Lectures on Hilbert schemes of points on surfaces

Lectures on Hilbert schemes of points on surfaces Preliminary Version (October 11, 1996) Hiraku Nakajima Author address: Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153, Japan E-mail address: [email protected]

iii

Preface In the spring of 1996, I gave a series of lectures on the Hilbert schemes of points on surfaces at Department of Mathematical Sciences, University of Tokyo. The purpose of the lectures was to discuss various properties of the Hilbert schemes of points on surfaces. Although it was not noticed until recently, the Hilbert schemes have relationship with many other branch of mathematics, such as topology, hyper-K¨ahler geometry, symplectic geometry, singularities, and representation theory. This is reflected to this note: each chapter, which roughly corresponds to one lecture, discusses different topics. These lectures were intended for graduate students who have basic knowledge on algebraic geometry and ordinary topology. The only results which will be used but not proved in this note are Grothendieck’s construction of the Hilbert scheme (Theorem 1.1) and results on intersection cohomology (§6.1). I recommend to the reader to accept these results when he/she is not familiar with them. I have tried to make it possible to read each chapter independently. I believe that it is almost successful. The interdependence of chapters is figured in the next page. The broken arrows mean that we need only the statement of results in the outgoing chapter, and do not need its proof. Sections 9.1,9.3 are based on A. Matsuo’s lectures. His lectures contained Monster and Macdonald polynomials. I regret omitting these subjects. I hope to understand these by Hilbert schemes in future. The note was prepared by T. Gocho and N. Nakamura. I would like to thank them for their efforts. I am also grateful to A. Matsuo and H. Ochiai for their comments throught the lectures. Particular thanks are due to G. Ellingsrud, I. Grojnowski, K. Hasegawa, N. Hitchin, Y. Ito, A. King, G. Kuroki, G. Segal, and S. Strømme for discussions on results in this note. August, 1996 Hiraku Nakajima

iv

Interdependence of the Chapters Chapter 2 §2.1

Chapter 1

Chapter 2 §2.2

Chapter 3

Chapter 4

Chapter 5

Chapter 6

Chapter 7

Chapter 8

Chapter 9

Contents Preface

iii

Introduction

1

Chapter 1. Hilbert scheme of points 1.1. General Results on the Hilbert scheme 1.2. Hilbert scheme of points on a surface 1.3. Symplectic structure 1.4. Hilbert scheme of points on the plane

4 4 6 8 9

Chapter 2. Framed moduli space of torsion free sheaves on CP2 2.1. Monad 2.2. Rank 1 case [n]

13 13 20

Chapter 3. Hyper-K¨ahler metric on (C2 ) 3.1. Geometric invariant theory and the moment map 3.2. Hyper-K¨ahler quotients

24 24 32

Chapter 4. Resolution of simple singularities 4.1. General Statement 4.2. Dynkin diagrams 4.3. Tautological vector bundles

42 42 44 47

Chapter 5. Poincar´e polynomials of the Hilbert schemes (1) 5.1. Perfectness of the Morse function arising from the moment map [n] 5.2. Poincare polynomial of (C2 )

52 52 57

Chapter 6. Poincar´e polynomials of Hilbert schemes (2) 6.1. Results on intersection cohomology 6.2. Proof of the formula

65 65 67

Chapter 7. Hilbert scheme on the cotangent bundle of a Riemann surface 7.1. Morse theory on holomorphic symplectic manifolds 7.2. Hilbert scheme of T ∗ Σ 7.3. Analogy with the moduli space of Higgs bundles

70 70 71 77

Chapter 8. Homology group of the Hilbert schemes and the Heisenberg algebra

80

v

vi

CONTENTS

8.1. 8.2. 8.3. 8.4.

Heisenberg algebra and Clifford algebra Correspondences Main construction Proof of Theorem 8.13

80 81 83 86

Chapter 9. Symmetric products of an embedded curve, symmetric functions and vertex operators 94 9.1. Symmetric functions 94 9.2. Symmetric products of an embedded curve 97 9.3. Vertex algebra 102 9.4. Moduli space of rank 1 sheaves 110 Bibliography

113

Introduction Moduli spaces parameterizing objects associated with a given space X are rich source of spaces with interesting structures. They usually inherits structures of X, but sometimes even more: they have more structures than X, or pull out hidden structures of X. The purpose of this note is to add an example of these phenomena. We study the moduli space parameterizing 0-dimensional subschemes of length n in a nonsingular quasi-projective surface X over C. It is called the Hilbert scheme of points, and denoted by X [n] . An easy example of a 0-dimensional subscheme is a collection of distinct points. In this case, the length is equal to the number of points. When some points collide, more complicated subschemes appear. For example, when two points collide, we get infinitely near points, that is a pair of a point x and and a 1-dimensional subspace of the tangent space Tx X. This shows difference between X [n] and the n-th symmetric product S n X on which the information of the 1-dimensional subspace is lost. However, when X is 1-dimensional, we have unique 1-dimensional subspace in Tx X. In fact, the Hilbert scheme X [n] is isomorphic to S n X when dim X = 1. When X is 2-dimensional, X [n] is smooth and there is a morphism π : X [n] → S n X which is a resolution of singularities by a result of Fogarty [17]. This present a contrast to Hilbert schemes for dim X > 2. As we mentioned at the beginning, X [n] inherits structures from X. First of all, it is a scheme. It is projective if X is projective. These follows from Grothendieck’s construction of Hilbert schemes. A nontrivial example is a result by Beauville [6]: X [n] has a holomorphic symplectic form when X has one. When X is projective, X has a holomorphic symplectic form only when X is a K3 surface or an abelian surface by the classification theory. We also have interesting noncompact examples: X = C2 or X = T ∗ Σ where Σ is a Riemann surface. These examples are particularly nice because of the existence of a C∗ -action, which naturally induces an action on X [n] . (See Chapter 7.) When X = C2 , X [n] can be identified with the set of GLn (C)-orbits of (B1 , B2 , i) where B1 , B2 are commuting n × n-matrices and i is a cyclic vector (Theorem 1.14). Many [n] properties of (C2 ) are derived from this description. In Chapter 3, we shall regard the description as a geometric invariant theory quotient and a hyper-K¨ahler quotient. This description is very similar to the definition of quiver varieties which were studied in [62]. These structures of X [n] , discussed in the first half of this note, are inherited from X. We shall begin to study newly arising structures in latter chapters. They appear when we consider components all together. We will encounter this phenomenon first in G¨ottche’s 1

2

INTRODUCTION

formula for the Poincare polynomials in Chapter 6. Their generating function is given by ∞ X

n

[n]

q Pt (X ) =

n=0

∞ Y

(1 + t2m−1 q m )b1 (X) (1 + t2m+1 q m )b3 (X) , (1 − t2m−2 q m )b0 (X) (1 − t2m q m )b2 (X) (1 − t2m+2 q m )b4 (X) m=1

where bi (X) is the i-th Betti number of X. Each individual term Pt (X [n] ) has no nice expression. It is even more apparent if we consider the generating function of Euler numbers. The answer is the power of Dedekind’s η-funtion (the term q 1/24 is missing), which has a nice modular property. Recall that the q-expansion coefficients of modular forms often have number theoretic meaning, but structures appear after they are treated all together. In Chapter 8 we shall push forward this thought. We shall construct a representation of products of Heisenberg and Clifford algebras on the direct sum of homology groups of all L [n] components n H∗ (X ). Hence the generating fucntion can be interpreted as a character of the Heisenberg algebra. Its modular property can be explained if one interpret it as a partition function of the conformal field theory on an elliptic curve Eτ = C/Z + Zτ . Since it depends only on the complex structure of Eτ , it is invariant under τ 7→ −1/τ . In Chapter 9, our construction is proceeded further. We shall construct a representation of an affine Lie algebra (or a structure of the vertex algebra) on the homology groups. These study will lead us to wonder the reason why we encounter objects, such as modular forms, affine Lie algebras, the conformal field theory etc. Usually we think these objects live with elliptic curves, not with surfaces. Formally, we have two possibilities: one is that these objects are so universal (like Dynkin diagrams) that they appear everywhere. The other possiblity is that elliptic curves are hidden in the Hilbert schemes. We do not know which is correct at this moment, but we believe that the second one is correct. In fact, we think the “space” which is really relevant is the generating function of the Hilbert schemes: ∞ X X [n] q n . n=0

[n]

This is because X should not be studied for indivisual n. We must study components all together. Moreover, the generating function should have some kind of modular properties. Unfortunately, there is no theory for generating functions (or generating spaces ?) of manifolds at this moment. But the author think the above object exists. Or, it might be true that an appropriate language here is the string theory. The relation between X [n] and S n X have many similarities with the relation between the string and field theory. In fact, one of our motivation comes from physics. It is the duality. In [79], Vafa and Witten have pointed out that the S-duality (or Mantonen-Olive duality) conjecture implies that the generating function of Euler numbers of moduli spaces of instantons has a modular property. When the base manifold is a K3 surface, the Euler numbers of moduli spaces of instantons are the same as those of Hilbert schemes of points (strictly speaking, we must consider moduli spaces of stable sheaves instead of moduli spaces of instantons, which are usually noncompact). Then G¨ottsche’s formula gives us the desired answer. Moreover, from results in [62], we can see that homology groups of moduli spaces of sheaves on an ALE

INTRODUCTION

3

space (the minimal resolution of a simple singularity) form an integrable representation of an affine Lie algebra. The modular property of the characters of the representation (or the string functions) were observed by Kac-Peterson (see [43, Chapter 13]), and explained in the language of the conformal field theory as above. More recently, Vafa [78] suggested that our Heisenberg algebra and the above affine Lie algebra could be understood in the frame work of the heterotic-type IIA duality. (See also Harvey-Moore [35].) Let us briefly explain the construction of the action of the Heisenberg/Clifford algebra on homology groups of the Hilbert schemes, which leads the author to write this note. Motivated by constructions of lower triangular part of the quantized enveloping algebra by Ringel and Lusztig ([73, 52]), the author constructed integrable highest weight representations of the affine Lie algebra on the homology group of moduli spaces of torsion-free sheaves on an ALE space. (The construction [62, 67] was given in terms of moduli spaces of representations of quivers, whose identification with moduli spaces of torsion-free sheaves is proved by a modification of [50].) The generators of the affine Lie algebra (as a Kac-Moody algebra) are given by moduli spaces of parabolic sheaves regarded as correspondences in products of two moduli spaces. Thus it is important to treat moduli spaces with different Chern classes all together. Parabolic sheaves consist of pairs of sheaves which are isomorphic outside a given curve. Generators of the Heisenberg/Clifford algebra acting on homology groups of the Hilbert schemes (Chapter 8) are again given by certain correspondences. They consist of pairs of ideals which are isomorphic outside a point, which may move. Thus two constuctions are have little differences, but also many common features. In the first chapter, we collect basic facts on the Hilbert schemes of points on surfaces, which will be needed in later chapters. In Chapter 2, the Hilbert scheme (more generally moduli spaces of torsion free sheaves) on the affine plane C2 is shown to be isomorphic to a certain space of quadruple of matrices. This description is useful to study the Hilbert scheme. In Chapter 3, the Hilbert scheme of points on C2 is identified with a hyperK¨ahler quotient of a certain quaternion vector space. In particular, it has a hyper-K¨ahler metric. Relationship between the moment map in symplectic geometry and the geometric invariant theory is discussed briefly. In Chapter 4, we construct the minimal resolution of the simple singularity, using the Hilbert scheme on C2 . Some properties of the minimal resolution are discussed from this point of view. For example, the minimal resolution inherits the hyper-K¨ahler metric on the Hilbert scheme. In Chapter 5, we compute the Poincar´e polynomial of the Hilbert scheme of points on C2 using the natural torus action. The corresponding moment map is a natural Morse function on the Hilbert scheme. In Chapter 6, we generalize this formula to the case of a general surface X. In Chapter 7, the Hilbert scheme is studied when the base space is the cotangent bundle of a Riemann surface. In Chapter 8, we shall construct a representation of products of the Heisenberg algebras and the Clifford algebras on the homology group of the Hilbert scheme. Finally in Chapter 9, we shall study various homology classes arising from an embedded curve. They have close relation to symmetric functions and vertex operators.

CHAPTER 1

Hilbert scheme of points In this chapter, we collect basic facts on the Hilbert scheme of points on a surface. We do not assume the field k is C unless mentioned. 1.1. General Results on the Hilbert scheme First, we recall the definition of the Hilbert scheme in general (not necessarily of points, nor on a surface). Let X be a projective scheme over an algebraically closed field k and OX (1) an ample line bundle on X. We consider the contravariant functor HilbX from the category of schemes to the category of sets which is given by

HilbX : [Schemes] → [Sets],     

 Z is a closed subscheme,     i Z ֒→ X × U HilbX (U) = Z ⊂ X × U .  π ↓  ↓ p : π is flat   2     U = U

Namely, HilbX is a functor which associates a scheme U with a set of families of closed subschemes in X parameterized by U. Let π : Z → U be the projection. For u ∈ U, the Hilbert polynomial in u is defined by Pu (m) = χ(OZu ⊗ OX (m)),

where Zu = π −1 (u). Since Z is flat over U, Pu is independent of u ∈ U if U is connected. Conversely, for each polynomial P , let HilbPX be the subfunctor of HilbX which associates U with a set of families of closed subschemes in X parameterized by U which has P as its Hilbert polynomial. Now the basic fact proved by Grothendieck is the following theorem. Theorem 1.1 (Grothendieck [34]). The functor HilbPX is representable by a projective scheme HilbPX . This means that there exists a universal family Z on HilbPX , and that every family on U is induced by some morphism φ : U → HilbPX . The proof of this theorem is not given in this note. But we shall give a concrete description when P is a constant polynomial and X is an affine plane A2 . The reader who has no interests on the proof should accept the above theorem, and proceed. We do not need the proof of the theorem in this note. 4

1.1. GENERAL RESULTS ON THE HILBERT SCHEME

5

Moreover, if we have an open subscheme Y of X, we have the corresponding open subscheme HilbPY of HilbPX parameterizing subschemes in Y . In particular, HilbPY is defined for a quasi-projective scheme Y . Definition 1.2. Let P be a constant polynomial given by P (m) = n, for all m ∈ Z. We denote by X [n] = HilbPX the corresponding Hilbert scheme and call it the Hilbert scheme of n points in X. Let x1 , x2 , . . . , xn ∈ X be n distinct points and consider Z = {x1 , x2 , . . . , xn } ⊂ X as a closed subscheme. Since the structure sheaf of Z is given by OZ =

n M

the skyscraper sheaf at xi ,

i=1

we have OZ ⊗ OX (m) = OZ , for all m ∈ Z, and hence Z ∈ X [n] . This is the reason why X [n] is called the Hilbert scheme of n points in X. Let S n X be the symmetric product of X, i.e. S nX = X · · × X} /Sn , | × ·{z n times

where Sn is the symmetric group of degree n. It parameterizes effective zero-dimensional P cyclesPof degree n in X. Its element is written as a formal sum ni [xi ], where ni ∈ N with ni = n. Roughly speaking, X [n] is “the moduli space of n points in X”, but in general it is much more complicated and interesting than the symmetric product S n X as we shall see in the course of these lectures. In order to see the difference, we suppose X nonsingular and consider X [2] . As mentioned above, {x1 , x2 } can be considered as a point in X [2] if x1 and x2 are distinct points. What happens when x1 and x2 collide ? For each point x ∈ X, a vector v 6= 0 ∈ Tx X defines an ideal J ⊂ OX given by (1.3)

J = {f ∈ OX | f (x) = 0, dfx(v) = 0}.

Then J has colength 2, and hence OX /J defines a zero-dimensional subscheme Z in X [2] . We can regard Z as a set of 2 points in X infinitesimally attached with each other in the direction of v, and X [2] contains a set of this kind of infinitely near points in X. Actually, this gives a complete description of X [2] . Namely, (1.4)

X [2] = Blow∆ (X × X)/S2,

where Blow∆ (X × X) is the blowup of X × X along the diagonal ∆. We shall show this explicitly when X is A2 later in this section. When dim X = 1, it is known that X [n] = S n X, and X [n] is actually the moduli space of n points in X. This is roughly because we have only one direction in which two different points collide, and hence only the positions of points and their multiplicities are relevant.

