x(t), ηt ) ∈ Tf∗| U. We wish to show that (x, η) ∈ TC∗ U. ◦ Y

◦

For t 6= 0, x(t) ∈ Y , and thus ηt can be written uniquely as ηt = ωt + λt dxt f˜, where ωt ∈ T ◦∗ U Y

and λt ∈ C. As we saw in Theorem 5.2, this implies that either |λt | → ∞ or that λt → λ0 , for some λ0 ∈ C. If |λt | → ∞, then ληtt → 0 and, therefore, − ωλtt → dx f˜; however, this implies that x ∈ Σcnr (f|Y ), contrary to our assumption. Thus, we must have that λt → λ0 . It follows at once that ωt converges to some ω0 . By Whitney’s condition a), (x, ω0 ) ∈ TC∗ U. As C ⊆ V (f − v), (x, dx f˜) ∈ TC∗ U. Hence, (x, η) ∈ TC∗ U and we have finished with case b). We must now prove the results in case a). The main step is to prove that ˜bk−1 (Ffb ,x ) 6= 0. We may refine our stratification, if necessary, so that V (t − b) is a union of strata. By the first part of Theorem 6.9, ˜bk−1 (Ffb ,x ) = µx (fb ; kP•b ). Hence, by Lemma 6.6.iii, ˜bk−1 (Ffb ,x ) would be
unequal to zero if we knew, for some Sβ for which mβ kP• 6= 0, that (x, dx f˜) ∈ Tt∗| U. However, Sβ

our fixed Sα is such a stratum, for bk+1−dα (Nα , Lα ) 6= 0 and, since x ∈ ΣNash (f|Y ), x ∈ Σcnr (f|Y ) and so (x, dx f˜) ∈ TS∗α U ⊆ Tt∗| U. Sα

Now, applying the first part of 6.9 again, we have that µr(a) (fa ; kP•a ) = ˜bk−1 (Ffa ,r(a) ) for all small a. The conclusions in case a) follow from Corollary 6.11. We must still demonstrate the last statement of corollary. Suppose that if ˜bi (Ffa ,r(a) ) is independent of a for all small a and for all i. Let tˆ denote the restriction of t to V (f − v). We will work in a small neighborhood of x. Applying the last two sentences of Theorem 6.10, we find that φtˆ−b [−1]φf −v [−1]iP• = 0 and φtˆ−b [−1]ψf −v [−1]iP• = 0 for all i. Commuting nearby and vanishing cycles with perverse cohomology, we find that
H 0 φtˆ−b [−1]φf −v [−1]C•X [i + 1] = 0

µ

and


H 0 φtˆ−b [−1]ψf −v [−1]C•X [i + 1] = 0,

µ

for all i. Therefore, φtˆ−b [−1]φf −v [−1]C•X = 0 and φtˆ−b [−1]ψf −v [−1]C•X = 0. It follows from the existence of the distinguished triangle (relating nearby cycles, vanishing cycles, and restriction to the hypersurface) that φtˆ−b [−1]C•V (f −v) [−1] = 0. This proves the last statement of the corollary. 

Remark 6.13. If X is a connected l.c.i., then each Lα has (possibly) non-zero cohomology concentrated in middle degree. Hence, for each visible Sα , ˜bj−1 (Lα ) 6= 0 only when j = codimX Sα ; this corresponds to k = d − 1. Therefore, the degree d − 2 reduced Betti number of Ffa ,r(a) controls the af condition between all visible strata and C.

Corollary 6.14. Let W be an analytic subset of an open subset of Cn . Let Z be a d-dimensional ◦

irreducible component of W . Let X := D × W be the product of an open disk about the origin with ◦

◦

W , and let Y := D × Z. Let f : (X, D × {0}) → (C, 0) be an analytic function such that f|Y 6≡ 0, and let ft (z) := f (t, z).

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DAVID B. MASSEY

Suppose that 0 is an isolated point of ΣC (f0 ), and that the reduced Betti number ˜bd−1 (Ffa ,(a,0) ) is independent of a for all small a. ◦

If either a) 0 ∈ ΣNash (f|Y ) or af condition at 0.

b) 0 6∈ Σcnr (f|Y ), then the pair (Yreg , D×{0}) satisfies Thom’s ◦

Moreover, in case a), ˜bd−1 (Ffa ,(a,0) ) 6= 0 and, near 0, Σ(f|Yreg ) ⊆ D × {0}. Remark 6.15. A question naturally arises: how effective is the criterion appearing in Corollary 6.14 that ˜bd−1 (Ffa ,(a,0) ) is independent of a? By Proposition 4.10, if {Rβ } is a Whitney stratification of W , then (using the notation from 4.10) ˜bd−1 (Ff ,(a,0) ) = a  X

 ˜bd−1 (L{0} ) + ˜bd−1−d (Lβ ) Γ1 (Rβ ) · V (fa ) − Γ1 (Rβ ) · V (L) , β fa ,L fa ,L 0 0 Rβ visible dim Rβ >1

where L{0} denotes the complex link of the origin. As the Betti numbers do not vary with a,

˜bd−1 (Ff ,(a,0) ) will be independent of a provided that Γ1 (Rβ ) · V (fa ) − Γ1 (Rβ ) · V (L) a fa ,L fa ,L 0 0 is independent of a for all visible strata, Rβ , of dimension at least one. This condition is certainly very manageable to check if the dimension of the singular set of X at the origin is zero or one.

The final statement of Corollary 6.12 has as its conclusion that the constant sheaf on X∩V (f −v), parametrized by the restriction of t, is continuous at x; this is useful for inductive arguments, since the hypothesis on the ambient space in Corollary 6.12 is that the constant sheaf, parametrized by t, should be continuous at x. For instance, we can prove the following corollary. Corollary 6.16. Suppose that f 1 , . . . , f k are analytic functions from U into C which define a sequence of local complete intersections at the origin, i.e., are such that, for all i with 1 6 i 6 k, the space X n+1−i := V (f 1 , . . . , f i ) is a local complete intersection of dimension n + 1 − i at the origin. If, for all i, Xtn+1−i has an isolated singularity at the origin and the restrictions fti+1 : Xtn+1−i → C are such that dim0 Σcan fti+1 6 0 and have Milnor numbers (in the sense of [Loo]) which are

n+1−(k−1) independent of t, then Σ f| n+1−(k−1) ⊆ C × {0} and the pair Xreg , C × {0} satisfies the Xreg

af k condition at the origin. Proof. Recall that C•X [d] is a perverse sheaf if X is a local complete intersection. The “ordinary” Milnor number of fti+1 at the origin is equal to µ0 (fti+1 ; C•X n+1−i [n − i]). Hence, using Proposition t

4.10.ii, this Milnor number is equal to the degree n − i − 1 (the “middle” degree) reduced Betti number of the Milnor fibre of fti+1 at the origin – the only possible non-zero reduced Betti number. Now, use Corollary 6.12 and induct; the inductive requirement on the Milnor fibre of z0 follows from the last statement of the corollary.  Remark 6.17. In [G-K], Gaffney and Kleiman deal with families of local complete intersections as above. In this setting, they obtain the result of Corollary 6.16 using multiplicities of modules.

Part IV. NON-ISOLATED CRITICAL POINTS OF FUNCTIONS ON SINGULAR SPACES

Chapter 0. INTRODUCTION In Part II, we generalized many results from the study of isolated critical points to the case of non-isolated critical points. We accomplished this by developing the Lˆe cycles and Lˆe numbers of a non-isolated critical point; the Lˆe numbers are a generalization of the Milnor number of an isolated critical point. However, throughout Part II, the domains of our analytic functions were required to be open subsets of affine space. In Part III, we investigated what an “isolated critical point” of a function on an arbitrarily singular space should mean, and we developed a theory of Milnor numbers. In Part IV, we wish to use our construction of the Lˆe cycles and numbers as a guide in order to decide how to generalize our work in Part III to the non-isolated case. We will produce Lˆe-Vogel cycles and numbers, and use them to generalize many previous results.

159

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ˆ Chapter 1. LE-VOGEL CYCLES We will adopt some notation that we will use throughout Part IV; much of this notation was used in Part III. We let U be an open subset of Cn+1 , and let X be a (not necessarily purely) d-dimensional analytic subset of U. Let f˜ : U → C be an analytic function, and let f := f˜|X . Let {Sα } denote a Whitney stratification, with connected strata, of X. We let dα := dim Sα . As in Part III, Chapter 3, we let dˆv := 1 + dim V (f − v), and will usually denote dˆ0 by simply ˆ If we work locally, or assume that X is pure-dimensional, and require f not to vanish on a d. component of X, then dˆ will have attain its “expected” value of d. We use z0 , . . . , zn as coordinates on U. We let η : T ∗ U → U denote the cotangent bundle, and we identify the cotangent space T ∗ U with U × Cn+1 by using dz0 , . . . , dzn as a basis. We use w0 , . . . , wn as coordinates for the cotangent vectors, i.e., a cotangent vector is w0 dz0 + · · · + wn dzn . ˆ is a linear change of coordinates applied to z, then we let w b denote cotangent coordinates with If z ˆ := Az, then w = At w. b respect to the new basis dˆ z0 , . . . , dˆ zn , i.e., if A is in Gln+1 (C) and z ∗ n+1 ∼ along (n + 1)-tuples. This blow-up will We shall be blowing-up subspaces of T U = U × C lie in U × Cn+1 × Pn ; we let π : U × Cn+1 × Pn → U × Pn , τ : U × Cn+1 × Pn → U × Cn+1 , and ν : U × Pn → U denote the projections. Note that η ◦ τ = ν ◦ π. We remind the reader that we slightly modified the definition of the reduced Betti number ˜bj () in III.4.4, so that the empty set has a non-zero reduced Betti number precisely in degree −1. Recall, from Part III, that we defined kP• := µH 0 (C•X [k + 1]). In Part IV, a different shift will be of more use to us. Thus, we define kQ• := d−k−1P• = µH 0 (C•X [dim X − k]). We wish to produce Lˆe-Vogel (LˆeVo) cycles in much the same way that we produce the Lˆe cycles: by taking the Vogel cycles of the Jacobian tuple. We immediately run into the problem of what ideal we should use. Theorem III.3.10 provides us with a clue: the vanishing cycles of the constant sheaf along f are integrally related to blowing-up im df˜ in TS∗α U for various strata Sα . Hence, we make the following definition.

Definition 1.1. The conormal Jacobian tuple of f˜ with respect to z (and the corresponding choice of w) is given by ! ∂ f˜ ∂ f˜ n+1 ∗ ˜ Jz (f ) := w0 − , . . . , wn − ∈ (OT ∗ U ) . ∂z0 ∂zn Thus, im df˜ is the zero-locus of Jz∗ (f˜). We shall normally be blowing-up TS∗α U along the restriction of Jz∗ (f˜) to (OT ∗

Sα

U

)n+1 ; we will

follow the standard practice of simply writing Blim df˜ TS∗α U.

In Part I, we defined the gap cycles and Vogel cycles with respect to given cycle M ; we developed the theory in this generality precisely so that we could now make the appropriate choice(s) for M . Our choice of M is guided by our work in Part III; in particular, we use III.4.3.

PART IV. NON-ISOLATED CRITICAL POINTS

161

Definition 1.2. Let kM be the cycle in T ∗ U given by h i
X k ∗ U . ˜b M := (−1)d Ch kQ• = (L ) T α d−k−1−dα Sα α

Note that kM will be zero unless 0 6 k 6 d. Note also that if X is purely d-dimensional, then the final expression for kM above can be written more simply as i h X ˜b (Lα ) TS∗α U , (dim Lα )−k α

where we mean that dim Lα = −1 if Lα = ∅. We define kmα := ˜bd−k−1−dα (Lα ), and so kM =

P

α

h i mα TS∗α U .

k

It is also convenient to define kX, the image in X of kM , i.e., [ k X := |η∗ (kM )| = Sα . ˜ bd−k−1−d (Lα )6=0 α

We can now define polar and Lˆe-Vogel cycles in the cotangent space by using the theory of gap and Vogel cycles developed in Part I. We can then push-forward these “conormal” Lˆe-Vogel cycles to arrive at the Lˆe-Vogel cycles in X. We will get one set of Lˆe-Vogel cycles for each kM ; note that each kM > 0 and that all the components of kM have dimension n+1.

ˇ i , is Definition 1.3. The i-th k-shifted conormal polar cycle of f˜ with respect to z in T ∗ U, k Γ f˜,z k ∗ ˜ k bi defined to be Π ( M ), the i-th inductive gap cycle of J ( f ) with respect to M . z Jz∗ (f˜) ˇ i , is defined The i-th k-shifted conormal Lˆe-Vogel (LˆeVo) cycle of f˜ with respect to z in T ∗ U, k Λ f˜,z

to be ∆iJ ∗ (f˜) (kM ), the i-th Vogel cycle of Jz∗ (f˜) with respect to kM , provided that these Vogel z

cycles exist (see I.2.14). ˇ i | is contained in im df˜. If the k-shifted LˆeVo cycles of f˜ with respect to z exist, then each |k Λ f˜,z ˇ i and its image in U; we Therefore, the proper push-forward η∗ induces an isomorphism between k Λ f˜,z
ˇi . define the i-th k-shifted Lˆe-Vogel (LˆeVo) cycle of f˜ with respect to z, k Λif˜,z , by k Λif˜,z := η∗ k Λ f˜,z

We have Proposition 1.4. Suppose that the k-shifted Lˆe-Vogel cycles of f with respect to z exist. Then, 0)

k ˇi ˇi , Γf˜,z , k Λ f˜,z

i)

[ ˇ i˜ = |kM | ∩ im df˜; k Λ f ,z i

and k Λif˜,z are non-negative and purely i-dimensional;

162

ii)

DAVID B. MASSEY

[ k Λi˜ = Σ f ; and kQ• f ,z i

iii)

[ k Λi˜ = Σ f . C f ,z i,k

Proof. As kM is non-negative, all of the cycles defined in 1.3 are non-negative. I.2.2.i implies that k ˇi ˇ i and k Λi are purely i-dimensional. Γf˜,z is purely i-dimensional, and I.2.15 implies that k Λ f˜,z f˜,z ˇ i is i) follows from I.2.4 (and I.2.15). By applying η to each side of i), and using that each k Λ ˜ [ 
f ,z k Λi˜ = x ∈ X | (x, dx f˜) ∈ |kM | = Ch kQ• . ii) purely i-dimensional, we obtain that f ,z i,k

now follows immediately from III.4.2. Finally, iii) follows from ii) by Theorem III.4.6. 

Example 1.5. Consider the case where
dimx ΣC f 6 0. By III.4.6, this is equivalent to requiring that, for all k, dim(x,dx f˜) |kM | ∩ im df˜ 6 0. ˇ i and k Λi are defined and are zero near Now, fix k. If (x, dx f˜) 6∈ |kM |, then, for all i, k Λ f˜,z f˜,z (x, dx f˜) and x, respectively. Suppose, then, that (x, dx f˜) is an isolated point of |kM | ∩ im df˜. ˇ i , are defined Then, as we saw in I.2.8 and I.2.16, near (x, dx f˜), the conormal LˆeVo cycles, k Λ f˜,z

ˇ i = 0 for i > 1, and, at (x, dx f˜), for all i, k Λ f˜,z k ˇ0 Λf˜,z

  ∂ f˜  ∂ f˜  · . . . · V w0 − = = kM · V wn − ∂zn ∂z0

   ∂ f˜ ∂ f˜  M · V w0 − , . . . , wn − = kM · im df˜ = ˜bd−1−k (Ff,x ) (x, dx f˜) , ∂z0 ∂zn

k

˜

˜

where the second equality follows from the fact that w0 − ∂z0f , . . . , wn − ∂znf must determine a regular sequence in OU at points in |kM |, and the last equality follows from III.4.10.   Thus, the LˆeVo cycles are all defined, k Λif˜,z = 0 for i > 1, and, at x, k Λ0f˜,z = ˜bd−1−k (Ff,x ) x .

Remark 1.6. In Remark 2.16, we discussed how one actually calculates Vogel cycles in practice; we wish to do this again in our present setting, in order to describe how one calculates the LˆeVo cycles. By definition, h i X k ˇi ˜b Λf˜,z = ∆iJ ∗ (f˜) (kM ) = (Lα ) ∆iJ ∗ (f˜) (TS∗α U) . d−k−1−dα z

z

α

So, how does one calculate ∆iα := ∆iJ ∗ (f˜) (TS∗α U)? One begins with z

b n+1 Π := α

b n+1 (T ∗ U) Π Jz∗ (f˜) Sα

= TS∗α U ¬ V

∂ f˜ wn − ∂zn

! ;

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163

  ∂ f˜ b n+1 (T ∗ U) is either 0 or T ∗ U. Next, one calculates the intersection Π b n+1 thus, Π · V w − n α ∗ ˜ ∂zn . Sα Jz ( f ) S α   ∂ f˜ ∂ f˜ This intersection cycle has components contained in W := V wn − ∂zn , . . . , wn − ∂zn and components which are not contained in W . By I.2.12, the sum of the components which are not b nα and the sum of the components which are contained in W is contained in W is precisely Π   ∂ f˜ n b n+1 bn · V w − ∆nα := ∆nJ∗ (f˜) (TS∗α U). Having calculated Π n α ∂zn = Πα + ∆α , we use our newly found z   b n in the next step: the calculation of Π b n · V wn−1 − ∂ f˜ b n−1 + ∆n−1 . One proceeds Π =Π α α α α ∂zn−1 downward inductively. As we pointed out in I.2.15, if one is working near a point of TS∗α U ∩ im df˜, the slightly subtle point here is that – to know that this method of calculation is valid– one has only to verify that each ∆iα is purely i-dimensional as one performs the calculations. We demonstrate such a calculation in Example 1.14, after we have discussed when the LˆeVo cycles are independent of the extension of f . If X is a pure-dimensional l.c.i. (e.g., a connected l.c.i.), then, by [Lˆ e9], the complex links of strata of X have the homotopy-types of bouquets of spheres of middle dimension. Consequently, ˇ i , 0Λ ˇi , for such a space, the only kM which can be non-zero is 0M and, therefore, it is only 0Γ f˜,z f˜,z and 0Λif˜,z which are of interest. Note that, at this point, there is no claim that the LˆeVo cycles of f˜ are independent of the extension of f . However, we will now use the Segre-Vogel Relation of I.2.22 to find a manageable criterion guaranteeing the existence of the LˆeVo cycles, to relate the LˆeVo cycles to our work in Part III, and to see that, under reasonable hypotheses, the LˆeVo cycles are independent of the extension f˜. In fact, we shall prove an analog of Theorem II.1.26, and so we need to define analogs of good stratifications and prepolar coordinates in our current, more general, setting.

Definition 1.7 Let R := {Rβ } be a Whitney stratification of X with connected strata. Then, R is a Lˆe-Vogel stratification for f provided that, for all visible Rβ ∈ R, i)

Σcnr (f|R ) is a union of strata; and β

ii)

for all Rγ ⊆ Σcnr (f|R ), the pair (Rβ , Rγ ) satisfies Thom’s af condition. β

Since, near a point x, Σcnr (f|R ) ⊆ f −1 f (x) (see III.1.7), the condition that the pair (Rβ , Rγ ) β
satisfies Thom’s af condition in ii) is equivalent to: Tf∗ U | ⊆ TR∗γ U. |R

β

Rγ

We call the strata comprising Σcnr (f|R ) the good strata of R associated to Rβ (with respect to β [
f ). As ΣC f = Σcnr f|R , we refer to any stratum contained in ΣC f as a good stratum visible Rβ

β

(of R with respect to f ). Let x ∈ X, and let {Rβ } be a Lˆe-Vogel stratification for f in an open neighborhood of x. The tuple (z0 , . . . , zk ) is a Lˆe-Vogel tuple for f at x with respect to {Rβ } provided that, for all i with 0 6 i 6 k, if Rβ ⊆ ΣC f and dim Rβ > i+1, then V (z0 −z0 (x), . . . , zi −zi (x)) transversely intersects

164

DAVID B. MASSEY

Rβ near x; as we saw in II.1.24, this is equivalent to there existing an open neighborhood Ω of x such that
P(TR∗ Ω) ∩ V (z0 − z0 (x), . . . , zi − zi (x)) × Pi × {0} = ∅. β

Naturally, we define a Lˆe-Vogel tuple for f at x to be a tuple (z0 , . . . , zk ) such that there exists a Lˆe-Vogel stratification for f near x with respect to which (z0 , . . . , zk ) is a Lˆe-Vogel tuple at x.

Proposition 1.8. Lˆe-Vogel stratifications always exist and, for all x ∈ X, for a generic linear ˆ, of z, z ˆ is a Lˆe-Vogel tuple for reorganization, z
f at x. In particular, if dimx ΣC f ∩ V (z0 − z0 (x)) 6 0, then z is a Lˆe-Vogel tuple for f at x. Proof. By [Hi], Thom stratifications always exist (since f has codomain C); if we now refine a Thom stratification, {Rβ }, so that, for all visible Rβ , Σcnr (f|R ) is a union of strata, then we will have a β

Lˆe-Vogel stratification. Now that we know that we can always produce a Lˆe-Vogel stratification, it is completely trivial, and standard, to conclude that a generic linear reorganization will be a Lˆe-Vogel tuple. We leave it as an exercise for the reader.
The last sentence follows from the fact that if dimx ΣC f ∩V (z0 −z0 (x)) 6 0, then dimx ΣC f 6 1 and V (z0 − z0 (x)) does not contain a 1-dimensional component of ΣC f through x. If Ci are the irreducible components of ΣC f of dimension 1 through x, then, in a neighborhood of x, we may refine any Lˆe-Vogel stratification so that Ci − {x} and {x} are the good strata. The conclusion follows. 

Recalling the notation that we used in I.2.22, we let X


k Blim df˜(kM ) := (−1)d Blim df˜ Ch kQ• := mα Blim df˜ TS∗α U ⊆ U × Cn+1 × Pn α

and Eim df˜(kM ) := (−1)d Eim df˜ Ch

Q•

k





:=

X

k


mα Eim df˜ TS∗α U ,

α

where Bl denotes the blow-up and E denotes the corresponding exceptional divisor.

The following theorem is analogous to the first part of Theorem II.1.2. It tells us that z being a Lˆe-Vogel tuple implies the correct hypotheses hold for us to apply the Segre-Vogel Relation of Part I.

Theorem 1.9. Let x ∈ X, and suppose that z is a Lˆe-Vogel tuple
for f at x. Then, there exists an open neighborhood, Ω, of x such that, for all i, |π∗ Eim df˜(kM ) | properly intersects Ω × (Pi × {0}). Proof. Our goal is to reduce the proof to the point where it precisely follows the proofs of Lemma II.1.25 and the first part of Theorem II.1.26.

PART IV. NON-ISOLATED CRITICAL POINTS

165

As kM is independent of the Whitney stratification, we may assume that {Sα } is a Lˆe-Vogel stratification for f at x, and that z is a Lˆe-Vogel tuple for f at x with respect to {Sα }. Fix an Sα such that TS∗α U appears in kM ; such an Sα is necessarily visible. Let Eα := Eim df˜(TS∗α U). We need to show that π(Eα ) properly intersects Ω × (Pi × {0}), for all i.
By III.5.2, π(Eα ) ⊆ P Tf∗| U . Since we are assuming that we have a Lˆe-Vogel stratification, if Sα
Sβ ⊆ Σcnr (f|S ), then Tf∗ U | ⊆ TS∗ U. Therefore, if Sβ ⊆ Σcnr (f|S ), then ν −1 (Sβ )∩π(Eα ) ⊆ |S

α

α

Sβ

β

α

P(TS∗ U). As ν(π(Eα )) = Σcnr (f|S ), it follows that β α [ π(Eα ) ⊆ Sβ ⊆Σcnr (f|

P(TS∗ U). β

Sα

)

The proof is now exactly the arguments of Lemma II.1.25 and the first part of Theorem II.1.26. 

Theorem 1.10. The analytic set |Eim df˜(kM )| properly intersects U × Cn+1 × (Pi × {0}) for all
P i if and only if |π∗ Eim df˜(kM ) | = | v P(Ch(φf −v (kQ• )))| properly intersects U × (Pi × {0}) for all i, and whenever these equivalent conditions hold: i)

the k-shifted LˆeVo and conormal LˆeVo cycles of f˜ with respect to z exist;

ii)

k ˇ i = τ∗ E for all i, k Λ im df˜( M ) · f˜,z



U × Cn+1 × (Pi × {0}) ;

iii) for all i, k



Λif˜,z = η∗ τ∗ Eim df˜(kM ) · U × Cn+1 × (Pi × {0})  
ν∗ π∗ Eim df˜(kM ) · (U × (Pi × {0})) =  X ˆ ν∗ (−1)dv P(Ch(φf −v (kQ• ))) · (U × (Pi × {0})) ; v

and there exists a neighborhood Ω of |kM | ∩ im df˜ such that iv)

the k-shifted conormal polar cycles of f˜|Ω with respect to z exist inside Ω;

v)

for all i, | Blim df˜(kM )| properly intersects Ω × (Pi × {0}) in Ω × Pn ;

vi)

k ˇ i+1 = τ∗ Bl inside Ω, for all i, k Γ im df˜( M ) · f˜,z



U × Cn+1 × (Pi × {0}) .

Note that iii) implies that the k-th shifted LˆeVo cycles are independent of the extension f˜.
P ˆ Proof. That π∗ Eim df˜(kM ) = v (−1)dv P(Ch(φf −v (kQ• ))) follows from Theorem III.3.10. The equivalence of the two intersection conditions is a trivial consequence of the fact that the points in Eim df˜(kM ) lie above the graph im df˜.

166

DAVID B. MASSEY

Now, i), ii), iv), v), and vi) follow immediately from the Segre-Vogel Relation (I.2.20). iii) follows
by applying η∗ to each side of ii), using that ν ◦ π = η ◦ τ , and using again that π∗ Eim df˜(kM ) = P dˆv k • v (−1) P(Ch(φf −v ( Q ))). 

Remark 1.11. Statement iii) of 1.10 tells us that – under the hypotheses of the theorem – k Λif˜,z only depends on f and the choice of w0 , . . . , wi . As w0 , . . . , wi are determined by z0 , . . . , zi , one might be tempted to reference only z0 , . . . , zi in the notation for k Λif˜,z , e.g., k Λ0f˜,z . Note, however, 0 the hypotheses of the theorem put conditions on all the z’s. This should not be surprising – the Vogel cycles are defined in terms of the inductive gap cycles, which are defined by downward induction. Thus, the higher-dimensional data needs to behave well before we can work with the lower-dimensional data. Of course, we could use 1.10.iii to define k Λif˜,z , and thereby avoid needing to impose conditions on all the coordinates and also avoid referring to f˜ at all. However, we prefer the algorithmic, Vogel cycle definition, because it is the most useful for calculation. Moreover, in general, we will not be interested in working in isolation with individual LˆeVo cycles, but, rather, will want to require that all of them are well-behaved. On the other hand, it is desirable to have the LˆeVo cycles be independent of the extension of f . Therefore, we make the following definition.

Definition 1.12. If |Eim df˜(kM )| properly intersects U × Cn+1 × (Pi × {0}) for all i, then we say that the k-shifted Lˆe-Vogel (LˆeVo) cycles of f with respect to z exist; we write k Λif,z in place of k i Λf˜,z and refer to it as the i-th k-shifted Lˆe-Vogel (LˆeVo) cycle of f with respect to z (that is, we eliminate the reference to the extension f˜). If x ∈ X and the k-shifted Lˆe-Vogel cycles of f with respect to z exist in a neighborhood of x, then we say that i-th k-shifted Lˆe-Vogel (LˆeVo) number of f at x with respect to z exists provided that |k Λif,z | properly intersects V (z0 − x0 , . . . , zi−1 − xi−1 ) at x, and then we define this Lˆe-Vogel
number to be k λif,z (x) := k Λif,z · V (z0 − x0 , . . . , zi−1 − xi−1 ) x . When i = 0, we mean that k 0 λf,z (x) = (k Λ0f,z )x . Note that the LˆeVo cycles and numbers are only (possibly) non-zero for 0 6 k 6 d and 0 6 i 6 d, and they exist near a point x provided that z is a Lˆe-Vogel tuple for f at x.

Example 1.13. As in Example 1.5, consider the case where dimx ΣC f 6 0. Thus, for all

k, dim(x,dx f˜) |kM | ∩ im df˜ 6 0. It follows that, in a neighborhood of x, |π∗ Eim df˜(kM ) | = ∗ P(T{x} U) = {x} × Pn , which certainly properly intersects U × (Pi × {0}) for all i. Therefore, the equivalent hypotheses of Theorem 1.10 hold, and so the LˆeVo cycles of f (not f˜) exist, and they equal the LˆeVo cycles of f˜ as given in 1.5.

Example 1.14. We return to the underlying space of Example III.4.14, the simplest non-l.c.i., but use a function with non-isolated critical points.

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167

Use (u, x, y, z) as coordinates for U := C4 , and let X := V (u, x) ∪ V (y, z). Let f˜ := (uα + xβ )τ + y γ + z δ , where α, β, γ, δ, τ > 2. Since d = 2, kQ• := k = 0 or 1, and

P• and our calculation in III.4.14 tells us that Ch(kQ• ) = 0 unless

1−k

Ch(0Q• ) = [V (u, x, w2 , w3 )] + [V (y, z, w0 , w1 )] , while Ch(1Q• ) = [V (u, x, y, z)] . One easily shows that im df˜ = V (w0 − τ (uα + xβ )τ −1 αuα−1 , w1 − τ (uα + xβ )τ −1 βxβ−1 , w2 − γy γ−1 , w3 − δz δ−1 ), im df˜ ∩ V (u, x, w2 , w3 ) = {0}, im df˜ ∩ V (y, z, w0 , w1 ) = V (uα + xβ , y, z, w0 , w1 , w2 , w3 ), im df˜ ∩ | Ch(0Q• )| = V (uα + xβ , y, z, w0 , w1 , w2 , w3 ) and im df˜ ∩ | Ch(1Q• )| = im df˜ ∩ V (u, x, y, z) = {0}. Thus, ΣC f is the 1-dimensional set V (uα + xβ , y, z), and we calculate in the manner discussed in Remark 1.6. Let C0 := V (u, x, y, z), C1 := V (u, x, w2 , w3 ), and C2 := V (y, z, w0 , w1 ). We need to calculate bi Π and ∆iJ ∗ (f˜) for each Cj ; let us denote the corresponding inductive gap cycles and Vogel Jz∗ (f˜) z b i and ∆i . cycles by simply Π j j As im df˜ intersects C0 and C1 in the isolated point 0, ∆i0 and ∆i1 are easy to calculate – they are both 0 unless i = 0 and, then, ∆00 = (im df˜ · V (u, x, y, z))0 [0] = [0], and ∆01 = (im df˜ · V (u, x, w2 , w3 ))0 [0] = (γ − 1)(δ − 1)[0]. Now, b 4 = [V (y, z, w0 , w1 )], Π 2 b 4 · V (w3 − δz δ−1 ) = [V (y, z, w0 , w1 , w3 )] = Π b 3, Π 2 2 b 3 · V (w2 − γy γ−1 ) = [V (y, z, w0 , w1 , w2 , w3 )] = Π b 2, Π 2 2 b 2 · V (w1 − τ (uα + xβ )τ −1 βxβ−1 ) = Π 2 (β − 1)[V (x, y, z, w0 , w1 , w2 , w3 )] + (τ − 1)[V (uα + xβ , y, z, w0 , w1 , w2 , w3 )] =

168

DAVID B. MASSEY

b 1 + ∆1 , Π 2 2 b 12 · V (w0 − τ (uα + xβ )τ −1 αuα−1 ) = Π (β − 1)[α(τ − 1) + (α − 1)][0] = (β − 1)(ατ − 1)[0] = ∆02 . Therefore, the only non-zero LˆeVo cycles are 0

0

Λ1f,z = (τ − 1)[V (uα + xβ , y, z)],


Λ0f,z = (γ − 1)(δ − 1) + (β − 1)(ατ − 1) [0],

and 1

Λ0f,z = [0].

The non-zero LˆeVo numbers at the origin are 0 1 λf,z (0) 0 0 λf,z (0)

= β(τ − 1),

= (γ − 1)(δ − 1) + (β − 1)(ατ − 1),

and 1 0 λf,z (0)

= 1.

We have the following analog of Theorem II.7.2. Theorem 1.15. Let x ∈ X, let Ω be an open neighborhood of x, and let {Rβ } be a Lˆe-Vogel stratification for f|Ω . Suppose that z is a Lˆe-Vogel tuple for f with respect to {Rβ } at all points of Ω. Then, inside Ω, the k-shifted LˆeVo cycles of f , k Λif,z , exist and |k Λif,z | ⊆

[

Rβ .

Rβ ⊆ΣC f dim Rβ 6i

Proof. The existence of k Λif,z follows from 1.7 and 1.8. As we saw in the proof of 1.7, if Rβ is visible, [
π Eim df˜(TR∗ β U) ⊆ P(TR∗γ U). Rγ ⊆Σcnr (f|

Rβ

)

The assumption on the coordinates z is that, if Rβ is visible, Rγ ⊆ Σcnr (f|R ), and dim Rγ > i + 1, β

then P(TR∗γ U) ∩


Ω × (Pi × {0}) = ∅.

The result now follows at once from 1.8.iii. 

PART IV. NON-ISOLATED CRITICAL POINTS

169

ˆ Chapter 2. LE-IOMDINE FORMULAS AND THOM’S CONDITION We developed the Lˆe-Iomdine-Vogel formulas in extreme generality in I.3.4 specifically so that we would be able to apply them to the LˆeVo cycles at this point. As we saw in Part II, Chapter 4, such formulas allow us to reduce questions concerning non-isolated singularities to questions about the isolated case. Hence, we will be able to use III.6.12 in order give conditions which imply that Thom’s af condition holds. We continue with our notation from the previous chapter. For simplicity, we assume in the following two results that x ∈ V (z0 ); clearly, this causes no loss of generality. Theorem 2.1. Let x ∈ V (z0 ) ∩ |η∗ (kM )| = V (z0 ) ∩ kX. Let a be a non-zero complex number, and ˆ denote the “rotated” coordinates (z1 , . . . , zn , z0 ). let j > 1 be an integer. Let z Suppose that the k-shifted LˆeVo cycles of f , k Λif,z , exist in a neighborhood of x and that, for all i > 1, V (z0 ) properly intersects each k Λif,z at x. Then, k λ0f,z (x) and k λ1f,z (x) exist and, if j > 1 + k λ0f,z (x), then, in a neighborhood of x, i)

ΣkQ• (f + az0j+1 ) = V (z0 ) ∩ ΣkQ• f ;

ii)


dimx ΣkQ• (f + az0j+1 ) = dimx ΣkQ• f − 1, provided that dimx ΣkQ• f > 1;

iii)

ˆ exist; and the k-shifted LˆeVo cycles of f + az0j+1 with respect to z

iv)

k 0 λf +azj+1 ,ˆz (x) 0


= k λ0f,z (x)+j(k λ1f,z (x)) and, for 1 6 i 6 n−1, k Λif +azj+1 ,ˆz = j k Λi+1 f,z ·V (z0 ) . 0

Proof. This a translationI.3.4 in our current situation. We have also used that (p, dp f˜) ∈  is simply ˜ ∂f ∂ f˜ , . . . , wn − ∂z if and only if x ∈ ΣkQ• f .  |kM | ∩ V w0 − ∂z 0 n

We immediately conclude Corollary 2.2 (Lˆ e-Iomdine Formulas). Let x ∈ V (z0 ) ∩ |η∗ (kM )| = V (z0 ) ∩ kX. Let a be a ˆ denote the “rotated” coordinates non-zero complex number, and let j > 1 be an integer. Let z (z1 , . . . , zn , z0 ). Suppose that the k-shifted LˆeVo numbers at x of f , k λif,z (x), exist and that j > 1 + k λ0f,z (x). Then, in a neighborhood of x, i)

ΣkQ• (f + az0j+1 ) = V (z0 ) ∩ ΣkQ• f ;

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DAVID B. MASSEY

ii)


dimx ΣkQ• (f + az0j+1 ) = dimx ΣkQ• f − 1, provided that dimx ΣkQ• f > 1;

iii)

ˆ exist; and the k-shifted LˆeVo numbers of f + az0j+1 with respect to z

iv)

k 0 λf +azj+1 ,ˆz (x) 0


= k λ0f,z (x)+j(k λ1f,z (x)) and, for 1 6 i 6 n−1, k λif +azj+1 ,ˆz (x) = j k λi+1 f,z (x) . 0

Just as our generalized Lˆe-Saito Theorem of II.6.5 followed immediately by applying the LˆeIomdine formulas to the actual result of Lˆe and Saito on families of isolated affine hypersurface singularities, so too does a “super” general Lˆe-Saito result follow by applying 2.2 above to Corollary III.6.12. ◦

Since we wish to apply 2.2 to families, we first need to introduce some new notation. Let D ◦ ◦ be an open disc about the origin in C, let Ω := D × U, let t˜ : Ω → D denote the projection, let g˜ : Ω → C be an analytic function, let X be a (d + 1)-dimensional analytic subset of Ω, let t denote ◦ the restriction of t˜ to X , and let g denote the restriction of g˜ to X . We suppose that D × {0} ⊆ X ◦

◦

and that g(D × {0}) = 0. For a ∈ D, use z as coordinates on each t˜−1 (a) ∼ = U, let Xa := t−1 (a), and let ga : Xa → C be given by ga := g|Xa .

Theorem 2.3 (General Lˆ e-Saito Theorem). Suppose that 0 6∈ ΣC t. Let Sα be a visible stratum of X of dimension dα such that g|Sα 6= 0, and let j be an integer such that ˜bj−1 (Lα ) 6= 0. Let Y := Sα and let k := d − dα − j. In particular, Y could be any irreducible component of X , j could be zero, and k would be 0. Suppose that, for all i, k λiga ,z (0) is independent of a, for all small a, and that either a)

0 ∈ ΣNash (g|Y ); or that

b)

0 6∈ Σcnr (g|Y ), and (Yreg , D × {0}) satisfies Whitney’s condition a) at 0.

◦

◦

Then, the pair (Yreg , D × {0}) satisfies the ag condition at 0. Moreover, in case a), there exists an i such that k λiga ,z (0) 6= 0. Proof. The argument in case b) is exactly that of III.6.12; so, assume that we are in case a). Let s := dim0 ΣC (g0 ). Let 0  j0  · · ·  js , and let h := g + z0j0 + · · · + zsjs . Certainly, h|Sα 6≡ 0, since g|Sα 6≡ 0. Consider the family ha := ga + z0j0 + · · · + zsjs . By 2.2.ii, dim0 ΣC (h0 ) = 0 and so, by III.6.7, dim0 ΣC (ha ) 6 0 for all small a. By 1.5 and 1.13, k λ0ha ,z (0) = ˜bdα +j−2 (Fha ,0 ). By an inductive  application of the Lˆe-Iomdine formulas, k λ0ha ,z (0) is a function of only k λiga ,z (0) i (and the fixed j’s); thus, ˜bdα +j−2 (Fha ,0 ) is independent of a for small a. It is trivial to show that, since 0 ∈ ΣNash (g|Y ), 0 ∈ ΣNash (h|Y ). Therefore, we may apply III.6.2 ◦

to conclude that (Yreg , D × {0}) satisfies the ah condition at 0 and ˜bdα +j−2 (Fha ,0 ) 6= 0.

PART IV. NON-ISOLATED CRITICAL POINTS

171

As k λ0ha ,z (0) = ˜bdα +j−2 (Fha ,0 ) 6= 0, the Lˆe-Iomdine formulas imply that there exists an i such that k λiga ,z (0) 6= 0. As in II.6.5, by inducting, we would be finished if we could show that: 0 ∈ ΣNash (g|Y ) and [η] ∈ P Tg˜∗|

Ω




Yreg

0

implies that, for all j sufficiently large,

∗ [η] ∈ P T(˜ g +z j )

0 |Yreg


Ω 0.

By III.5.2, what we
need to show is that [η] ∈ p(Eg˜ )0 implies that, for all j sufficiently large, ∗ Ω 0 , where Eg˜ denotes the exceptional divisors of Blim d˜g TY∗reg Ω, and p denotes [η] ∈ P T(˜ g +z j ) 0 |Yreg

the projection Ω × Cn+2 × Pn+1 → Ω × Pn+1 . Now, [η] ∈ p(Eg˜ )0 if and only if there exists an analytic path γ(u) = (x(u), ω(u)) in TY∗reg Ω such that γ(0) = (0, d0 g˜), γ(u) ∈ TY∗reg Ω − im d˜ g , and [ω(u) − dx(u) g˜] → [η] as u → 0. That [ω(u) − dx(u) g˜] → [η] is equivalent to ξ·

ω(u) − dx(u) g˜ η → , |ω(u) − dx(u) g˜| |η|

for some root of unity ξ. One concludes easily that |η|ξ |ω(u) − dx(u) (˜ g+

z0j )|


ω(u) − dx(u) (˜ g + z0j ) → η,

for all large j.
However, the terms on the left side of the above expression are clearly elements of ∗ T(˜ Ω x(u) whose projective class approaches that of η.  g +z j ) 0 |Yreg

172

DAVID B. MASSEY

Chapter 3. LE-VOGEL CYCLES AND THE EULER CHARACTERISTIC In this final chapter, we will relate the Lˆe-Vogel cycles to the Euler characteristic of the Milnor fibre, in a way that generalizes our result in II.10.3. In order to accomplish this, we must recall a definition and a result from [Mas11]. We continue with the notation from the previous two chapters.

Proposition/Definition 3.1. Let p ∈ X and let F• be a bounded, constructible complex on X. Then, for a generic choice of the coordinates z, there exists an open neighborhood, W, of p and cycles ΛiF• ,z in W such that each ΛiF• ,z is purely i-dimensional, ΛiF• ,z properly intersects V (z0 − z0 (x), . . . , zi−1 − zi−1 (x)) and, for all x ∈ W, X
χ(F• )x = (−1)d (−1)i ΛiF• ,z · V (z0 − z0 (x), . . . , zi−1 − zi−1 (x)) x , i

(here, when i = 0, we mean that the intersection number is simply Λ0F• ,z Moreover, whenever such cycles exist, they are unique.


x

).


In the case where dimp supp F• = 1, such cycles exist if dimp V (z0 − z0 (p)) ∩ supp F• 6 0. We call ΛiF• ,z the i-dimensional characteristic polar cycle of F• with respect to z, and refer to
λiF• ,z (x) := ΛiF• ,z · V (z0 − z0 (x), . . . , zi−1 − zi−1 (x)) x as the i-th characteristic polar number of F• with respect to z at x. Proof. The statement about the case where dimp supp F• 6 1 is trivial. The remaining statements are a combination of Propositions 2.4 and 3.1 from [Mas11]. 

Below, we refer to the absolute polar varieties of Lˆe and Teissier ([L-T2], [Te4], [Te5]); however, we need to explain our notation. We let Γiz (Sα ) denote the i-dimensional polar variety of Sα with respect to the flag {0} ⊆ V (z0 , z1 , . . . , zn−1 ) ⊆ . . . V (z0 , z1 ) ⊆ V (z0 ) ⊆ Cn+1 . Thus, as a set,
. See also our treatment in Section 7 of [Mas11]. Γiz (Sα ) = crit (z0 , . . . zi )|(S ) α reg

Theorem 3.2. Let p ∈ X and let F• be a bounded h icomplex on X, which is constructible with P • ∗ respect to {Sα }. Suppose that Ch(F ) = α mα TSα U . Then, for a generic choice of the coordinates z, there exists an open neighborhood, W, of p in which all of the ΛiF• ,z exist, such that, for all i, P(Ch(F• )) and W × (Pi × {0}) intersect properly in P(T ∗ W), and the restriction of ν to W × Pn yields the equalities X
ΛiF• ,z = mα Γiz (Sα ) = ν∗ P(Ch(F• )) · (W × (Pi × {0})) . α

In the case where Y := supp
F• is 1-dimensional at p, the conclusions hold if z0 is finite at p, i.e., if dimp (V (z0 − z0 (p)) ∩ Y p = 0.

PART IV. NON-ISOLATED CRITICAL POINTS

173

Proof. Aside from the statement about the case where dimp supp F• = 1, this follows immediately from Theorem 7.5 of [Mas11]. However, when dimp supp F• = 1, the hypotheses of 7.5 of [Mas11] are strictly stronger than saying that z0 is finite at p, so we must provide a proof of this portion. Assume that Y := supp F• is 1-dimensional at p and that z0 is finite at p. Let C denote a component of Y through p, and let xC denote a point in C − {p}. By shrinking our neighborhood, we may assume that C − {p} is smooth, and that χ(F• )xC is independent of the choice of xC . As z0 is not constant along C, we may also assume that (x, dx z0 ) 6∈ TC∗reg U for x 6= p. It follows at once that P(Ch(F• )) and W × (Pi × {0}) intersect properly for all i. Also, since the hypotheses of 7.5 of [Mas11] are satisfied at all x 6= p, 7.5 of [Mas11] implies that
X ν∗ P(Ch(F• )) · (W × (P1 × {0})) = χ(F• )xC [C]. C

Moreover, by applying Theorem III.3.10 to the function z0 , we find that
ν∗ P(Ch(F• )) · (W × (P0 × {0})) = (Ch(F• ) · im dz0 )p [p] = χ(φz0 −z0 (p) F• )p [p]. As χ(φz0 −z0 (p) F• )p = χ(ψz0 −z0 (p) F• )p − χ(F• )p = −χ(F• )p +

X

(C · V (z0 − z0 (p)))p χ(F• )xC ,

C

the conclusion follows. 

We relate the LˆeVo cycles and numbers to the Euler characteristic by the following theorem.

Theorem 3.3. Let p ∈ X and assume that f (p) = 0. Fix a k and let F• := φf (kQ• ). Then, for a generic choice of the coordinates z, there exists an open neighborhood, W, of p in which all of the ΛiF• ,z exist, all of the k-shifted LˆeVo numbers, k λif,z , exist, and such that k Λif,z = (−1)d ΛiF• ,z and k λif,z (x) = (−1)d λiF• ,z (x) for all x ∈ W, i.e., X
χ φf [−1](kQ• ) x = (−1)i k λif,z (x). i

In the case where Y := supp F• is 1-dimensional at p, the conclusions hold if dimp (V (z0 − z0 (p)) ∩ Y


p

= 0.

Proof. Throughout the proof, we will work in an arbitrarily small neighborhood of p. From 1.10.iii, we have k

  ˆ Λif,z = (−1)d ν∗ P(Ch(φf (kQ• ))) · (W × (Pi × {0})) .

174

DAVID B. MASSEY ˆ

ˆ

Hence, by 3.2, k Λif,z = (−1)d ΛiF• ,z and k λif,z (x) = (−1)d λiF• ,z (x). ˆ

By definition, χ(F• )x = (−1)d−1

i i i (−1) λF• ,z (x).

P

χ(F• )x = −

X

Therefore,

(−1)i k λif,z (x),

i

or, equivalently, χ(F• [−1])x =

X

(−1)i k λif,z (x). 

i

Before we can connect Theorem 3.3 to the ordinary cohomology of the Milnor fibre with constant coeeficients, we need to prove a lemma. While we suspect that this lemma is well-known, we can find no reference.

Lemma 3.4. Let F• be a bounded, constructible complex on X. Then, for all x ∈ X, X
χ(F• )x = (−1)k χ µH 0 (F• [k]) x . k

Proof. For convenience, assume that x = 0. The proof is by induction on the dimension of X at 0. If dim0 X = 0, then H i (µH 0 (F• [k]))0 = 0 unless i = 0, and then H 0 (µH 0 (F• [k]))0 ∼ = H 0 (F• [k])0 = H k (F• )0 . Thus, the lemma holds if dim0 X = 0. Now, assume the lemma for spaces of dimension j, and suppose that dim0 X = j + 1. Let L be a generic linear form. Consider the distinguished triangle
H 0 (F• [k]) |

[1]

µ

V (L)

[−1] −→ ψL [−1]µH 0 (F• [k]) −→ φL [−1]µH 0 (F• [k]) −→;

it yields the equality
χ ψL [−1]µH 0 (F• [k]) 0 = χ


H 0 (F• [k]) |

µ

V (L)



[−1] 0 + χ φL [−1]µH 0 (F• [k]) 0 .

Thus,


χ µH 0 (ψL [−1]F• [k]) 0 = −χ µH 0 (F• [k]) 0 + χ µH 0 (φL [−1]F• [k]) 0 , and so X X X


(−1)k χ µH 0 (φL [−1]F• [k]) 0 . (−1)k χ µH 0 (F• [k]) 0 + (−1)k χ µH 0 (ψL [−1]F• [k]) 0 = − k

k

Applying our inductive hypothesis twice, we obtain

k

PART IV. NON-ISOLATED CRITICAL POINTS

X

175






(−1)k χ µH 0 (F• [k]) 0 = χ φL [−1]F• 0 − χ ψL [−1]F• 0 = −χ φL F• 0 + χ ψL F• 0 = χ(F• )0 ,

k

and we are finished. 

The relationship between the characteristic polar cycles of φf C•X and the k-shifted LˆeVo cycles is given in

Theorem 3.5. Let p ∈ X and suppose that f (p) = 0. Then, for a generic choice of the coordinates z, for all k, there exists an open neighborhood, W, of p in which all of the Λiφ C• ,z exist, all of the f X k ˆP k-shifted LˆeVo numbers, k λif,z , exist, and such that Λiφ C• ,z = (−1)d−d k (−1) k Λif,z and, for all f X

x ∈ W ∩ V (f ), X
X d−i−1 k i χ e Ff,x = (−1)k (−1) λf,z (x), i

k

where Ff,x denotes the Milnor fibre of f at x. In the case where Y :=
ΣC f has dimension 1 at p, the conclusions hold for all k provided that dimp V (z0 − z0 (p)) ∩ Y p = 0. Proof. Using 3.3 and 3.4, X X


χ e Ff,x = χ(φf C•X )x = (−1)k χ µH 0 (φf C•X [k]) x = (−1)k χ φf [−1]µH 0 (C•X [k + 1]) x = k

X

k

X X
(−1)k χ φf [−1] d−k−1Q• x = (−1)k (−1)i

k

k

X

(−1)k

• f CX ,z

=

=

i

(−1)d−i−1 k λif,z (x).

i

k

That Λiφ

X

d−k−1 i λf,z (x)

k ˆP (−1)d−d k (−1) k Λif,z

follows immediately. 

Example 3.6. Recall Example 1.14 where X = V (u, x) ∪ V (y, z) ⊆ C4 , and f˜ := (uα + xβ )τ + y γ + z δ , where α, β, γ, δ, τ > 2. Applying Theorem 3.5, we obtain
χ e Ff,0 = −0λ0f,z (0) + 0λ1f,z (0) + 1λ0f,z (0) = −(γ − 1)(δ − 1) − (β − 1)(ατ − 1) + β(τ − 1) + 1 = −(γ − 1)(δ − 1) + τ (−αβ + α + β).

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DAVID B. MASSEY

We can verify this calculation. One easily sees that Ff,0 is the disjoint union of F1 , the Milnor fibre of (uα + xβ )τ restricted to V (x, y), and F2 , the Milnor fibre of y γ + z δ restricted to V (u, x). Thus, F1 is homotopy-equivalent to the disjoint union of τ copies of a bouquet of (α − 1)(β − 1) circles, and F2 is homotopy-equivalent to a bouquet of (γ − 1)(δ − 1) circles. Therefore,
χ e Ff,0 = −τ (α − 1)(β − 1) − (γ − 1)(δ − 1) + τ + 1 − 1, where the last −1 is due to the fact that we use the reduced cohomology. One sees, then, that the calculations agree.