6

1. HILBERT SCHEME OF POINTS

For example, the Hilbert scheme of n points in the affine line A is A[n] = {I ⊂ k[z] | I is an ideal, dimk k[z]/I = n}

= {f (z) ∈ k[z] | f (z) = z n + a1 z n+1 + · · · + an , ai ∈ k} = S n A.

In general, the relation between X [n] and S n X is given by the following theorem. Theorem 1.5 ([60, 5.4]). There exists a morphism [n]

π : Xred → S n X defined by π(Z) =

X

length(Zx )[x].

x∈X

This morphism π is called the Hilbert-Chow morphism which associates a closed subscheme with its corresponding cycle. For example, OX /J ∈ X [2] is mapped to 2[x] in the example (1.3). Let ν = (ν1 , . . . , νk ) be a partition of n, namely, ni ∈ Z>0 with ν1 ≥ ν2 ≥ · · · ≥ νk > 0 P and ki=1 νi = n. For each partition ν of n, we define ) ( k X Sνn X = νi [xi ] ∈ S n X xi 6= xj for i 6= j . i=1

Then Sνn X has dimension k dim X, where the number k is called the length of ν and denoted by l(ν). We have the stratification of S n X, [ (1.6) S nX = Sνn X. ν

n The open stratum S(1,...,1) is the nonsingular locus of S n X.

Exercise 1.7. (1) Suppose that X is nonsingular with dim X = 1. Show that X [n] = S X is nonsingular. (2) Show S n (P1 ) = Pn . n

1.2. Hilbert scheme of points on a surface In these lectures, we are interested in the case of dim X = 2, and in the following we assume that X is nonsingular and dim X = 2 whenever otherwise stated. In this case, X [n] has especially nice properties as the next theorem shows. Theorem 1.8 (Fogarty [17]). Suppose X is nonsingular and dim X = 2, then the following holds. (1) X [n] is nonsingular, and has dimension 2n. (2) π : X [n] → S n X is a resolution of singularities.

1.2. HILBERT SCHEME OF POINTS ON A SURFACE

7

Proof. (1) Let Z ∈ X [n] and JZ the corresponding ideal. The Zariski tangent space of X [n] at Z is given by TZ X [n] = HomOX (JZ , OX /JZ ).

To prove the smoothness of X [n] , it is enough to show that the dimension of TZ X [n] does not depend on Z ∈ X [n] . From the exact sequence 0 → JZ → OX → OX /JZ = OZ → 0, we have 0 → Hom(OZ , OZ ) → Hom(OX , OZ ) → Hom(JZ , OZ ) → Ext1 (OZ , OZ ) → Ext1 (OX , OZ ) → Ext1 (JZ , OZ )

→ Ext2 (OZ , OZ ) → Ext2 (OX , OZ ) → Ext2 (JZ , OZ ) → 0. P Since the Euler characteristic 2i=0 dim Exti (JZ , OZ ) is independent of Z, it is enough to check that dim Exti (JZ , OZ ) is independent of Z only for i = 1 and 2. Since Exti (OX , OZ ) ∼ = H i (X, OZ ) ∼ = H i (X, OZ (n)) = 0 for sufficiently large n ∈ N and i ≥ 1 by the Serre vanishing theorem, we have ( Ext1 (JZ , OZ ) ∼ = Ext2 (OZ , OZ ) Ext2 (JZ , OZ ) = 0.

By the duality theorem, we have Ext2 (OZ , OZ ) ∼ = (Hom(OZ , OZ ⊗KX ))∨ = (Hom(OZ , OZ ))∨ . Since Hom(OZ , OZ ) ∼ = Hom(OX , OZ ), we have dim Ext2 (OZ , OZ ) = n. This shows that dim Hom(JZ , OZ ) is independent of Z. Note that we also proved Hom(JZ , OZ ) ∼ = Ext1 (OZ , OZ ). n (2) The nonsingular locus of S n X is the open stratum S(1,...,1) X. Suppose π(Z) = n [n] {x1 , . . . , xn } ∈ S(1,...,1) X, for Z ∈ X . Since length(Zxi ) = 1, for i = 1, . . . , n, we must have n M OZ = the skyscraper sheaf at xi , i=1

n and hence Z = {x1 , . . . , xn } ∈ X [n] . This shows that π|π−1 (S(1,...,1) )X is an isomorphism. It

n n remains to show π −1 (S(1,...,1) X) = X [n] , which follows from dim(X [n] \π −1 (S(1,...,1) X)) < 2n. −1 It will be shown that dim π (n[x]) ≤ n − 1 in Theorem 1.16. If C is contained in the stratum Sνn X for ν = (n1 , n2 , . . . , nk ), then

π −1 (C) ∼ = π −1 (n1 [x1 ]) × · · · × π −1 (nk [xk ]). Hence we have dim π −1 (C) ≤ n − k. It implies that dim π −1 (Sνn X) ≤ n + k < 2n, unless ν = (1, . . . , 1). (More precise result will be proved in Lemma 6.10.) Remark 1.9. π : X [n] → S n X can be considered as an analogy of the Springer resolution π : T ∗ flag → N for the nilpotent variety N . This analogy will become clearer in Chapter 6.

8

1. HILBERT SCHEME OF POINTS

1.3. Symplectic structure In the case of k = C and KX = 0, X [n] has a further nice structure. Suppose X has a holomorphic symplectic form ω, i.e. ω is an element in H 0 (X, Ω2X ) which is nondegenerate at every point x ∈ X. Usually, the moduli space inherits a nice property of the base space, and in the case of X [n] , this is the case as the next theorem shows. Theorem 1.10 (Fujiki (n = 2) [21], Beauville (n ≥ 2) [6]). Suppose X has a holomorphic symplectic form ω. Then X [n] also has a holomorphic symplectic form. Proof. We give the proof following [6]. P Let S∗n X be the subset of S n X consisting of νi [xi ] (xi distinct) with νi ≤ 2. Its inverse image by the Hilbert-Chow morphism π : X [n] → S n X (resp. the quotient map [n] X n → S n X) is denoted by X∗ (resp. X∗n ). Let us denote by ∆ ⊂ X n the “big diagonal” consisting of elements (x1 , . . . , xn ) with xi = xj for some i 6= j. Then ∆ ∩ X∗n is smooth of codimension 2 in X∗n . Moreover, generalizing (1.4), we have the following commutative diagram η

Blow∆ (X∗n ) −−−→ X∗n     ρy y [n]

X∗

π

−−−→ S∗n X,

where η : Blow∆ (X∗n ) → X∗n denotes the blow-up of X∗n along ∆, and ρ is the map given by taking the quotient by the action of Sn . It is a covering ramified along the exceptional divisor E of η. The holomorphic symplectic form ω on X induces one on X n , which we still denote by ω. Its pull-back η ∗ ω is invariant under the action of Sn , hence defines a holomorphic [n] 2-form ϕ on X∗ with ρ∗ ϕ = η ∗ ω. Then we have div(ρ∗ φn ) = ρ∗ div(ϕn ) + E. On the other hand, the left hand side is equal to div(η ∗ ω n ) = η ∗ div(ω n ) + E = E. [n]

Therefore we have div ϕn = 0, hence ϕ is a holomorphic symplectic form on X∗ . Now, [n] X [n] \X∗ is of codimension 2 in X [n] , hence ϕ extends to the whole X as a holomorphic form by the Hartogs theorem. We still have div ϕn = 0 in X [n] , hence ϕ is non-degenerate. Note that the above construction is “local”. It works for a quasi-projective surface X. For a projective surface, Mukai obtained more general results. Theorem 1.11 (Mukai [56]). Let X be a K3 surface or an abelian surface and Mr,c1,c2 be the moduli space of stable sheaves on X with fixed rank r and Chern classes c1 , c2 , then Mr,c1,c2 has a holomorphic symplectic form.

1.4. HILBERT SCHEME OF POINTS ON THE PLANE

9

The symplectic form on the moduli space Mr,c1,c2 is described as follows. Let E be a stable sheaf on X, then the tangent space of Mr,c1,c2 at E is given by TE Mr,c1,c2 = Ext1 (E, E), and the symplectic structure is defined by Yoneda product tr Ext1 (E, E) × Ext1 (E, E) −−−−−−−−→ Ext2 (E, E) −−−→ H 2 (X, OX ) ∼ = C.

In the last part, we use the fact that X is a K3 surface or an abelian surface. For the proof of the non-degeneracy and the closedness, we refer to [56]. In Chapter 3, we shall show that the framed moduli space of torsion free sheaves on C2 has a holomorphic symplectic form. The relation between the two theorems is as follows. Let E be a rank 1 torsion free sheaf on X, then its double dual E ∨∨ is locally free. There exists a natural inclusion E ֒→ E ∨∨ and the cokernel E/E ∨∨ defines an element in X [n] . If we associate E with (E ∨∨ , E ∨∨ /E), we have an identification of the component of M1,c1,c2 with Pic0 (X) × X [n] . Note that if X is a K3 surface, Pic0 (X) is a point. In Chapter 3, we shall explain the hyper-K¨ahler structure, which is closely related with the holomorphic symplectic structure. Actually, a hyper-K¨ahler manifold has a holomorphic symplectic structure as we shall explain in Chapter 3. The converse is also true if X is compact. Proposition 1.12. Let X be a compact K¨ahler manifold which admits a holomorphic symplectic structure. Then X has a hyper-K¨ahler metric. Proof. Since X has a holomorphic symplectic form ω, we can identify T X and T ∗ X by using ω. This implies c1 (X) = 0. By the Calabi conjecture proved by Yau [81], there exists a Ricci-flat K¨ahler metric on X. Using the Ricci-flatness, the Bochner-Weitzenb¨ock formula gives ∆|ω|2 = |∇ω|2,

where ∆ is the Laplacian and ∇ is the Levi-Civita connection. Integrating both handside over X, we have ∇ω ≡ 0, which means that ω is parallel. This shows that the holonomy group is contained in SU(2n) ∩ Sp(n, C) = Sp(n), where n = 12 dimC X. Since the hyper-K¨ahler manifold can be defined as a Riemannian manifold whose holonomy group is contained in Sp(n), this completes the proof. It follows that X [n] has a hyper-K¨ahler metric if X is a K3 surface or an abelian surface. Exercise 1.13. Compare Beauville’s symplectic form and Mukai’s symplectic form n n on the open set π −1 (S(1,...,1) X), where S(1,...,1) X is the open stratum of S n X and π is the Hilbert-Chow morphism. 1.4. Hilbert scheme of points on the plane Now we give an explicit example of X [n] , which is a very good toy model for the whole subjects of these lectures. This is the case where X is the affine plane A2 . In this case, (A2 )[n] can be described as in the next theorem.

10

1. HILBERT SCHEME OF POINTS

Theorem 1.14. There exists an isomorphism   (i) [B1 , B2 ] = 0 ,     (ii) (stability) There exists no subspace [n] (A2 ) ∼ GLn (k), = (B1 , B2 , i) S ( k n such that Bα (S) ⊂ S (α =    1, 2) and im i ⊂ S

where Bα ∈ End(k n ) and i ∈ Hom(k, k n ) with the action given by g · (B1 , B2 , i) = (gB1g −1 , gB2 g −1, gi),

for g ∈ GLn (k). [n]

Proof. From the definition, (A2 ) can be written as   I is an ideal 2 [n] (A ) = I ⊂ k[z1 , z2 ] . dimk (k[z1 , z2 ]/I) = n [n]

Suppose we have I ∈ (A2 ) , then we define V = k[z1 , z2 ]/I, Bα ∈ End(V ) the multiplication by zα mod I for α = 1, 2, and i ∈ Hom(k, V ) by i(1) = 1 mod I. It follows that [B1 , B2 ] = 0, and the stability condition also holds since 1 multiplied by products of z1 ’s and z2 ’s span the whole of k[z1 , z2 ]. Conversely, if we have (B1 , B2 , i) as in the theorem, we can define the map φ : k[z1 , z2 ] → k n by φ(f ) = f (B1 , B2 )i(1). Since im φ is Bα -invariant and contains im i, it must be k n by the stability condition. Hence φ is surjective, and if we define I = ker φ, I is an ideal in k[z1 , z2 ] and dimk (k[z1 , z2 ]/I) = n. As shown in the proof of the theorem, the ideal I corresponding to (B1 , B2 , i) is given by I = {f (z) ∈ k[z1 , z2 ] | f (B1 , B2 )i(1) = 0}.

Note that this can be written as

I = {f (z) ∈ k[z1 , z2 ] | f (B1 , B2 ) = 0}, by the stability condition. Now let us examine few examples. Example 1.15. (1) First, we consider the case of n = 1. We write B1 = λ, B2 = µ ∈ k. From the stability condition, i must be non-zero. Hence we may assume i = 1, after applying the action of GL1 (k) if necessary. The corresponding ideal is given by I = {f (z) ∈ k[z1 , z2 ] | f (λ, µ) = 0}. Therefore it corresponds to the point (λ, µ) ∈ k 2 , and this gives the description (A2 )[1] ∼ = A2 . (2) Next, we consider the case of n = 2. Supposeeither B differenteigen1 or B2 has  λ1 0 µ1 0 values. It is easy to see that we can assume B1 = and B2 = with 0 λ2 0 µ2

1.4. HILBERT SCHEME OF POINTS ON THE PLANE

11

  1 (λ1 , µ1 ) 6= (λ2 , µ2 ), and that i(1) = by the stability condition. The corresponding 1 ideal is given by I = {f (z) ∈ k[z1 , z2 ] | f (λ1 , µ1 ) = f (λ2 , µ2 ) = 0}.

Therefore it corresponds to a set of two distinct points in A2 .



λ 0 0 λ



(3) Suppose both B1 and B2 have single eigenvalues. We cannot have B1 =   µ 0 together with B2 = since then S = im i violates the stability condition. We 0 µ     µ α λ 1 assume B1 = and B2 = for some α ∈ k. It is easy to see that we may 0 µ 0 λ   0 assume i(1) = . The corresponding ideal is given by 1 ∂f ∂f +α )(λ, µ) = 0} ∂z1 ∂z2 = (α(z1 − λ) − (z2 − µ), (z1 − λ)2 ).

I = {f (z) ∈ k[z1 , z2 ] | f (λ, µ) = (

Therefore it corresponds to a set of two points at (λ, µ) ∈ k 2 infinitesimally attached    with  λ α µ β ∂ ∂ each other in the direction ∂z1 + α ∂z2 . This shows that if B1 = and B2 = , 0 λ 0 µ we cannot have (α, β) = (0, 0), and that [α : β] corresponds to the homogeneous coordinates in the projectified tangent space P(T(λ,µ) A2 ). Hence we have an explicit identification [2] (A2 ) ∼ = Blow∆ (A2 × A2 )/S2 . [n] (4) It is also easy to describe the Hilbert-Chow morphism π : (A2 ) → S n (A2 ). Let [n] [(B1 , B2 , i)] ∈ (A2 ) . Since [B1 , B2 ] = 0, we can make B1 and B2 simultaneously into upper triangular matrices as     λ1 . . . ∗ µ1 . . . ∗ B1 =  ... . . . ...  , B2 =  ... . . . ...  . 0 . . . λn 0 . . . µn

The Hilbert-Chow morphism is given by π(B1 , B2 , i) = {(λ1 , µ1 ), . . . , (λn , µn )}. If all (λi , µi )’s are distinct, B1 and B2 must be semisimple. It follows that π is an isomorphism away from the diagonal. The complication along the diagonal comes from a point [n] [(B1 , B2 , i)] ∈ (A2 ) where either B1 or B2 is not semisimple. We have already encountered this complication in the example of n = 2 above. [n]

Suppose k = C. Let π : (C2 ) → S n (C2 ) be the Hilbert-Chow morphism. The parallel [n] [n] translation of C2 gives us the factorization (C2 ) = C2 × ((C2 ) /C2 ). In the description [n] in Theorem 1.14, a point in (C2 ) /C2 corresponds to (B1 , B2 , i) with tr(B1 ) = tr(B2 ) = 0. [n] We regard π −1 (n[0]) as a subvariety of (C2 ) /C2 . In Chapter 5, we shall see that π −1 (n[0])

12

1. HILBERT SCHEME OF POINTS

have exactly one (n − 1)-dimensional irreducible component. As a preliminary result, we prove the following: Theorem 1.16. The subvariety π −1 (n[0]) is isotropic with respect to the holomorphic [n] symplectic form on (C2 ) /C2 . In particular, dim π −1 (n[0]) ≤ n−1. Moreover, there exists at least one n − 1-dimensional component. Proof. Let us consider the torus action on C2 given by Φt1 ,t2 : (z1 , z2 ) 7→ (t1 z1 , t2 z2 ) [n]

for (t1 , t2 ) ∈ C∗ × C∗ .

This action lifts to (C2 ) and π −1 (n[0]) is preserved under the action. We use the same notation Φt1 ,t2 for the lifted action. As t1 , t2 goes to ∞, any point Z in π −1 (n[0]) converges to a fixed point of the torus action. In particular, if Z is a nonsingular point of π −1 (n[0]) and v, w is a tangent vector at Z, (Φt1 ,t2 )∗ (v), (Φt1 ,t2 )∗ (w) converges as t1 , t2 → ∞. On the other hand, if we pull back the symplectic form ω by Φt1 ,t2 it is multiplied by t1 t2 . Hence t1 t2 ω(v, w) = ω((Φt1 ,t2 )∗ (v), (Φt1 ,t2 )∗ (w)). When t1 , t2 goes to ∞, the above converges only when ω(v, w) = 0. Now we give an (n − 1)-dimensional component explicitly. In the description in Theorem 1.14, it is given by 

   B1 =    

0

0

1 0

 0 ... 0 0 1 0 . . . 0 . . . . . . ..  . . . . , 0 1 0  0 1 0



   B2 =    

0

0

 a1 a2 . . . an−2 an−1 0 a1 a2 . . . an−2  ..  .. .. .. . . . .  , 0 a1 a2   0 a1  0

  0  ...   i= 0 , 1

where a1 , . . . , an−1 are parameters. As an ideal, it is give by

J = (z1n , z2 − (a1 z1 + a2 z12 + · · · + an−1 z1n−1 )).

[n]

Exercise 1.17. (1) Using the description given in Theorem 1.14, show that (A2 ) is nonsingular. Hint: First show that the differential of the map (B1 , B2 , i) 7→ [B1 , B2 ] has constant rank. Next prove that the stabilizer of GLn (k)-action at (B1 , B2 , i) is trivial and define the slice for the action. (2) Generalize Theorem 1.14 to higher dimensions.

CHAPTER 2

Framed moduli space of torsion free sheaves on P2 Throughout this note from this chapter, we assume k = C. Let Mr,n be the framed moduli space of torsion free sheaves on P2 with rank r and c2 = n, i.e.  , E : torsion free sheaf,   rank E = r, c (E) = n Mr,n = (E, Φ) isomorphism, 2 ∼   Φ : E|ℓ∞ → Oℓ⊕r : framing at infinity ∞

where ℓ∞ = {[0 : z1 : z2 ] ∈ P2 } ⊂ P2 is the line at infinity. Notice that in order for the existence of the framing Φ we must have c1 (E) = 0. 2.1. Monad The main purpose of this section is to show the following description of Mr,n which is essentially due to Barth [4]. Theorem 2.1 (see [72]). There exists an isomorphism   (i) [B1 , B2 ] + ij = 0 ,     (ii) (stability) there exists no subspace Mr,n ∼ GLn (C), = (B1 , B2 , i, j) S ( Cn such that Bα (S) ⊂ S (α =    1, 2) and im i ⊂ S

where B1 , B2 ∈ End(Cn ), i ∈ Hom(Cr , Cn ) and j ∈ Hom(Cn , Cr ) with the action given by for g ∈ GLn (C).

g · (B1 , B2 , i, j) = (gB1 g −1, gB2 g −1, gi, jg −1)

The data (B1 , B2 , i, j) is expressed as in the figure.

B1

B2

V i

j W

Figure 2.1. data 13

2. FRAMED MODULI SPACE OF TORSION FREE SHEAVES ON P2

14

In the case of r = 1, we have E ֒→ E ∨∨ ∼ = OP2 since the double dual E ∨∨ is locally free of rank 1 with c1 (E ∨∨ ) = c1 (E) = 0. By noticing that E is locally free on ℓ∞ , the correspondence E → E ∨∨ /E gives the following isomorphism, [n] [n] M1,n ∼ = (P2 \ ℓ∞ ) = (C2 ) ,

and the theorem gives a description of the Hilbert scheme of n points in C2 as a special [n] case. Later in this section we shall show that this is exactly the description of (C2 ) which we encountered in Chapter 1. The difference in those descriptions is the appearance of j, which is 0 when r = 1 (see Proposition 2.7). The reason why we present the above description is to explain that the auxiliary data j is not artificial at all. The data j will [n] play a crucial role when we construct a hyper-K¨ahler structure on (C2 ) . The above description is very similar to the definition of quiver varieties [62]. In fact, quiver varieties were modeled after the ADHM description of moduli spaces of vector bundles over ALE spaces [50], which was found as a generalization of the above description. In turn, the above description could be considered as a quiver variety corresponding a quiver consisting of one vertex and one allow starting from the vertex and returning to the same vertex itself. The main idea of the proof of the theorem is the Beilinson spectral sequence which gives the monad description of a torsion free sheaf E on P2 . The Beilinson spectral sequence. Let ∆ be the diagonal in P2 × P2 . First, we shall construct a resolution of O∆ which has certain nice properties. Let pi : P2 × P2 → P2 be the projection to the i-th factor. We denote by Q the locally free sheaf on P2 of rank 2 which is defined by the following Euler sequence, 0 → OP2 (−1) → OP⊕3 2 → Q → 0.

We define the section s of the holomorphic bundle OP2 (1) ⊠ Q = p∗1 OP2 (1) ⊗ p∗2 Q on P2 × P2 as follows. (We consider P2 as a set of 1-dimensional subspaces in C3 .) The fiber of OP2 (1) ⊠ Q at (x, y) ∈ P2 × P2 is given by OP2 (1)x ⊗ Qy ∼ = Hom(OP2 (−1)x , Qy ) ∼ = Hom(Cx, C3 /Cy), and we define s(x, y) : λx → λx mod Cy for λ ∈ C. One can check that s is a transversal section and that s−1 (0) = ∆. Therefore we have the corresponding Koszul complex 2 ^

t (∧s)

ts

(OP2 (−1) ⊠ Q∨ ) −→ OP2 (−1) ⊠ Q∨ −→ OP2 ×P2 → O∆ → 0, V which is a resolution of O∆ . In other words, [O∆ ] = [ • (OP2 (−1) ⊠ Q∨ ] in the derived V category. We put C i = −i (OP2 (−1) ⊠ Q∨ ). Now we explain the Beilinson spectral sequence. For any coherent sheaf E on P2 , we have a trivial identity, p1∗ (p∗2 E ⊗ O∆ ) = p1∗ (p∗2 E|∆ ) = E. Replacing O∆ with the complex {C • }, we get the double complex for the hyperdirect image R• p1∗ (p∗2 E ⊗ C • ) for which we can consider two spectral sequences as usual. If we take the (2.2)

0→

2.1. MONAD

15

cohomology of {C • } first, we have a spectral sequence {Erp,q } whose E2 -term is given by E2p,q = Rq p1∗ (H p (p∗2 E ⊗ C • )) ( E for (p, q) = (0, 0) = . 0 for (p, q) 6= (0, 0)

This means that the spectral sequence degenerates at E2 -term and converges to E. This is nothing but the trivial identity mentioned above. The point is that taking the direct image first we can consider the second spectral sequence {Er′p,q } whose E1 -term is given by E1′p,q = Rq p1∗ (p∗2 E ⊗ C p ).

Since each C p is of the form C p = p∗1 F1 ⊗ p∗2 F2 , E1 -term can be written as E1′p,q = F1 ⊗ H q (P2 , E ⊗ F2 ). Of course, {Er′p,q } must also converge to E, and this gives us a different description of E. We call the second spectral sequence the Beilinson spectral sequence and examine it closely. The vanishing theorem. For our purpose, it is better to replace E with E(−1) because of the following lemma. Lemma 2.3. Let E be a torsion free sheaf on P2 which is locally free on ℓ∞ , then ( H q (P2 , E(−p)) = 0 for p = 1, 2, q = 0, 2 q 2 ∨ H (P , E(−1) ⊗ Q ) = 0 for q = 0, 2. Before proving the lemma, we shall complete the computation of the Beilinson spectral sequence. The complex p∗2 E(−1) ⊗ C • is written as 2 ^

0 → OP2 (−2) ⊠ (E(−1) ⊗ Q∨ ) → OP2 (−1) ⊠ (E(−1) ⊗ Q∨ ) → OP2 ⊠ E(−1) → 0. V V2 ∨ ∼ Since 2 Q∨ ∼ Q = E(−2), and E1 -term of the Beilinson = OP2 (−1), we have E(−1) ⊗ spectral sequence is given by 0 → OP2 (−2) ⊗ H q (P2 , E(−2))

→ OP2 (−1) ⊗ H q (P2 , E(−1) ⊗ Q∨ ) → OP2 ⊗ H q (P2 , E(−1)) → 0.

From the above lemma, E1 -term vanishes unless q = 1. It follows that the spectral sequence must degenerate at E2 -term. Furthermore, since the spectral sequence must converge to E(−1) which sits only on degree 0, we have ker a = 0, coker b = 0, and E(−1) ∼ = ker b/ im a. 2 Therefore, by tensoring with OP (1) we have the following monad description of E. Namely, there exists a complex a

b

OP2 (−1) ⊗ H 1 (P2 , E(−2)) − → OP2 ⊗ H 1 (P2 , E(−1) ⊗ Q∨ ) − →OP2 (1) ⊗ H 1 (P2 , E(−1)), where a is injective, b surjective and ker b/ im a ∼ = E. Now we give the proof of the lemma.

16

2. FRAMED MODULI SPACE OF TORSION FREE SHEAVES ON P2

Proof of Lemma. We consider the exact sequence mult. by z0

0 → OP2 (−1) −−−−−−→ OP2 → Oℓ∞ → 0. By tensoring with E(−k) we have 0 → E(−k − 1) → E(−k) → E(−k) ⊗ Oℓ∞ → 0, and hence the long exact sequence 0 → H 0 (P2 , E(−k − 1)) →H 0 (P2 , E(−k)) → H 0 (ℓ∞ , E(−k)|ℓ∞ )

→ H 1 (P2 , E(−k − 1)) →H 1 (P2 , E(−k)) → H 1 (ℓ∞ , E(−k)|ℓ∞ ) → H 2 (P2 , E(−k − 1)) →H 2 (P2 , E(−k)) → 0.

Since E|ℓ∞ ∼ , we have = Oℓ⊕r ∞ ( H 0 (ℓ∞ , E(−k)|ℓ∞ ) = 0 for k ≥ 1 H 1 (ℓ∞ , E(−k)|ℓ∞ ) = 0 for k ≤ 1, and the exact sequence gives ( H 0 (P2 , E(−k − 1)) ∼ = H 0 (P2 , E(−k)) for k ≥ 1 H 2 (P2 , E(−k − 1)) ∼ = H 2 (P2 , E(−k)) for k ≤ 1 By the Serre vanishing theorem, we have H 2 (P2 , E(k)) = 0 for sufficiently large k ∈ N. If E is torsion free, the double dual E ∨∨ is locally free, and we have an inclusion E ֒→ E ∨∨ . Hence we have H 0 (P2 , E(−k)) ֒→ H 0 (P2 , E ∨∨ (−k)) ∼ = H 2 (P2 , (E ∨∨ )∨ (k − 3))∨ by the Serre duality. Again by the Serre vanishing theorem, we have H 0 (P2 , E(−k)) = 0 for sufficiently large k ∈ N. Therefore we have ( 0 2 H (P , E(−1)) ∼ = H 0 (P2 , E(−2)) ∼ = H 0 (P2 , E(−3)) ∼ = ··· = 0 H 2(P2 , E(−2)) ∼ = H 2 (P2 , E) ∼ = · · · = 0. = H 2 (P2 , E(−1)) ∼

Notice that we also have H 1 (P2 , E(−2)) ∼ = H 1 (P2 , E(−1)). The same argument gives the exact sequence, 0 → H 0 (P2 , E(−k − 1) ⊗ Q∨ ) →H 0 (P2 , E(−k) ⊗ Q∨ ) → H 0 (ℓ∞ , E(−k) ⊗ Q∨ |ℓ∞ )

→ H 1 (P2 , E(−k − 1) ⊗ Q∨ ) →H 1 (P2 , E(−k) ⊗ Q∨ ) → H 1 (ℓ∞ , E(−k) ⊗ Q∨ |ℓ∞ ) → H 2 (P2 , E(−k − 1) ⊗ Q∨ ) →H 2 (P2 , E(−k) ⊗ Q∨ ) → 0.

It is easy to see that Q|ℓ∞ ∼ = Oℓ∞ ⊕ Oℓ∞ (1). Hence we have ( H 0 (ℓ∞ , E(−k) ⊗ Q∨ |ℓ∞ ) = 0 for k ≥ 1 H 1 (ℓ∞ , E(−k) ⊗ Q∨ |ℓ∞ ) = 0 for k ≥ 0,

2.1. MONAD

17

and the exact sequence gives ( H 0 (P2 , E(−k − 1) ⊗ Q∨ ) ∼ = H 0 (P2 , E(−k) ⊗ Q∨ ) for k ≥ 1 H 2 (P2 , E(−k − 1) ⊗ Q∨ ) ∼ = H 2 (P2 , E(−k) ⊗ Q∨ ) for k ≤ 0. Therefore we have ( 0 2 H (P , E(−1) ⊗ Q∨ ) ∼ = H 0 (P2 , E(−2) ⊗ Q∨ ) ∼ = H 0 (P2 , E(−3) ⊗ Q∨ ) ∼ = ··· = 0 H 2 (P2 , E(−1) ⊗ Q∨ ) ∼ = H 2 (P2 , E ⊗ Q∨ ) ∼ = H 2 (P2 , E(1) ⊗ Q∨ ) ∼ = · · · = 0. This completes the proof of the lemma. The triviality on the line at infinity. We now impose the triviality of the restriction of E to ℓ∞ . This identify the monad description with the description using matrices (B1 , B2 , i, j) as in Theorem 2.1. f = H 1 (P2 , E(−1) ⊗ Q∨ ), V = H 1 (P2 , E(−2)), and V ′ = H 1 (P2 , E(−1)), then Let W f = 2c2 (E) + by the Riemann-Roch formula we have dim V = dim V ′ = c2 (E) and dim W rank E. We have the complex, a

b

f −→ OP2 (1) ⊗ V ′ , OP2 (−1) ⊗ V −→ OP2 ⊗ W

with ker a = 0, coker b = 0, and ker b/ im a ∼ = E. f )) ∼ f ), we Since a ∈ H 0 (P2 , Hom(OP2 (−1) ⊗ V, OP2 ⊗ W = H 0 (P2 , OP2 (1)) ⊗ Hom(V, W f ), and similarly b = z0 b0 + z1 b1 + z2 b2 can write a = z0 a0 + z1 a1 + z2 a2 with ai ∈ Hom(V, W ′ f, V ). Since ba = 0, we have the following six equations: with bi ∈ Hom(W b0 a0 = 0 b1 a1 = 0 b2 a2 = 0

(2.4)

b0 a1 + b1 a0 = 0 b1 a2 + b2 a1 = 0 b0 a2 + b2 a0 = 0.

Restricting the complex to ℓ∞ , we have b|ℓ

a|ℓ

where

∞ ∞ f −→ Oℓ∞ (1) ⊗ V ′ , Oℓ∞ (−1) ⊗ V −→ Oℓ∞ ⊗ W

( a|ℓ∞ b|ℓ∞

= z1 a1 + z2 a2 = z1 b1 + z2 b2 .

From the exact sequence a|ℓ

∞ ker b|ℓ∞ → E|ℓ∞ → 0, 0 → Oℓ∞ (−1) ⊗ V −→

we have 0 →H 0 (ℓ∞ , Oℓ∞ (−1)) ⊗ V → H 0 (ℓ∞ , ker b|ℓ∞ ) → H 0 (ℓ∞ , E|ℓ∞ )

→H 1 (ℓ∞ , Oℓ∞ (−1)) ⊗ V → H 1 (ℓ∞ , ker b|ℓ∞ ) → H 1 (ℓ∞ , E|ℓ∞ ) → 0.