Appendix A: Analytic Cycles and Intersections We wish to consider schemes, cycles, and sets. Frequently, we will be in the algebraic setting and, hence, we may use algebraic schemes, cycles, and sets. However, as we wish to treat the more general analytic case, we should clarify what we mean by the terms scheme and cycle. In the analytic setting, by scheme, we actually mean a (not necessarily reduced) complex analytic space, (X, OX ), in the sense of [G-R1] and [G-R2]. By the irreducible components of X, we mean simply the irreducible components of the underlying analytic set X. If we concentrate our attention on the germ of X at some point p, then we may discuss embedded subvarieties and (non-embedded, or isolated) components of the germ of X at p – these correspond to non-minimal and minimal primes, respectively, in the set of associated primes of the Noetherian local ring OX,p . If X is a complex space and α is a coherent sheaf of ideals in OX , then we write V (α) for the possibly non-reduced analytic subspace defined by the vanishing of α. By the intersection of a collection of closed subschemes, we mean the scheme defined by the sum of the underlying ideal sheaves. By the union of a finite collection of closed subschemes, we mean the scheme defined by the intersection (not the product) of the underlying ideal sheaves. We say that two subschemes, V and W , are equal up to embedded subvariety provided that, in each stalk, the isolated components of the defining ideals (those corresponding to minimal primes) are equal. Our main concern with this last notion is that it implies that the cycles [V] and [W] are equal (see below). A.1 P Given an analytic space X (with its reduced structure), an analytic cycle in X is a formal sum mV [V ], where the V ’s are irreducible analytic subsets of X, the mV ’s are integers, and the collection {V } is a locally finite collection of subsets of X. As a cycle is a locally finite sum, and as we will normally be concentrating on the germ of an analytic space at a point, usually we can safely assume that a cycle is actually a finite formal sum. P Throughout this book, whenever we write a cycle mV [V ], we shall assume that the V ’s are distinct and that none of the mV ’s are zero. This is the same as saying that the presentation is minimal, in the sense that no further cancellations are possible. P We say that a cycle mV [V ] is positive if mV > 0 for all V ; a cycle is non-negative if it is the zero-cycle or is positive. A.2 Given an analytic space, (X, OX ), we wish to define the (positive) cycle associated to (X, OX ). In the algebraic context, this is given by Fulton in [Fu, 1.5] as [X] :=

X

mV [V ],

where the V ’s run over all the irreducible components of X, and mV equals the length of the ring OX,V , the local ring of X along V . In the analytic context, we wish to use the same definition, but we must be more careful in defining the mV . Define mV as follows. Take a point p in V . The germ of V at p breaks up into irreducible germ components (Vp )i . Take any one of the (Vp )i and let mV equal the length of the ring (OX,p )(Vp )i (that is, the local ring of X at p localized at the prime corresponding to (Vp )i ). This number is independent of the point p in V and the choice of (Vp )i . 177

178

DAVID B. MASSEY

Note that any embedded subvarieties of a scheme do not contribute to the associated cycle. One can easily show that, if f, g ∈ OX , then [V (f g)] = [V (f )] + [V (g)]; in particular, [V (f m )] = m[V (f )]. If Y is an analytic subset of X and C is a cycle in Y , then we may naturally consider C as a cycle in X. We shall be dealing with analytic schemes, cycles, and analytic sets. For clarification of what structure we are considering, we shall at times enclose cycles in square brackets, [ ], and analytic sets in a pair of vertical lines,||. Occasionally, when the notation becomes cumbersome, we shall simply state explicitly whether we are considering V as a scheme, a cycle, or a set. We say that two cycles are equal at a point, p, provided that the portions of each cycle which pass through p are equal. When we say that a space, X, is purely k-dimensional at a point, p, we mean to allow for the vacuous case where X has no components through p. We wish to describe some aspects of intersection theory. Of course, [Fu] is the definitive reference for this subject. However, we deal only with cycles, not cycle classes, and we deal only with proper intersections inside complex manifolds; this makes much of the theory fairly trivial to describe. A.3 If V and W are irreducible subschemes of a connected complex manifold, M , and Z is an irreducible component of V ∩ W such that codimM Z = codimM V + codimM W , then we say that V and W intersect properly along Z, or that Z is a proper component of V ∩ W . Two irreducible subschemes V and W in a connected complex manifold, M , are said to intersect properly in M provided that they intersect properly along each component of V ∩ W ; when this is the case, the intersection product, ([V ] · [W ]; M ), of [V ] and [W ] in M is characterized axiomatically by four properties listed below: openness, transversality, projection, and continuity (see [Fu], Example 11.4.4). A.4 If α is a coherent sheaf of ideals in OM and f ∈ OM is such that V (f ) contains no embedded subvarieties or irreducible components of V (α), then V (α) and V (f ) intersect properly in M and [V (α)] · [V (f )] = [V (α + hf i)] (see [Fu], 7.1.b). This statement immediately implies one which, a priori, seems stronger: if α and f are as before and V (f ) contains no irreducible component of V (α) and contains no embedded subvariety which is of codimension one inside some irreducible component of V (α), then V (α) and V (f ) intersect properly in M and [V (α)] · [V (f )] = [V (α + hf i)] More generally, if W := V (α) is a subscheme of M and f1 , . . . , fk ∈ OM determine regular sequences in the stalks OW,p and OM,p at all points p ∈ W ∩ V (f1 , . . . , fk ), then [V (α)] · [V (f1 , . . . , fk )] = [V (α + hf1 , . . . , fk i)] . The two paragraphs above allow one to define the intersection with a hypersurface (or, more generally, a Cartier divisor) without having to refer to an ambient manifold. Suppose that V (α) is a subscheme of an analytic space X, that X is contained in an analytic manifold M , and that f ∈ OX . Then, locally, OX ∼ ˜ ⊆ OM be a coherent = OM /γ for some coherent sheaf of ideals γ ⊆ OM . Let α sheaf of ideals such that γ ⊆ α ˜ and such that α ˜ /γ corresponds to α, i.e., α ˜ is such that V (α) = V (˜ α). Let f˜ be an extension of f to M . If V (f ) contains no embedded subvarieties or isolated components of V (α), then V (f˜) contains no embedded subvarieties or isolated components of V (˜ α) and so, by
˜ ˜ the previous paragraph, [V (˜ α)] · [V (f )]; M = [V (˜ α+ < f >)], which defines the same cycle in X

APPENDIX A: ANALYTIC CYCLES AND INTERSECTIONS

179

as does [V (α+ < f >)]. Therefore, we may unambiguously define ([V (α)] · [V (f )] ; X) by setting it equal to [V (α+ < f >)]. P P A.5 Two cycles mi [Vi ] and nj [Wj ] are said to intersect properly if Vi and Wj intersect properly for all i and j; when this is the case, the intersection product is extended bilinearly by defining X X X mi [Vi ] · nj [Wj ] = mi nj ([Vi ] · [Wj ]) . Occasionally it is useful to include the ambient manifold in the notation; in these cases we write (C1 · C2 ; M ) for the proper intersection of cycles C1 and C2 in M . P If two cycles C1 and C2 intersect properly and C1 · C2 = nk [Zk ], where the Zk are irreducible, then the intersection number of C1 and C2 at Zk , (C1 · C2 )Zk , is defined to be nk ; that is, the number of times Zk occurs in the intersection, counted with multiplicity. Note that, when C1 and C2 have complementary codimensions, all the Zk are merely points. If V is irreducible at p, then the multiplicity of V at p, multp V , is the minimum value of ([V ] · [W ])p , where W ranges over all analytic subsets which are irreducible at p and which have p as a component of the proper intersection of V and W ; in fact, when working in affine space, W may be chosen to be a generic affine linear subspace through p of dimension complementary to that of V . Suppose that p is an isolated point in the
proper intersection of V and W , where V and W
are irreducible. Then, ([V ] · [W ])p > multp V multp W with equality holding if and only if the projectivized tangent cones P(Tp V ) and P(Tp W ) are disjoint. A.6 It is fundamental that (C1 · C2 )Zk can be calculated locally; that is, if U is an open subset of M such that Zk ∩ U = 6 ∅, then (openness)

(C1 ∩ U · C2 ∩ U; U)Zk ∩U = (C1 · C2 ; M )Zk

(see [Fu], 11.4.4). A.7 If V and W are two irreducible subvarieties of M and P is an irreducible component of V ∩ W , we say that V and W are generically transverse along P in M provided that V and W are reduced and, at generic points of P , V and W are smooth and intersect transversely in M ; naturally, we say that V and W are generically transverse in M provided they are generically transverse along every component of the intersection. Another fundamental property of intersection numbers is the transversality characterization: if V and W are irreducible subschemes of M which intersect properly along an irreducible component P , then (V · W )P = 1 if and only if V and W are generically transverse along P in M ([Fu], 8.2.c and 11.4.4). A.8 If C1 , C2 , and C3 are positive cycles such that C1 and C2 intersect properly, and C3 properly intersects C1 · C2 , then C2 and C3 intersect properly, C1 properly intersects C2 · C3 , and (associativity)

(C1 · C2 ) · C3 = C1 · (C2 · C3 ) .

We wish to introduce a slight generalization of proper intersections of cycles. If V and W are irreducible subschemes of a connected complex manifold, M , and Z is an irreducible component of

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V ∩ W along which V and W intersect properly, then, for every open neighborhood U ⊆ M such that V ∩ W ∩ U = Z ∩ U, the value of (V ∩ U · W ∩ U; U)Z∩U is independent of U; we define (V · W )Z to be this common value. If Z is a proper component of V ∩ W , then, by [Fu], 8.2.a, (V · W )Z 6 [V ∩ W ]Z . P We define [V ] ·p [W ] := Z (V · W )Z , where the sum is over all Z along which V and W intersect properly. We extend bilinearly X

mi [Vi ] ·p

X

nj [Wj ] =

X

mi nj ([Vi ] ·p [Wj ]) .

We refer to this as the proper intersection of the two cycles. One easily verifies that, if C1 , C2 , and C3 are positive cycles, then (C1 ·p C2 ) ·p C3 = C1 ·p (C2 ·p C3 ) .

A.9 Given a point p ∈ M , a curve W = V (α) in M which is reduced and irreducible at p, and a hypersurface V (f ) ⊆ M which intersects W properly at p, there is a very useful way to calculate the intersection number ([W ] · [V (f )])p . One takes a local parameterization φ(t) of W which takes 0 to p, and then ([W ] · [V (f )])p = multt f (φ(t)), the degree of the lowest non-zero term. This is easy to see, for composition with φ induces an isomorphism C{t} OM,p ◦φ −−−→ . α+ < f > f (φ(t)) Of course, if c is small and unequal to zero, multt f (φ(t)) is precisely the number of roots of f (φ(t)) − c which occur near zero. More generally, given a point p ∈ M , a curve W = V (α) in M (which need not be reduced or irreducible at p), and a hypersurface V (f ) ⊆ M which intersects W properly at p, consider the map given by multiplication by f OM,p ·f OM,p −−→ ; α α the intersection number ([W ] · [V (f )])p = dimC (coker(·f )) − dimC (ker(·f )). A.10

Combining this with transversality, we obtain the following dynamic intersection property: X

(V (α) · V (f ))p =


V (α) · V (f − c) q ,

◦

q∈B ∩V (α)∩V (f −c) ◦

where  > 0 is sufficiently small, B is an open ball of radius  centered at p,
and |c|  . This formula may seem ridiculously complex, since all the V (α) · V (f − c) q equal 1; however, it is the form which generalizes nicely: if C is a purely one-dimensional cycle and V (f ) properly intersects |C| at p, then X

(C · V (f ))p =


C · V (f − c) q .

◦

q∈B ∩|C|∩V (f −c)

This is a special case of conservation of number, which we shall discuss more generally below.

APPENDIX A: ANALYTIC CYCLES AND INTERSECTIONS

181

A.11 The projection formula ([Fu], 11.4.4.iii) allows us to calculate intersections inside normal P slices. Let C = ni [Vi ] be a cycle in M and let N be a closed submanifold of M such that N generically transversely intersects each Vi in M (this is equivalent to: for each component Z of Vi ∩ N , (Vi · N )Z = 1). e Let B be a cycle in N ; We may consider (C · N ; M ) as a cycle in N ; denote this cycle by C. we may also consider B as a cycle in M . Then, the projection formula states: e ·p B; N ) = (C ·p B; M ). (C

(projection formula)

The projection formula lets us reduce the problem of calculating intersection numbers to the case where the intersection consists of isolated points. P To see this, suppose that C1 and C2 are two cycles in M which intersect properly and let C1 · C2 = ni [Vi ]. To calculate ni0 , first let p be a smooth point of Vi0 which is not contained in any other Vi of dimension less than or equal to that of Vi0 . Now, take a normal slice, N , to Vi0 at a smooth point, p, of Vi0 ; that is, in an open neighborhood, U, of p in M , N is a closed submanifold of U of complementary codimension to Vi0 such that N transversely intersects Vi0 inside U in the single point p and such that N is generically transverse to all other Vi and to all components of C1 and C2 in U. By locality ni0 = (C1 ∩ U · C2 ∩ U; U)Vi0 ∩U . As Vi0 ∩ U is the only component of (C1 ∩ U · C2 ∩ U; U) whose intersection with N gives {p}, the transversality characterization yields that ni0 = (C1 ∩ U · C2 ∩ U · N ; U)p . But, we wish to calculate this intersection inside of the normal slice N – this is what we get from the projection formula. Replace the M , N , C, and B in the projection formula as stated above by letting M = U, N = N , C = C1 ∩ U, and B = (C2 ∩ U) · N (consider B as a cycle in N ). Then, the formula yields that
(C1 ∩ U) · N ) · ((C2 ∩ U) · N ); N = (C1 ∩ U · C2 ∩ U · N ; U). Thus, we see that taking normal slices reduces calculating proper intersections of cycles to the case where the dimension of the intersection is zero. A.12 There is one last property of intersections of cycles that we need – continuity ([Fu], 11.4.4.iii). This property is what makes intersections dynamic; one can move the intersections in a family. ◦

Let M be a analytic manifold and let D be an open disc centered at the origin in C. Then, ◦

π

◦

the projection M × D − → D determines a one-parameter family of spaces, and any subscheme ◦

W ⊆ M × D determines a one-parameter family of schemes Wt := W ∩ (M × {t}). Hence, any ◦ P P cycle C := ni [Vi ] in M × D determines a family of cycles Ct := ni [(Vi )t ] in M ∼ = M × {t}. ◦

If a cycle C in M × D has a component contained in M × {t} for some t, then that component does not “propagate” through the family; we wish to eliminate such “bad” components. For any ◦ ◦ P cycle C = ni [Vi ] in M × D and any analytic set W ⊆ M × D, let X C¬W := ni [Vi ], Vi 6⊆W

and let Ct∗ :=



◦
C¬(M × {t}) · (M × {t}); M × D =

X

ni [(Vi )t ] .

Vi 6⊆M ×{t} ◦

Continuity of intersections states that, if C is a cycle in M × D with no component contained in M × {0}, and E is a cycle in M such that C0 properly intersects E in M , then there exists a

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DAVID B. MASSEY ◦

◦

◦

(possibly) smaller disk centered at the origin D0 ⊆ D such that C properly intersects E × D0 in ◦

◦

M × D0 and, for all t ∈ D0 , 

◦

◦

0

(E × D ) · C; M × D

(continuity)

0

∗ = (E · Ct ; M ). t

A.13 We saw earlier how the projection formula allows us to reduce the calculation of intersection multiplicities to the case where the intersection is zero-dimensional. We wish to see now how continuity allows us to deform in a family in order to calculate zero-dimensional intersection multiplicities. We will prove a dynamic formula for intersection multiplicities; this formula is known as conk servation of number. Let E be a k-dimensional cycle in M and let f := (f1 , . . . , fk ) ∈ (OM ) be such that E and V (f ) intersect properly in the single point p. This implies that V (f ) is purely k-codimensional inside M at p. In what follows, we assume that we are always working in an arbitrarily small neighborhood of p. Let g1 (z, t), . . . , gk (z, t) ∈ O Let C be the cycle ◦ be such that gi (z, 0) = fi (z) for all i. M ×D ◦     in M × D given by V (g1 (z, t), . . . , gk (z, t)) . Note that C0 = V (f ) (in M ) and that C has no components contained in M × {0}, for otherwise V (f ) would have a component of dimension at least (dim M ) + 1 − k. Applying continuity at t = 0, we find that  (E · V (f ); M ) =

◦

◦

0

(E × D ) · C; M × D

0

∗ 0

◦

◦

for a smaller disc D0 . Note that (E × D0 ) · C is a purely 1-dimensional cycle, say ◦

P

j

mj [Wj ].

0

Applying continuity at general t, we find that, for all t ∈ D , X (E · Ct ; M ) = mj [(Wj )t ] . Wj 6⊆M ×{0}

Hence, by openness and transversality, we find that m j = (E · Ct ; M )q for sufficiently small t 6= 0 X and q ∈ (Wj )t . Now, since (E · V (f ); M ) = mj [(Wj )0 ], we may apply our earlier Wj 6⊆M ×{0}

special case of dynamic intersections between curves and hypersurfaces to conclude the general conservation of number formula X
(E · V (f ))p = E · Ct q , ◦

q∈B ∩|E∩Ct | ◦

where  > 0 is sufficiently  small, B is an open ball of radius  centered at p, |t|  , and Ct equals V (g1 (z, t), . . . , gk (z, t)) . A.14 Finally, we need to define the proper push-forward of cycles (see [Fu, 1.4]). Let f : X → Y be a proper morphism of analytic spaces. Then, for each irreducible subvariety V ⊆ X, W := f (V ) is an irreducible subvariety of Y . There is an induced embedding of rational function fields

APPENDIX A: ANALYTIC CYCLES AND INTERSECTIONS

183

R(W ) ,→ R(V ), which is a finite field extension if V and W have the same dimension. Define the degree of V over W by  deg(V /W ) :=

[R(V ) : R(W )]

if dim W = dim V

0

if dim W 6= dim V,

where [R(V ) : R(W )] denotes the degree of the field extension, which equals the number of points in V ∩ f −1 (p) for a generic choice of p ∈ W . Define f∗ (V ) by f∗ (V ) = deg(V /W )[W ]. This extends linearly to a homomorphism which is called the proper push-forward of cycles: f∗

X

 X mV [V ] = mV f∗ (V ).

We will need the following special case of the more general push-forward formula (see [Fu], 2.3.c). Let π : M → N be a proper map between analytic manifolds. Let f ∈ ON . Let C be a cycle in M which intersects V (f ◦ π) properly in M . Then, V (f ) properly intersects π∗ (C) in N and (push-forward formula)


π∗ V (f ◦ π) · C = V (f ) · π∗ (C).

We need one other formula involving the push-forward and graphs of morphisms. Suppose that we have an analytic map f : M → N between analytic manifolds. Then, for any irreducible subvariety V ⊆ M , the graph of f|V , Gr(f|V ), is isomorphic to V . Thus, one would expect that intersecting with V in M could be identified with intersecting with Gr(f|V ) in M × N ; this is, in fact the case. P mi [Vi ] and B are properly intersecting cycles in M . Let Gr(A) := P Suppose that A := mi [Gr(f|Vi )] in M × N , and let pr : M × N → M denote the projection. Then, Gr(A) properly intersects B × N in M × N , and we have the graph formula: (graph formula)

(A · B; M ) = pr∗ (Gr(A) · (B × N ); M × N ).

This is easy to see: Gr(A) = (A × N ) · Gr(f ), since Gr(f ) determines a regular sequence in each OVi ×N (see A.4). Hence, Gr(A) · (B × N ) = (A × N ) · (B × N ) · Gr(f ) = ((A · B) × N ) · Gr(f ), where the last equality follows from normal slicing (see A.11). The graph formula follows easily by using A.4 again, together with the definition of the proper push-forward.

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APPENDIX B: THE DERIVED CATEGORY AND VANISHING CYCLES

This appendix contains a number of basic results on the derived category, perverse sheaves, and vanishing cycles. Primary sources for most of these results are [BBD], [Br], [De], [G-M3], [K-S2], [Mac2], [M-V], and [Ve].

This appendix is organized as follows: §1. Constructible Complexes – This section contains general results on bounded, constructible complexes of sheaves and the derived category. §2. Perverse Sheaves – This section contains the definition and basic results on perverse sheaves. Here, we also give the axiomatic characterization of the intersection cohomology complex. Finally in this section, we also give some results on the category of perverse sheaves. This categorical information is augmented by that in section 5. §3. Nearby and Vanishing Cycles – In this section, we define and examine the complexes of sheaves of nearby and vanishing cycles of an analytic function. These complexes contain hypercohomological information on the Milnor fibre of the function under consideration. §4. Some Quick Applications – In this section, we give three easy examples of results on Milnor fibres which follow from the machinery described in the previous three sections. §5. Truncation and Perverse Cohomology – This section contains an informal discussion on t-structures. This enables us to describe truncation functors and the perverse cohomology of a complex. It also sheds some light on our earlier discussion of the categorical structure of perverse sheaves.

§1. Constructible Complexes

Much of this section is lifted directly from Goresky and MacPherson’s paper “Intersection Homology II” [G-M3]. In this appendix, we are primarily interested in sheaves on complex analytic spaces, and we make an effort to state most results in this context. However, as one frequently wishes to do such things as intersect with a closed ball, one really needs to consider at least the real semi-analytic case (that is, spaces locally defined by finitely many real analytic inequalities). In fact, one can treat the subanalytic case. Generally, when we leave the analytic category we shall do so without comment, assuming the natural generalizations of any needed results. However, the precise statements in the subanalytic case can be found in [G-M2], [G-M3], and [K-S2]. Let R be a regular Noetherian ring with finite Krull dimension (e.g., Z, Q, or C). A complex (A• , d• ) (usually denoted simply by A• if the differentials are clear or arbitrary) d−1

d0

d1

d2

· · · → A−1 −−→ A0 −→ A1 −→ A2 −→ · · · 185

186

DAVID B. MASSEY

of sheaves of R-modules on a complex analytic space, X, is bounded if Ap = 0 for |p| large. The cohomology sheaves Hp (A• ) arise by taking the (sheaf-theoretic) cohomology of the complex. The stalk of Hp (A• ) at a point x is written Hp (A• )x and is isomorphic to what one gets by first taking stalks and then taking cohomology, i.e., H p (A•x ). The complex A• is constructible with respect to a complex analytic stratification, S = {Sα }, of X provided that, for all α and i, the cohomology sheaves Hi (A• |Sα ) are locally constant and have finitely-generated stalks; we write A• ∈ DS (X). If A• ∈ DS (X) and A• is bounded, we write A• ∈ DbS (X). If A• ∈ DbS (X) for some stratification (and, hence, for any refinement of S) we say that A• is a bounded, constructible complex and write A• ∈ Dbc (X). (Note, however, that Dbc (X) actually denotes the derived category and, while the objects of this category are, in fact, the bounded, constructible complexes, the morphisms are not merely maps between complexes. We shall return to this.) When it is important to indicate the base ring in the notation, we write DS (RX ), DbS (RX ), and b Dc (RX ). A single sheaf A on X is considered a complex, A• , on X by letting A0 = A and Ai = 0 for i 6= 0; thus, R•X denotes the constant sheaf on X. The shifted complex A• [n] is defined by (A• [n])k = An+k and differential dk[n] = (−1)n dk+n . A map of complexes is a graded collection of sheaf maps φ• : A• → B• which commute with the differentials. The shifted sheaf map φ•[n] : A• [n] → B• [n] is defined by φk[n] := φk+n (note the lack of a (−1)n ). A map of complexes is a quasi-isomorphism provided that the induced maps Hp (φ• ) : Hp (A• ) → Hp (B• ) are isomorphisms for all p. We use the term “quasi-isomorphic” to mean the equivalence relation generated by “existence of a quasi-isomorphism”; this is sometimes refered to as “generalized” quasi-isomorphic. If φ• : A• → I• is a quasi-isomorphism and each Ip is injective, then I• is called an injective resolution of A• . Injective resolutions always exist (in our setting), and are unique up to chain homotopy. However, it is sometimes important to associate one particular resolution to a complex, so it is important that there is a canonical injective resolution which can be associated to any complex (we shall not describe the canonical resolution here). If A• is a complex on X, then the hypercohomology module, Hp (X; A• ), is defined to be the p-th cohomology of the global section functor applied to the canonical injective resolution of A• . Note that if A is a single sheaf on X and we form A• , then Hp (X; A• ) = H p (X; A) = ordinary sheaf cohomology. In particular, Hp (X; R•X ) = H p (X; R). Note also that if A• and B• are quasi-isomorphic, then H∗ (X; A• ) ∼ = H∗ (X; B• ). If Y is a subspace of X and A• ∈ Dbc (X), then one usually writes H∗ (Y ; A• ) in place of H (Y ; A• |Y ). ∗

The usual Mayer-Vietoris sequence is valid for hypercohomology; that is, if U and V form an open cover of X and A• ∈ Dbc (X), then there is an exact sequence · · · → Hi (X; A• ) → Hi (U ; A• ) ⊕ Hi (V ; A• ) → Hi (U ∩ V ; A• ) → Hi+1 (X; A• ) → . . . . Of course, hypercohomology is not a homotopy invariant. However, it is true that: if S is a real analytic Whitney stratification of X, A• ∈ DbS (X), and r : X → [0, 1) is a proper real analytic

APPENDIX B

187

map such that, for all S ∈ S, r|S has no critical values in (0, 1), then the inclusion r−1 (0) ,→ X induces an isomorphism Hi (X; A• ) ∼ = Hi (r−1 (0); A• ). If R is a principal ideal domain, we may talk about the rank of a finitely-generated R-module. In this case, if A• ∈ Dbc (X), then the Euler characteristic, χ, of the stalk cohomology is defined as the P alternating sum of the ranks of the cohomology modules, i.e., χ(A• )x = (−1)i rank Hi (A• )x . • b If the hypercohomology modules are finitely-generated
– for instance, if A ∈ Dc (X) and X is ∗ • compact – then the Euler characteristic χ H (X; A ) is defined analogously. If Hi (A• ) = 0 for all but, possibly, one value of i - say, i = p, then A• is quasi-isomorphic to the complex that has Hp (A• ) in degree p and zero elsewhere. We reserve the term local system for a locally constant single sheaf or a complex which is concentrated in degree zero and is locally constant. If M is the stalk of a local system L on a path-connected space X, then L is determined up to isomorphism by a monodromy representation π1 (X, x) → Aut(M ), where x is a fixed point in X. For any A• ∈ Dbc (X), there is an E2 cohomological spectral sequence: p+q

E2p,q = H p (X; Hq (A• )) ⇒ H

(X; A• ).

If A• ∈ Dbc (X), x ∈ X, and (X, x) is locally embedded in some Cn , then for all  > 0 small, the ◦ ◦  restriction map Hq (B  (x); A• ) → Hq (A• )x is an isomorphism here, B  (x) = z ∈ Cn |z − x| < ◦

 . If, in addition, R is a principal ideal domain, the Euler characteristic χ H∗ (B  (x) − x; A• ) is defined and ◦

χ H∗ (B  (x) − x; A• ) = χ H∗ (S0 (x); A• ) = 0, where 0 < 0 <  and S0 (x) denotes the sphere of radius 0 centered at x. We now wish to say a little about the morphisms in the derived category Dbc (X). The derived category is obtained by formally inverting the quasi-isomorphisms so that they become isomorphisms in Dbc (X). Thus, A• and B• are isomorphic in Dbc (X) provided that there exists a complex C• and quasi-isomorphisms A• ← C• → B• ; A• and B• are then said to be incarnations of the same isomorphism class in Dbc (X). More generally, a morphism in Dbc (X) from A• to B• is an equivalence class of diagrams of maps of complexes A• ← C• → B• where A• ← C• is a quasi-isomorphism. Two such diagrams, f1

g1

f2

A• ←− C•1 −→ B• ,

g2

A• ←− C•2 −→ B• f

g

are equivalent provided that there exists a third such diagram A• ← − C• − → B• and a diagram • C1 f1

.

↑

f

& g1 g

A• ←−− C• −−→ B• f2

-

↓ C•2

% g2

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DAVID B. MASSEY

which commutes up to (chain) homotopy. Composition of morphisms in Dbc (X) is not difficult to describe. If we have two representatives of morphisms, from A• to B• and from B• to D• , respectively, f1

g1

A• ←− C•1 −→ B• ,

f2

g2

B• ←− C•2 −→ D•

then we consider the pull-back C•1 ×B• C•2 (in the category of chain complexes) and the projections π1 and π2 to C•1 and C•2 , respectively. As f2 is a quasi-isomorphism, so is π1 , and the composed f1 ◦π1

g2 ◦π2

morphism from A• to D• is represented by A• ←−−− C•1 ×B• C•2 −−−→ D• . If we restrict ourselves to considering only injective complexes, by associating to any complex its canonical injective resolution, then morphisms in the derived category become easy to describe – they are chain-homotopy classes of maps between the injective complexes. The moral is: in Dbc (X), we essentially only care about complexes up to quasi-isomorphism. Note, however, that the objects of Dbc (X) are not equivalence classes – this is one reason why it is important that to each complex we can associate a canonical injective resolution. It allows us to talk about certain functors in Dbc (X) being naturally isomorphic. When we write A• ∼ = B• , b b we mean in Dc (X). As we shall discuss later, Dc (X) is an additive category, but is not Abelian. Warning: While morphisms of complexes which induce isomorphisms on cohomology sheaves become isomorphisms in the derived category, there are morphisms of complexes which induce the zero map on cohomology sheaves but are not zero in the derived category. The easiest example of such a morphism is given by the following. Let X be a space consisting of two complex lines L1 and L2 which intersect in a single point e L denote the C-constant sheaf on Li extended by zero to all of X. There is p. For i = 1, 2, let C i eL ⊕ C e L , which on L1 − p a canonical map, α, from the sheaf CX to the direct sum of sheaves C 1 2 is id ⊕0, on L2 − p is 0 ⊕ id, and is the diagonal map on the stalk at p. Consider the complex, eL ⊕ C e L in degree 1, zeroes elsewhere, and the coboundary map A• , which has CX in degree 0, C 1 2 from degree 0 to degree 1 is α. This complex has cohomology only in degree 1. Nonetheless, the morphism of complexes from A• to C•X which is the identity in degree 0 and is zero elsewhere determines a non-zero morphism in the derived category. We now wish to describe derived functors; for this, we will need the derived category of an arbitrary Abelian category C. Let C be an Abelian category. Then, the derived category of bounded complexes in C is the category whose objects consist of bounded differential complexes of objects of C, and where the morphisms are obtained exactly as in the case of Dbc (X) – namely, by inverting the quasi-isomorphisms as we did above. Naturally, we denote this derived category by Db (C). We need some more general notions before we come back to complexes of sheaves. If C is an Abelian category, then we let Kb (C) denote the category whose objects are again bounded differential complexes of objects of C, but where the morphisms are chain-homotopy classes of maps of differential complexes. A triangle in Kb (C) is a sequence of morphisms A• → B• → C• → A• [1], which is usually written in the more“triangular” form A• −→ B• . [1] • C

APPENDIX B

189

A triangle in Kb (C) is called distinguished if it is isomorphic in Kb (C) to a diagram of maps of complexes φ e • −→ e• A B . [1] M•

e • → M• → A e • [1] are the canonical maps. where M• is the algebraic mapping cone of φ and B (Recall that the algebraic mapping cone is defined by e k+1 ⊕ B e k −→ A e k+2 ⊕ B e k+1 =: Mk+1 Mk := A (a, b) 7−→ (−∂a, φa + δb) e • and B e • respectively.) Note that if φ = 0, then we have an where ∂ and δ are the differentials of A • • • equality M = A [1] ⊕ B (recall that the shifted complex A• [1] has as its differential the negated, shifted differential of A• ). Now we can define derived functors. Let C denote the Abelian category of sheaves of R-modules on an analytic space X, and let C 0 be another Abelian category. Suppose that F is an additive, covariant functor from Kb (C) to Kb (C 0 ) such that F ◦[1] = [1]◦F and such that F takes distinguished triangles to distinguished triangles (such an F is called a functor of triangulated categories). Suppose also that, for all complexes of injective sheaves I• ∈ Kb (C) which are quasi-isomorphic to 0, F (I• ) is also quasi-isomorphic to 0. Then, F induces a morphism RF – the right derived functor of F – from Db (X) to Db (C 0 ); for any A• ∈ Db (X), let A• → I• denote the canonical injective resolution of A• , and define RF (A• ) := F (I• ). The action of RF on the morphisms is the obvious associated one. A morphism F : Kb (C) → Kb (C 0 ) as described above is frequently obtained by starting with a left-exact functor T : C → C 0 and then extending T in a term-wise fashion to be a functor from Kb (C) to Kb (C 0 ). In this case, we naturally write RT for the derived functor. This is the process which is applied to: Γ(X; ·) (global sections); Γc (X; ·) (global sections with compact support); f∗ (direct image); f! (direct image with proper supports); and f ∗ (pull-back or inverse image), where f : X → Y is a continuous map (actually, in these notes, we would need an analytically constructible map; e.g., an analytic map). If the functor T is an exact functor from sheaves to sheaves, then RT (A• ) ∼ = T (A• ); in this • b • case, we normally suppress the R. Hence, if f : X → Y , A ∈ D (X), and B ∈ Db (Y ), we write: f ∗ B• ; f! A• , if f is the inclusion of a subspace and, hence, f! is extension by zero;

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DAVID B. MASSEY

f∗ A• , if f is the inclusion of a closed subspace. Note that hypercohomology is just the cohomology of the derived global section functor, i.e., H∗ (X; ·) = H ∗ ◦ RΓ(X; ·). The cohomology of the derived functor of global sections with compact support is the compactly supported hypercohomology and is denoted H∗c (X; A• ). If f : X → Y is the inclusion of a subset and B• ∈ Db (Y ), then the restriction of B• to X is defined to be f ∗ (B• ), and is usually denoted by B• |X .

If f : X → Y is continuous and A• ∈ Dbc (X), there is a canonical map Rf! A• → Rf∗ A• . For f : X → Y continuous, there are canonical isomorphisms RΓ(X; A• ) ∼ = RΓ(Y ; Rf∗ A• ) and RΓc (X; A• ) ∼ = RΓc (Y ; Rf! A• ) which lead to canonical isomorphisms H∗ (X; A• ) ∼ = H∗ (Y ; Rf∗ A• ) and H∗c (X; A• ) ∼ = H∗c (Y ; Rf! A• ) for all A• in Dbc (X). If f : X → Y is continuous, A• ∈ Dbc (X), and B• ∈ Dbc (Y ) , there are natural maps induced by restriction of sections B• → Rf∗ f ∗ B• and f ∗ Rf∗ A• → A• . (X × Ck ), and π : X × Ck → X is the projection, If {Sα } is a stratification of X, A• ∈ Db {Sα ×Ck } then restriction of sections induces a quasi-isomorphism π ∗ Rπ∗ A• → A• . It follows easily that if j : X ,→ X ×Ck is the zero section, then π ∗ j ∗ A• ∼ = A• . This says exactly • k what one expects: the complex A has a product structure in the C directions. An important consequence of this is the following: let S = {Sα } be a Whitney stratification of X and let A• ∈ DbS (X). Let x ∈ Sα ⊆ X. As Sα is a Whitney stratum, X has a product structure along Sα near x. By the above, A• itself also has a product structure along Sα . Hence, by taking a normal slice, many problems concerning the complex A• can be reduced to considering a zero-dimensional stratum. Let A• , B• ∈ Dbc (X). Define A• ⊗ B• to be the single complex which is associated to the double L

complex Ap ⊗ Bq . The left derived functor A• ⊗ ∗ is defined by L

A • ⊗ B• = A • ⊗ J • , where J• is a flat resolution of B• , i.e., the stalks of J• are flat R-modules and there exists a quasi-isomorphism J• → B• . L L For all A• , B• ∈ Db (X), there is an isomorphism A• ⊗ B• ∼ = B• ⊗ A • . c

APPENDIX B

191

For any map f : X → Y and any A• , B• ∈ Dbc (Y ), L

L

f ∗ (A• ⊗ B• ) ∼ = f ∗ A • ⊗ f ∗ B• .

Fix a complex B• on X. There are two covariant functors which we wish to consider: the functor Hom• (B• , ∗) from the category of complexes of sheaves to complexes of sheaves and the functor Hom• (B• , ∗) from the category of complexes of sheaves to the category of complexes of R-modules. These functors are given by Y n (Hom• (B• , A• )) = Hom(Bp , An+p ) p∈Z

and n

(Hom• (B• , A• )) =

Y

Hom(Bp , An+p )

p∈Z

with differential given by [∂ n f ]p = ∂ n+p f p + (−1)n+1 f p+1 ∂ p (there is an indexing error in [Iv, 12.4]). The associated derived functors are RHom• (B• , ∗) and RHom• (B• , ∗), respectively. If P• → B• is a projective resolution of B• , then, in Dbc (X), RHom• (B• , A• ) is isomorphic to Hom• (P• , A• ). For all k, RHom• (B• , A• [k]) = RHom• (B• , A• )[k]. The functor RHom• (B• , ∗) is naturally isomorphic to the derived global sections functor applied to RHom• (B• , ∗), i.e., for any A• ∈ Dbc (X), RHom• (B• , A• ) ∼ = RΓ (X; RHom• (B• , A• )) . H 0 (RHom• (B• , A• )) is naturally isomorphic as an R-module to the derived category homomorphisms from B• to A• , i.e., H 0 (RHom• (B• , A• )) ∼ = HomDb (X) (B• , A• ). c

If B• and A• have locally constant cohomology sheaves on X then, for all x ∈ X, RHom• (B• , A• )x is naturally isomorphic to RHom• (B•x , A•x ). For all A• , B• , C• ∈ Dbc (X), there is a natural isomorphism L

RHom• (A• ⊗ B• , C• ) ∼ = RHom• (A• , RHom• (B• , C• )). Moreover, if C• has locally constant cohomology sheaves, then there is an isomorphism L

L

RHom• (A• , B• ⊗ C• ) ∼ = RHom• (A• , B• ) ⊗ C• .

For all j, we define Extj (B• , A• ) := Hj (RHom• (B• , A• )) and define Extj (B• , A• ) := H j (RHom• (B• , A• )).

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It is immediate that we have isomorphisms of R-modules Extj (B• , A• ) = H 0 (RHom• (B• , A• [j])) ∼ = HomDb (X) (B• , A• [j]). c

L If X = point, B• ∈ Dbc (X), and the base ring is a PID, then B• ∼ = k Hk (B• )[−k] in Dbc (X); if we also have A• ∈ Dbc (X), then L

Hi (A• ⊗ B• ) ∼ =

 M

 Hp (A• ) ⊗ Hq (B• ) ⊕

M

p+q=i


Tor(Hr (A• ), Hs (B• )) .

r+s=i+1

If, in addition, the cohomology modules of A• are projective (hence, free), then M
HomDb (X) (A• , B• ) ∼ Hom Hk (A• ), Hk (B• ) . = c

k

If we have a map f : X → Y , then the functors f ∗ and Rf∗ are adjoints of each other in the derived category. In fact, for all A• on X and B• on Y , there is a canonical isomorphism in Dbc (Y ) RHom• (B• , Rf∗ A• ) ∼ = Rf∗ RHom• (f ∗ B• , A• ) and so HomDb (Y ) (B• , Rf∗ A• ) ∼ = H 0 (RHom• (B• , Rf∗ A• )) ∼ = H0 (Y ; RHom• (B• , Rf∗ A• )) c

∼ = H0 (X; RHom• (f ∗ B• , A• )) ∼ = H 0 (RHom• (f ∗ B• , A• )) ∼ = HomDb (X) (f ∗ B• , A• ). c

We wish now to describe an analogous adjoint for Rf! Let I• be a complex of injective sheaves on Y . Then, f ! (I• ) is defined to be the sheaf associated to the presheaf given by Γ(U ; f ! I• ) = Hom• (f! K•U , I• ), for any open U ⊆ X, where K•U denotes the canonical injective resolution of the constant sheaf R•U . For any A• ∈ Dbc (X), define f ! A• to be f ! I• , where I• is the canonical injective resolution of A• . Now that we have this definition, we may state: (Verdier Duality) If f : X → Y, A• ∈ Dbc (X), and B• ∈ Dbc (Y ), then there is a canonical b isomorphism in Dc (Y ): Rf∗ RHom• (A• , f ! B• ) ∼ = RHom• (Rf! A• , B• ) and so

HomDb (X) (A• , f ! B• ) ∼ = HomDb (Y ) (Rf! A• , B• ). c

c

If B• and C• are in Dbc (Y ), then we have an isomorphism f ! RHom• (B• , C• ) ∼ = RHom• (f ∗ B• , f ! C• ).

APPENDIX B

193

Let f : X → point. Then, the dualizing complex, D•X , is f ! applied to the constant sheaf, i.e., D•X = f ! R•pt . For any complex A• ∈ Dbc (X), the Verdier dual (or, simply, the dual) of A• is RHom• (A• , D•X ) and is denoted by DX A• ( or just DA• ). There is a canonical isomorphism between D•X and the dual of the constant sheaf on X, i.e., D•X ∼ = DR•X .

Let A• ∈ Dbc (X). The dual of A• , DA• , is well-defined up to quasi-isomorphism by: for any open U ⊆ X, there is a natural split exact sequence: 0 → Ext(Hq+1 (U ; A• ), R) → H−q (U ; DA• ) → Hom(Hqc (U ; A• ), R) → 0. c

In particular, if R is a field, then H−q (U ; DA• ) ∼ = Hqc (U ; A• ), and so ◦

◦

• Hq (DA• )x ∼ = Hq (B  (x); DA• ) ∼ = H−q c (B  (x); A ).

If, in addition, X is compact, H−q (X; DA• ) ∼ = Hq (X; A• ). Dualizing is a local operation, i.e., if i : U ,→ X is the inclusion of an open subset and A• ∈ then i∗ DA• ∼ = Di∗ A• . If L is a local system on a connected real m-manifold, N , then (DL• )[−m] is quasi-isomorphic to a local system; if, in addition, N is smooth and oriented, and L is actually locally free with stalks Ra and monodromy representation η : π1 (N, p) → Aut(Ra ), then DL• [−m] is quasiisomorphic to a local system with stalks Ra and monodromy t η : π1 (N, p) → Aut(Ra ), where t η(α) = transpose of η(α). Dbc (X),

If A• ∈ Dbc (X), then D(A• [n]) = (DA• )[−n]. If π : X × Cn → X is projection, then D(π ∗ A• )[−n] ∼ = π ∗ (DA• )[n]. The dualizing complex, DX , is quasi-isomorphic to the complex of sheaves of singular chains on X which is associated to the complex of presheaves, C• , given by Γ(U ; C−p ) := Cp (X, X − U ; R). The cohomology sheaves of D•X are non-zero in negative degrees only, with stalks H−p (D•X )x = Hp (X, X − x; R). If X is a smooth, oriented, real m-manifold, then D•X [−m] is quasi-isomorphic to R•X ; hence, for any A• ∈ Dbc (X),
DA• ∼ = RHom• (A• , R•X [m]) = RHom• (A• , R•X ) [m]. L

D•V ×W is naturally isomorphic to π1∗ D•V ⊗ π2∗ D•W , where π1 and π2 are the projections onto V and W , respectively. H∗ (X; D•X ) ∼ = homology with closed supports = Borel-Moore homology. If X is a real, smooth, oriented m-manifold and R = R, then D•X [−m] is naturally isomorphic to the complex of real differential forms on X.

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D•X is constructible with respect to any Whitney stratification of X. It follows that if S is a Whitney stratification of X, then A• ∈ DbS (X) if and only if DA• ∈ DbS (X). The functor D from Dbc (X) to Dbc (X) is contravariant, and DD is naturally isomorphic to the identity. For all A• , B• ∈ Dbc (X), we have isomorphisms   L • • • ∼ • • RHom• (A• , B• ) ∼ RHom (DB , DA ) D DB ⊗ A . = =

If f : X → Y is continuous, then we have natural isomorphisms Rf! ∼ = DRf∗ D and f ! ∼ = Df ∗ D.

If Y ⊆ X and f : X − Y ,→ X is the inclusion, we define Hk (X, Y ; A• ) := Hk (X; f! f ! A• ).

Excision has the following form: if Y ⊆ U ⊆ X, where U is open in X and Y is closed in X, then Hk (X, X − Y ; A• ) ∼ = Hk (U, U − Y ; A• ). If x ∈ X and A• ∈ Dbc (X), then for all  > 0 sufficiently small, ◦

◦

◦

Hqc (B  (x); A• ) ∼ = Hq (B  (x), B  (x) − x; A• ) ∼ = Hq (X, X − x; A• ) and so, if R is a field, ◦

◦

H−q (DA• )x ∼ = Hq (B  (x), B  (x) − x; A• ) ∼ = Hq (X, X − x; A• ).

If f : X → Y and g : Y → Z, then there are natural isomorphisms R(g ◦ f )∗ ∼ = Rg∗ ◦ Rf∗

R(g ◦ f )! ∼ = Rg! ◦ Rf! and

(g ◦ f )∗ ∼ = f ∗ ◦ g∗

(g ◦ f )! ∼ = f ! ◦ g! .

Suppose that f : Y ,→ X is inclusion of a subset. Then, if Y is open, f ! = f ∗ . If Y is closed, then Rf! = f! = f∗ = Rf∗ . If f : Y ,→ X is the inclusion of one complex manifold into another and B• ∈ Dbc (X) has locally constant cohomology on X, then f ! B• has locally constant cohomology on Y and f ! B• ∼ = f ∗ B• [−2 codimX Y ].

APPENDIX B

195

(Here, we mean the complex codimension. There is an error here in [G-M3]; they have the negation of the correct shift.) If π : X × Cn → X is projection and A• ∈ Dbc (X), then (π ! A• )[−2n] ∼ = π ∗ A• . If f : X → Y is continuous, A• ∈ Dbc (X), and B• ∈ Dbc (Y ) , then dual to the canonical maps B• → Rf∗ f ∗ B• and f ∗ Rf∗ A• → A• are the canonical maps Rf! f ! B• → B• and A• → f ! Rf! A• .

If fˆ

Z −−−−→   π ˆy

W  π y

f

X −−−−→ S ˆ ∗ A• ∼ is a pull-back diagram (fibre square, Cartesian diagram), then for all A• ∈ Dbc (X), Rfˆ! π = ∗ • π Rf! A (there is an error in [G-M3]; they have lower ∗’s, not lower !’s, but see below for when these agree) and, dually, Rfˆ∗ π ˆ ! A• ∼ = π ! Rf∗ A• . In particular, if f is proper (and, hence, fˆ is proper) or π is the inclusion of an open subset (and, hence, so is π ˆ , up to homeomorphism), then Rfˆ∗ π ˆ ∗ A• ∼ = π ∗ Rf∗ A• ; this is also true if W = S × Cn and π : W → S is projection (and, hence, up to homeomorphism, π ˆ is projection from X × Cn to X). L

L

If we have A• ∈ Dbc (X) and B• ∈ Dbc (W ), then we let A• S B• := π ˆ ∗ A• ⊗ fˆ∗ B• , assuming that the maps π ˆ and fˆ are clear. If S is a point, so that Z ∼ = X × W , then we omit the S in the L

notation and write simply A•  B• . There is a K¨ unneth formula, which we now state in its most general form, in terms of maps over a base space S. Suppose that we have two maps f1 : X1 → Y1 and f2 : X2 → Y2 over S, i.e., there are commutative diagrams f1

f2

X1 −→ Y1 r1 & . t1 S

X2 −→ Y2 r2 & . t 2 S

and

. •

Then, there is an induced map f = f1 ×S f2 : X1 ×S X2 → Y1 ×S Y2 . If A ∈ B• ∈ Dbc (X2 ), there is the K¨ unneth isomorphism L

Dbc (X1 )

and

L

Rf! (A• S B• ) ∼ = Rf1 ! A• S Rf2 ! B• . Using the above notation, if S is a point and F• ∈ Dbc (Y1 ) and G• ∈ Dbc (Y2 ), then there is a natural isomorphism (the adjoint K¨ unneth isomorphism) L

L

f ! (F•  G• ) ∼ = f1 ! F•  f2 ! G• .

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DAVID B. MASSEY

If we let q1 and q2 denote the projections from Y1 × Y2 onto Y1 and Y2 , respectively, then the adjoint K¨ unneth formula can be proved by using the following natural isomorphism twice L

DF•  G• ∼ = RHom• (q1∗ F• , q2! G• ).

Let Z be a locally closed subset of an analytic space X. There are two derived functors, associated to Z, that we wish to describe: the derived functors of restricting-extending to Z, and of taking the sections supported on Z. Let i denote the inclusion of Z into X. If A is a (single) sheaf on X, then the restriction-extension of A to Z, (A)Z , is given by i! i∗ (A). Thus, up to isomorphism, (A)Z is characterized by ((A)Z )|Z ∼ = A|Z and ((A)Z )|X−Z = 0. This functor is exact, and so we also denote the derived functor by ()Z . Now, we want to define the sheaf of sections of A supported by Z, ΓZ (A). If U is an open subset of X which contains Z, then we define  ΓZ (U; A) := ker Γ(U; A) → Γ(U − Z; A) . Up to isomorphism, ΓZ (U; A) is independent of the open set U (this uses that A is a sheaf, not just a presheaf). The sheaf ΓZ (A) is defined by, for all open U ⊆ X, Γ(U; ΓZ (A)) := ΓU ∩Z (U; A). One easily sees that supp ΓZ (A) ⊆ Z. It is also easy to see that, if Z is open, then ΓZ (A) = i∗ i∗ (A). The functor ΓZ () is left exact; of course, we denote the right derived functor by RΓZ (). There is a canonical isomorphism i! ∼ = i∗ ◦ RΓZ . It follows that, if Z is closed, then RΓZ ∼ = i! i! . ∗ In addition, if Z is open, then RΓZ ∼ Ri i . = ∗ Avoiding Injective Resolutions: To calculate right derived functors from the definition, one must use injective resolutions. However, this is inconvenient in many proofs if some functor involved in the proof does not take injective complexes to injective complexes. There are (at least) four “devices” which come to our aid, and enable one to prove many of the isomorphisms described earlier; these devices are fine resolutions, flabby resolutions, c-soft resolutions, and injective subcategories with respect to a functor. If T is a left-exact functor on the category of sheaves on X, then the right derived functor RT is defined by applying T term-wise to the sheaves in a canonical injective resolution. The importance of saying that a certain subcategory of the category of sheaves on X is injective with respect to T is that one may take a resolution in which the individual sheaves are in the given subcategory, then apply T term-wise, and end up with a complex which is canonically isomorphic to that produced by RT . Recall that a single sheaf A on X is: fine, if partitions of unity of A subordinate to any given locally finite open cover of X exist; flabby, if for every open subset U ⊆ X, the restriction homomorphism Γ(X; A) → Γ(U; A) is a surjection; c-soft, if for every compact subset K ⊆ X, the restriction homomorphism Γ(X; A) → Γ(K; A) is a surjection;

APPENDIX B

197

Injective sheaves are flabby, and flabby sheaves are c-soft. In addition, fine sheaves are c-soft. The subcategory of c-soft sheaves is injective with respect to the functors Γ(X; ∗), Γc (X; ∗), and f! . The subcategory of flabby sheaves is injective with respect to the functor f∗ . If A• is a bounded complex of sheaves, then a bounded c-soft resolution of A• is given by A → A• ⊗ S• , where S• is a c-soft, bounded above, resolution of the base ring (which always exists in our context). •

Triangles: Dbc (X) is an additive category, but is not an Abelian category. In place of short exact sequences, one has distinguished triangles, just as we did in Kb (C). A triangle of morphisms in Dbc (X) A• −→ B• . [1] • C (the [1] indicates a morphism shifted by one, i.e., a morphism C• → A• [1]) is called distinguished if it is isomorphic in Dbc (X) to a diagram of sheaf maps φ ˜ • −→ ˜• A B . [1] M•

where M• is the algebraic mapping cone of φ and B• → M• → A• [1] are the canonical maps. [1]

The “in-line” notation for a triangle is A• → B• → C• → A• [1] or A• → B• → C• −→. Any short exact sequence of complexes becomes a distinguished triangle in Dbc (X). Any edge of a distinguished triangle determines the triangle up to (non-canonical) isomorphism in Dbc (X); more α

β

γ

specifically, we can “turn” the distinguished triangle: A• − → B• − → C• − → A• [1] is a distinguished triangle if and only if β

−α[1]

γ

B• − → C• − → A• [1] −−−→ B• [1] is a distinguished triangle. Given two distinguished triangles and maps u and v which make the left-hand square of the following diagram commute A• → B• → C• → A• [1] ↓u

↓v

↓ u[1]

f• → B f• → C f• → A f• [1], A f• such that there exists a (not necessarily unique) w : C• → C A• → B• → C• → A• [1] ↓u

↓v

↓w

↓ u[1]

f• → B f• → C f• → A f• [1] A

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DAVID B. MASSEY

also commutes. We say that the original commutative square embeds in a morphism of distinguished triangles. We will now give the octahedral lemma, which allows one to realize an isomorphism between mapping cones of two composed maps. Suppose that we have two distinguished triangles f

g

h

β

γ

δ

A• − → B• − → C• − → A• [1] and → B• [1]. → F• − B• − → E• − Then, there exists a complex M• and two distinguished triangles β◦f

τ

ω

A• −−→ E• − → M• − → A• [1] and σ

g[1]◦δ

ν

C• − → M• − → F• −−−→ C• [1]. such that the following diagram commutes f

g

h

A• −−−→ B• −−−→ C• −−−→ A• [1] id ↓

β↓ β◦f

σ↓ τ

id ↓ ω

A• −−−→ E• −−−→ M• −−−→ A• [1] f↓

id ↓ β

ν↓ γ

f [1] ↓ δ

B• −−−−→ E• −−−→ F• −−−→ B• [1] g↓

τ↓ σ

id ↓ ν

g[1] ↓ g[1]◦δ

C• −−−→ M• −−−→ F• −−−→ C• [1]. It is somewhat difficult to draw this in its octahedral form (and worse to type it); moreover, it is no easier to read the relations from the octahedron. However, the interested reader can give it a try: the octahedron is formed by gluing together two pyramids along their square bases. One pyramid has B• at its top vertex, with A• , C• , E• , and F• at the vertices of its base, and has the original two distinguished triangles as opposite faces. The other pyramid has M• at its top vertex, with A• , C• , E• , and F• at the vertices of its base, and has the other two distinguished triangles (whose existence is asserted in the lemma) as opposite faces. The two pyramids are joined together by matching the vertices of the two bases, forming an octahedron in which the faces are alternately distinguished and commuting. A distinguished triangle determines long exact sequences on cohomology and hypercohomology: · · · → Hp (A• ) → Hp (B• ) → Hp (C• ) → Hp+1 (A• ) → · · · · · · → Hp (X; A• ) → Hp (X; B• ) → Hp (X; C• ) → Hp+1 (X; A• ) → · · · . L

If f : X → Y and F• ∈ Dbc (X), then the functors Rf∗ , Rf! , f ∗ , f ! , and F• ⊗ ∗ all take distinguished triangles to distinguished triangles (with all arrows in the same direction and the shift in the same place).

APPENDIX B

199

As for RHom• , if F• ∈ Dbc (X) and A• → B• → C• → A• [1] is a distinguished triangle in then we have distinguished triangles

Dbc (X),

RHom• (F• , A• ) −→ RHom• (F• , B• ) RHom• (A• , F• ) ←− RHom• (B• , F• ) . and % [1] [1] & • • • • RHom (F , C ) RHom (C• , F• ). By applying the right-hand triangle above to the special case where F• = D•X , we find that the dualizing functor D also takes distinguished triangles to distinguished triangles, but with a reversal of arrows, i.e., if we have a distinguished triangle A• → B• → C• → A• [1] in Dbc (X), then, by dualizing, we have distinguished triangles DA• ←− DB• % [1] & • DC

DC• −→ DB• . [1] • DA .

or

There are (at least) six distinguished triangles associated to the functors ()Z and RΓZ . Let F• be in Dbc (X), U1 and U2 be open subsets of X, Z1 and Z2 be closed subsets of X, Z be locally closed in X, and Z 0 be closed in Z. Then, we have the following distinguished triangles: [1]

RΓU1 ∪U2 (F• ) → RΓU1 (F• ) ⊕ RΓU2 (F• ) → RΓU1 ∩U2 (F• ) −→

[1]

RΓZ1 ∩Z2 (F• ) → RΓZ1 (F• ) ⊕ RΓZ2 (F• ) → RΓZ1 ∪Z2 (F• ) −→

[1]

(F• )U1 ∩U2 → (F• )U1 ⊕ (F• )U2 → (F• )U1 ∪U2 −→

[1]

(F• )Z1 ∪Z2 → (F• )Z1 ⊕ (F• )Z2 → (F• )Z1 ∩Z2 −→

[1]

RΓZ 0 (F• ) → RΓZ (F• ) → RΓZ−Z 0 (F• ) −→

[1]

(F• )Z−Z 0 → (F• )Z → (F• )Z 0 −→ .