18

2. FRAMED MODULI SPACE OF TORSION FREE SHEAVES ON P2

Since H 0 (ℓ∞ , Oℓ∞ (−1)) = H 1 (ℓ∞ , Oℓ∞ (−1)) = 0, we have ( ∼ H 0 (ℓ∞ , ker b|ℓ∞ ) → H 0 (ℓ∞ , E|ℓ∞ ) ∼ H 1 (ℓ∞ , ker b|ℓ∞ ) → H 1 (ℓ∞ , E|ℓ∞ ). Moreover, since E|ℓ∞ ∼ , we have = Oℓ⊕r ∞ ( 1 H (ℓ∞ , ker b|ℓ∞ ) = 0

∼ H 0 (ℓ∞ , ker b|ℓ∞ ) → H 0 (ℓ∞ , E|ℓ∞ ) ∼ = Ep ,

where Ep is the fiber of E at p ∈ ℓ∞ . The last isomorphism shows that H 0 (ℓ∞ , ker b|ℓ∞ ) gives the trivialization of E on ℓ∞ , and the choice of a basis in H 0 (ℓ∞ , ker b|ℓ∞ ) corresponds to the framing of E at infinity. Similarly, from the exact sequence f → Oℓ∞ (1) ⊗ V ′ → 0, 0 → ker b|ℓ∞ → Oℓ∞ ⊗ W

we have

f → H 0 (ℓ∞ , Oℓ∞ (1)) ⊗ V ′ 0 →H 0 (ℓ∞ , ker b|ℓ∞ ) → H 0 (ℓ∞ , Oℓ∞ ) ⊗ W

f → H 1 (ℓ∞ , Oℓ∞ (1)) ⊗ V ′ → 0. →H 1 (ℓ∞ , ker b|ℓ∞ ) → H 1 (ℓ∞ , Oℓ∞ ) ⊗ W

Since H 0 (ℓ∞ , Oℓ∞ ) = C and H 1 (ℓ∞ , ker b|ℓ∞ ) = 0, we have the exact sequence f → H 0 (ℓ∞ , Oℓ∞ (1)) ⊗ V ′ → 0. 0 → H 0 (ℓ∞ , ker b|ℓ∞ ) → W

If we identify H 0 (ℓ∞ , Oℓ∞ (1)) ⊗ V ′ = (Cz1 ⊕ Cz2 ) ⊗ V ′ ∼ = V ′ ⊕ V ′ , we have the exact sequence “ ” b1

b2 f −− 0 → H 0 (ℓ∞ , ker b|ℓ∞ ) → W → V ′ ⊕ V ′ → 0. We put W = H 0 (ℓ∞ , ker b|ℓ∞ ) = ker b1 ∩ ker b2 . If we apply the same argument to the dual complex t b|

t a|

∞ ∞ f ∗ −−−→ Oℓ∞ (1) ⊗ V ∗ → 0, Oℓ∞ ⊗ W 0 → Oℓ∞ (−1) ⊗ (V ′ )∗ −−−→ ℓ

ℓ

we have the exact sequence

t

(a1 ,a2 ) f ∗ −− 0 → H 0 (ℓ∞ , kert a|ℓ∞ ) → W −−→ V ∗ ⊕ V ∗ → 0, f is injective, and hence that a|ℓ∞ is injective on and it follows that (a1 , a2 ) : V ⊕ V → W each fiber. In particular, if we consider the fiber at p0 = [0 : 1 : 0] ∈ ℓ∞ , we have a1 f b1 V −→ W − → V ′,

∼

and ker b1 / im a1 = Ep0 . Since we know W = ker b1 ∩ ker b2 → Ep0 , we have im a1 ∩ ker b2 = im a1 ∩ W = 0. It follows that b2 a1 : V → V ′ is bijective since dim V = dim V ′ . f Now we are in a position to put a|ℓ∞ and b|ℓ∞ in a simple form. We first identify W with V ⊕ V ⊕ W by the sequence (a1 ,a2 )

“ ” b1 b2

f −−→ V ′ ⊕ V ′ . V ⊕ V −−−−→ W

2.1. MONAD

19

We also identify V ′ and V through b1 a2 = −b2 a1 . Then, we have     −1V 0
a1 =  0  , a2 = −1V  , b1 = 0 −1V 0 , b2 = 1V 0 0


0 0 .

Now it is easy to see that from the condition ba = 0, we can write   B1
a0 = B2  , b0 = −B2 B1 i , j with [B1 , B2 ] + ij = 0. Therefore, we have the monad description in a simple form,

V ⊗ OP2 ⊕ V ⊗ OP2 (−1) −−−−−−−−−→ −−−−−−−−−−−−−−−−−→ V ⊗ OP2 (1). ! V ⊗ OP2 − z0 B1 −z1 b = ( −(z0 B2 −z2 ) z0 B1 −z1 z0 i ) ⊕ a = z0 B2 −z2 z0 j W ⊗ OP2 In particular, restricting to C2 = P2 \ ℓ∞ (i.e. setting z0 = 1), we have

(2.5)

V ⊗ OC2

V ⊗ OC2 ⊕ V ⊗ OC2 −−−−−−−−−−−−−−→V ⊗ OC2 . −−−−−−−−→ ! B1 −z1 b = ( −(B2 −z2 ) B1 −z1 i ) ⊕ a = B2 −z2 j W ⊗ OC2

The next lemma completes the proof of the theorem. Lemma 2.6. Suppose that a quadruplet (B1 , B2 , i, j) satisfying the equation (i) in Theorem 2.1 is given, and define endomorphisms a, b as above. Then (1) ker a = 0. (2) b is surjective if and only if there exists no subspace S ( V such that Bα (S) ⊂ S, (α = 1, 2) and im i ⊂ S. Proof. It is easy to see that a is injective and b surjective on ℓ∞ . Hence it is sufficient to prove the lemma on C2 . On the fiber at z = (z1 , z2 ) ∈ C2 , a and b induce the homomorphisms σ

τ

z z V −→ V ⊕ V ⊕ W −→ V,

where

  B1 − z1 σz = B2 − z2  , j


τz = −(B2 − z2 ) B1 − z1 i .

20

2. FRAMED MODULI SPACE OF TORSION FREE SHEAVES ON P2

(1) If σz is not injective, there exists v 6= 0 ∈ V such that   B1 v = z1 v B2 v = z2 v .  jv = 0

Hence (z1 , z2 ) must be a simultaneous eigenvalues of (B1 , B2 ), and it follows that σz is not injective only at finitely many points in C2 . This shows that a is injective. (2) Notice that b is surjective ⇐⇒ τz is surjective for all z ∈ C2

⇐⇒ t τz : V ∗ → W ∗ ⊕ V ∗ ⊕ V ∗ is injective for all z ∈ C2 .

For any subspace S ⊂ V , We put S ⊥ = {φ ∈ V ∗ | φ(S) = 0}. It is easy to see that ( ( t Bα (S) ⊂ S Bα (S ⊥ ) ⊂ S ⊥ ⇐⇒ im i ⊂ S S ⊥ ⊂ ker t i. Suppose there exists such a subspace S ( V . Since [ t B2 , t B1 ] + t j t i = 0 and t i|S ⊥ = 0, we have [ t B2 , t B1 ]|S ⊥ = 0. Therefore, there exists φ 6= 0 ∈ S ⊥ such that t Bα φ = λk φ, for some λ1 , λ2 ∈ C. If we put λ = (λ1 , λ2 ), then τλ is not surjective. Conversely, suppose there exists z ∈ C2 such that τz is not surjective, then t τz is not injective. Hence there exists φ 6= 0 ∈ ker t τz . If we take S = ker φ ( V , we have Bα (S) ⊂ S and im i ⊂ S. 2.2. Rank 1 case As remarked before, we can identify the framed moduli space M1,n of rank 1 torsion free [n] sheaves with (C2 ) . Hence our description should be the same as that in Theorem 1.14. The difference in those descriptions is the appearance of j in Theorem 2.1. In fact, this is not the difference because we have Proposition 2.7. Assume r = 1. Suppose a quadruple (B1 , B2 , i, j) satisfying conditions (i), (ii) in Theorem 2.1 is given. Then (1) j = 0. (2) The sheaf ker b/ im a is isomorphic to the ideal given in the description in Theorem 1.14. Lemma 2.8. Take (B1 , B2 , i, j) as above, but suppose that it satisfies the equation (i) in Theorem 2.1, not necessarily (ii). Let S ⊂ Cn be a subspace defined by X S= Bα1 Bα2 · · · Bαk i(C) X = (products of B1 ’s and B2 ’s)i(C).

Then the restriction j|S of j to S vanishes.

2.2. RANK 1 CASE

21

b = 0 for B b = Bα1 Bα2 · · · Bα (α1 , . . . , αk ∈ {1, 2}) by the Proof. We show that j Bi k b = 1, and we have ji = tr(ji) = tr(ij) = − tr([B1 , B2 ]) = 0. induction on k. If k = 0, then B b contains a sequence · · · B2 B1 · · · , we have Suppose the claim is true for k ≤ m − 1. If B b = jBα1 · · · B2 B1 · · · Bαm jB

= jBα1 · · · ([B2 , B1 ] + B1 B2 ) · · · Bαm

= (jBα1 · · · i)j · · · Bαm + jBα1 · · · B1 B2 · · · Bαm = jBα1 · · · B1 B2 · · · Bαm ,

where we have used the induction hypothesis in the last equality. Hence we have jBα1 Bα2 · · · Bαm = jB1m1 B2m2 with m1 + m2 = m and ms = #{l|αl = s}, s = 1, 2, and it is b = B m1 B m2 . In this case, we have sufficient to show the claim for B 1 2 b = tr(Bij) b = − tr(B1m1 B2m2 [B1 , B2 ]) j Bi

= − tr([B1m1 B2m2 , B1 ]B2 ) = − tr(B1m1 [B2m2 , B1 ]B2 ) =− =−

=− =−

m 2 −1 X

tr(B1m1 B2l [B2 , B1 ]B2m2 −l−1 B2 )

l=0

m 2 −1 X

tr(B2m2 −l B1m1 B2l [B2 , B1 ])

l=0

m 2 −1 X

tr(B2m2 −l B1m1 B2l ij)

l=0

m 2 −1 X

jB2m2 −l B1m1 B2l i.

l=0

b we have Since jB2m2 −l B1m1 B2l = jB1m1 B2m2 = j B, b = 0. Hence j Bi

b = −m2 j Bi. b j Bi

Proof of Proposition 2.7. (1) Let (B1 , B2 , i, j) and S be as in the above lemma. If (B1 , B2 , i, j) satisfies the stability condition (ii) in Theorem 2.1, the subspace S is Bα invariant and im i ∈ S, we must have S = Cn . It follows from the lemma that j = 0. (2) Since j = 0, the complex (2.5) becomes

Cn ⊗ OC2

Cn ⊗ OC2 ⊕ n n −−−−−−−−→ ! C ⊗ OC2 −−−−−−−−−−−−−−→ C ⊗ OC2 . b = ( −(B2 −z2 ) B1 −z1 i ) B1 −z1 ⊕ a = B2 −z2 0 OC2

22

2. FRAMED MODULI SPACE OF TORSION FREE SHEAVES ON P2

Hence the projection to the third factor gives the inclusion ker b/ im a ֒→ OC2 . We denote its image by J . Notice that by the stability condition we cannot have i = 0, otherwise we can take S = 0. It is easy to see that for f (z) ∈ OC2 f (z) ∈ J ⇐⇒∃u1 (z), u2 (z) ∈ OC⊕n 2 such that

f (z)i(1) = (z1 − B1 )u1 (z) + (z2 − B2 )u2 (z)

=⇒f (B1 , B2 )i(1) = 0.

Since the vectors of the form B1l B2m i(1), l, m ∈ Z≥0 span the whole Cn by the stability condition, it follows that f (z) ∈ J implies f (B1 , B2 ) = 0. Conversely, suppose f (B1 , B2 ) = 0, then f (z1 , z2 )1n = f (z1 − B1 + B1 , z2 − B2 + B2 )

= f (B1 , B2 ) + (z1 − B1 )C(z) + (z2 − B2 )D(z) = (z1 − B1 )C(z) + (z2 − B2 )D(z),

for some C(z), D(z) ∈ Mn (OC2 ). If we write u1 (z) = C(z)i(1), u2 (z) = D(z)i(1), we have f (z)i(1) = (z1 − B1 )u1 (z) + (z2 − B2 )u2 (z). Hence f (z) ∈ J . Therefore we have ∼

ker b/ im a → J = {f (z) ∈ OC2 |f (B1 , B2 ) = 0}. The right hand side is exactly the description of (C2 )

[n]

explained in Theorem 1.14. [n]

We have a description of the symmetric product S n C2 similar to that of (C2 ) . Proposition 2.9. There exists an isomorphism S n C2 ∼ = {(B1 , B2 , i, j) | [B1 , B2 ] + ij = 0}// GLn (C), where // means the affine algebro-geometric quotient. Here we just consider the set of closed points in the both hand side. Hence, the affine algebro-geometric quotient is considered as the set of closed GLn (C)-orbits. This subject will be explained in Chapter 3 in more detail. Proof. Suppose GLn (C) · (B1 , B2 , i, j) be a closed orbit. Let S be as in the above lemma, then we have Bα (S) ⊂ S, im i ⊂ S, and j|S = 0. If we decompose Cn = S ⊕ S ⊥ , we can write    
∗ ∗ ∗ Bα = ,i = ,j = 0 ∗ . 0 ∗ 0

2.2. RANK 1 CASE

23

  1 0 Let g(t) = ∈ GLn (C) for t 6= 0 ∈ C, then we have 0 t    
∗ t∗ ∗ −1 −1 g(t) Bα g(t) = , g(t) i = , jg(t) = 0 t∗ . 0 ∗ 0

Since GLn (C)·(B1 , B2 , i, j) is a closed g(t)·(B1, B2 , i, j) must be in the same t→0  orbit,  lim ∗ 0 ∗ orbit. Hence we may assume Bα = ,i= and j = 0. Let g(t) = t−1 1n , then 0 ∗ 0   t∗ −1 −1 we have g(t) Bα g(t) = Bα , g(t) i = . Letting t → 0, we may assume i = j = 0. In 0 this case [B1 , B2 ] = 0, and hence we may assume that both B1 and B2 are upper-triangular matrices,     λ1 . . . ∗ µ1 . . . ∗ B1 =  ... . . . ...  , B2 =  ... . . . ...  . 0 . . . λn 0 . . . µn   t   t2 −1 , we may assume that B1 Taking the limit lim g(t) Bα g(t) with g(t) =  ..   t→0 . tn

and B2 are semisimple. Therefore it is shown that each closed orbit contains an element (B1 , B2 , 0, 0) such that both B1 , B2 are semisimple satisfying [B1 , B2 ] = 0. Associating the orbit with the set of simultaneous eigenvalues of (B1 , B2 ), we have the claimed isomorphism.

As is shown in this section, the main idea to obtain the ADHM description is the Beilinson spectral sequence, and the reason why it is so efficient depends on the fact that it comes from a resolution of the structure sheaf O∆ of the diagonal · · · → C −2 → C −1 → C 0 → O∆ → 0, L ′ such that each C p is of the form C p = i p∗1 Fi,p ⊠ p∗2 Fi,p (see (2.2)). This property is called decomposable diagonal class. (See [15, Theorem 2.1] for the consequence of this condition.) Therefore we end this section with the following question. Question 2.10. Which variety has a decomposable diagonal classes ? [n]

We shall see that (C2 ) and the minimal resolution of the simple singularities has decomposable diagonal class in Chapter 4.