If j : Y ,→ X is the inclusion of a closed subspace and i : U ,→ X the inclusion of the open complement, then for all A• ∈ Dbc (X), the last two triangles above give us distinguished triangles Ri! i! A• −→ A• . [1] ∗ • Rj∗ j A

and

Rj! j ! A• −→ A• . [1] ∗ • Ri∗ i A ,

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DAVID B. MASSEY

where the second triangle can be obtained from the first by dualizing. (Note that Ri! = i! , Rj∗ = j∗ = j! = Rj! , and i! = i∗ .) The associated long exact sequences on hypercohomology are those for the pairs H∗ (X, Y ; A• ) and H∗ (X, U ; A• ), respectively. By applying these two triangles to Ri∗ i∗ A• and Ri! i! A• , respectively, we obtain a natural isomorphism Rj! j ! Ri! i! A• [1] ∼ = Rj∗ j ∗ Ri∗ i∗ A• . As in our earlier discussion of the octahedral lemma, all of the morphisms of the last two paragraphs fit into the fundamental octahedron of the pair (X, Y ). The four distinguished triangles making up the fundamental octahedron are the top pair Ri! i! A• → A• → Rj∗ j ∗ A• → Ri! i! A• [1] and A• → Ri∗ i∗ A• → Rj! j ! A• [1] → A• [1] and the bottom pair Ri! i! A• → Ri∗ i∗ A• → M• → Ri! i! A• [1] and Rj∗ j ∗ A• → M• → Rj! j ! A• [1] → Rj∗ j ∗ A• [1], where M• ∼ = Rj∗ j ∗ Ri∗ i∗ A• . = Rj! j ! Ri! i! A• [1] ∼

§2. Perverse Sheaves Suppose that P• ∈ Dbc (X). There are two non-equivalent definitions of what it means for P• to be perverse. The first one (which is actually the definition of perverse) is a purely local definition and, when the base ring is a field, is symmetric with respect to dualizing. This definition is definitely the more elegant of the two, but it gives cohomology groups only in negative dimensions; this seems non-intuitive from the topologist’s point of view. The second definition of perverse - which differs from the first only by a shift - has the advantage that the cohomology groups appear in non-negative dimensions only. Also, the constant sheaf on a local complete intersection is such a sheaf and, with this definition of perverse, the nearby and vanishing cycles (see §3) of a perverse sheaf are again perverse. Finally, if one wants intersection cohomology with its usual indexing (that is, the indexing that gives cohomology in non-negative dimensions) to be a perverse sheaf, then one must use this second definition of perverse. Despite these advantages of this second definition of perverse, the fact that it does not localize well on non-pure-dimensional spaces complicates general statements in almost every case. Statements tend to be much cleaner using the first definition. Hence, below, we use the term perverse sheaf for this first definition, and use positively perverse sheaf for the second definition. We shall give most statements in terms of perverse sheaves only; the reader may do the necessary shifts to obtain the positively perverse statements. The exceptions to this are those few statements which seem cleaner using positively perverse. Definition: Let X be a complex analytic space, and for each x ∈ X, let jx : x ,→ X denote the inclusion. If F• ∈ Dbc (X), then the support of Hi (F• ) is the closure in X of {x ∈ X| Hi (F• )x 6= 0} = {x ∈ X| Hi (jx∗ F• ) 6= 0};

APPENDIX B

201

we denote this by suppi F• . The i-th cosupport of F• is the closure in X of ◦

◦

{x ∈ X| Hi (jx! F• ) 6= 0} = {x ∈ X| Hi (B  (x), B  (x) − x; F• ) 6= 0}; we denote this by cosuppi F• . If the base ring, R, is a field, then cosuppi F• = supp-i DF• . Definition: Let X be a complex analytic space (not necessarily pure dimensional). Then, P• ∈ Dbc (X) is perverse provided that for all i: (support)

dim(supp-i P• ) 6 i;

(cosupport)

dim(cosuppi P• ) 6 i,

where we set the dimension of the empty set to be −∞. This definition is equivalent to: let {Sα } be any Whitney stratification of X with respect to which P• is constructible, and let sα : Sα ,→ X denote the inclusion. Then, (support)

Hk (s∗α P• ) = 0

for k > −dimC Sα ;

(cosupport)

Hk (s!α P• ) = 0

for k < −dimC Sα .

(There is a missing minus sign in [G-M2, 6.A.5].) If X is an n-dimensional space, then P• is positively perverse if and only if P• [n] is perverse.

From the definition, it is clear that being perverse is a local property. If the base ring R is, in fact, a field, then the support and cosupport conditions can be written in the following form, which is symmetric with respect to dualizing: (support)

dim(supp-i P• ) 6 i;

(cosupport)

dim(supp-i DP• ) 6 i.

Suppose that P• is perverse on X, (X, x) is locally embedded in Cn , S is a stratum of a Whitney stratification with respect to which P• is constructible, and x ∈ S. Let M be a normal slice of X at x; that is, let M be a smooth submanifold of Cn of dimension n − dim S which transversely intersects S at x. Then, for some open neighborhood U of x in X, P•|X∩M ∩U [−dim S] is perverse on X ∩ M ∩ U . Let P• ∈ Dbc (X); one can use this normal slicing proposition to prove: if P• is perverse, then Hi (P• ) = 0 for all i < −dim X; and so, if P• is positively perverse, then Hi (P• ) = 0 for all i < 0.

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A converse to the normal slicing proposition is: if π : X × Cs → X is projection and P• is positively perverse on X, then π ∗ P• is also positively perverse. Thus, if P• is perverse on X, then π ∗ P• [s] is perverse. Suppose P• ∈ Dbc (X n ). Let Σ = supp H∗ (P• ). Then, P• is perverse on X if and only if P•|Σ is perverse on Σ. Hence, P• is positively perverse on X if and only if P•|Σ [codimX Σ] is positively perverse on Σ. Another way of saying this is: if j : Σ → X is the inclusion of a closed subspace, then Q• is perverse on Σ if and only if j! Q• is perverse on X; hence, Q• is positively perverse on Σ if and only if j! Q• [−codimX Σ] is positively perverse on X. It follows that if P• is perverse, then Hi (P• ) = 0 unless −dim Σ 6 i 6 0; and so, if P• is positively perverse, then Hi (P• ) = 0 unless codimX Σ 6 i 6 n. In particular, on an n-dimensional space, a positively perverse sheaf which is supported only at isolated points has cohomology only in dimension n (i.e., the middle dimension). The constant sheaf R•X is positively perverse provided that X is a pure-dimensional local complete intersection. More generally, if X is a pure-dimensional local complete intersection, and M is a locally free sheaf of R-modules, then M• is a positively perverse sheaf on X. The other basic example of a perverse sheaf that we wish to give is that of intersection cohomology with local coefficients (with the perverse indexing, i.e., cohomology in degrees less than or equal to zero). Note that the definition below is shifted by − dimC X from the definition in [GM3], and yields a perverse sheaf which has possibly non-zero cohomology only in degrees between − dimC X and −1, inclusive. Let X be a n-dimensional complex analytic set, let X (n) = X1 ∪ . . . Xk be the union of the n-dimensional components of X, and let L be a local system on a smooth, open dense subset, ◦

X, of X (n) . Then, in Dbc (X), there is an object, IC•X (L), called the intersection cohomology with coefficients in L which is uniquely determined up to quasi-isomorphism by: 0)

IC•X (L)|X−X (n) = 0;

1)

IC•X (L)| ◦ = L• [n];

2)


H ICX (L) = 0

X

3) 4)

i

•

for i < −n;
dim supp−i IC•X (L) < i for all i < n;
dim cosuppi IC•X (L) < i for all i < n.

Note the strict inequalities in 3) and 4). The uniqueness assertion implies that IC• (L) ∼ = j 1 IC• (L| X

!

X1

◦ X1 ∩X

) ⊕ · · · ⊕ j!k IC•X (L| k

◦ Xk ∩X

),

where j m denotes the inclusion of Xm into X. In many sources, IC•X (L) is only defined if X is pure-dimensional. We find it convenient to have the intersection cohomology complex defined in the general situation – though, condition 0) above

APPENDIX B

203

says that our intersection cohomology complex is precisely the intersection cohomology complex on the pure-dimensional space X (n) extended by zero to all of X. See section 5 for more on IC•X (L). The uniqueness assertion which accompanied our axioms for intersection cohomology implies that IC•X (L) is semi-simple. More precisely, suppose that X is a n-dimensional complex analytic set, let be the union of the n-dimensional components of X, and let L be a local system on a ◦

smooth, open dense subset, X, of X (n) .

The Category of Perverse Sheaves (see, also, section 5) The category of perverse sheaves on X, P erv(X), is the full subcategory of Dbc (X) whose objects are the perverse sheaves. Given a Whitney stratification, S, of X, it is also useful to consider the category P ervS (X) := P erv(X) ∩ DbS (X) of perverse sheaves which are constructible with respect to S. P erv(X) and P ervS (X) are both Abelian categories in which the short exact sequences 0 → A• → B• → C• → 0 are precisely the distinguished triangles A• −→ B• . [1] C• . If we have complexes A• , B• , and C• in Dbc (X) (resp. DbS (X)), a distinguished triangle A• → B → C• → A• [1], and A• and C• are perverse, then B• is also in P erv(X) (resp. P ervS (X)). •

If the Whitney stratification S has a finite number of strata, then P ervS (X) is actually an Artinian category, which means that every perverse sheaf which is constructible with respect to S has a finite composition series in P ervS (X) with uniquely determined simple subquotients. If X is compact, then P erv(X) is also Artinian. The simple objects in P erv(X) (resp. P ervS (X)) are extensions by zero of intersection cohomology sheaves on irreducible analytic subvarieties (resp. connected components of strata) of X with coefficients in irreducible local systems. To be precise, let M be a connected analytic submanifold (resp. a connected component of a stratum) of X and let LM be an irreducible local system on M ; then, the pair (M , LM ) is called an irreducible enriched subvariety of X (where M denotes the closure of M ). Let j : M ,→ X denote the inclusion. Then, the simple objects of P erv(X) (resp. P ervS (X)) are those of the form j! IC• (LM ), where (M , LM ) is an irreducible M enriched subvariety (again, we are indexing intersection cohomology so that it is non-zero only in non-positive dimensions). Finally, we wish to state the decomposition theorem of Beilinson, Bernstein, Deligne, and Gabber. For this statement, we must restrict ourselves to R = Q. We give the statement as it appears in [Mac2], except that in [Mac2] intersection cohomology is defined as a positively perverse sheaf, and we must adjust by shifting. Note that, in [Mac2], the setting is algebraic; the analytic version appears in [Sai].

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An algebraic map f : X → Y is called projective if it can be factored as an embedding X ,→ Y × Pm (for some m) followed by projection Y × Pm → Y . The Decomposition Theorem [BBD, 6.2.5]: If f : X → Y is proper, then there exists a unique set of irreducible enriched subvarieties {(M α , Lα )} in Y and Laurent polynomials {φα = −2 −1 α α 2 · · · + φα + φα + φα −2 t −1 t 0 + φ1 t + φ2 t + . . . } such that there is a quasi-isomorphism Rf∗ IC•X (Q•◦ ) ∼ = X

M α,i

α

IC• (Lα )[−i] ⊗ Qφi , Mα

(here, IC• (Lα ) actually equals jα ! IC• (Lα ), where jα : M α ,→ Y is the inclusion). Mα

Mα

Moreover, if f is projective, then the coefficients of φα are palindromic around 0 (i.e., φα (t−1 ) = α α φ (t)) and the even and odd terms are separately unimodal (i.e., if i 6 0, then φα i−2 6 φi ). Applying hypercohomology to each side, we obtain: IH k (X; Q) =

M

α

(IH k−i (M α ; Lα ))φi .

α,i

We now wish to describe the category of perverse sheaves on a one-dimensional space; this is a particularly nice case of the results obtained in [M-V]. Unfortunately, we will use the notions of vanishing cycles and nearby cycles, which are not covered until the next section. Nonetheless, it seems appropriate to place this material here. We actually wish to consider perverse sheaves on the germ of a complex analytic space X at a point x. Hence, we assume that X is a one-dimensional complex analytic space with irreducible analytic components X1 , . . . , Xd which all contain x, such that Xi is homeomorphic to a complex line and Xi − {x} is smooth for all i. We wish to describe the category, C, of perverse sheaves on X with complex coefficients which are constructible with respect to the stratification {X1 − {x}, . . . , Xd − {x}, {x}}. Since perverse sheaves are topological in nature, we may reduce ourselves to considering exactly the case where X consists of d complex lines through the origin in some CN . Let L denote a linear form on CN such that X ∩ L−1 (0) = {0}. Suppose now that P• is in C, i.e., P• is perverse on X and constructible with respect to the stratification which has {0} as the only zero-dimensional stratum. Then P•|X−{0} consists of a collection of local systems, L1 , . . . Ld , in degree −1. These local systems are completely determined by monodromy isomorphisms hi : Cri → Cri representing looping once around the in Xi . In L origin 0 • ri ∼ terms of nearby cycles, the monodromy automorphism on H (ψ P [−1]) C is given by = L 0 i L h . i i The vanishing cycles φL P• [−1] are a perverse sheaf on a point, and so have possibly non-zero cohomology only in degree 0; say, H 0 (φL P• [−1])0 ∼ = Cλ . We have the canonical map r : H 0 (ψL P• [−1])0 → H 0 (φL P• [−1])0 and the variation map var : H 0 (φL P• [−1])0 → H 0 (ψL P• [−1])0 , and var ◦r = id −

L

i

hi .

APPENDIX B

205

ri Thus, an object in C determines a vector space W := H 0 (φL P• [−1])0 , a vector space L Vi := C for each irreducible L component Xi , an automorphism L hi on Vi , and two linear maps α : i Vi → W and β : W → V such that β ◦ α = id − i i i hi . This situation is nicely represented by a commutative triangle L id −

hi

i ⊕i Vi −−−−−−− −−→ ⊕i Vi

α&

%β W

.

The category C is equivalent to the category of such triangles, where a morphism of triangles is defined in the obvious way: a morphism is determined by linear maps τi : Vi → Vi0 and η : W → W 0 such that β α ⊕i Vi −−→ W −−→ ⊕i Vi ⊕i τi ↓

η↓ α0

⊕i τi ↓ β0

⊕i Vi0 −−→ W 0 −−→ ⊕i Vi0 commutes.

§3. Nearby and Vanishing Cycles Historically, there has been some confusion surrounding the terminology nearby (or neighboring) cycles and vanishing cycles; now, however, the terminology seems to have stabilized. In the past, the term “vanishing cycles” was sometimes used to describe what are now called the nearby cycles (this is true, for instance, in [A’C], [BBD], and [G-M1].) The two different indexing schemes for perverse sheaves also add to this confusion in statements such as “the nearby cycles of a perverse sheaf are perverse”. Finally, a new piece of confusion has been added in [K-S2], where the sheaf of vanishing cycles is shifted by one from the usual definition (we will not use this new, shifted definition). The point is: one should be very careful when reading works on nearby and vanishing cycles. Let S = {Sα } be a Whitney stratification of X and suppose F• ∈ DbS (X). Given an analytic map f : X → C, define a (stratified) critical point of f (with respect to S) to be a point x ∈ Sα ⊆ X such that f|Sα has a critical point at x; we denote the set of such critical points by ΣS f . We wish to investigate how the cohomology of the level sets of f with coefficients in F• changes at a critical point (which we normally assume lies in f −1 (0)). Consider the diagram f∗ E −−−−→ C   π  π ˆy y fˆ

X − f −1 (0) −−−−→ C∗ i↓ f where:

−1

(0) ,→ X j

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DAVID B. MASSEY

j : f −1 (0) ,→ X is inclusion; i : X − f −1 (0) ,→ X is inclusion; fˆ = restriction of f ; f∗ = cyclic (universal) cover of C∗ ; C and E denotes the pull-back. The nearby (or neighboring) cycles of F• along f are defined to be ψf F• := j ∗ R(i ◦ π ˆ )∗ (i ◦ π ˆ )∗ F• . Note that this is
a sheaf on
f −1 (0). As ψf F• [k] = ψf F• [k], we may write ψf F• [k] unambiguously. In fact, it is frequently useful to consider the functor where one first shifts the complex by k and then takes the nearby cycles; thus, we introduce the notation ψf [k] to be the functor such that ψf [k]F• = ψf F• [k] (and which has the corresponding action on morphisms). The functor ψf takes distinguished triangles to distinguished triangles. If P• is a perverse sheaf on X, then ψf [−1]P• is perverse on f −1 (0). (Actually, to conclude that ψf [−1]P• is perverse, we only need to assume that P• |X−f −1 (0) is perverse.) Because ψf [−1] takes perverse sheaves to perverse sheaves, it is useful to include the shift by −1 in many statements about ψf . Consequently, we also want to shift j ∗ F• by −1 in many statements, and so we write j ∗ [−1] for the functor which first shifts by −1 and then pulls-back by j. As there is a canonical map F• → Rg∗ g ∗ F• for any map g : Z → X, there is a map F• → R(i ◦ π ˆ )∗ (i ◦ π ˆ )∗ F• and, hence, a canonical map, called the comparison map: c

j ∗ [−1]F• −−→ j ∗ [−1]R(i ◦ π ˆ )∗ (i ◦ π ˆ )∗ F• = ψf [−1]F• .

For x ∈ f −1 (0), the stalk cohomology of ψf F• at x is the cohomology of the Milnor fibre of f at x with coefficients in F• , i.e., for all  > 0 small and all ξ ∈ C∗ with |ξ| << , ◦

Hi (ψf F• )x ∼ = Hi (B  (x) ∩ X ∩ f −1 (ξ); F• ), ◦

where the open ball B  (x) is taken inside any local embedding of (X, x) in affine space. The sheaf ψf F• only depends on f and F• |X−f −1 (0) . While the above definition of the nearby cycles treats all angular directions equally, it is perhaps more illuminating to fix an angle θ and describe the nearby cycles in terms of moving out slightly along the ray eiθ [0, ∞). Consider the three inclusions kθ : f −1 (eiθ (0, ∞)) ,→ f −1 (eiθ [0, ∞)), mθ : f −1 (0) ,→ f −1 (eiθ [0, ∞)), and lθ : f −1 (eiθ [0, ∞)) ,→ X.

APPENDIX B

207

Then, one can define the nearby cycles at angle θ to be ψfθ F• := m∗θ Rkθ ∗ kθ∗ lθ∗ F• . For each θ, there is a canonical isomorphism ψf F• ∼ = ψfθ F• . By letting θ travel around a θ+2π • θ • ∼ full circle, we obtain isomorphisms ψf F = ψf F . These isomorphisms correspond to the • monodromy automorphism Tf : ψf [−1]F → ψf [−1]F• , which comes from the deck transformation obtained in our definition of ψf F• (and, hence, ψf [−1]F• ) by traveling once around the origin in C. Actually, Tf is a natural automorphism from the functor ψf [−1] to itself; thus, strictly speaking, when we write Tf : ψf [−1]F• → ψf [−1]F• , we should include F• in the notation for Tf – however, we shall normally omit the explicit reference to F• if the complex is clear. There is a natural distinguished triangle j ∗ [−1]Ri∗ i∗ F• −→ ψf [−1]F• . Tf − id ψf [−1]F• .

[1]

The associated long exact sequences on stalk cohomology are the Wang sequences. c

The comparison map j ∗ [−1]F• −−→ ψf [−1]F• is Tf -equivariant, i.e., c = Tf ◦ c. Since we have a map c[1] : j ∗ F• → ψf F• , the third vertex of a distinguished triangle is defined up to quasi-isomorphism. We define the sheaf of vanishing cycles, φf F• , of F• along f to be this third vertex, i.e., there is a distinguished triangle j ∗ F• −→ ψf F• [1]

-

. φf F• .

Letting φf [−1] denote the functor which first shifts by −1 and then applies φf , we can write the triangle above as c j ∗ [−1]F• −−−→ ψf [−1]F• [1]

-

. φf [−1]F• .

Note that this is a triangle of sheaves on f −1 (0). Note also that, by replacing F• with i! i! F• , we conclude that there is a natural isomorphism ψf [−1]F• ∼ = φf [−1](i! i! F• ). There is another natural • ∼ ∗ • isomorphism ψf [−1]F = φf [−1](Ri∗ i F ). The functor φf takes distinguished triangles to distinguished triangles. If P• is a perverse sheaf on X, then φf [−1]P• is a perverse sheaf on f −1 (0). For x ∈ f −1 (0), the stalk cohomology of φf F• at x is the relative cohomology of the Milnor fibre of f at x with coefficients in F• and with a shift by one, i.e., for all  > 0 small and all ξ ∈ C∗ with |ξ| << , ◦

◦

Hi (φf F• )x ∼ = Hi+1 (B  (x) ∩ X, B  (x) ∩ X ∩ f −1 (ξ); F• ).

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DAVID B. MASSEY

As an example, if X = Cn+1 and F• = C•X , then for all x ∈ f −1 (0), H i (ψf C•X )x = i-th cohomology of the Milnor fibre of f at x (with C coefficients) = H i (Ff,x ; C), while H i (φf C•X )x = e i (Ff,x ; C). reduced i-th cohomology of the Milnor fibre of f at x = H Just as we defined the nearby cycles at angle θ to be ψfθ F• := m∗θ Rkθ ∗ kθ∗ lθ∗ F• , we can define the vanishing cycles at angle θ to be φθf F• := m∗θ mθ ! m!θ lθ∗ F• [1] = m!θ lθ∗ F• [1]. Then, φθf F• ∼ = φf F• , and again there is a monodromy automorphism Tef : φf [−1]F• → φf [−1]F• . The monodromy Tef is actually a natural automorphism of the
functor φf [−1]. If we let Zθ := {z ∈ X | Re eiθ f (z) 6 0}, then there is a canonical isomorphism
φθf F• ∼ = RΓZθ (F• ) |

f −1 (0)

[1].

Thus, there is a monodromy automorphism on the distinguished triangle c

r

j ∗ [−1]F• −−→ ψf [−1]F• −−→ φf [−1]F• −→ j ∗ F• given by (id, Tf , Tef ), i.e., a commutative diagram c

r

j ∗ [−1]F• −−→ ψf [−1]F• −−→ φf [−1]F• −→ j ∗ F• id ↓

Tf ↓ c

Tef ↓

id ↓

r

j ∗ [−1]F• −−→ ψf [−1]F• −−→ φf [−1]F• −→ j ∗ F• . From this, it follows formally that there exists a variation morphism, var : φf [−1]F• → ψf [−1]F• such that r ◦ var = id −Tef and var ◦r = id −Tf . (Note that, if we are not using field coefficients, then the variation morphism does not necessarily exist on the level of chain complexes – the derived category structure is necessary here.) The monodromy isomorphisms Tf and Tef are natural automorphisms of the (shifted) nearby cycle and vanishing cycle functors, respectively, and the maps c, r, and var above are all natural maps. The variation map can be described in a more concrete fashion. There is the canonical map from F• to Ri∗ i∗ F• . Applying the shifted vanishing cycle functor, we obtain a natural map from φf [−1](F• ) to φf [−1](Ri∗ i∗ F• ), and as we mentioned above, there is a natural isomorphism φf [−1](Ri∗ i∗ F• ) ∼ = ψf [−1]F• . The variation map is the composition of these two natural maps. To make this more clear, we will describe the variation map on the stalk cohomology; this should also help clarify how one obtains the isomorphism φf [−1](Ri∗ i∗ F• ) ∼ = ψf [−1]F• . −1 We follow the construction in [G-M1]. Let x be a point in f (0), let N denote the intersection of X with a sufficiently small open ball around x (for some Reimannian metric), and let Dη be a complex disk of sufficiently small radius, η, centered at the origin so that, for all ξ with 0 < ξ 6 η, f N ∩ f −1 (∂Dξ ) −−→ ∂Dξ represents the Milnor fibration of f at x with coefficients in F• . Let W := N ∩f −1 ({v ∈ Dη | Re v > 0} − {0}), let Z := N ∩f −1 ({v ∈ Dη | Re v 6 0} − {0}), let A := N ∩ f −1 ({v ∈ Dη | Re v = 0, Im v > 0}), and let B := N ∩ f −1 ({v ∈ Dη | Re v = 0, Im v < 0}). Then, we have isomorphisms: H i (φf F• )x ∼ = Hi+1 (N ∩ f −1 (Dη ), N ∩ f −1 (η); F• ) ∼ = Hi+1 (N ∩ f −1 (Dη ), W ; F• );

APPENDIX B

209

the map induced by inclusion of pairs: Hi+1 (N ∩ f −1 (Dη ), W ; F• ) → Hi+1 (N ∩ f −1 (Dη − {0}), W ; F• ); and isomorphisms: Hi+1 (N ∩ f −1 (Dη − {0}), W ; F• ) ∼ = H i (ψf F• )x , = Hi+1 (Z, A ∪ B; F• ) ∼ where the first isomorphism is by excision, and the second is from the long exact sequence of the pair. The map induced by the (shifted) variation on the stalk cohomology is the composition of the above maps.

Applying the shifted vanishing cycle functor to the distinguished triangle j! j ! F• −→ F• [1]

. ∗ • Ri∗ i F ,

noting that φf [−1](j! j ! F• ) ∼ = j ! F• , and using the natural isomorphism ψf [−1]F• ∼ = φf [−1](Ri∗ i∗ F• ), we obtain the distinguished triangle j ! F• −→ φf [−1]F• [1]

. var ψf [−1]F• .

Starting with the two distinguished triangles var

φf [−1]F• −−−→ ψf [−1]F• −→ j ! [1]F• −→ φf F• and c

ψf [−1]F• −−−→ φf [−1]F• −→ j ∗ F• −→ ψf F• , we may apply the octahedral lemma to conclude that there exists a complex wf F• and two distinguished triangles id −Tef

φf [−1]F• −−−−−→ φf [−1]F• −→ wf F• −→ φf F• and τ

j ! [1]F• −→ wf F• −→ j ∗ F• −−→ j ! [2]F• . We refer to the morphism ωf := τ [−1] from j ∗ [−1]F• to j ! [1]F• as the Wang morphism of f . The application of the octahedral lemma above tells us that the mapping cone of id −Tef is isomorphic to the mapping cone of ωf . Note that, while j ∗ [−1]F• and j ! [1]F• depend only on f −1 (0) (and F• ), ωf may change if f (or some factor of f ) is raising to a power.

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DAVID B. MASSEY

For any Whitney stratification, S, with respect to which F• is constructible, the support of H (φf F• ) is contained in the stratified critical locus of f , ΣS f . In addition, if S is a Whitney stratification with respect to which F• is constructible and such that f −1 (0) is a union of strata, then – by [BMM] and [P2] – it follows that S also satisfies Thom’s af condition; by Thom’s second isotopy lemma, this implies that the entire situation locally trivializes over strata, and hence both ψf F• and φf F• are constructible with respect to {S ∈ S | S ⊆ f −1 (0)}. ∗

f

π

Suppose we have X − →Y − → C where π is proper and π ˆ : π −1 f −1 (0) → f −1 (0) is the restriction • b of π. Then, for all A ∈ Dc (X), Rˆ π∗ (ψf ◦π A• ) ∼ π∗ (φf ◦π A• ) ∼ = ψf (Rπ∗ A• ) and Rˆ = φf (Rπ∗ A• ).

The Sebastiani-Thom Isomorphism Let f : X → C and g : Y → C be complex analytic functions. Let π1 and π2 denote the projections of X × Y onto X and Y , respectively. Let A• and B• be bounded, constructible L

complexes of sheaves of R-modules on X and Y , respectively. In this situation, A•  B• := L

π1∗ A• ⊗ π2∗ B• . Let us adopt the similar notation f
g := f ◦ π1 + g ◦ π2 . Let p1 and p2 denote the projections of V (f ) × V (g) onto V (f ) and V (g), respectively, and let k denote the inclusion of V (f ) × V (g) into V (f
g). Theorem (Sebastiani-Thom Isomorphism). There is a natural isomorphism L

k ∗ φf
g [−1] A•  B•




L

∼ = φf [−1]A•  φg [−1]B• ,

and this isomorphism commutes with the corresponding monodromies. Moreover, if we let p := (x, y) ∈ X × Y be such that f (x) = 0 and g(y) = 0, then, in an open L
neighborhood of p, the complex φf
g [−1] A•  B• has support contained in V (f ) × V (g), and, in any open set in which we have this containment, there are natural isomorphisms L

φf
g [−1] A•  B•




L

L

∼ = k∗ (φf [−1]A•  φg [−1]B• ). = k! (φf [−1]A•  φg [−1]B• ) ∼

If Q• is perverse on X − f −1 (0) and i : X − f −1 (0) → X is the inclusion, then it is easy to see that Ri∗ Q• satisfies the cosupport condition; moreover, by combining the fact that ψf (Ri∗ Q• )[−1] is perverse on f −1 (0) with the Wang sequences on stalk cohomology, one can prove that Ri∗ Q• also satisfies the support condition - hence, Ri∗ Q• is perverse. In an analogous fashion, one obtains that Ri! Q• is perverse (if the base ring is a field, this can be obtained by dualizing). If R is a field, then the operators ψf [−1] and D commute, as do φf [−1] and D; i.e., D(ψf A• [−1]) ∼ = ψf (DA• )[−1] and D(φf A• [−1]) ∼ = φf (DA• )[−1].

APPENDIX B

211

These isomorphisms in Dbc (X) are non-canonical. Tf

Let A• ∈ Dbc (X) and f : X → C. The monodromy automorphism ψf [−1]A• −→ ψf [−1]A• induces a map on cohomology sheaves which is quasi-unipotent, i.e., letting Tf also denote the map on cohomology, this means that there exist integers k and j such that (id −Tfk )j = 0. Suppose that the base ring is a field; if mx denotes the maximal ideal of X at x and f ∈ m2x , Tf

then the Lefschetz number of the map H∗ (ψf [−1]A• )x −→ H∗ (ψf [−1]A• )x equals 0, i.e., X Tf (−1)i Trace{Hi (ψf [−1]A• )x −→ Hi (ψf [−1]A• )x } = 0. i

If ψf [−1]A• is a perverse sheaf, then we may use the Abelian structure of the category P erv(X) Tf

to investigate the map ψf [−1]A• −→ ψf [−1]A• . This morphism can be factored into Tf = F · (1 + N ), where F has finite order and N is nilpotent. It follows that there is a unique increasing filtration W i on ψf [−1]A• such that N sends W i to W i−2 and N i takes Gri ψf [−1]A• isomorphically to Gr−i ψf [−1]A• , where Gri is the associated graded to the filtration W • . This is called the nilpotent filtration of ψf [−1]A• . (The existence of such a filtration is just linear algebra; the interesting result is the following theorem, due to Gabber.) Theorem: If we have f : X → C, S a Whitney stratification of X with a finite number of strata, and A• ∈ DbS (X) such that A• |X−f −1 (0) ∼ = IC•X−f −1 (0) (C•◦ ), X •

then the graded pieces of the nilpotent filtration of ψf [−1]A are semi-simple in P ervS (f −1 (0)), i.e., they are direct sums of intersection cohomology sheaves of irreducible enriched subvarieties of f −1 (0) (extended by zero). In particular, if X = Cn+1 , then each Gri ψf [−1]C•X [n + 1] is semi-simple.

§4. Some Quick Applications The applications of perverse sheaves are widespread and are frequently quite deep - particularly for those applications which rely on the decomposition theorem. For beautiful discussions of these applications, we highly recommend [Mac1] and [Mac2]. We shall not describe any of these applications here; rather we shall give some fairly easy results on general Milnor fibres. These results are “easy” now that we have all the machinery of the first three sections at our disposal. While the applications below could undoubtedly be proved without the general theory of perverse sheaves, with this theory in hand, the results and their proofs can be presented in a unified manner and, what is more, the proofs become mere exercises. Consider the classical case of the Milnor fibre of a non-zero map f : (Cn+1 , 0) → (C, 0). Let X = Cn+1 and let s = dim Σf . Then, as X is a manifold, C•X is a positively perverse sheaf and so φf C•X is positively perverse on f −1 (0) with support only on Σf . It follows that the stalk cohomology of φf C•X is non-zero only for dimensions i with n − s 6 i 6 n; that is, we recover the well-known result that the reduced cohomology of the Milnor fibre is non-zero only in these dimensions.

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DAVID B. MASSEY

A much more general case is just as easy to derive from the machinery that we have. Suppose that X is a purely (n + 1)-dimensional local complete intersection with arbitrary singularities. Let S be a Whitney stratification of X. Let p ∈ X be such that dimp f −1 (0) = n, and let Ff,p denote the Milnor fibre of f at p. Then, as X is a local complete intersection, C•X is a positively perverse sheaf and so φf C•X is positively perverse on f −1 (0) with support only on ΣS f . It follows that the stalk cohomology of φf C•X is non-zero only for dimensions i with n − dimp ΣS f 6 i 6 n. Hence, the reduced cohomology of Ff,p is non-zero only in these dimensions. While this general statement could no doubt be proved by induction on hyperplane sections, the above proof via general techniques avoids the re-working of many technical lemmas on privileged neighborhoods and generic slices. Another application relates to the homotopy-type of the complex link of a space at a point; for instance, for an s-dimensional local complete intersection, the complex link has the homotopy-type of a bouquet of spheres of real dimension s − 1. In terms of vanishing cycles and perverse sheaves, we only obtain this result up to cohomology: let (X, x) be a germ of an analytic space embedded in some Cn , and assume s := dim X = dimx X. Suppose that we have a positively perverse sheaf, P• , on X (e.g., the constant sheaf, if X is a local complete intersection). Let l be a generic linear form, and consider φl−l(x) P• ; this is a perverse sheaf on an s − 1 dimensional space and, as l is generic, it is supported at the single point x (because the hyperplane slice l = l(x) can be chosen to transversely intersect all the strata of any stratification with respect to which P• is constructible - except, possibly, the point-stratum x itself). Hence, H ∗ (φl−l(x) P• )x is (possibly) non-zero only in dimension s − 1. In the case of the constant sheaf on a local complete intersection, this gives the desired result. For our final application, we wish to investigate functions with one-dimensional critical loci; we must first set up some notation. n+1 Let U be an open neighborhood of the origin in C and suppose that f : (U, 0) → (C, 0) has a one-dimensional critical locus at the origin, i.e., dim0 Σf = 1. The reduced cohomology of the Milnor fibre, Ff,0 , of f at the origin is possibly non-zero only in dimensions n − 1 and n. We wish to show that the n − 1-st cohomology group embeds inside another group which is fairly easy to describe; thus, we obtain a bound on the n − 1-st Betti number of the Milnor fibre of f . For each component ν of Σf , one may consider a generic hyperplane slice, H, at points p ∈ ν −0 close to the origin; then, the restricted function, f|H , will have an isolated critical point at p. By shrinking the neighborhood U if necessary, we may assume that the Milnor number of this isolated ◦ singularity of f|H at p is independent of the point p ∈ ν − 0; denote this value by µν . As ν − 0 is homotopy-equivalent to a circle, there is a monodromy map from the Milnor fibre of f|H at p ∈ ν −0 ◦ µν

◦ µν

to itself, which induces a map on the middle dimensional cohomology, i.e., a map hν : Z → Z . We wish to show that H n−1 (Ff,0 ) (with integer coefficients) injects into ⊕ν ker(id − hν ). Let j denote the inclusion of the origin into X = V (f ), let i denote the inclusion of X − 0 into X, and let K• denote φf (Z•U ). As Z•U is positively perverse, φf (Z•U ) is positively perverse with one-dimensional support (as we are assuming a one-dimensional critical locus). Also, we always have the distinguished triangle Rj∗ j ! K• −→ K• . [1] ∗ • Ri∗ i K We wish to examine the associated stalk cohomology exact sequence at the origin.

APPENDIX B

213

First, we have that H n−1 ((Rj∗ j ! K• )0 ) = H n−1 (j ! K• ) and so, by the cosupport condition for perverse sheaves, H n−1 ((Rj∗ j ! K• )0 ) = 0. Now, we need to look more closely at the sheaf Ri∗ i∗ K• . i∗ K• is the restriction of K• to X − 0; near the origin, this sheaf has cohomology only in degree n − 1 with support on Σf − 0. Moreover, the cohomology sheaf Hn−1 (i∗ K• ) is locally constant when restricted to Σf − 0. It follows that i∗ K• is naturally isomorphic in the derived category to the extension by zero of a local system of coefficients in dimension n − 1 on Σ − 0. To be more precise, let p denote the inclusion of the closed subset Σf − 0 into X − 0. Then, there exists a locally constant (single) sheaf, L, on Σf − 0 such that when L is considered as a complex, L• , we have that p! L• [−(n − 1)] ∼ = p∗ L• [−(n − 1)] is naturally isomorphic to i∗ K• . For ◦ µν

each component ν of Σf , the restriction of L to ν − 0 is a local system with stalks Z ◦ µν

completely determined by the monodromy map hν : Z ◦

which is

◦ µν

→Z .

Therefore, inside a small open ball B, ◦

H 0 ((Ri∗ i∗ L• )0 ) ∼ = ⊕ν H0 (B ∩ (ν − 0); L) and these global sections are well-known to be given by ker(id − hν ). It follows that H n−1 ((Ri∗ i∗ K• )0 ) ∼ = ⊕ν ker(id − hν ). Thus, when we consider the long exact sequence on stalk cohomology associated to our distinguished triangle, we find – starting in dimension n − 1 – that it begins 0 → H n−1 (Ff,0 ) → ⊕ν ker(id − hν ) → . . . . The desired conclusion follows.

§5. Truncation and Perverse Cohomology

This section is taken entirely from [BBD], [G-M3], and [K-S2]. There are (at least) two forms of truncation associated to an object F• ∈ Dbc (X) – one form of truncation is related to the ordinary cohomology of the complex, while the other form leads to something called the perverse cohomology or perverse projection. These two types of truncation bear little resemblance to each other, except in the general framework of a t-structure on Dbc (X). 60

Loosely speaking, a t-structure on Dbc (X) consists of two full subcategories, denoted D (X) >0 60 >0 and D (X), such that for any F• ∈ Dbc (X), there exist E• ∈ D (X), G• ∈ D (X), and a distinguished triangle E• −→ F• . [1] • G [−1]

;

moreover, such E• and G• are required to be unique up to isomorphism in Dbc (X).

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DAVID B. MASSEY

Given a t-structure as above, and using the same notation, we write E• = τ60 F• (the truncation of F• below 0) and G• = τ >0 (F• [1]) (the truncation of F• [1] above 0); these are the basic truncation functors associated to the t-structure. 6n In addition, we write D (X) for n o 60 60 D (X)[−n] := F• [−n] | F• ∈ D (X) , >n

and we analogously write D Also, we define τ6n F• by

>0

(X) for D (X)[−n].

τ6n F• = (τ60 (F• [n])) [−n] = ([−n] ◦ τ60 ◦ [n])F• , and we analogously define τ >n F• as ([−n] ◦ τ >0 ◦ [n])F• . 6n

Note that τ6n F• ∈ D

>n

(X), τ >n F• ∈ D

(X) and, for all n, we have a distinguished triangle

τ6n F• −→ F• . [1] τ

>n+1

.

•

F

Writing ' to denote natural isomorphisms between functors: for all a and b, τ6b ◦ τ >a ' τ >a ◦ τ6b , τ6b ◦ τ6a ' τ6a ◦ τ6b , and τ >b ◦ τ >a ' τ >a ◦ τ >b . If a > b, then τ6b ◦ τ6a ' τ6b , and τ >a ◦ τ >b ' τ >a . Also, if a > b, then τ6b ◦ τ >a = τ >a ◦ τ6b = 0. 60

>0

The heart of the t-structure is defined to be the full subcategory C := D (X) ∩ D (X); this is always an Abelian category. We wish to describe the kernels and cokernels in this category. Let E• , F• ∈ C and let f be a morphism from E• to F• . We can form a distinguished triangle in Dbc (X) f

E• −−−→ F• . , [1] • G where G• need not be in C. Then, up to natural isomorphism, coker f = τ >0 G• and ker f = τ60 (G• [−1]).

APPENDIX B

215

We define cohomology associated to a t-structure as follows. Define t H 0 (F• ) to be τ >0 τ60 F• ; this is naturally isomorphic to τ60 τ >0 F• . Now, define t H n (F• ) to be
t 0 H (F• [n]) = τ >n τ6n F• [n]. Note that this cohomology does not give back modules or even sheaves of modules, but rather gives back complexes which are objects in the heart of the t-structure. If F• ∈ Dbc (X), then the following are equivalent: 60

>0

1)

F• ∈ D (X) (resp. D (X));

2)

the morphism τ60 F• → F• is an isomorphism (resp. the morphism F• → τ >0 F• is an isomorphism);

3)

τ >1 F• = 0 (resp. τ6−1 F• = 0));

4)

τ >i F• = 0 for all i > 1 (resp. τ6i F• = 0 for all i 6 −1);

5)

there exists a such that F• ∈ D (X) and t H i (F• ) = 0 for all i > 1 (resp. there exists a such >a that F• ∈ D (X) and t H i (F• ) = 0 for all i 6 −1).

6a

It follows that, if F• ∈ Dbc (X), then the following are equivalent: 1)

F• ∈ C;

2)

t

3)

there exist a and b such that F• ∈ D for all n 6= 0.

H 0 (F• ) is isomorphic to F• ; >a

6b

(X), F• ∈ D (X), and t H n (F• ) = 0

As the heart is an Abelian category, we may talk about exact sequences in C. Any distinguished triangle in Dbc (X) determines a long exact sequence of objects in the heart of the t-structure; if E• −→ F• . [1] G• is a distinguished triangle in Dbc (X), then the associated long exact sequence in C is · · · → t H −1 (G• ) → t H 0 (E• ) → t H 0 (F• ) → t H 0 (G• ) → t H 1 (E• ) → . . . . We are finished now with our generalities on t-structures and wish to, at last, give our two primary examples. The “ordinary” t-structure The “ordinary” t-structure on Dbc (X) is given by 60

D (X) = {F• ∈ Dbc (X) | Hi (F• ) = 0 for all i > 0}

216

DAVID B. MASSEY

and >0

D (X) = {F• ∈ Dbc (X) | Hi (F• ) = 0 for all i < 0}. The associated truncation functors are the ordinary ones described in [G-M3]. If F• ∈ Dbc (X), then  n  F • n (τ6p F ) = ker dp   0

if n < p if n = p if n > p

and   0
>p • n τ F = coker dp−1   n F

if n < p if n = p if n > p.

These truncated complexes are naturally quasi-isomorphic to the complexes  n  F n (˜ τ6p F• ) = Im dp   0

if n 6 p if n = p + 1 if n > p + 1

and   0
n τ˜>p F• = Im dp−1   n F

if n < p − 1 if n = p − 1 if n > p.

If A• , B• ∈ Dbc (X), then 1. (τ6p A• )x = τ6p (A•x ); 2. Hk (τ6p A• )x =



Hk (A• )x

if k 6 p

0

for k > p.

3. If φ : A• → B• is a morphism of complexes of sheaves which induces isomorphisms on the associated cohomology sheaves φ∗ : Hn (A• ) ∼ = Hn (B• ) for all n 6 p, then τ6p φ : τ6p A• → τ6p B• is a quasi-isomorphism. 4. If f : X → Y is a continuous map and C• is a complex of sheaves on Y , then τ6p f ∗ (C• ) ∼ = f ∗ τ6p (C• ).

APPENDIX B

217

5. If R is a field and A• is a complex of sheaves of R-modules on X with locally constant cohomology sheaves, then there are natural quasi-isomorphisms τ >−p RHom• (A• , R•X ) → τ >−p RHom• (τ6p A• , R•X ) ← RHom• (τ6p A• , R•X ).

The heart of this t-structure consists of those complexes which have non-zero cohomology sheaves only in degree 0; such complexes are quasi-isomorphic to complexes which are non-zero only in degree 0. The t-structure cohomology of a complex F• is essentially the sheaf cohomology of F• ; t H n (F• ) is quasi-isomorphic to a complex which has Hn (F• ) in degree 0 and is zero in all other degrees. With this identification, the t-structure long exact sequence associated to a distinguished triangle is merely the usual long exact sequence on sheaf cohomology.

We are now going to give the construction of the intersection cohomology complexes as it is presented in [G-M3]. Our indexing will look different from that of [G-M3] for several reasons. First, we are dealing only with complex analytic spaces, X, and we are using only middle perversity; this accounts for some of the indexing differences. In addition, in this setting, the intersection cohomology complex defined in [G-M3] would have possibly non-zero cohomology only in degrees between −2 dimC X and −(dimC X) − 1, inclusive. The definition below is shifted by − dimC X from the [G-M3] definition, and yields a perverse sheaf which has possibly non-zero cohomology only in degrees between − dimC X and −1, inclusive. Let X be a complex analytic n-dimensional space with a complex analytic Whitney stratification S = {Sα }. While we do not explicitly require that X is pure-dimensional, it will follow from the construction that components of X of dimension less than n will essentially be ignored. For all k, let X k denote the union of the strata of dimension less than or equal to k. By convention, we set X −1 = ∅. Hence, we have a filtration ∅ = X −1 ⊆ X 0 ⊆ X 1 ⊆ · · · ⊆ X n−1 ⊆ X n = X. For all k, let Uk := X − X n−k , and let ik denote the inclusion Uk ,→ Uk+1 . Let L•U1 be a local system on the top-dimensional strata. Then, the intersection cohomology complex on X with coefficients in L•U1 , as described in section 2, is given by
IC•X (L•U1 ) := τ6−1 Rin∗ . . . τ61−n Ri2∗ τ6−n Ri1∗ L•U1 [n] . Up to quasi-isomorphism, this complex is independent of the stratification. Note that the cohomology sheaves of IC•X (L•U1 ) are supported only in degrees k for which −n 6 k 6 −1 (unless X is 0-dimensional, and then IC•X (L•U1 ) ∼ = L•U1 ). Also note that it follows from the construction that there is always a canonical map from the shifted constant sheaf R•X [n] to IC•X (R•U1 ) which induces an isomorphism when restricted to U1 . To see this, consider the canonical morphism R•U [n] → Rik∗ i∗k R•U [n] for each k > 1. As k+1 k+1 i∗k R•U [n] ∼ R•U [n], we have a canonical map R•U [n] → Rik∗ R•U [n] and, hence, a canonical = k+1 k k
k+1
map between the truncations τ6k−n−1 R•U [n] → τ6k−n−1 Rik∗ R•U [n] . But, k+1 k
τ6k−n−1 R•U [n] ∼ = R•U [n] k+1

k+1

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DAVID B. MASSEY


and so we have a canonical map R•U [n] → τ6k−n−1 Rik∗ R•U [n] . By piecing all of these maps k+1 k together, one obtains the desired morphism. The perverse t-structure The perverse t-structure (with middle perversity µ) on Dbc (X) is given by µ

and

µ

60

D (X) = {F• ∈ Dbc (X) | dim supp-j F• 6 j for all j} >0

D (X) = {F• ∈ Dbc (X) | dim cosuppj F• 6 j for all j}.

Note that the heart of this t-structure is precisely P erv(X). Thus, every distinguished triangle in Dbc (X) determines a long exact sequence in the Abelian category P erv(X). We naturally call the t-structure cohomology associated to the perverse t-structure the perverse cohomology or perverse projection and denote it in degree n by µH n (F• ). Let d be an integer, and let f : Y → X be a morphism of complex spaces such that dim f −1 (x) 6 d, for all x ∈ X. Let dim Y /X := dim Y − dim X. Then, µ

60

µ

6d

1) f ∗ sends D (X) to D µ

>0

µ

>−d

2) f ! sends D (X) to D µ

60

µ

>0

µ

µ

>0

>dim Y /X

(Y ), and sends D (X) to D µ

µ

60

(Y ), and sends D (X) to D µ

6d

3) if F• ∈ D (Y ) and Rf! F• ∈ Dbc (X), then Rf! F• ∈ D µ

(Y );

6− dim Y /X

(Y );;

(X);

>−d

4) if F• ∈ D (Y ) and Rf∗ F• ∈ Dbc (X), then Rf∗ F• ∈ D

(X).

Let f : Y → X be a morphism of complex spaces such that each point in X has an open neighborhood U such that f −1 (U) is a Stein space (e.g., an affine map between algebraic varieties). Then, µ

60

µ

>0

µ

60

1) if F• ∈ D (Y ) and Rf∗ F• ∈ Dbc (X), then Rf∗ F• ∈ D (X); µ

>0

2) if F• ∈ D (Y ) and Rf! F• ∈ Dbc (X), then Rf! F• ∈ D (X). If f : X → C is an analytic map, then the functors ψf [−1] and φf [−1] are t-exact with respect µ 60 µ >0 to the perverse t-structures; this means that if E• ∈ D (X) and F• ∈ D (X), then ψf E• [−1] µ >0 µ 60 and φf E• [−1] are in D (f −1 (0)), and ψf F• [−1] and φf F• [−1] are in D (f −1 (0)). In particular, ψf [−1] and φf [−1] take perverse sheaves to perverse sheaves and, for any F• ∈ b Dc (X), H n (ψf F• [−1]) ∼ = ψf µH n (F• )[−1] and µH n (φf F• [−1]) ∼ = φf µH n (F• )[−1].

µ

If the base ring is a field, then the functor µH 0 also commutes with Verdier dualizing; that is, there is a natural isomorphism D ◦ µH 0 ∼ = µH 0 ◦ D.

APPENDIX B

219

Let F• be a bounded complex of sheaves on X which is constructible with respect to a connected Whitney stratification {Sα }
of X, and let dα := dim Sα . Then, µH 0 (F• ) is also constructible with respect to S, and µH 0 (F• ) | [−dα ] is naturally isomorphic to µH 0 (F•|N [−dα ]), where Nα denotes Nα α a normal slice to Sα . • Let Smax Smax . Then,
be a maximal stratum contained in the support of F , and let m = dim−m µ 0 • H (F ) | is isomorphic (in the derived category) to the complex which has (H (F• ))|S max Smax in degree −m and zero in allSother degrees. In particular, supp F• = i supp µH i (F• ), and if F• is supported on an isolated point, q, then 0 µ 0 H ( H (F• ))q ∼ = H 0 (F• )q . From this, and the fact that perverse cohomology commutes with nearby and vanishing cycles shifted by −1, one easily concludes that, at all points x ∈ X, X
χ(F• )x = (−1)k χ µH k (F• ) x . k

Switching Coefficients Suppose that the base ring R is a p.i.d. For each prime ideal p of R, let kp denote the field of fractions of R/p, i.e., k0 is the field of fractions of R, and for p 6= 0, kp = R/p. There are the obvious L

functors δp : Dbc (RX ) → Dbc ((kp )X ), which sends F• to F• ⊗ (kp )•X , and p : Dbc ((kp )X ) → Dbc (RX ), which considers kp -vector spaces as R-modules. If A• is a complex of kp -vector spaces, we may consider the perverse cohomology of A• , µ i Hkp (A• ), or the perverse cohomology of (A• ), which we denote by µHRi (A• ). If A• ∈ Dbc ((kp )X ) and Smax is a maximal stratum contained in the support of A• , then there is a canonical isomorphism
 (µHkip (A• ))|Sα ∼ = (µHRi (A• ))|Sα ; in particular, supp µHkip (A• ) = supp µHRi (A• ). If F• ∈ Dbc (RX ), Smax is a maximal stratum contained in the support of F• , and x ∈ Smax , then for some prime ideal p ⊂ R and for some integer i, H i (F• )x ⊗ kp 6= 0; it follows that Smax is L

•

also a maximal stratum in the support of F• ⊗ (kp )X . Thus, supp F• =

[

L

•

supp(F• ⊗ (kp )X )

p

and so supp F• =

[

L

•

supp µHkip (F• ⊗ (kp )X ),

i,p

where the boundedness and constructibility of F• imply that this union is locally finite.