CHAPTER 3

Hyper-K¨ ahler metric on (C2 )

[n]

The purpose of this chapter is to construct a hyper-K¨ahler metric on the Hilbert scheme [n] [n] (C2 ) of n points on C2 . This will be accomplished by identifying (C2 ) with a hyperK¨ahler quotient (see Theorem 3.23). Our approach relies on the description in Theorem 2.1, rather than one given in Theorem 1.14. However, we do not need the proof of Theorem 2.1. Hence the reader who skips Chapter 2 should read the statement of Theorem 2.1 and §2.2. 3.1. Geometric invariant theory and the moment map First we need some generalities on the relationship between the geometric invariant theory and the moment map. General reference of this section is [60, Chapter 8]. (See also [71] for reference to the geometric invariant theory.) Let V be a vector space over C with a hermitian metric, G a connected Lie subgroup of U(V ), and GC its complexification. The Lie algebra of G is denoted by g, and its dual by g∗ . We want to define the quotient space of V by the action of GC . But the set theoretical quotient V /GC usually behaves very bad, is not even Hausdorff in general. The idea of the geometric invariant theory is to consider the ring of functions on the quotient space. Let A(V ) be the coordinate ring of V , i.e. the symmetric power of the dual space of V . The C GC -action on V induces a GC -action on A(V ). Let A(V )G be the ring of invariants. Then C the affine algebro-geometric quotient of V by GC is defined as Spec(A(V )G ). We denote it by V //GC . The principal result of the geometric invariant theory says Theorem 3.1 ([71, 3.5],[60, 1.1]). (1) There exists a surjective morphism φ : V → V //GC C

induced by the inclusion A(V )G ⊂ A(V ). Moreover, φ(x) = φ(y) if and only if (3.2)

GC · x ∩ GC · y 6= ∅,

where “ ” denotes the closure. (2) The underlying space of V //GC is the set of closed GC -orbits modulo the equivalence relation defined by x ∼ y if and only if (3.2) holds. The first statement follows from the following: 24

3.1. GEOMETRIC INVARIANT THEORY AND THE MOMENT MAP

25

Theorem 3.3 ([60, 1.2]). Let W1 and W2 be two closed invariant subsets of V . Then C W1 and W2 are disjoint if and only if there exists an invariant function f ∈ A(V )G such that f |W1 = 1 and f |W2 = 0. Proof. The “if” direction is clear. Let W1 and W2 be two disjoint closed invariant subsets of V . There exists f ∈ A(V ) (not necessarily invariant) such that f |W1 = 1 and f |W2 = 0. Averaging by the action of the compact group G, we may assume that f is invariant under the action of G. Since f is holomorphic, it is automatically invariant under the action of GC . The second statement of Theorem 3.1 follows from the following fact: the closure of a GC -orbit is a union of orbits of smaller dimensions. Hence any orbit contains a closed GC -orbit in its closure. (Moreover, it is unique by Theorem 3.3.) In order to illustrate how the closedness of the orbit is determined, we consider the case GC is the complex torus T C = (C∗ )r . We choose a basis {x1 , . . . , xn } of V so that T C is contained in the group of nonsingular diagonal matrices. Then we have distinguished characters χi : T C → C∗ for i = 1, . . . , n given by   t1   t2  7→ ti . TC ∋  .   .. tn

Let M be the weight lattice of T C , i.e. M = Hom(T C , C∗ ). An element in the dual lattice M ∗ can be identified with a one-parameter subgroup λ : C∗ → T C by χ ◦ λ(t) = thχ,λi for all t ∈ C∗ and χ ∈ M. A fundamental role will be played by the following rational polyhedral convex cone def.

σ = R≥0 χ1 + R≥0 χ2 + · · · + R≥0 χn ⊂ M ⊗ R. Its dual cone is defined by def.

σ ∨ = {λ ∈ M ∗ ⊗ R | hχ, λi ≥ 0 for all χ ∈ σ}

={λ ∈ M ∗ ⊗ R | hχi , λi ≥ 0 for i = 1, . . . , n}.

If σ ∨ 6= {0}, it is also rational polyhedral cone by Farkas’ theorem (see e.g. [74]). Hence there exists a set of generators λ1 , . . . , λs such that (3.4)

σ ∨ = R≥0 λ1 + R≥0 λ2 + · · · + R≥0 λs .

Theorem 3.5. If x = t (1, . . . , 1) ∈ V , then T C · x is closed if and only if σ = M ⊗ R. Proof. Suppose σ 6= M ⊗R, and hence σ ∨ 6= {0}. Then we can take a set of generators λ1 , . . . , λs of σ ∨ as (3.4). The one-parameter subgroup λj : C∗ → T C satisfies hχi , λj i ≥ 0 for all i = 1, . . . , n. Then, as t → 0, λj (t) · x converges to a point which is not contained in the orbit of x. Hence T C · x is not closed.

¨ 3. HYPER-KAHLER METRIC ON (C2 )

26

[n]

Suppose T C · x is not closed. There exists a divergent sequence gk in T C such that gk x converges to a point in V as k → ∞. We choose χ ∈ M ⊗ R so that χ(gk ) goes to ∞ as k → ∞. If σ = M ⊗ R, we can write χ=

n X

with mi ≥ 0.

mi χi

i=1

Then

χ(gk ) =

n Y

χi (gk )

i=1

mi

=

n Y

yi(gk )mi ,

i=1

where we write gk x as t (y1 (gk ), . . . , yn (gk )) in the coordinate system of V . Hence χ(gk ) converges. This is a contradiction. Hence σ 6= M ⊗ R. As an application of the above discussion, we give a following version of the HilbertMumford criterion. Theorem 3.6 (Birkes [8], Kempf [44]). If x ∈ V and Y is a nonempty closed GC invariant subset contained in the closure of GC · x, then there exists a one-parameter subgroup λ : C∗ → GC such that limt→0 λ(t)x = y for some y ∈ Y . Proof. First suppose GC is a complex torus T C . Taking a basis of V , we may suppose we are in the above situation. (If some coordinates are 0, we replace V by a subspace.) We may also assume that Y is the unique closed orbit in the closure of T C · x. The uniqueness follows from the existence of T C -invariant polynomial which separates two disjoint T C invariant closed subsets (Theorem 3.3). Let def.

I = {i ∈ {1, . . . , n} | hχi , λi = 0 for all λ ∈ σ ∨ }. If I = {1, . . . , n}, then σ ∨ = 0. Hence T C · x is closed by Theorem 3.5. Thus we are done. Otherwise, we can choose λ ∈ σ ∨ such that hχi , λi > 0 for i ∈ {1, . . . , n} \ I. Further we may assume λ ∈ σ ∨ ∩ M ∗ ⊗ Q, and hence λ ∈ σ ∨ ∩ M ∗ by multiplying by a positive integer. We then have lim λ(t) · x = y = t (y1 , y2 , . . . , yn ), t→0

and yi 6= 0 if and only if i ∈ I. Consider the rational polyhedral cone X X def. σ e = R≥0 χi + Rχi , i∈I

i∈I /

which contains σ. If λ is an element of its dual cone σ e∨ , then hχi , λi = 0 for i ∈ / I. On ∨ ∨ the other hand, since σ e ⊂ σ , we have hχi , λi = 0 for i ∈ I by the definition of I. Thus λ = 0, hence σ e∨ = {0}. By Theorem 3.5, y has a closed orbit. The proof is completed when GC = T C .

3.1. GEOMETRIC INVARIANT THEORY AND THE MOMENT MAP

27

Now return to the case of general GC . Following [8, 4.2], we reduce the general case to the torus case as follows. First we fix a maximal torus T C . If Y and T C g · x intersect for some g ∈ GC , then we find a one-parameter subgroup λ : C∗ → T C such that limt→0 λ(t)g · x = y for some y ∈ Y by the first part of the proof. Then g −1λg is a desired one-parameter subgroup connecting x and g −1 · y ∈ Y . Hence we may assume that Y ∩ T C g · x = ∅ for any g ∈ GC . By Theorem 3.3, there exists a T C -invariant function fg such that fg |Y = 0 and fg |T C g·x = 1. Let Ug be the open set {v ∈ V | fg (v) 6= 0}. Now we use the fact GC = GT C G. Since G · x is compact, we can take g1 , . . . , gn in G such that Ug1 , . . . , Ugn cover G · x. Then def.

F = |fg1 | + · · · + |fgn | is a T C -invariant function which never vanishes on T C G · x. On the other hand, we have F |Y = 0, therefore Y ∩ T C G · x = ∅. Here “ ” denotes the closure with respect to the ordinary topology (instead of the Zariski topology). Since Y is G-invariant, we have ∅ = Y ∩ GT C G · x = Y ∩ GC · x. Since GC · x = GC · x (see [59, p.84]), this contradicts with the assumption. Let us define a map µ : V → g∗ by (3.7)

1 √ hµ(x), ai = ( −1ξx, x) for x ∈ V , ξ ∈ g. 2

This is a special case of the moment map which is defined for an action on a general symplectic manifold. Consider a function px : GC → R>0 given by (3.8)

def.

px (g) = kgxk2

for g ∈ GC .

This function plays a fundamental role in the relationship between the moment map and the geometric invariant theory as seen in the following proposition. Proposition 3.9. The map px has following properties. (1) (2) (3) (4) (5) (6)

px descends to a function on G\GC /GCx , where GCx denotes the stabilizer of x. px is convex as a function on the non-compact type symmetric space G\GC . g is a critical point if and only if µ(gx) = 0. All critical points are minima of px . If px attains its minimum, it does so on exactly one double coset G\g/GCx . px attains its minimum if and only if the orbit GC · x is closed in V.

Proof. Assertion (1) is clear from the definition.

¨ 3. HYPER-KAHLER METRIC ON (C2 )

28

[n]

For A ∈ g, we have (3.10)

(3.11)

√ √ 1d k(exp t −1ξ)gxk2 = (iξ(exp tiξ)gx, (exp t −1ξ)gx) 2 dt √
= 2hµ (exp t −1ξ)gx , ξi, √ √ 1 d2 k(exp t −1ξ)gxk2 = 2kξ exp t −1ξgxk2 ≥ 0. 2 2 dt

(3.10) proves Assertion (3), and (3.11) proves Assertion (2). Assertion (4) follows from the convexity of px and the fact that any two points in G\GC can be joined by a geodesic. To prove Assertion (5), suppose px attains minimum at g and exp iξ · g. Then the convexity of px implies √ √ k exp(t −1ξ)gxk2 = px (exp t −1ξg) = const . √ Therefore we get ξgx = 0 by setting t = 0 in (3.11). Hence we have exp −1ξgx = gx, i.e. √ g −1 exp −1ξg ∈ GCx . Now we prove Assertion (6). If the orbit GC · x is closed, then px attains its minimum at the nearest point to origin. Conversely suppose px attains a minimum. We may assume it does so at g = e, replacing x if necessary. Let g⊥ x be the orthogonal complement of gx in g. By (3.11), we have √ d2 p (exp t −1ξ) > 0 x dt2 for any 0 6= ξ ∈ g⊥ x . Hence we can choose a positive constant ε so that √ d2 p (exp t −1ξ) ≥ ε x dt2

for ξ ∈ g⊥ x with kξk = 1 and t ∈ [0, 1]. Therefore, we have √ d px (exp t −1ξ) ≥ ε dt

for ξ ∈ g⊥ x with kξk = 1 and t = 1. The same inequality holds for t ≥ 1 since px is convex. It implies √ √ px (exp t −1ξ) ≥ ε(t − 1) + px (exp −1ξ) for t ≥ 1. √ Thus px (exp t −1A) diverges as t → ∞. This implies the orbit GC · x is closed. Now we can describe the set of closed orbits via the moment map. Theorem 3.12 (Kempf and Ness [45]). There is a bijection between the set of closed GC -orbits in V and µ−1 (0)/G.

3.1. GEOMETRIC INVARIANT THEORY AND THE MOMENT MAP

29

Proof. If x ∈ µ−1 (0), then GC · x is closed by Assertion (3)(4)(6) of Proposition 3.9. Therefore we have a natural map µ−1 (0)/G −→ {closed GC -orbits}.

Surjectivity of this map follows from Assertion (6) and (3). Injectivity follows from (5). The quotient space µ−1 (0)/G is called a symplectic quotient (or Marsden-Weinstein reduction). It has a complex structure and natural K¨ahler metric (cf. Theorem 3.30) on points where G acts freely. On the other hand, the set of closed GC -orbits is the affine algebrogeometric quotient and denoted by V //GC . In fact, it is known that the above identification intertwines the complex structures. Example 3.13. Consider the vector space of matrices V = End(Cn ). We have an adjoint action of g ∈ U(n) on V given by V = End(Cn ) ∋ B 7→ g −1Bg. √

Then the corresponding moment map is given by µ(B) = 2−1 [B, B † ]. Hence Theorem 3.12 implies {closed GL(n, C)-orbits} ∼ = {B | [B, B † ] = 0}/ U(n). It is easy to see a matrix has a closed orbit if and only if it is diagonalizable. Hence the set of closed orbits can be identified with the set of eigenvalues. On the other hand, a matrix B with [B, B † ] = 0 (i.e. a normal matrix) can be diagonalizable by a unitary matrix. Hence the quotient space is also identified with the set of eigenvalues. The identification can be seen directly in this example.

Example 3.14 (Application to description given in Proposition 2.9). Let us consider Hermitian vector spaces V and W whose dimensions are n and 1 respectively. Then M = End(V )⊕End(V )⊕Hom(W, V )⊕Hom(V, W ) becomes a vector space with a Hermitian product. We consider an action of G = U(V ) on M given by (3.15)

M ∋ (B1 , B2 , i, j) 7→ (g −1 B1 g, g −1B2 g, g −1i, jg).

The moment map µ1 : V → u(V ) is defined by √  −1  † † † † (3.16) µ1 (B1 , B2 , i, j) = [B1 , B1 ] + [B2 , B2 ] + ii − j j . 2 Looking at Theorem 2.1, we introduce a map µC given by (3.17)

µC (B1 , B2 , i, j) = [B1 , B2 ] + ij.

This is a holomorphic map from M to gl(V ). (More detailed explanation of µC will be given later.) Then µC −1 (0) is a GL(V )-invariant set. Hence by Theorem 3.12, we have {closed GL(V )-orbits in µ−1 (0)} ∼ = µ−1 (0) ∩ µ−1 (0)/ U(V ). C

1

C

In Proposition 2.9, we have seen that the left hand side is identified with the symmetric product S n (C2 ).

¨ 3. HYPER-KAHLER METRIC ON (C2 )

30

[n]

With this example understood, the next question is “Is there a similar description [n] [n] for (C2 ) ?”. In order to describe (C2 ) as S n (C2 ) in Example 3.14, we need some modification of Theorem 3.12. The following construction is due to A. D. King [46]. Let χ : G → U(1) be a character, and χ also denotes its complexification, χ : GC → C∗ .

by

Consider the trivial line bundle V × C over V . Using χ, we lift the GC -action to V × C g · (x, z) = (g · x, χ(g)−1 z) for (x, z) ∈ V × C.

(3.18) C

Let A(V )G ,χ be the space of polynomials satisfying f (g · v) = χ(g)f (v). It can be identified with the space of GC -invariant sections of the above line bundle. Then the direct sum M C n A(V )G ,χ n≥0

is a graded algebra. Hence we can define def.

V //χ GC = Proj

M

A(V )G

n≥0

C

The inclusion A(V )G ⊂ (3.19)

L

n≥0

A(V )G

C ,χn

C ,χn

!

.

induces a projective morphism

V //χ GC → V //GC .

In the geometric language, V //χ GC can be described as follows. We say x ∈ V is χC n semistable if there exists f ∈ A(V )G ,χ with n ≥ 1 such that f (x) 6= 0. Let V ss (χ) be the set of χ-semistable points. We introduce an equivalence relation ∼ on V ss (χ) by defining x ∼ y if and only if G · x and G · y intersects in V ss (χ). Then V //χ GC is V ss (χ)/∼ by [60, 1.10],[71, 3.14]. Instead of px in the previous situation, we consider its modified version p(x,z) : GC → R given by p(x,z) (g) = log N (g · (x, z))

(3.20) 1

for z 6= 0,

2

where N(x, z) = |z|e 2 kxk . For this p(x,z) , the following holds, (cf. Proposition 3.9) Proposition 3.21. For z 6= 0, (1) p(x,z) is a convex function on G\GC . √ (2) g is a critical point if and only if hµ(gx), ξi = −1dχ(ξ). (3) All critical points are minima of p(x,z) . (4) If p(x,z) attains its minimum, it does so on exactly one double coset G\g/GC(x,z). (5) p(x,z) attains minimum if and only if GC · (x, z) is closed in V × C.