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DAVID B. MASSEY

APPENDIX C: PRIVILEGED NEIGHBORHOODS AND LIFTING MILNOR FIBRATIONS

In this appendix, we prove a number of very technical results. These results tell us when we can use certain types of “nice” neighborhoods to define the Milnor fibre (at least, up to homotopy), and give conditions under which Milnor fibrations remain constant in a parameterized family. Lˆe numbers and cycles do not appear here, though we will use the relative polar curve. Throughout, for convenience, we concentrate our attention at the origin. Let U be an open neighborhood of the origin in some Cn+1 and let h : (U, 0) → (C, 0) be an analytic function. In what sense the Milnor fibre and Milnor fibration of h are well-defined has been discussed in a number of places (see, for instance, [Se-Th]). If one is primarily interested in the ambient, local topology of the hypersurface V (h) defined by h, then “the” Milnor fibre is only well-defined up to homotopy-type [Lˆ e6]. Thus, we may make the weakest possible definition of the Milnor fibre of h at the origin as a homotopy-type:

Definition/Proposition C.1. A system of Milnor neighborhoods for h at the origin is a fundamental system of neighborhoods, {Cα }, at the origin in U such that for all Cα ⊆ Cβ , there exists  > 0 such that for all complex ξ with 0 < |ξ| < , we have that the inclusion Cα ∩V (h−ξ) ,→ Cβ ∪V (h−ξ) is a homotopy-equivalence. The standard system of Milnor neighborhoods for h at the origin is just the set of closed balls of sufficiently small radius centered at the origin (this system is independent of h except for how small the radii must be). If {Cα } is a system of Milnor neighborhoods for h at the origin, then for each Cα there exists  > 0 such that the homotopy-type of Cα ∩ V (h − ξ) is independent of the complex number ξ chosen as long as 0 < |ξ| < . Moreover, this homotopy-type is independent of the choice of the particular Cα and is, in fact, independent of the choice of the system of Milnor neighborhoods. Proof. The proof is standard. Let {Cα } be a system of Milnor neighborhoods for h at the origin. We shall compare it with the standard system. Select any Cβ . Now, pick Cα , Bη , and Bδ such that Bη and Bδ are in the standard system of Milnor neighborhoods for h at the origin and such that Bδ ⊆ Cα ⊆ Bη ⊆ Cβ . We may certainly pick  > 0 such that, for all complex ξ with 0 < |ξ| < , the inclusion Cα ∩ V (h − ξ) ,→ Cβ ∩ V (h − ξ) and the inclusion Bδ ∩ V (h − ξ) ,→ Bη ∩ V (h − ξ) are both homotopy-equivalences. It follows that the inclusion Cα ∩ V (h − ξ) ,→ Bη ∩ V (h − ξ) is a homotopy-equivalence for all small ξ 6= 0 and thus, as the homotopy-type of Bη ∩ V (h − ξ) is independent of ξ, so is that of Cα ∩ V (h − ξ). The conclusion follows immediately.  It is sometimes more convenient to prove that Cα ∩ V (h − ξ) ,→ Cβ ∩ V (h − ξ) is a homotopyequivalence whenever Cα is contained in the interior of Cβ . It is easy to see by the proof above that this is enough to show that the system is a system of Milnor neighborhoods. A system of Milnor neighborhoods allows one to discuss the Milnor fibre up to homotopy. However, one frequently wishes to use stratified, differential techniques to study the Milnor fibre and, hence, one would like for the Milnor fibre to have the structure of a smooth, compact manifold with (stratified) boundary and would also like to have some control over what happens on the boundary as one moves through a family of singularities. Furthermore, one would like to have a 221

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notion of the Milnor fibration – at least up to fibre-homotopy-type. To gain this additional structure, we will use two types of (complex analytic) stratifications. One is the well-known Whitney stratification [G-M2], [Mat], [Th]. The second is a good stratification, as defined in 1.24. Note that any refinement of a good stratification which does not refine the smooth stratum is automatically a good stratification. This fact will be very useful when combined with the following proposition, which is Theorem 18.11 of [W] (or just a small portion of Theorem 1.7 of [G-M2]). Proposition C.2. Let X be an analytic subset of CN and let Y be an analytic subset of X. Suppose that D and F are analytic stratifications for X and Y , respectively. Then, there exists an analytic stratification, L, of X which is a common refinement of both D and F, i.e. every stratum of L is contained in a stratum of D, Y is a union of strata of L, and every stratum of L which is contained in Y is contained in a stratum of F. The L above is sometimes referred to as a refinement of D adapted to Y . (The reader should note that when Goresky and MacPherson use the term “stratification”, they mean that the Whitney conditions are satisfied. Hence, their Theorem 1.7 actually allows us to pick a common analytic, Whitney refinement.) We now generalize the notion of a privileged polydisc as given in [Lˆ e3]. This definition should be compared with [L-T1, 2.2.3].

Definition C.3. Let G be a good stratification for h at the origin. A fundamental system of neighborhoods, {Cα }, at the origin in U is a system of privileged neighborhoods for h at 0 with respect to G if and only if i)

{Cα } is a system of compact, Milnor neighborhoods for h at the origin;

and, for each Cα , there is an associated Whitney stratification, Sα , of Cα such that ◦

ii)

the interior of Cα , C α , in U is a stratum in Sα ;

iii)

Cα equals the closure of C α in U;

◦

By ii) and iii) and the condition of the frontier, the boundary of Cα , ∂Cα , is a union of Whitney strata, and we make the final requirement: iv)

the boundary strata of each Cα transversely intersect all the strata of G.

A fundamental system of neighborhoods, C = {Cα }, is a system of privileged neighborhoods for h at 0 if and only if there exists a good stratification, G, for h at 0 such that C is a system of privileged neighborhoods for h at 0 with respect to G. A fundamental system of neighborhoods, C = {Cα }, satisfying i), ii), and iii) above is a system of weakly privileged neighborhoods for h at 0 if and only if for each Cα , for all small ξ 6= 0, V (h − ξ) transversely intersects the boundary strata of Cα . We shall see below that a system of privileged neighborhoods is automatically a system of weakly privileged neighborhoods. A fundamental system of neighborhoods, C = {Cα }, is a universal system of privileged neighborhoods for h at 0 if and only if for every good stratification, G, for h at 0, there exists an open neighborhood W of the origin such that {Cα ∈ C | Cα ⊆ W } is a system of privileged neighborhoods for h at 0 with respect to G.

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One should note that the set of closed balls centered at the origin is a universal system of privileged neighborhoods for h, regardless of the function h – this is a very “universal” system, and this may seem like the more natural notion. This, however, seems to be too restrictive. Universal for h simply means that, locally, the fundamental system is privileged independent of the choice of good stratification for the particular function h.

Proposition C.4. Suppose that C = {Cα } is a system of privileged neighborhoods for h at 0. Then, C = {Cα } is a system of weakly privileged neighborhoods for h at 0 and, hence, for all Cα , h

for all small δ > 0, Cα ∩ h−1 (∂Dδ ) − → ∂Dδ is a proper, stratified submersion and is thus a locally trivial fibration. The fibre-homotopy-type of this fibration is independent of the choice of the system of weakly privileged neighborhoods, C, for h, the choice of Cα , and the choice of small δ > 0. Proof. The proof is essentially that of Lˆe in [Lˆ e 4]. Let G be a good stratification for h at the origin with respect to which C is a system of privileged neighborhoods. Pick a Cα in C. We shall ◦

h

◦

actually show that there exists  > 0 such that Cα ∩ h−1 (D − 0) − → D − 0 is a proper, stratified h

submersion. It follows that, for all δ with 0 < δ < , Cα ∩ h−1 (∂Dδ ) − → ∂Dδ is a proper, stratified submersion and, hence, a locally trivial fibration with fibre-homotopy-type independent of the choice of δ. This certainly shows that C is a system of weakly privileged neighborhoods for h at 0. Suppose to the contrary that no matter how small we choose  > 0 it is not the case that ◦

h

◦

Cα ∩ h−1 (D − 0) − → D − 0 is a proper, stratified submersion. As each Cα is compact, clearly this map is always proper. So, by the local finiteness of the stratification, there must exist a single Whitney stratum, S, of Cα and a sequence of points pi ∈ S such that the pi converge to some point p ∈ V (h), Tpi S converges to some T , Tpi V (h − h(pi )) converges to some T , and Tpi S ⊆ Tpi V (h − h(pi )). Let G denote the good stratum of G which contains p and let R denote the Whitney stratum of Cα which contains p. As Tpi S ⊆ Tpi V (h − h(pi )), we must have that T ⊆ T . By the Thom condition, Tp G ⊆ T . By Whitney’s condition a), Tp R ⊆ T . Hence, Tp R and Tp G are both contained in T – a contradiction as R and G intersect transversely. h

Thus, there exists  > 0 such that for all δ with 0 < δ < , Cα ∩ h−1 (∂Dδ ) − → ∂Dδ is a proper, stratified submersion and, hence, a locally trivial fibration with fibre-homotopy-type independent of the choice of δ. To see that the fibre-homotopy-type is independent of the choice of C and the choice of Cα , one may once again compare with the standard system of Milnor neighborhoods and then use the theorem of Dold [Hu, p.209], since we know that the inclusion of each fibre is a homotopy-equivalence by the proof of C.1. We leave the details to the reader. 

Note that we have the implications: {Cα } is a universal system ⇒ for all good stratifications G, {Cα } is a privileged system with respect to G ⇒ {Cα } is a privileged system ⇒ {Cα } is a weakly privileged system ⇒ {Cα } is a Milnor system.

Definition C.5. If C = {Cα } is a system of Milnor neighborhoods for h at 0, then a Milnor pair ◦

◦

for h at 0 is a pair (Cα , Dδ ) such that for all ξ ∈ Dδ − 0, Cα ∩ V (h − ξ) has the homotopy-type of the Milnor fibre. If, in addition, C is a system of weakly privileged neighborhoods, then we also h make the requirement that Cα ∩ h−1 (∂Dδ ) − → ∂Dδ is a proper, stratified submersion.

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DAVID B. MASSEY ◦

◦

◦

We now wish to consider an analytic function f : (D × U, D × 0) → (C, 0) where D is an open ◦

complex disc centered at the origin and U ⊆ Cn+1 . We use the coordinates (t, z0 , . . . , zn ) for D × U. We distinguish the t-coordinate because we will either be considering the particular hyperplane slice V (t) or because we will be interested in the family ft (z0 , . . . , zn ) := f (t, z0 , . . . , zn ). Proposition C.6. Suppose that V (t) is prepolar for f at the origin with respect to a good stratification G, and let {Cα } be a system of privileged neighborhoods with respect to the good stratification G ∩ V (t) for f|V (t) . Then, there exits an open neighborhood, W , of the origin in V (t) such that, for all Cα ⊆ W , there exists τα > 0 such that i)

there exists ω > 0 such that ◦ ◦
◦
Dτα × ∂Cα ∩ Ψ−1 Dω − 0 × Dτα   yΨ := (f, t) ◦
◦ Dω − 0 × Dτα

is a proper, stratified submersion; ii)

for all δ with 0 < δ < τα , there exists ξ > 0 such that ◦

Dδ × Cα ∩ f −1 (Dξ − 0)   yf ◦

Dξ − 0 ◦

is a proper, stratified submersion, where the strata are the cross-product strata of Dδ × Cα together with those of ∂Dδ × Cα ; and

iii) {Dδ × Cα | 0 < δ < τα } is a system of Milnor neighborhoods for f at the origin and hence, by ii), is in fact a system of weakly privileged neighborhoods. ◦

◦

Proof. There exists an open neighborhood of the origin in D × U of the form Dη × W such that ◦

◦

(Dη × W ) ∩ Σf ⊆ V (f ). As V (t) is prepolar, we may assume that G is defined inside Dη × W ◦

and that V (t) transversely intersects all strata of G, other than the origin, inside Dη × W . Finally, 1 as V (t) is prepolar, we may use Theorem 1.28 to conclude that γf,t (0) exists and, hence, we may ◦

select Dη × W so that (0 × W ) ∩ Γ1f,t ⊆ {0}. Let Cα ⊆ W . i) This follows the proof of Proposition 2.1 of [Lˆ e1], applied to each stratum of ∂Cα . Suppose the contrary. Then, we would have a stratum S of Cα and a sequence of points pi not in V (f ) but in C × S such that pi = (ti , qi ) → p = (0, q) ∈ V (t) ∩ V (f ) and such that (*)

Tpi V (f − f (pi ), t − ti ) + Tpi (C × S) 6= Cn+2 .

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225

(That Tpi V (f − f (pi ), t − ti ) exists is not completely trivial – it follows from the assumptions made in the preceding paragraph.) Let G denote the good stratum of V (f ) containing p. Note that G cannot be the point-stratum {0} as p is contained in 0 × ∂Cα . Let R denote the stratum of ∂Cα containing q. By taking a subsequence if necessary, we may assume that Tpi V (f − f (pi )) converges to some T and that Tqi S converges to some T . By the Thom condition, Tp G ⊆ T and, by Whitney’s condition a), Tq R ⊆ T . Furthermore, as V (t) is prepolar, V (t) transversely intersects G at p. Thus, Tpi (C × S) → C × T and Tpi V (f − f (pi ), t − ti ) → T ∩ Tp V (t). Also, we have that Tp (G ∩ V (t)) = Tp G ∩ Tp V (t) ⊆ T ∩ Tp V (t) and we know that Tp (G ∩ V (t)) + Tp (0 × S) = 0 × Cn+1 , as {Cα } is a system of privileged neighborhoods with respect to G ∩ V (t). It follows at once that T ∩ Tp V (t) + C × T = Cn+2 , but this contradicts (∗ ). This proves i). ◦

ii)

◦

◦

◦

◦

That f can be made a submersion on Dδ × C α follows from the fact that Dδ × C α ⊆ Dη × W ◦

and (Dη × W ) ∩ Σf ⊆ V (f ).

◦

That f can be made a stratified submersion on Dδ × ∂Cα and on ∂Dδ × ∂Cα is exactly the argument of i). ◦

Thus, what remains to be shown is that f can be made a submersion on the stratum ∂Dδ × C α . By Theorem 1.28 and Proposition 1.23, dim0 (Γ1f,t ∩ V (f )) 6 0 and thus we may assume that Γ1f,t ∩ V (f ) ∩ (∂Dδ × Cα ) is empty. As Γ1f,t ∩ (∂Dδ × Cα ) is compact, |f | obtains a minimum, ξ > 0, on Γ1f,t ∩ (∂Dδ × Cα ). Now, ◦

◦

consider the critical points of f restricted to ∂Dδ × C α that occur in f −1 (Dξ − 0). These points occur precisely on ◦

◦

Γ1f,t ∩ (∂Dδ × C α ) ∩ f −1 (Dξ − 0) which we know is empty. This proves ii). iii) a)

We first need two results. for all ω1 , ω2 with 0 < ω1 < ω2 < τα , there exists ξ > 0 such that ◦

C × Cα ∩ Φ−1 ((Dξ − 0) × [ω12 , ω22 ])   yΦ := (f, |t|2 ) ◦

(Dξ − 0) × [ω12 , ω22 ]

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DAVID B. MASSEY ◦

is a proper, stratified submersion and thus, for all η ∈ Dξ − 0, the inclusion (Dω1 × Cα ) ∩ V (f − η) ,→ (Dω2 × Cα ) ∩ V (f − η) is a homotopy-equivalence; and ◦

◦

◦

if Cα ⊆ C β , then there exist τ, ξ > 0 such that, for all δ ∈ Dτ −0 and η ∈ Dξ −0, the inclusion

b)

(Dδ × Cα ) ∩ V (f − η) ,→ (Dδ × Cβ ) ∩ V (f − η) is a homotopy-equivalence. ◦

Assuming a) and b) for the moment, we proceed with the proof. Suppose that Dσ ×Cα ⊆ Dρ ×C β . By b), for all small, non-zero δ and η, (Dδ × Cα ) ∩ V (f − η) ,→ (Dδ × Cβ ) ∩ V (f − η) is a homotopy-equivalence. If we select δ so small that Dδ is contained in both Dσ and Dρ , then we may apply a) twice to obtain that, for all small, non-zero η, (Dδ × Cα ) ∩ V (f − η) ,→ (Dσ × Cα ) ∩ V (f − η) and (Dδ × Cβ ) ∩ V (f − η) ,→ (Dσ × Cβ ) ∩ V (f − η) are homotopy-equivalences. The conclusion that (Dδ × Cα ) ∩ V (f − η) ,→ (Dρ × Cβ ) ∩ V (f − η) is a homotopy-equivalence now follows immediately by combining the three previous homotopyequivalences. We now prove a) and b). Proof of a):

That Φ is a stratified submersion on C × ∂Cα is once again exactly the proof of i). ◦

That Φ is a submersion on C × C α is similar to our argument in ii): as dim0 (Γ1f,t ∩ V (f )) 6 0, we may assume that ◦

Γ1f,t ∩ V (f ) ∩ ((Dω2 − Dω1 ) × Cα ) ◦

is empty. Therefore, by compactness, |f | obtains a minimum, ξ > 0, on Γ1f,t ∩ ((Dω2 − Dω1 ) × Cα ). ◦

◦

Now, consider the critical points of Φ restricted to C × C α that occur in Φ−1 ((Dξ − 0) × [ω12 , ω22 ]). These points occur precisely in ◦

◦

Γ1f,t ∩ f −1 (Dξ − 0) ∩ ((Dω2 − Dω1 ) × Cα ) which we know is empty. This proves a). ◦

Proof of b): ◦

Let Cα ⊆ C β . Let τ be so small that inside Dτ × Cβ all points of Γ1f,t occur in

Dτ × C α . Further, choose τ < min{τα , τβ } so that we may apply i) in both cases. Choose ξ so

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227

small that (Cα , Dξ ) and (Cβ , Dξ ) are Milnor pairs for f|V (t) and so small that we may apply i) to ◦

◦

◦

◦

both Dτ × ∂Cα and Dτ × ∂Cβ over (Dξ − 0) × Dτ . Fix some δ ∈ Dτ − 0 and η ∈ Dξ − 0. By i) or ii), V (f − η) transversely intersects all the strata of C × ∂Cα and C × ∂Cβ , so may Whitney stratify (C × Cβ ) ∩ V (f − η) by taking as strata the intersection of V (f − η) with each ◦

◦

of C × C α , C × (C β − Cα ), and the strata of C × ∂Cα and C × ∂Cβ . As Cα ∩ V (f|V (t) − η) ,→ Cβ ∩ V (f|V (t) − η) is a homotopy-equivalence and V (f|V (t) ) − η) transversely intersects ∂Cα and ∂Cβ , for all small Dµ we must have that (Dµ × Cα ) ∩ V (f − η) ,→ (Dµ × Cβ ) ∩ V (f − η) is also a homotopy-equivalence. We wish to pass from Dµ to Dδ by considering the function |t|2 on the stratified space (C × Cβ ) ∩ V (f − η) (with the stratification given above). By i), |t|2 has no critical points on the strata of (C × ∂Cα ) ∩ V (f − η) and (C × ∂Cβ ) ∩ V (f − η) ◦

when |t| < δ. In addition, the critical points on the interior strata, (C × C α ) ∩ V (f − η) and ◦

(C × (C β − Cα )) ∩ V (f − η), occur on the polar curve and, hence, by our earlier requirement, these ◦

critical points all occur in C × C α . Therefore, using stratified Morse theory [G-M2] together with the homotopy-equivalence lemma 3.7 of [Mi2], we find that the inclusion (Dδ × Cα ) ∩ V (f − η) ,→ (Dδ × Cβ ) ∩ V (f − η) is a homotopy-equivalence.  For a family of analytic functions ft : (U, 0) → (C, 0), we are interested in how the Milnor fibre and fibration “jump” as we move from small non-zero t to t = 0. Hence, we make the following definition. Definition C.7. If we are considering the family ft : (U, 0) → (C, 0), we refer to i) of C.6 by saying that the family satisfies the conormal condition with respect to {Cα }. The point of this condition is that it says that the Milnor fibration of f0 lifts trivially in the family ft on the boundary of the neighborhoods Cα . Definition C.8. The Thom set at the origin, Tf , is the set of (n + 1)-planes which occur as limits at the origin of the tangent spaces to level hypersurfaces of f , i.e. T ∈ Tf if and only if there exists ◦

a sequence of points pi in D × U − Σf such that pi → 0 and T = lim Tpi V (f − f (pi )). Equivalently, Tf is the fibre over the origin in the Jacobian blow-up of f (see [H-L]). Tf is thus a closed algebraic subset of the Grassmanian Gn+1 (Cn+2 ) = the projective space of (n + 1)-planes in Cn+2 . Proposition C.9. Suppose that V (t) is a prepolar slice for f at 0 or that V (t) = T0 V (t) 6∈ Tf . Then, i)

dim0 Γ1f,t 6 1, and

ii) the family ft satisfies the conormal condition with respect to any universal system of privileged neighborhoods, C, for f0 at 0.

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Moreover, whenever i) and ii) are satisfied, there is an inclusion of the Milnor fibre Fft0 ,0 into the Milnor fibre Ff0 ,0 for all small non-zero t0 ; the homotopy-type of this inclusion is independent of the choice of t0 and the choice of the universal system of privileged neighborhoods, C. Proof. That there is such an inclusion whenever i) and ii) are satisfied is standard. One considers the map Ψ := (f, t) and its restriction (Dτ × C) ∩ Ψ−1



Dξ − 0 × Dτ

  yΨ
Dξ − 0 × Dτ for appropriately small choices of C ∈ C, ξ, and τ . By the conormal condition,
this is a stratified submersion on the boundary. As dim0 Γ1f,t 6 1, the discriminant of Ψ, Ψ Γ1f,t , is also at most one-dimensional. Thus, we may lift a path in the base which avoids the discriminant to get a diffeomorphism between the Milnor fibre of f0 and C ∩ V (ft0 − η) for all small t0 and for all η with 0 < |η|  |t0 |. And, though we do not know that C is a system of privileged neighborhoods for ft0 , we may still take a small enough ball inside C to obtain the desired inclusion, which is clearly independent of the choice of t0 . That the inclusion is independent of the choice of privileged neighborhoods follows similarly. Suppose that C 0 is second universal system of privileged neighborhoods for f0 . Let C ∈ C and let ◦

C 0 ∈ C 0 be such that C 0 ⊆ C, and such that C and C 0 are small enough to give the Milnor fibre, i.e. for all small non-zero ξ, the inclusion of C 0 ∩ V (f0 − ξ) into C ∩ V (f0 − ξ) is a homotopy-equivalence where both spaces are homotopy-equivalent to the Milnor fibre of f0 at the origin. Then, as above, over a curve which avoids the discriminant, we have a proper, stratified submersion – where the strata are those of Dτ × ∂C together with those of Dτ × ∂C 0 plus the interior. Hence, the homotopy-equivalence C 0 ∩V (f0 −ξ) → C ∩V (f0 −ξ) lifts to a homotopy-equivalence 0 C ∩ V (ft0 − ξ) → C ∩ V (ft0 − ξ). The independence statement now follows easily. We must still show that if V (t) is a prepolar slice for f at 0 or V (t) = T0 V (t) 6∈ Tf , then i) and ii) hold. If V (t) is prepolar for f at 0, then i) follows from Theorem 1.28 and ii) follows from C.6.i. If V (t) 6∈ Tf , then clearly Γ1f,t is empty near the origin. It remains for us to show that if V (t) 6∈ Tf , then the family ft satisfies the conormal condition with respect to any universal system of privileged neighborhoods, C, for f0 . If V (t) 6∈ Tf , then V (t) certainly transversely intersects the smooth part of V (f ) in a neighborhood of the origin. Hence, we may use Proposition C.2 to conclude that there exists a good stratification, G, for f at the origin such that the strata of G which are contained in V (t) form a good stratification for f0 at the origin. The proof now proceeds like that of C.6.i. Suppose to the contrary that, for arbitrarily small Cα in C, there exists a stratum S of ∂Cα and a sequence of points pi not in V (f ) but which are in C × S such that pi = (ti , qi ) → p := (0, q) ∈ V (t) ∩ V (f ) and such that (*)

Tpi V (f − f (pi ), t − ti ) + Tpi (C × S) 6= Cn+2 .

Let G denote the good stratum of V (f ) which contains p. Note that G is contained in V (t) by the nature of our good stratification and that G cannot be simply the stratum consisting of the origin since p is contained in 0 × ∂Cα . Let R denote the stratum of ∂Cα containing q.

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229

By taking a subsequence if necessary, we may assume that Tpi V (f − f (pi )) converges to some T and that Tqi S converges to some T . By the Thom condition, Tp G ⊆ T and, by Whitney’s condition a), Tq R ⊆ T . Furthermore, as V (t) 6∈ Tf , we may assume that p is close enough to the origin that T 6= V (t). Thus, Tpi (C × S) → C × T and Tpi V (f − f (pi ), t − t − i) → T ∩ Tpi V (t). Also, we have that Tp G = Tp G ∩ Tp V (t) ⊆ T ∩ Tp V (t), and we know that Tp G + Tp (0 × S) = 0 × Cn+1 , as {Cα } is a system of privileged neighborhoods with respect to G ∩ V (t). It follows at once that T ∩ Tp V (t) + C × T = Cn+2 – which contradicts (∗).  If the polar curve, Γ1f,t , is empty, then the map Ψ which appears in the proof of Proposition C.9 is a stratified submersion over the entire base space and so, for all small t0 6= 0, we have a fibre-prerserving inclusion of the total space of the Milnor fibration of ft0 into the total space of the Milnor fibration of f0 . Moreover, exactly as above, this inclusion is independent – up to homotopy – of all of the choices made. By the theorem of Dold (see [Hu, p. 209]), this inclusion is a fibre homotopy-equivalence if and only if the inclusion of each fibre is a homotopy-equivalence. Therefore, we make the following definitions.

Definition C.10 Whenever i) and ii) of C.9 hold, we say that the family, ft , satisfies the universal conormal condition. If ft satisfies the universal conormal condition, we say that ft has the homotopy Milnor fibre lifting property if and only if the inclusion of C.9 is a homotopy-equivalence. If ft satisfies the universal conormal condition, we say that ft has the homology Milnor fibre lifting property if and only if the inclusion of C.9 induces isomorphisms on all integral homology groups. The family, ft , has the homotopy Milnor fibration lifting property if and only if ft has the homotopy Milnor fibre lifting property and Γ1f,t = ∅ in a neighborhood of the origin. This definition makes sense in light of our above discussion concerning the result of Dold.

One may also discuss the Milnor fibre and Milnor fibration up to diffeomorphism if one is willing to restrict consideration to the standard universal system of Milnor neighborhoods, namely the set of closed balls centered at the origin. In this case, we may use the h-cobordism Theorem and the pseudo-isotopy result of Cerf [Ce] to translate the homotopy information into smooth information – provided that we are in a sufficiently high dimension and that the Milnor fibre and its boundary are sufficiently connected. More specifically, if U is an open neighborhood of the origin in Cn+1 , ft : (U, 0) → (C, 0) has the homotopy Milnor fibration lifting property, n > 3, and the Milnor fibre and its boundary are simply-connected for each ft for all small t, then the diffeomorphism-type of the Milnor fibrations is constant in the family near t = 0. This connectedness condition can be realized by requiring n − dim0 Σf0 > 3 (see [K-M] and [Ra]). We wish to state the diffeomorphism results discussed above precisely. First, we give without proof Cerf’s pseudo-isotopy result in the form that we shall need it.

˙ 1 and let π : X → S 1 be Lemma C.11. Let X be a smooth manifold with boundary ∂X = X0 ∪X a smooth locally trivial fibration over a circle with fibre diffeomorphic to M × [0, 1], where M is a closed, simply-connected, smooth manifold of dimension > 5. Then, the restriction of π to X0 is a smooth locally trivial fibration with fibre diffeomorphic to

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DAVID B. MASSEY

M , and there exists a commutative diagram ∼ =

(X, X0 ) −−−−→ (X0 × [0, 1], X0 × {0}) diffeo.

π&

. π|X0 ◦ pr1 S1

where the diffeomorphism is the identity on X0 = X0 × {0}. Proposition C.12. Let U be an open neighborhood of the origin in Cn+1 . Suppose that the family ft : (U, 0) → (C, 0) has the homotopy Milnor fibration lifting property and n − dim0 Σf0 > 3. Then, the diffeomorphism-type of the Milnor fibrations of ft at the origin is independent of t for all small t. Proof. We shall use the notation from the proof of Proposition C.9. We fix the universal system of privileged neighborhoods to be the collection of closed balls centered at the origin. As ft has the homotopy Milnor fibration lifting property, the polar curve Γ1f,t is empty and so the map Ψ in the proof of Proposition C.9 is a proper stratified submersion. Hence, for 0 < ξ, |t0 |  , ft

0 the Milnor fibration of f0 is diffeomorphic to B ∩ ft−1 (∂Dξ ) −−→ ∂Dξ . The problem, of course, is 0 that B may be too large a ball in which to define the Milnor fibration of ft0 . Let F and E denote the fibre and the total space, respectively, of this previous fibration. Let F 0 denote the Milnor fibre of ft0 at the origin, where we again use closed balls for the Milnor neighborhoods. Let E 0 denote the total space of the Milnor fibration ft0 . As ft has the homotopy Milnor fibration lifting property, the inclusion of E 0 into E induces an inclusion F 0 ,→ F which is a homotopy-equivalence. Since n − dim0 Σf0 > 3, F , F 0 , ∂F , and ∂F 0 are simply-connected (see [Ra]). Combining this with the fact that F 0 ,→ F is a homotopy-equivalence, we may duplicate the argument of Lˆe and

◦

Ramanujam [L-R] to conclude that ∆T := E − E 0 is the total space of a differentiable fibration over ◦

∂Dξ with projection ft0 and fibre F − F 0 which is diffeomorphic to ∂F × [0, 1] via the h-cobordism theorem. ft0 Now, by Lemma C.11, ∆T −−→ ∂Dξ is diffeomorphic to ft ◦pr1

0 ∂E 0 × [0, 1] −−− −−→ ∂Dξ × {0}

by a diffeomorphism which is the identity on ∂E 0 = ∂E 0 × {0}. Combining this with a fibred collar ft

ft

0 0 of ∂E 0 in E 0 , we conclude that E 0 −−→ ∂Dξ is diffeomorphic to E −−→ ∂Dξ , which we already know is diffeomorphic to the Milnor fibration of f0 at 0. 

We now wish to prove a fundamental result – namely, that if we have a family ft in which the Milnor fibrations of a hyperplane slice are independent of t and the number of handles attached in passing from the Milnor fibre of the hyperplane slice to the entire Milnor fibre is constant, then the Milnor fibrations are constant in the family. Despite the fact that the dimension of the critical loci is allowed to be arbitrary, the argument is exactly that which we used in [Mas3] where the critical loci were all one-dimensional.

APPENDIX C

231

Theorem C.13. Let W be an open neighborhood of the origin in Cn+2 and let gt : (W, 0) → (C, 0) be an analytic family. Let s denote dim0 Σg0 . Assume that gt satisfies the universal conormal condition and that L is a linear form such that V (L) is prepolar for ft at the origin for all small t and
such that gt |V (L) satisfies the universal conormal condition. Suppose further that Γ1gt ,L · V (gt ) 0 is constant for all small t. Under the above assumptions, if gt |V (L) has the homology Milnor fibre lifting property, then gt has the homology Milnor fibre lifting property. Moreover, if s 6 n − 1 and gt |V (L) has the homotopy Milnor fibre lifting property, then gt has the homotopy Milnor fibre lifting property. Proof. This is actually quite trivial. Let F0 and Ft0 denote the Milnor fibre of g0 and gt0 for small non-zero t0 , respectively. The Milnor fibres of g0 |V (L) and gt0 |V (L) are then F0 ∩ V (L) and
Ft0 ∩ V (L), respectively. Let γ denote the constant value of Γ1gt ,L · V (gt ) 0 . As gt and gt |V (L) satisfy the universal conormal condition, we may repeat the argument of Proposition C.9 – lifting a path in the base which avoids the discriminants of both (gt , t) and (gt |V (L) , t) – to obtain compatible inclusions Ft0 ,→ F0 and Ft0 ∩ V (L) ,→ F0 ∩ V (L). Suppose that gt |V (L) has the homology Milnor fibre lifting property, i.e. Ft0 ∩ V (L) ,→ F0 ∩ V (L) induces isomorphisms on homology. We wish to show that Ft0 ,→ F0 induces isomorphisms on homology. We will accomplish this by showing that H∗ (F0 , Ft0 ) = 0. By considering the homology long exact sequence of the triple (F0 , F0 ∩ V (L), Ft0 ∩ V (L)), we find that Hi (F0 , Ft0 ∩ V (L)) ∼ = Hi (F0 , F0 ∩ V (L)) for all i. By Lˆe’s attaching result (Theorem 0.9) or Theorem 3.1, Hi (F0 , F0 ∩ V (L)) = 0 unless i = n + 1 and Hn+1 (F0 , F0 ∩ V (L)) ∼ = Zγ . Now, we are going to consider the homology long exact sequence of the triple (F0 , Ft0 , Ft0 ∩V (L)). From the last paragraph, we know that Hi (F0 , Ft0 ∩ V (L)) = 0 unless i = n + 1. In addition, Hi (Ft0 , Ft0 ∩ V (L)) = 0 unless i = n + 1. Moreover, Hn+1 (F0 , F0 ∩ V (L)) ∼ = Hn+1 (Ft0 , Ft0 ∩ V (L)) ∼ = Zγ . Thus, in the long exact sequence of the triple (F0 , Ft0 , Ft0 ∩ V (L)), all terms are zero except in the portion 0 → Zγ → Zγ → Hn+1 (F0 , Ft0 ) → 0. But, as in the proof of the result of Lˆe and Ramanujam [L-R], Hn+1 (F0 , Ft0 ) is free Abelian, since F0 is obtained from Ft0 by attaching handles of index less than or equal to n + 1. (One considers the function distance squared from the origin and lets the function grow from the small ball used to define Ft0 out to the ball used to define F0 . One hits no critical points of index greater than or equal to n + 2.) Thus, Hn+1 (F0 , Ft0 ) = 0 and we have proved the first claim. The second claim follows from the first, since s 6 n − 1 guarantees that F0 and Ft0 are simplyconnected, and then we apply the Whitehead Theorem. 

There are two more big results which we need to prove in this appendix – both deal with suspending singularities (see Chapter I.8). The first result is that there exists a universal system of privileged neighborhoods of a particularly nice form for the function h + wj , where w is a variable disjoint from those of h. The second result says, with a few extra assumptions, that the constancy of the Milnor fibrations in the family ft implies the constancy of the Milnor fibrations in the family ft + wj , where, again, w is disjoint from the variables of ft . This second result seems reasonable

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since the result of Proposition II.8.1 is that the Milnor fibre of ft + wj at the origin is homotopyequivalent to one-point union of j − 1 copies of the Milnor fibre of ft at the origin. However, both of these results are technical nightmares. Let U be an open neighborhood of the origin in Cn+1 and let h : (U, 0) → (C, 0) be an analytic ˜ function. Let j > 2 and define h(w, z) := h(z)+wj . We wish to show that the set {Dω ×B2n+2 | 0 < ˜ at the origin. Note that we may ω  } is a universal system of privileged neighborhoods for h not use C.6 to conclude that {Dω × B2n+2 | 0 < ω  } is even weakly privileged since the slice V (w) contains the entire critical locus of h + wj and, hence, is certainly not prepolar for h + wj . Of course, the actual argument is very similar to the proof of Proposition C.6. Proposition C.14. The set {Dω × B2n+2 | 0 < ω  } is a universal system of privileged neighborhoods for h + wj at the origin. Proof. As j > 2, Σ(h + wj ) = {0} × Σh. Fix any good stratification, G, for h + wj at the origin in Cn+2 . Let 0 > 0 be so small that the critical locus of the map h inside B2n+2 is contained in V (h) 0 and so small that, for all  with 0 <  6 0 , {0} × ∂B2n+2 transversely intersects all strata of {0} × ΣV (h) inside {0} × Cn+1 , ∂B2n+4 transversely intersects all strata of G (we write ∂B2n+4 t G), and ∂B2n+2 transversely intersects all strata of some good stratification for h at 0. This last condition guarantees, for all small non-zero ζ, that ∂B2n+2 t V (h − h(ζ)).

(*)

Now fix an  between 0 and 0 . We wish to show that there exists ω > 0 such that, for all ω with 0 < ω 6 ω , we have: ◦

a)

Dω × ∂B2n+2 t G;

b)

t G; ∂Dω × B 2n+2 

c)

∂Dω × ∂B2n+2 t G.

◦

After we show this, it will still remain to show that, if Dω1 × B1 ⊆ Dω2 × B2 , then, for all small non-zero t, the inclusion Dω1 × B1 ∩ V (h + wj − t) ,→ Dω2 × B2 ∩ V (h + wj − t) is a homotopy-equivalence. Proof of a): Clearly, as {0} × ∂B2n+2 transversely intersects all strata of {0} × ΣV (h) inside {0} × Cn+1 , C × ∂B2n+2 transversely intersects all singular strata of V (h + wj ). Suppose, however, that no matter how small we pick ω > 0, we still have a point in the smooth stratum, S := ◦

V (h + wj ) − {0} × ΣV (h), where S does not transversely intersect Dω × ∂B2n+2 . Then, we would have a sequence pi := (wi , qi ) ∈ C × ∂B contained in S such that pi → p := (0, q) ∈ {0} × ∂B , Tpi S ⊆ Tpi (C × ∂B ), Tpi S converges to some T , and Tpi (C × ∂B ) → C × Tq (∂B ). Let S 0 denote the stratum of G containing p.

APPENDIX C

233

By the Thom condition, Tp S 0 ⊆ T (this is true because T comes from the smooth stratum – we are not assuming Whitney conditions hold between the strata). Hence, Tp S 0 ⊆ T ⊆ C × Tq (∂B ) = Tp (∂B2n+4 ), where this last equality is true because the w-coordinate of p is 0. But, this contradicts the fact that ∂B2n+4 t G. This proves a). Before we prove b) and c), note that if w ∈ ∂Dω , then w 6= 0 and, hence, the only stratum of G ◦

which ∂Dω ×B 2n+2 and ∂Dω ×∂B2n+2 intersect is the smooth stratum S := V (h+wj )−{0}×ΣV (h).  ◦

Proof of b): Actually, we show, regardless of the size of ω > 0, that ∂Dω × B 2n+2 t S. 

◦

For if not, we would have p := (w, q) ∈ S such that w 6= 0 and Tp V (h+wj ) ⊆ Tp (∂D|w| ×B 2n+2 ).  This implies that ∂h ∂h = ··· = = 0, ∂z0 |q ∂zn |q i.e. that q ∈ Σh. Recalling that we chose  such that B ∩ Σh ⊆ V (h), we see that h(q) = 0. However, this contradicts that h(q) = −wj 6= 0. This proves b). Proof of c): Suppose not. Then, we would have a sequence pi := (wi , qi ) ∈ S ∩ (C × ∂B2n+2 ) with wi 6= 0, pi → p = (0, q) ∈ {0} × ∂B , and such that Tpi V (h + wj ) + Tpi (∂D|wi | × ∂B ) 6= Cn+2 . This implies that Tqi V (h − h(qi )) ⊆ Tqi (∂B2n+2 ), while h(qi ) = −wij approaches – but is unequal to – zero. This, however, is impossible by (∗). This proves c). We must still prove the homotopy-equivalence statement. In a manner completely similar to the proofs of a), b), and c) above, one can easily show, using the Thom condition, that the following statements are true: d) for all  with 0 <  6 0 , if ω1 is between 0 and ω , then for all ω2 with 0 < ω2 6 ω1 , there exists ξ > 0 such that

C × B2n+2 ∩ Ψ−1 (Dξ − 0) × [ω22 , ω12 ]   y Ψ := (h + wj , |w|2 ) (Dξ − 0) × [ω22 , ω12 ] is a proper, stratified submersion and therefore

d0 ) Dω2 × B ∩ (h + wj )−1 (Dξ − 0) ,→ Dω1 × B ∩ (h + wj )−1 (Dξ − 0) is a fibre-homotopy equivalence between total spaces (where the projection in each case is the obvious map h + wj ). e)

if 0 < 2 < 1 6 0 , then for all small, non-zero ω, there exists ξ > 0 such that

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DAVID B. MASSEY



Dω × Cn+1 ∩ Φ−1 (Dξ − 0) × [22 , 21 ]   y Φ := (h + wj , |z|2 ) (Dξ − 0) × [22 , 21 ] is a proper, stratified submersion and therefore

e0 ) Dω × B2 ∩ (h + wj )−1 (Dξ − 0) ,→ Dω × B1 ∩ (h + wj )−1 (Dξ − 0) is a fibre-homotopy equivalence. Now, suppose that we have Dω1 × B1 ⊆ Dω1 × B1 , where 0 < 2 < 1 6 0 , 0 < ω2 < ω1 6 ω1 , and ω2 < ω2 . We shall show that, for all small ξ > 0,

Dω2 × B2 ∩ (h + wj )−1 (Dξ − 0) ,→ Dω1 × B1 ∩ (h + wj )−1 (Dξ − 0) is a fibre-homotopy equivalence. By applying e), we know that, for all small ω > 0, there exists ξ 6= 0 such that e0 ) holds. On the other hand – by applying d) twice – for all small ω > 0, there exists ξ 6= 0 such that

Dω × B2 ∩ (h + wj )−1 (Dξ − 0) ,→ Dω2 × B2 ∩ (h + wj )−1 (Dξ − 0) and

Dω2 × B1 ∩ (h + wj )−1 (Dξ − 0) ,→ Dω1 × B1 ∩ (h + wj )−1 (Dξ − 0) are fibre-homotopy equivalences. The desired conclusion follows from the two homotopy-equivalences above together with e0 ).  For the final results of this appendix, we return to the setting of families of analytic functions. Again, U will denote an open neighborhood of the origin in Cn+1 and ft : (U, 0) → (C, 0) will be an analytic family. We continue with w being a variable disjoint from those of ft and with j > 2. Recall from C.8 that Tf denotes the Thom set of f at the origin. We need the following easy lemma: Lemma C.15. If V (t) 6∈ Tf , then V (t) 6∈ Tf +wj . Proof. This is completely trivial. We leave it as an exercise.  Proposition C.16. Suppose that V (t) 6∈ Tf and that the family ft + wj has the homology Milnor fibre lifting property. Then, Γ1f,t = ∅ near the origin and ft has the homology Milnor fibre lifting property. Moreover, if dim0 Σf0 6 n − 2, V (t) 6∈ Tf , and the family ft + wj has the homotopy Milnor fibre lifting property, then ft has the homotopy Milnor fibration lifting property. Proof. The second claim follows immediately from the first claim, since the condition dim0 Σf0 6 n − 2 implies that the Milnor fibres are simply-connected. Also, since V (t) 6∈ Tf , we immediately

APPENDIX C

235

have that Γ1f,t = ∅ near the origin. What we need to prove is that ft has the homology Milnor fibre lifting property. Fix a good stratification G for f0 at the origin. We must now make many choices. 1) 2) 3)

Let (B0 , Dλ0 ) be a Milnor pair for f0 such that B0 ∩ Σf0 ⊆ V (f0 ), and ∂B0 transversely intersects the strata of G.

From C.9, we know that the conormal condition holds, and so we may pick η, τ > 0 such that the map G := (f, t) restricted to C × ∂B0 has no critical values in (Dη − 0) × Dτ .

4)

Using C.14, we may also choose ω0 , ξ0 > 0 such that 5) (Dω0 × B0 , Dξ0 ) is a Milnor pair for f0 + wj , where 6) ω0j < η, and 7) all of the obvious Whitney strata of Dω0 × B0 transversely intersect all of the strata in the good stratification for f0 + wj which is induced by G (as given in Proposition 8.3). Now, as V (t) 6∈ Tf , Lemma C.15 tells us that V (t) 6∈ Tf +wj . Hence, ft + wj satisfies the universal conormal condition and so, for all small ν 6= 0 and all small t1 ,
8) Dω0 × B0 ∩ V (ft1 + wj − ν) is diffeomorphic to Ff0 +wj ,0 . We select t1 so that 9) t1 is in Dτ ,
10) Γ1f,t ∩ D|t1 | × B0 = ∅, and
11) Σf ∩ D|t1 | × B0 ⊆ V (f ). As t1 is in Dτ , there exists λ00 such that 12) 13) 14)

for all γ with 0 < γ < λ00 , B0 ∩ V (ft1 − γ) is diffeomorphic to Ff0 ,0 . Now, let (B , Dλ ) be a Milnor pair for ft1 with  < 0 and λ < λ00 .

Then, there exist ω, ξ > 0 such that 15) 16) 17)

(Dω × B , Dξ ) is a Milnor pair for ft1 + wj , where we assume that ω j < min{λ, λ00 , ω0j } and ξ < min{ξ0 , ω j , η − ω0j }, where η − ω0j > 0 by 6).

Finally, we select ν in 8) so small that 18)

0 < |ν| < min{η − ω0j , λ − ω j , ξ}.

Now that we have made all of these choices, we are ready to begin the intuitive part of the proof. We have the inclusions i Fft1 +wj ,0 ∼ → (Dω × B0 ) ∩ V (ft1 + wj − ν) = (Dω × B ) ∩ V (ft1 + wj − ν) − l − → (Dω0 × B0 ) ∩ V (ft1 + wj − ν) ∼ = Ff0 +wj ,0 ,

236

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where we are assuming that l ◦ i induces isomorphisms on homology. We will first show that l induces isomorphisms on homology and, hence, so does i. Actually, we will show that l is a homotopy-equivalence. We accomplish this by showing that ◦
(Dω0 − Dω ) × B0 ∩ V (ft1 + wj − ν)

(*)

  yw ◦

Dω0 − Dω is a proper, stratified submersion. ◦

◦

Critical points of the map in (Dω0 − Dω ) × B0 occur where grad(ft1 ) = 0; that is, at points (w, t1 , z) such that (t1 , z) is in Γ1f,t or in Σf . By 10), Γ1f,t ∩ (Dt1 × B0 ) is empty and, by 11), Σf ∩ (Dt1 × B0 ) ⊆ V (f ). But, if ft1 = 0, then wj − ν = 0. However, this is impossible since ◦

w ∈ Dω0 − Dω and thus we would have to have |wj | > ω j – but we know that ω j > ξ > |ν| by 17) and 18). ◦

Now, we consider critical points of (∗) which occur on (Dω0 − Dω ) × ∂B0 . These occur at points (w, p) where Tp V (ft1 − ft1 (p)) ⊆ Tp ∂B0 . However, 0 < |ft1 (p)| = |wj − ν| 6 |w|j + |ν|, where 0 < |wj − ν| by the argument of the preceding paragraph. But, w ∈ Dω0 and so |w|j + |ν| 6 ω0j + |ν| which is 6 ω0j + η − ω0j by 18). Hence, 0 < |ft1 (p)| 6 η, t1 ∈ Dτ , and Tp V (ft1 − ft1 (p)) ⊆ Tp ∂B0 ; this contradicts 4). Therefore, the map (∗) is a proper, stratified submersion and, hence, is a locally trivial fibration. It follows at once that the inclusion, l, is a homotopy-equivalence and, thus, it follows that our earlier map i (Dω × B ) ∩ V (ft1 + wj − ν) − → (Dω × B0 ) ∩ V (ft1 + wj − ν) induces isomorphisms on homology. We wish now to show that i is obtained up to homotopy by wedging together j − 1 copies of the suspension of the inclusion map B ∩ V (ft1 − ν) ,→ B0 ∩ V (ft1 − ν) which, by 12), 13), and 18), is nothing more than the inclusion Fft1 ,0 ,→ Ff0 ,0 . It would then follow that Fft1 ,0 ,→ Ff0 ,0 induces isomorphisms on homology since i does. But, since |wj − ν| 6 |w|j + |ν| 6 ω j + |ν| 6 min{ξ, λ00 } by 18) and 14), we may proceed as in Proposition II.8.1 and find that projection by w realizes, up to homotopy: (Dω × B ) ∩ V (ft1 + wj − ν) as the wedge of j − 1 copies of the suspension of Fft1 ,0 , (Dω × B0 ) ∩ V (ft1 + wj − ν) as the wedge of j − 1 copies of the suspension of Ff0 ,0 , and the map i as the wedge of j − 1 copies of the suspension of the map Fft1 ,0 ∼ = B ∩ V (ft1 − ν) ,→ B0 ∩ V (ft1 − ν) ∼ = Ff0 ,0 . The conclusion follows. 

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242

Subject Index

Additivity result . . . . . . . . . . . . . . . . . . . . . 150 Agreeable reorganization . . . . . . . . . . . . . . 14 Aligned good stratification . . . . . . . . . . . 97 Aligned singularity . . . . . . . . . . . . . . . . . . . 97 Aligning coordinates . . . . . . . . . . . . . . . . . 97 Analytic cycle . . . . . . . . . . . . . . . . . . . . . . . 177 Characteristic cycle . . . . . . . . . . . . . . . . . 128 Characteristic polar cycle . . . . . . . . . . . 172 Conormal condition . . . . . . . . . . . . . . . . . 227 Conormal Jacobian tuple . . . . . . . . . . . . 160 Conormal polar cycle . . . . . . . . . . . . . . . . 161 Conormal Lˆe-Vogel cycle . . . . . . . . . . . . . 161 Continuous family of sheaves . . . . . . . . . 147 Coordinate planes example . 36, 37, 86-90 Correct dimension . . . . . . . . . . . . . . . . . . . . 12 Critical locus algebriac . . . . . . . . . . . . . . . . . . . . . . . . . . 118 C− . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 canonical stratified. . . . . . . . . . . . . . . . .119 conormal-regular . . . . . . . . . . . . . . . . . . . 118 F• − . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 Nash . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 regular . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 relative differential . . . . . . . . . . . . . . . . . 119 stratified. . . . . . . . . . . . . . . . . . . . . . . . . . .119 Derived category . . . . . . . . . . . Appendix B Essential arrangement . . . . . . . . . . . . . . . . 88 Exceptional pair . . . . . . . . . . . . . . . . . . . . . 28 Flat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75, 85 FM cone singularity . . . . . . . . . . . . . . 61, 62 Gap cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Gap ratio . . . . . . . . . . . . . . . . . . . . . . . . . 25, 26 Gap sheaf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Gap varieties . . . . . . . . . . . . . . . . . . . . . . . . 7, 8 Generic linear reorganization . . . . . . . . . . 14 Generic arrangement . . . . . . . . . . . . . . . . . 86 243

Generic Lˆe number . . . . . . . . . . . . . . . . . . 110 Global Lˆe number . . . . . . . . . . . . . . . . . . . . 80 Good stratification . . . . . . . . . . . . . . 54, 165 Handles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 Hyperplane arrangement . . . . . . . . . . 75, 85 Intersection theory . . . . . . . . . Appendix A K¨ unneth isomorphism . . . . . . . . . . . . . . . . 195 Lˆe’s attaching result . . . . . . . . . . . . . . 38, 39 Lˆe cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 Lˆe-Iomdine formulas . . . . . . . . . 73, 169-170 Lˆe-Iomdine-Vogel formulas . . . . . . . . . . . . 27 Lˆe number . . . . . . . . . . . . . . . . . . . . . . . 43, 44 Lˆe-Ramanujam result . . . . . . . . . . . . . . . . 39 Lˆe-Saito result . . . . . . . . 39, 40, 91-96, 170 Lˆe-Vogel cycle . . . . . . . . . . . . . . . . . . 161, 166 Lˆe-Vogel number . . . . . . . . . . . . . . . . . . . . .166 Lˆe-Vogel stratification . . . . . . . . . . . . . . . 163 Lˆe-Vogel tuple . . . . . . . . . . . . . . . . . . . 163-164 Milnor fibration . . . . . . . . . . . . . . . . . . 35, 36 Milnor fibre . . . . . . . . . . . . . . . . . . . . . . . . . . 36 Milnor fibre lifting property . . . . . . . . . 229 Milnor neighborhood . . . . . . . . . . . . . . . . 221 Milnor number . . . . . . . . . . . . . . . . . . 36, 138 Milnor pair . . . . . . . . . . . . . . . . . . . . . . . . . 223 M¨obius function . . . . . . . . . . . . . . . . . . . . . . 88 Morse inequalities . . . . . . . . . . . . . . . . . . . . 68 Nearby cycles . . . . . . . . . . . . . . . . . . . 205-206 Non-reduced plane curve example . . . . 64 Perverse cohomology . . . . . . . . . . . . . . . . . 218 Perverse sheaf . . . . . . . . . . . . . . . . . . . 200-201 Pl¨ ucker formula . . . . . . . . . . . . . . . . . . . . . . 74 Polar curve, relative . . . . . . 39, 41, 123-127 Polar cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 Polar number . . . . . . . . . . . . . . . . . . . . . 42-43 Polar ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 Polar variety . . . . . . . . . . . . . . . . . . . . . . . . . 41

Positively perverse sheaf . . . . . . . . . . . . 201 Pre-aligning coordinates . . . . . . . . . . . . . . 98 Prepolar coordinates . . . . . . . . . . . . . . . . . 54 Prepolar deformation . . . . . . . . . . . . . . . 103 Prepolar slice . . . . . . . . . . . . . . . . . . . . . . . . 54 Prepolar tuple . . . . . . . . . . . . . . . . . . . . . . . 54 Privileged neighborhood . . . . . . . . . . . . 222 universal system of . . . . . . . . . . . . . . . 222 weakly . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 Pseudo-isotopy result . . . . . . . . . . . 229-230 Pseudo-Zariski topology . . . . . . . . . . . .13-14 Sebastiani-Thom result . . . . . . . . . . . 38, 210 Segre-Vogel relation . . . . . . . . . . . . . . . . . . . 22 Semi-continuity of Lˆe numbers . . . . . . . 82 Σ∗ f . . . . . . . . . . . . . . . . . . . . . see Critical locus Stability of Continuity . . . . . . . . . . . . . . . 148 Super aligned singularity . . . . . . . . . . . . . 99 Suspension result . . . . . . . . . . . . 37, 38, 102 Swallowtail singularity . . . . . . . . . . . . 76-79 Thom’s af condition . . . . . . . . . . . . . 91, 142 Thom reduction . . . . . . . . . . . . . . . . . . . . . 145 Thom set . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 Total exceptional divisor . . . . . . . . . . . . . 130 Uniform Lˆe-Iomdine formulas . . . . . . . . 81 Unifying reorganization . . . . . . . . . . . . . . . 23 Universal conormal condition . . . . . . . . 229 Upper-semicontinuity result . . . . . . . . . . 150 Vanishing cycles . . . . . . . . . . . . . . . . . . . . . . 207 Vanishing M¨obius function . . . . . . . . . . . . . 86 Visible stratum . . . . . . . . . . . . . . . . . . . . . . . 136 Vogel cycle. . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 Vogel reorganization . . . . . . . . . . . . . . . . . . . 23 Vogel set . . . . . . . . . . . . . . . . . . . . . . . . . . . 9, 10 Weighted homogeneous polynomial . . . . . . . . . . . . . . . . . . . 36, 74-76 Whitney umbrella . . . . . . . . . . . . . . . . 37, 38 Zariski multiplicity conjecture . . . . . . . 100

244