3.1. GEOMETRIC INVARIANT THEORY AND THE MOMENT MAP

31

Proof. For ξ ∈ g, the following is easy to check, √ √
1d p(x,z) (exp tiξg) = hµ (exp t −1ξ)gx , ξi − dχ( −1ξ), 2 dt √ √ 1 d2 p (exp t −1ξg) = 2kξ exp t −1ξgxk2. (x,z) 2 2 dt Assertion (1)(2)(3)(4) follows from above calculations by arguments similar to Proposition 3.9. Now we prove Assertion (5). Suppose GC · (x, z) is closed. Since GC · (x, z) and V × {0} are mutually disjoint, closed subsets, there exists an invariant polynomial P = zP1 (x) + · · · + z n Pn (x) which satisfies ( 1 on GC · (x, z) P ≡ 0 on V × {0}. 1

2

Suppose N(˜ x, z˜) = |˜ z |e 2 k˜xk ≤ C, then |˜ z | is bounded. And, 1

2

n

2

1

2

x)|e− 2 k˜xk ≤ C ′ e− 4 k˜xk 1 = |˜ z P1 (˜ x) + · · · + z˜n Pn (˜ x)| ≤ C|P1 (˜ x)|e− 2 k˜xk + · · · + C n |P1 (˜

Thus k˜ xk is bounded. Therefore p(x,z) attains minimum. The proof of the converse is similar to that of Proposition 3.9. √ Corollary 3.22. There exists a bijection between µ−1 ( −1dχ)/G and the set {x ∈ V | GC · (x, z) is closed for z 6= 0}. Now we apply Corollary3.22 to our situation. Let (B1 , B2 , i, j) be as in Example 3.14. Define χ by χ(g) = (det g)l , where l is an arbitrary positive integer. Theorem 3.23. [n]

(C2 )

√ −1 ∼ −1 = µ−1 C (0)//χ GLn (C) = µ1 ( −1dχ) ∩ µC (0)/ U(n).

To prove Theorem 3.23, the only thing we have to do is to prove the next. Lemma 3.24. (B1 , B2 , i, j) satisfies the stability condition in Theorem 2.1 if and only if GC · (x, z) is closed for z 6= 0. Proof. Suppose GC · (x, z) is closed for z 6= 0. And suppose there exists a subspace S ( V which satisfies the following, (i) S is Bα -invariant (α = 1, 2). (ii) im i ⊂ S. Taking a complementary subspace S ⊥ , we decompose W as S ⊕ S ⊥ . Then we have     ∗ ∗ ∗ Bα = (α = 1, 2), i = . 0 ∗ 0

¨ 3. HYPER-KAHLER METRIC ON (C2 )

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Let g(t) =



 1 0 , then we have 0 t−1 g(t)Bα g(t)

−1 ⊥

[n]

  ∗ t∗ = , g(t)i = i. 0 ∗

On the other hand (det g)−n z = tn dim S → 0 as t → 0. This contradicts the closedness of GC · (x, z). Next, suppose the stability condition is satisfied. If GC · (x, z) is not closed, then, by the Hilbert criterion Theorem 3.6, there exists a map λ : C∗ → GL(V ) which satisfies the C C condition, limt→0 λ(t) · (x, z) exists and this limit is contained within M G · (x, z) \ G · (x, z). Let us take weight decomposition of V with respect to λ, V = V (m). The existence of m

limit implies the following:

Bα (V (m)) ⊂ Im i ⊂ By the stability condition, we have V =

M

m≥0

M

V (l),

l≥m

M

V (m).

m≥0

V (m). Hence det λ(t) = tN for some N ≥ 0. If

N equals to zero, then V = V (0), i.e. λ ≡ 1. This is impossible because limt→0 λ(t)·(x, z) ∈ / GC · (x, z). If N > 0, then λ(t) · (x, z) = (λ(t)x, det λ(t)−1 z) = (λ(t)x, t−nN z)

diverges as t → 0. This is contradiction.

The Hilbert-Chow morphism π : (C2 ) (3.19): (C2 )

[n]

[n]

→ S n (C2 ) can be identified with the morphism

−1 n 2 = µ−1 C (0)//χ GLn (C) → µC (0)// GLn (C) = S C .

3.2. Hyper-K¨ ahler quotients In this section, we shall see that the quotient in Theorem 3.23 is, in fact, a hyper-K¨ahler quotient. Let us review on the hyper-K¨ahler structure and hyper-K¨ahler quotients briefly. The interested reader should read Hitchin’s book [38]. First, we recall the definition of K¨ahler manifolds. Definition 3.25. Let X be a 2n-dimensional manifold. A K¨ahler structure of X is a pair of a Riemmnian metric g and an almost complex structure I which satisfies the following conditions: (3.25.1) g is hermitian for I, i.e. g(Iv, Iw) = g(v, w) for v, w ∈ T X. (3.25.2) I is integrable.

¨ 3.2. HYPER-KAHLER QUOTIENTS

33

(3.25.3) If we define a 2-form ω by ω(v, w) = g(Iv, w) for v, w ∈ T X, then dω = 0. (This ω is called the K¨ahler form associated with (g, I).) It is well-known (see e.g. [48, Chapter IX, Theorem 4.3]) that the above conditions (3.25.2), (3.25.3) (under (3.25.1)) are equivalent to requiring that I is parallel with respect to the Levi-Civita connection of g, i.e. ∇I = 0. This is also equivalent to the condition: (the holonomy group of ∇) ⊂ U(n). The hyper-K¨ahler structure is a quaternionic version of the K¨ahler structure. However, there is no good definition of the integrability (i.e. the existence of local charts) for the almost hyper-complex structure. Hence we generalize the second equivalent definition explained in above. Definition 3.26. Let X be a 4n-dimensional manifold. A hyper-K¨ahler structure of X consists of a Riemmanian metric g and a triple of almost complex structures I, J, K which satisfy the following conditions: (3.26.1) g(Iv, Iw) = g(Jv, Jw) = g(Kv, Kw) = g(v, w) for v, w ∈ T X. (3.26.2) (I, J, K) satisfies a relation I 2 = J 2 = K 2 = IJK = −1. (3.26.3) I, J, K are parallel with respect to the Levi-Civita connection of g, i.e. ∇I = ∇J = ∇K = 0. We call a manifold with a hyper-K¨ahler structure simply a hyper-K¨ahler manifold. Remark 3.27. (1) The above conditions are equivalent to the condition: (the holonomy group of ∇) ⊂ Sp(n). (2) Each one of (g, I), (g, J), (g, K) defines a K¨ahler structure. We pick up √ a particular complex structure, say I, and combine the other K¨ahler forms as ωC = ω2 + −1ω3 . Then √ ωC (Iv, w) = g(JIv, w) + −1g(KIv, w) √ √ = −1(g(Jv, w) + −1g(Kv, w)) √ = −1ωC (v, w). This means ωC is of type (2, 0). It is clear that dωC = 0 and ωC is non-degenerate. Hence ωC is a holomorphic symplectic form. (3) One of the advantages of the hyper-K¨ahler structure is that one can identify two apparently different complex manifolds with one hyper-K¨ahler manifold. Namely, a hyperK¨ahler manifold (X, g, I, J, K) gives two complex manifolds (X, I) and (X, J), which are not isomorphic in general. For example, on a compact Riemann surface, the moduli space of Higgs bundles and the moduli space of flat PGLr (C)-bundles come from one hyper-K¨ahler manifold, namely moduli space of 2D-self-duality equation (see [36] for detail.)

¨ 3. HYPER-KAHLER METRIC ON (C2 )

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[n]

K¨ahler structures are easy to construct and flexible. For example, any complex submanifold of a K¨ahler manifold √ is again K¨ahler, and a K¨ahler metric is locally given by a K¨ahler potential, i.e. ω = −1∂∂u for a strictly pseudo convex function u. However, hyper-K¨ahler structures are neither easy to construct nor flexible (even locally). A hypercomplex submanifold of a hyper-K¨ahler manifold must be totally geodesic, and there is no good notion of hyper-K¨ahler potential. The following quotient construction, which was introduced by Hitchin et al.[39] as an analogue of Marsden-Weinstein quotients for symplectic manifolds, is one of the most powerful tool for constructing new hyper-K¨ahler manifolds. Let (X, g, I, J, K) be a hyper-K¨ahler manifold, and ω1 , ω2 , ω3 the associated K¨ahler forms corresponding to I, J, K. Suppose that a compact Lie group G acts on X preserving g, I, J, K. Definition 3.28. A map (3.29)

µ = (µ1 , µ2 , µ3) : X → R3 ⊗ g∗

is said to be a hyper-K¨ahler moment map if we have the following: (3.29.1) µ is G-equivariant, i.e. µ(g · x) = Ad∗g−1 µ(x). (3.29.2) hdµi(v), ξi = ωi (ξ ∗, v) for any v ∈ T X, any A ∈ g and i = 1, 2, 3, where ξ ∗ is a vector field generated by ξ. Take ζ = (ζ1 , ζ2, ζ3 ) ∈ R3 ⊗ g∗ which satisfies Ad∗g (ζi) = ζi for any g ∈ G, (i = 1, 2, 3). Then µ−1 (ζ) is invariant under the G-action. So we can consider the quotient space µ−1 (ζ)/G. Theorem 3.30 (Hitchin et al.[39]). Suppose G-action on µ−1 (ζ) is free. Then the quotient space µ−1 (ζ)/G is a smooth manifold and has a Riemannian metric and a hyperK¨ ahler structure induced from those on X. Remark 3.31. The meaning of “a hyper-K¨ahler structure induced from that on X” is as follows: Let i be the natural inclusion µ−1 (ζ) → X, and π the natural projection µ−1 (ζ) → µ−1 (ζ)/G. If ω1 , ω2 , ω3 (resp. ω1′ , ω2′ , ω3′ ) are K¨ahler forms associated with the hyper-K¨ahler structure on X (resp. µ−1 (ζ)/G), we have the relation π ∗ ωα′ = i∗ ωα (α = 1, 2, 3) hold. We call this quotient space a hyper-K¨ahler quotient. We shall give the proof of Theorem 3.30 following [38]. Another proof can be found in [27]. First we begin with the following useful lemma: Lemma 3.32. Let (X, g) a Riemannian manifold with skew adjoint endomorphisms I, J and K of the tangent bundle T X satisfying (3.26.1),(3.26.2). Then (g, I, J, K) is hyper-K¨ ahler if and only if the associated K¨ahler forms ω1 , ω2 and ω3 are closed. Proof. The “only if” direction is clear. Hence it is enough to show that I, J, K are integrable when ω1 , ω2 and ω3 are closed.

¨ 3.2. HYPER-KAHLER QUOTIENTS

35

We shall prove the integrability of I using the Newlander-Nirenberg theorem. (Replacing the argument with J and K, we find J and K are integrable.) If v and w are complex-valued vector field on X, we have ω2 (v, w) = g(Jv, w) = g(KIv, w) = ω3 (Iv, w). √ Hence we have Iv = −1v (i.e. v is of type (1, 0) with respect to I) if and only if (3.33)

i(v)ω C = 0, √

where ω C = ω2 − −1ω3 . Now suppose that v and w are (1, 0) vector fields with respect to I. Let us denote by Lv the Lie derivative with respect to the vector field v. Then i([v, w])ωC = Lv i(w)ω C − i(w)Lv ω C

by the naturality of the Lie derivative

=0

by (3.33) for v.

= − i(w)d(i(v)ω C )

by (3.33) for w and dω C = 0

This shows [v, w] is of type (1, 0) again by (3.33). The Newlander-Nirenberg theorem implies I is integrable. Proof of Theorem 3.30. Take x ∈ µ−1 (ζ) and consider the differential dµx : Tx X → R3 ⊗ g∗ .

Since the G-action is free on µ−1 (ζ), the tangent space of the orbit through x, denoted by Vx , is isomorphic g under the identification g ∋ ξ 7→ ξx∗ ∈ Vx ⊂ Tx X, where ξx∗ is the value of the vector field ξ ∗ at x. First we prove Claim. Vx , IVx , JVx , KVx are orthogonal to each other. Proof. Let ξ, η ∈ g. Since µi is equivariant, we have µ(exp(tη)x) = ζ for any t ∈ R. Differentiating with respect to t, we get dµx (ηx∗ ) = 0. Hence we have (g(Iξx∗, ηx∗ ), g(Jξx∗, ηx∗ ), g(Kξx∗, ηx∗ )) = (ω1 (ξx∗ , ηx∗ ), ω2 (ξx∗ , ηx∗ ), ω3 (ξx∗ , ηx∗ )) = hdµx (ηx∗ ), ξi = 0. This implies Vx and IVx (resp. JVx , KVx ) is orthogonal. Since I, J and K are hermitian, we get the assertion.

¨ 3. HYPER-KAHLER METRIC ON (C2 )

36

[n]

Take ξ ∈ g, and consider a tangent vector Iξx∗ ∈ Tx X. Then we have dhµx (Iξx∗ ), ηi = (ω1 (ηx∗ , Iξx∗), ω2 (ηx∗ , Iξx∗ ), ω3(ηx∗ , Iξx∗ ))

= (g(Iηx∗ , Iξx∗), g(Jηx∗ , Iξx∗ ), g(Kηx∗ , Iξx∗)) = (g(ηx∗ , ξx∗ ), 0, 0), where we have used the above claim in the last equality. Similarly, we have dhµx (Jξx∗ ), ηi = (0, g(ηx∗ , ξx∗), 0), dhµx (Kξx∗ ), ηi = (0, 0, g(ηx∗, ξx∗ )), which implies the surjectivity of dµx . Hence µ−1 (ζ) is a submanifold of X. Moreover, since we have dhµx (v), ηi = (ω1 (ηx∗ , v), ω2 (ηx∗ , v), ω3(ηx∗ , v)) = (g(Iηx∗ , v), g(Jηx∗ , v), g(Kηx∗, v)), the kernel of dµx is the orthogonal complement of IVx ⊕ JVx ⊕ KVx . Since the G-action on µ−1 (ζ) is free, the slice theorem implies that the quotient space −1 µ (ζ)/G has a structure of a C ∞ -manifold such that the tangent space TG·x (µ−1 (ζ)/G) at the orbit G · x is isomorphic to the orthogonal complement of Vx in Tx µ−1 (ζ). Hence the tangent space is the orthogonal complement of Vx ⊕ IVx ⊕ JVx ⊕ KVx in Tx X, which is invariant under I, J and K. Thus we have the induced almost hyper-complex structure. The restriction of the Riemannian metric g induces a Riemannian metric on the quotient µ−1 (ζ)/G. In order to show that these define a hyper-K¨ahler structure, it is enough to check that the associated K¨ahler forms ω1′ , ω2′ and ω3′ are closed by Lemma 3.32. Let i : µ−1 (ζ) ֒→ X be the inclusion, and π : µ−1 (ζ) → µ−1 (ζ)/G the projection. By definition, we have i∗ ωi = π ∗ ωi′ for i = 1, 2, 3. Hence, we have π ∗ (dωi′ ) = d(π ∗ ωi′ ) = d(i∗ ωi) = i∗ (dωi ) = 0. Since π is a submersion by construction, the above equation implies dωi′ = 0. Remark 3.34. There are two interpretation of the factor R3 in (3.29). First one is (3.35) R3 ∼ = RI ⊕ RJ ⊕ RK ∼ = Im H ∼ = sp(1). The second one is

V+ (3.36) R3 ∼ T. = Rω1 ⊕ Rω2 ⊕ Rω3 ∼ = V+ V Here T is as follows: Let T be the fundamental representation of SO(4). Then 2 T V2 ∼ is isomorphic to so(4). Then the Hodge star operator defines the decomposition T = V+ V− + T⊕ T . If we denote the positive (negative)Vspinor representation by S (resp. S − ), V ± + ± the Clifford multiplication induces isomorphisms T ∼ T (∼ = su(S ). Then = su(S + )) corresponds to sp(1) above in (3.35) via the isomorphism su(2) ∼ = sp(1). Now we apply this construction to the description of Theorem 3.23. Let (B1 , B2 , i, j) as in Example 3.14, i.e. consider Hermitian vector spaces V and W whose dimensions are n

¨ 3.2. HYPER-KAHLER QUOTIENTS

37

and 1, and let M = End(V ) ⊕ End(V ) ⊕ Hom(W, V ) ⊕ Hom(V, W ), then (B1 , B2 , i, j) ∈ M. The anti-linear endomorphism J(B1 , B2 , i, j) = (B2† , −B1† , j † , −i† )

makes M into a quaternion vector space. In particular, M is a (flat) hyper-K¨ahler manifold. The action considered in (3.15) preserves the hyper-K¨ahler structure. If we decompose the map µC given in (3.17) as √ µC = µ2 + −1µ3 ,

considering gln (C) as the complexification of u(V ), the map µ = (µ1 , µ2 , µ3 ) : M → R3 ⊗ u(V √ ) is a hyper-K¨ahler moment map. It is easy to check that the U(V )-action on µ−1 ( −1dχ, 0, 0) is free. Hence we have, √ −1 −1 2 [n] Corollary 3.37. µ−1 is a hyper-K¨ahler 1 ( −1dχ) ∩ µ2 (0) ∩ µ3 (0)/ U(V ) = (C ) 2 [n] quotient. In particular, (C ) has a hyper-K¨ahler structure.

−1 −1 Exercise 3.38. Show that the hyper-K¨ahler metric on µ−1 1 (0)∩µ2 (0)∩µ3 (0)/ U(V ) = S n C2 is the standard (flat) metric. √ Remark 3.39. (1) More generally, one can take −1ζ idV ∈ R3 ⊗ u(V ) for any √ ζ ∈ R3 . The U(V )-action is free on µ−1 ( −1ζ idV ) if ζ 6= 0. However, √ we get essen−1 tially the same hyper-K¨ a hler manifolds. There exists a map from µ ( −1ζ idV )/ U(V ) √ −1 −1 −1 to µ1 ( −1|ζ| idV ) ∩ µ2 (0) ∩ µ3 (0)/ U(V ) which is isometry and transforms the hyperK¨ahler structure (I, J, K) into  ′   I I  J′  = R  J  K′ K

for some R ∈ SO(3) satisfying t (|ζ|, 0, 0) = Rζ. More precisely, if (X, g, I, J, K) is a hyperK¨ahler manifold and R ∈ SO(3), then I ′ , J ′ , K ′ given by the above equation is also a hyper-K¨ahler structure. This transformation commutes with the hyper-K¨ahler quotient when ζ is transformed into Rζ. [n] However, µ−1 (ζ)/ U(V ) is not isomorphic to (C2 ) as a complex manifold in general. (2) As we have seen in §1.3, X [n] has a hyper-K¨ahler structure when X is a K3 surface or an abelian surface. This is proved by using the solution of the Calabi conjecture. It seems natural to conjecture that this is always true when X is a hyper-K¨ahler manifold. Although there are several extensions of the Calabi conjecture to noncompact manifolds (e.g. [3, 77]), it is not applicable to our problem. It is because these extensions always [n] give manifolds with quadratic curvature decay while (C2 ) does not satisfy this decay. Example 3.40. The moduli space of instantons on a 4-dimensional hyper-K¨ahler manifold X can be considered as a hyper-K¨ahler quotient. Let us take a smooth vector bundle E over X with a Hermitian metric. Let us denote the space of metric connections on E by A. Its tangent space at A ∈ A can be identified with TA A ∼ = Ω1 (u(E)).

¨ 3. HYPER-KAHLER METRIC ON (C2 )

38

We introduce a natural L2 -metric Z (α, β) = − tr(α ∧ ∗β) X

[n]

for α, β ∈ TA A ∼ = Ω1 (u(E)).

Almost complex structures I, J, K on X induce natural almost complex structures on TA A = Ω1 (u(E)). These make A an infinite dimensional flat hyper-K¨ahler manifold. The group of gauge transformations, denoted by G, acts on A by pull-back. The hyper-K¨ahler moment map of the action of G on A µ = (µ1 , µ2 , µ3) : A → R3 ⊗ Lie G ∗ ∼ = R3 ⊗ Ω4 (u(E)).

is given by (3.41)

µi (A) = FA ∧ ωi ∈ Ω4 (u(E)) (i = 1, 2, 3).

Here FA is the curvature 2-form of A, and ωi is the K¨ahler form associated with the complex V structure I, J, or K on X. Since ω1 , ω2 , ω3 span the space of self-dual 2-forms + T X (see 3.36), FA ∧ ωi = FA+ ∧ ωi = 0 for i = 1, 2, 3 implies FA+ = 0. Hence µ−1 (0) is the space of anti-self-dual connections. Thus µ−1 (0)/G ∼ = {A ∈ A | FA+ = 0}/G. This is the moduli space of the anti-self-dual connections. (Note that G-action is not necessarily free. Hence the moduli spaces may have singularities.) The spaces µ−1 (0) and G are both infinite dimensional, but its quotient, that is, the moduli space of the anti-selfdual connections is finite dimensional. The proof for Theorem 3.30 works even in this case if one uses the appropriate analytical packages, i.e. the Sobolev space, etc. This construction works even in the case X = C2 . Although C2 is non-compact, we also have an appropriate analytical package, i.e. the weighted Sobolev space (see e.g., [61] for detail). In this case, we must consider the framed moduli space, which means that we take a quotient by a group of gauge transformations converging to the identity at the end of X. In other words, if we consider the one point compactification S 4 = C2 ∪ {∞}, then the framed moduli space is the space of pairs: {(anti-self-dual connection A on E, isomorphism E∞ → Cr )}/isomorphism, where r is the rank of the vector bundle E. However there is another way to describe the moduli space which is called the ADHM description. It is quite relevant to us, so let us review on that. Let V , W be hermitian vector spaces whose dimensions are n, r. Define a complex vector space M by M = {(B1 , B2 , i, j) | B1 , B2 ∈ Hom(V, V ), i ∈ Hom(W, V ), j ∈ Hom(V, W )}. Let U(V ) act on M by (3.15). As explained in the case of dim W = 1, maps µ1 , µC given by the same formulae in (3.16) and (3.17) give rise a hyper-K¨ahler moment map −1 µ = (µ1 , µ2 , µ3 ). Let M0 (r, n) = µ−1 ahler quotient. 1 (0) ∩ µC (0)/ U(V ) be the hyper-K¨

¨ 3.2. HYPER-KAHLER QUOTIENTS

39

Since this space has singularities, we take the nonsingular locus given by Mreg 0 (r, n)

= { [(B1 , B2 , i, j)] ∈ M0 (r, n) | the stabilizer in U(V ) of (B1 , B2 , i, j) is trivial }.

A system (B1 , B2 , i, j) of matrices satisfying µ = 0 and having trivial stabilizer is called an ADHM datum. Then the ADHM description is the following, Theorem 3.42 (Atiyah et al. [2]). There is a bijective correspondence between the framed moduli space and Mreg 0 (r, n), where r is the rank of the vector bundle E, and n is the second Chern class of E. The complete proof of this theorem can be found in [13]. Here we see how to construct an anti-self-dual connection from an ADHM datum. Let T be the fundamental representation of SO(4), and S + be the positive half-spinor representation. Let M1 = T ⊗R u(V ) and M2 = S + ⊗C Hom(V, W ). If we choose a complex structure on R4 , in other words, a reduction of the symmetry group from SO(4) into SU(2), T could be identified with Λ0,1 . Hence M1 can be identified with Λ0,1 ⊗C End(V ). More explicitly, choosing a basis for T , we could write the identification as √ √ (A0 , A1, A2 , A3 ) ∼ = (B1 , B2 ) where B1 = A0 + −1A1 , B2 = A2 + −1A3 . V0,0 V2,0 Similarly, the identification R4 ∼ ⊕ . We = C2 induces a decomposition S + ∼ = + † decompose Φ ∈ S ⊗C Hom(V, W ) into i ⊕ j, where i ∈ Hom(W, V ) and j ∈ Hom(V, W ). In this way, we identify (B1 , B2 , i, j) ∈ M and (A, Φ) ∈ M1 ⊕ M2 . Moreover, we can rewrite (3.16) and (3.17) into a single equation µ = (µ1 , µ2 , µ3 ) = [A ∧ A]+ + {Φ† , Φ}. V Here [A∧A]+ is, as usual, an element in + T ⊗u(V ) ∼ = R3 ⊗u(V ) which is given by taking the Lie bracket in u(V )-part and the wedge product in T -part. The term V {Φ† , Φ} is the + trace-free part of Φ† Φ ∈ u(S + )⊗u(V ), hence an element of su(S + )⊗u(V ) ∼ T ⊗u(V ) ∼ = = 3 R ⊗ u(V ). Regarding A as a “connection”, we consider [A ∧ A]+ as the self-dual part of the “curvature” of A. Similarly, Φ can be regarded as a “Higgs field”, and the equation µ = 0 represents the Seiberg-Witten equation. Let (A, Φ) = (B1 , B2 , i, j) be an element of µ−1 (0). For z = (z1 , z2 ) ∈ C2 , we have (3.43)

(3.44)

V ⊕ V V −−−−−−−−−→ −−−−−−−−−−−−−−−→V. ! B1 −z1 ⊕ τz = ( −(B2 −z2 ) B1 −z1 i ) σz = B2 −z2 j W

It is not difficult to check Ker σz = 0 S and Coker τz = 0 when (B1 , B2 , i, j) ∈ µ−1 (0) and stabilizer in U(V ) is trivial. Therefore z∈C2 Ker τz / Im σz is a vector bundle on C2 . Since τz and σz vary holomorphically on z ∈ C2 , it is a holomorphic vector bundle. In fact, this is the vector bundle attached to the data (B1 , B2 , i, j) in Chapter 2. Moreover, we can

40

¨ 3. HYPER-KAHLER METRIC ON (C2 )

[n]

consider it as a subbundle of the trivial bundle V ⊕ V ⊕ W over C2 by the identification Ker τz / Im σz ∼ = Ker(τz ⊕ σz† ). Hence it has a metric and a connection induced from the trivial bundle. It is easy to see ([13, Lemma 3.1.20]) that this induced connection is compatible with the above mentioned holomorphic structure. In particular, the curvature 2-form FA is a (1, 1)-form. Now note that this connection is independent of the identification R4 with C2 . To see this, we identify V ⊕ V and V ⊕ V ⊕ W with S + ⊗ V and S − ⊗ V ⊕ W respectively. Then (σz , τz† ) is transformed into “

”

S− ⊗ V S ⊗ V −−−−−−−→ ⊕ . W +

A−x⊗1V Φ

Here x is regarded as an endomorphism from S + to S − by the Clifford multiplication. Now we change the complex structure which was used for the identification R4 ∼ = C2 . So we can see that FA is always of type (1, 1) with respect to arbitrary complex structure. Since V1,1 V1,1 V1,1 ∼ V− ⊗ C, we have FA+ ≡ 0, i.e. A is an anti-self-dual connection. I ∩ J ∩ K =

Remark 3.45. Let ζ = (ζR , ζC ) ∈ (R × C) ⊗ u(V ). Set Mζ (r, n) = µ−1 1 (ζR ) ∩ −1 µC (ζC )/ U(V ). There is an analogue of the Hilbert-Chow morphism π : M(√−1dχ,0,0) (r, n) → M0 (r, n) for general r, n by (3.19) (see [64]). The space M0 (r, n) can be considered as the moduli space of ideal instantons ([13, Lemma 3.4.8]), which contains the moduli space Mreg 0 (r, n) of genuine instantons as a nonsingular locus if r > 1. In fact, morphisms from moduli spaces of semistable torsion-free sheaves to moduli spaces of ideal instantons on general projective surfaces were constructed by J. Li [51]. Although Mreg 0 (1, n) = ∅, we consider 2 [n] (C ) as the resolution of the moduli space of “rank-1 ideal instantons”. Since the space M0 (r, n) can be defined without the identification R4 ∼ = C2 , it seems natural to ask the following question. [n]

Question 3.46. (1) Is there the definition of (C2 ) independent of the identification 4 ∼ 2 R =C ? [n] (2) Our hyper-K¨ahler metric on (C2 ) depends on the choice of the hermitian metric on V and W . This hermitian metric should be defined “naturally” under the identification V ∼ = H 0 (OZ ). Recall that the hyper-K¨ahler metric on the moduli space of instantons on a hyper-K¨ahler manifold is induced from the “natural L2 -metric”. Do we have a similar natural definition for the hermitian metric on V ? As we see above, µ = 0 is analogous to the Seiberg-Witten equation. It may be helpful to pursue this analogy to the above questions. We conclude this chapter by a historical comment. Remark 3.47. It has been well-known that the ADHM equation µ = 0 can be considered as a hyper-K¨ahler moment map equation after the introduction of the hyper-K¨ahler

¨ 3.2. HYPER-KAHLER QUOTIENTS

41

quotient [39]. The fact that the space µ−1 (ζ1 , 0, 0)/G is the resolution of µ−1 (0)/G was observed by Kronheimer in the case of ALE spaces (see Chapter 4). This motivated the author to study Mζ (r, n) in [64]. Although the geometric interpretation of Mζ (r, n) had not been given at that time, the author noticed later that Mζ (r, n) is the framed moduli space of torsion free sheaves as a complex manifold (see Chapter 2) in [69]. The space Mζ (r, n) was also studied by Valli independently [80].

CHAPTER 4

Resolution of simple singularities In this chapter, we shall construct minimal resolutions of simple singularities using Hilbert schemes of points on C2 . These resolutions inherit the hyper-K¨ahler structures from those on Hilbert schemes. This construction will give us a new interpretation Kronheimer’s construction of ALE spaces [49] in terms of Hilbert schemes. 4.1. General Statement A simple singularity is a quotient space C2 /Γ, where Γ is a finite subgroup of SU(2). This singularity has been studied from various point of view. See e.g. [5], [76]. Let us construct its resolution using the Hilbert scheme of points on C2 as follows. [N ] Consider the Hilbert scheme (C2 ) , where N is the order of Γ. The Γ-action on C2 [N ] naturally induces that on (C2 ) and the symmetric product S N (C2 ). Since Γ-action on C2 \ {0} is free, the Γ-orbit Γ · p of a point p in C2 \ {0} consists of N distinct points, [N ] hence defines a 0-dimensional subscheme Z ∈ (C2 ) . This Z is fixed by the Γ-action. N Conversely, any Γ-fixed point in the open stratum π −1 (S(1,...,1) (C2 )) comes from a Γ-orbit. Let X be the component of Γ-fixed point set which contains the set of Γ-orbits Γ·(C2 \{0}). It is the closure of Γ · (C2 \ {0}) and has dimension 2. Then we have Theorem 4.1 (Ginzburg and Kapranov [25], Y. Ito and I. Nakamura [42]). The re striction of the Hilbert-Chow morphism to X is the minimal resolution of singularities
Γ of C2 /Γ ∼ = S N (C2 ) .
Γ Proof. It is easy to check that the fixed point set S N (C2 ) in the symmetric product [N ] is isomorphic to C2 /Γ. Since (C2 ) is nonsingular, a fixed point component X of a finite group action is nonsingular. N 2 Since (C2 \ {0})/Γ is contained in S(1,... ,1) (C ) on which the Hilbert-Chow morphism π is an isomorphism, the restriction of π induces an isomorphism between X \ π −1 (0) and (C2 \ {0})/Γ. Hence π is a resolution. [N ] The holomorphic symplectic structure on (C2 ) restricts to that on X. In particular, it implies that the canonical bundle is trivial, i.e. KX = OX . Hence the resolution is minimal. Combining this construction with the hyper-K¨ahler structure given in Chapter 3, we obtain the following Corollary 4.2. X has a hyper-K¨ahler metric. 42

4.1. GENERAL STATEMENT

43

Proof. The Γ-action preserves the Riemannian metric and the hyper-K¨ahler structure [N ] on the Hilbert scheme (C2 ) constructed in Chapter 3. The restriction gives a hyperK¨ahler structure on X. Remark 4.3. The hyper-K¨ahler structures on the minimal resolutions of simple singularities were constructed by Kronheimer [49]. His construction will be explained later. Next we study the other fixed components [N ]

Theorem 4.4. Suppose that Z is a fixed point in (C2 ) of Γ-action, and let V denote H 0 (OZ ) considered as a Γ-module. Then (1) If Z is contained in the component X, then V is isomorphic to the regular representation R. (2) Conversely, if V ∼ = R, then Z is contained in X. [N ] (3) Other fixed point components of (C2 ) consist of points. [N ]

Proof. We shall use the description of (C2 ) in terms of matrices given in Theo[N ] rem 1.14. Suppose Z is a Γ-invariant 0-dimensional subscheme in (C2 ) , and corresponds to a triple of matrices (B1 , B2 , i). Recall that it is given as follows: Define a N-dimensional vector space V as H 0 (OZ ), and a 1-dimensional vector space W . Then the multiplications of coordinate functions z1 , z2 ∈ C define endomorphisms B1 , B2 . The natural map OC2 → OZ defines a linear map i : W → V . From this construction, V is a Γ-module, and W is the trivial Γ-module. The pair (B1 , B2 ) is Γ-equivariant, if it is considered as an element in Hom(V, Q ⊗ V ), where Q is 2-dimensional representation given by the inclusion Γ ⊂ SU(2). (This follows from that (z1 , z2 ) is an element in Q.) And i is also a Γ-equivariant homomorphism W → V . Let us prove Assertion (1). Since V is independent of the choice of a point of X, it is sufficient to see V = R when Z is given by the orbit Γ · x, where x = (z1 , z2 ) is a point in C2 \ {0}. In this case, V = H 0 (OZ ) is identified with the set of functions on Z = Γ · x. Hence we have a basis {eγ | γ ∈ Γ} of V = H 0 (OZ ), where eγ is the characteristic function of γx. The Γ-module structure on V is given by the pull-back of the function. For g ∈ Γ, the pull-back (g −1 )∗ eγ is the characteristic function of gγx, i.e. egγ . Hence the Γ-module structure is given by g · eγ = egγ . This shows that V is the regular representation as a Γ-module. It is also easy to write down the corresponding matrix data: Choose a numbering of elements in Γ so that Γ = {γ1, γ2 , . . . , γN }, we write γi x = (z1i , z2i ). Then Γ · x corresponds to matrices  1   1  z1 z2   1     2 2 z1 z2     B1 =  B2 =  i =  ...  . , , .. ..     . . 1 N z1 z2N

0

0

0

0

(See Example 1.15.) The basis of V , which gives the above matrix expression, is {eγ1 , . . . , eγN }. One can directly check that (B1 , B2 ) and i are Γ-equivariant.

44

4. RESOLUTION OF SIMPLE SINGULARITIES

(2) As in Theorem 1.14, we consider the space of matrices M0 = Hom(R, Q ⊗ R) ⊕ Hom(W, R). Then its Γ-fixed component is MΓ0 = HomΓ (R, Q ⊗ R) ⊕ HomΓ (W, R),

where HomΓ denotes the space of Γ-equivariant homomorphisms. Let GLΓ (R) be the group of Γ-equivariant automorphisms of V . It acts on MΓ0 . Then the set of Γ-invariant subschemes Z such that H 0 (OZ ) is isomorphic to the regular representation R is given by ( ), (i) [B1 , B2 ] = 0 (4.5) (B1 , B2 , i) ∈ MΓ0 (ii) (B1 , B2 , i) satisfies stability condition GLΓ (R). in Theorem 1.14 (ii)

By the stability condition (ii), one can show that the cokernel of the differential of the map (B1 , B2 ) 7→ [B1 , B2 ] is 1-dimensional for any (B1 , B2 ) (cf. Exercise 1.17). Hence the dimension of the above set is dim MΓ0 + 2 − 2 dim GLΓ (R),

which is independent of a point (B1 , B2 , i). Since it contains a 2-dimensional component X, it is 2-dimensional. Suppose that it has a component X ′ other than X. It has a symplectic structure as in the proof of Theorem 4.1. Since there is no other fixed points in π −1 (Γ · (C2 \ {0})) other than X, we must have X ′ ⊂ π −1 (N[0]). However, this is impossible, since π −1 (N[0]) is an isotropic subvariety by Theorem 1.16. (3) The proof is almost the same as that of the statement (2). Other components have symplectic structures, but are contained in the isotropic subvariety π −1 (N · [0]). This is possible only when components are 0-dimensional. Question 4.6 (Hitchin). Consider a finite group action on a K3 surface which preserves a hyper-K¨ahler structure. (Such actions were classified by Mukai [58].) It naturally induces the action on the Hilbert scheme of points on the K3 surface. Its fixed point component is a compact hyper-K¨ahler manifold as in 4.2. Is the component a new hyper-K¨ahler manifold ? The known compact irreducible hyper-K¨ahler manifolds are equivalent to the Hilbert scheme of points on a K3 surface, or the higher order Kummar variety (denoted by Kr in [6]) modulo deformation and birational modification. (cf. [57, p.168]) 4.2. Dynkin diagrams Now we give a brief review on finite subgroups of SU(2). It is known that there exists one-to-one correspondence between finite subgroups of SU(2) and the simply-laced Dynkin diagrams An , Dn , E6 , E7 and E8 . Indeed there are following types of the finite subgroup Γ of SU(2): (1) Type An : Cyclic group of order n + 1. (2) Type Dn : Binary dihedral group of order 4(n − 1). (3) Type E6 , E7 , E8 : Binary polyhedral groups. The correspondence can be given as follows (e.g. see [5]). Let E be the exceptional set of X, i.e. E = π −1S (0). Since X is a minimal resolution, E can be written as a sum of reduced divisors E = k Σk which satisfies Σk ∼ = P1 , Σk · Σk = −2. We draw a diagram by

4.2. DYNKIN DIAGRAMS

45

the following rule: assign a vertex corresponding to each irreducible component. Connect vertices with an edge if corresponding irreducible components intersect. (See Figure 4.1.) ⇐⇒ Figure 4.1. simple singularity of type An Recall that the Cartan matrix is determined from the Dynkin diagram by   if k = l, 2 ckl = −1 if vertices k and l are connected by an edge,  0 otherwise,

where we only consider the simply-laced cases. For example, the Cartan matrix of An is 

2

−1

 −1 2 −1  −1 2 −1   ..  .   −1

0



0 

  .   2 −1 −1 2

From the above discussion, the intersection matrix is given by −(Cartan matrix). Another correspondence between finite subgroups of SU(2) and Dynkin diagrams was given by McKay [55]. Let R0 , R1 , . . . , Rn be the irreducible representations of Γ with R0 the trivial representation. Let Q be the 2-dimensional representation L given by the inclusion Γ ⊂ SU(2). Let us decompose Q ⊗ Rk into irreducibles, Q ⊗ Rk = l akl Rl , where akl is the multiplicity. Then the matrix 2I − (akl )kl is an affine Cartan matrix of a simply-laced (1) ˜ (1) ˜ (1) ˜ (1) ˜ (1) extended Dynkin diagram, A˜n , D n , E6 , E7 or E8 . For example, consider the case when Γ is of type An , i.e.  k   √  γ 0 k = 0, 1, . . . , n, γ = exp 2π −1 Γ = Z/(n + 1)Z ∼ . = −k 0 γ n+1

The irreducible representations Rk of Γ are

  γ 0 Rk : 7→ γ k 0 γ −1

(k = 0, 1, . . . , n).

46

4. RESOLUTION OF SIMPLE SINGULARITIES

And we have Q = R1 ⊕ Rn . Then we have Q ⊗ Rk = Rk−1 ⊕ Rk+1 where the suffix is understood by modulo n + 1. Hence, we get   0 1 1 1 0 1     1 0 1   . (akl )k,l=1,...,n =  .. .. ..  . . .    1 0 1 1

1

0

e(1) On the other hand, the extended Dynkin diagram A n is Figure 4.2. The corresponding Cartan matrix is 2I − (akl )k,l . 0

1

2

n−1 n

3

Figure 4.2 In fact, it is also known that the Dynkin diagram given by the resolution graph is obtained by the extended Dynkin diagram by removing the vertex corresponding to the trivial representation R0 . Now we return to the description of X given in (4.5). Take the irreducible decomposition of V , W as Γ-module M W = W0 ⊗ R0 , V (∼ Vk ⊗ Rk . = R) = k

Here W0 , Vk are multiplicities considered as vector spaces. Since i is in HomΓ (W, V ), it is an element of Hom(V0 , W0 ). Similarly we have M (B1 , B2 ) ∈ HomΓ (V, Q ⊗ V ) = HomΓ (Vl ⊗ Rl , Vk ⊗ Rk ⊗ Q) =

k,l M k,l

akl Hom(Vl , Vk ),

L where we have used the decomposition Q ⊗ Rk = l akl Rl into irreducibles. With the aid of the McKay correspondence, we can visualize (B1 , B2 , i). For example, e(1) let us consider the case of A n . Put the vector space Vk on the k-th vertex of the diagram. Draw an arrow from Vl to Vk when akl 6= 0. It corresponds to the Hom(Vl , Vk )-component

4.3. TAUTOLOGICAL VECTOR BUNDLES

47

of (B1 , B2 ). At last, put W0 and an arrow representing i upon V0 . Thus we obtain Figure 4.3. W0 ? 1 V0 i

)

V1 

-

V2 

-

··· 

-

Vk 

-

··· 

q - Vn

Figure 4.3 Conversely, starting from this description, Kronheimer constructed resolutions of the simple singularities [49] as follows. In order to explain the hyper-K¨ahler structures, we use the description given in Theorem 2.1 rather than that in Theorem 1.14. Let M = Hom(V, Q ⊗ V ) ⊕ Hom(W, V ) ⊕ Hom(V, W ). Then its Γ-fixed point is M MΓ = ak,l Hom(Vl , Vk ) ⊕ Hom(W0 , V0 ) ⊕ Hom(V0 , W0 ). k,l

The hermitian metric and the quaternion module structure on M descends to MΓ . In particular,QMΓ is a hyper-K¨ahler manifold. There is a natural action on MΓ of a Lie group UΓ (V ) ∼ = k U(Vk ). This action preserves the hyper-K¨ahler structure. The corresponding hyper-K¨ahler moment map is p ◦ µ ◦ i, where i is the inclusion MΓ ⊂ M, µ is the L hyperK¨ahler moment map for U(V )-action on M, and p is the orthogonal projection to k u(Vk ) in u(V ). We denote this hyper-K¨ahler moment map also by µ = (µ1 , µ2 , µ3 ). This increases the flexibility of the choice of parameters. Take ζ i = (ζ0i , ζ1i , . . . , ζni ) (i = 1, 2, 3) such that ζki is a scalar matrix in u(Vk ). Then we can consider a hyper-K¨ahler quotient Y µ−1 (ζ 1 , ζ 2, ζ 3)/ U(Vk ) = X(ζ 1 ,ζ 2,ζ 3 ) . k

This is the Kronheimer’s construction. Since the hyper-K¨ahler manifolds which are obtained as the Γ-fixed component of the Hilbert scheme (or its deformation) satisfy ζ0 = ζ1 = · · · = ζn , not all X(ζ 1 ,ζ 2 ,ζ 3 ) are obtained as a fixed point component. 4.3. Tautological vector bundles

Many properties of simple singularities can be studied by using the description given in previous sections. Since these have been discussed in other papers (e.g. see [49, 50, 66]), we shall explain only briefly. [N ] [N ] Let Z be the universal family on (C2 ) , which is a subvariety of C2 × (C2 ) . Let p [N ] denote the projection to (C2 ) . Then p∗ OZ is a vector bundle (i.e. locally free sheaf) of

48

4. RESOLUTION OF SIMPLE SINGULARITIES [N ]

rank N over (C2 ) . We denote it by V. If we use the description 1.14, it can be obtained as the vector bundle associated with the principal bundle ) (i) [B1 , B2 ] = 0 [N ] (B1 , B2 , i) (ii) (B1 , B2 , i) satisfies stability condition −−−−−→ (C2 ) . / GLn (C) in Theorem 1.14 (ii)

(

We have tautological homomorphisms

i : O(C2 )[N] → V,

Bα : V → V

(α = 1, 2),

induced by the natural projection OC2 → OZ and multiplication of the coordinate function zα (α = 1, 2) respectively. In the description 1.14, it is induced by corresponding matrices i, Bα . (Hence we use the same notation.) [N ] [N ] [N ] Let pa : (C2 ) ×(C2 ) → (C2 ) be the projection to the a-th factor (a = 1, 2). Then we have tautological complex Hom(p∗1 V, p∗2 V) ⊕ Hom(p∗1 V, p∗2 V) b a ∗ ∗ ⊕ −→ Hom(p∗1 V, p∗2 V), Hom(p1 V, p2 V) −→ Hom(O(C2 )[N] ×(C2 )[N] , p∗2 V) ⊕ Hom(p∗1 V, O(C2 )[N] ×(C2 )[N] ) where  2  B1 ξ − ξB11 B 2 ξ − ξB 1  2 2 a(ξ) =   −ξi1  , 0



 C1  C2  2 1 1 2 2  b  I  = B1 C2 − C2 B1 + C1 B2 − B2 C1 + i J. J

Here the superscript indicates the factor from which the tautological homomorphism is pulled back. For example, B11 is a pull-back of B1 by p1 , hence is a section of Hom(p∗1 V, p∗1 V). Lemma 4.7. (1) ba = 0. (2) a is injective and b is surjective. Proof. (1) The equations [B11 , B21 ] = 0, [B12 , B22 ] = 0 imply ba = 0. (2) Suppose ξ is in Ker a. Then Ker ξ is invariant under Bα1 (α = 1, 2) and contains Im i1 . Hence the stability condition for (B11 , B21 , i1 ) implies Ker ξ = p∗1 V. Hence ξ = 0. The surjectivity can be proved exactly the same way by considering the transpose of b.

4.3. TAUTOLOGICAL VECTOR BUNDLES [N ]

Hence Ker b/ Im a forms a vector bundle over (C2 ) × (C2 ) s of Ker b/ Im a by   0  0   s= −i2  mod Im a. 0

[N ]

49

. Let us define a section

Then s vanishes when there exists ξ ∈ Hom(V, V ) such that B12 ξ = ξB11 ,

B22 ξ = ξB21 ,

i2 = ξi1 .

The stability condition for (B12 , B22 , i2 ) implies Im ξ = V , hence ξ is invertible. It means [N ] that (B11 , B21 , i1 ) and (B12 , B22 , i2 ) defines the same point in (C2 ) . Namely s−1 (0) is the [N ] [N ] diagonal ∆ in (C2 ) × (C2 ) . Lemma 4.8. s is transversal to the zero section. Proof. One can show that the differential ∇s is surjective over the diagonal ∆. The detail is left to the reader. See [67, 5.7]. This lemma implies the Koszul complex is exact: (4.9)

0→

2N ^

t (s∧)

ts

(Ker b/ Im a)∨ −−−→ · · · → (Ker b/ Im a)∨ −→ O(C2 )[N] ×(C2 )[N] → O∆ → 0

Now we restrict everything to the Γ-fixed component X. We restrict the tautological vector bundle V to X, which is still denoted by V. It has a structure of a Γ-module and decomposes into irreducibles: M (4.10) Vk ⊗ Rk∗ . V= k

The bundle Vk is the vector bundle associated with the principal bundle ( ) (i) [B1 , B2 ] = 0 (B1 , B2 , i) ∈ MΓ0 (ii) (B1 , B2 , i) satisfies stability condition −−−−−→ X, / GLΓ (V ) in Theorem 1.14 (ii) Q where the representation is the projection GLΓ (V ) = GL(Vk ) → GL(Vk ). Tautological homomorphisms (B1 , B2 ), i are now Γ-equivariant. From now, it is more convenient to consider the pair (B1 , B2 ) as a section of HomΓ (V, Q ⊗ V). We denote it simply by B. We take the Γ-equivariant part of the restriction of the tautological complex: HomΓ (p∗1 V, Q ⊗ p∗2 V) ⊕ a′ b′ ∗ ∗ HomΓ (p1 V, p2 V) −→ HomΓ (OX×X , p∗2 V) −→ HomΓ (p∗1 V, p∗2 V), ⊕ ∗ HomΓ (p1 V, OX×X )

where a′ (resp. b′ ) is the restriction of a (resp. b). Then a′ is injective and b′ is surjective, and Ker b′ / Im a′ is a vector bundle.

50

4. RESOLUTION OF SIMPLE SINGULARITIES

Instead of the Koszul complex (4.9), we introduce a slightly modified version, which is more convenient (4.11) HomΓ (p∗1 V, Q ⊗ p∗2 V) HomΓ (p∗1 V, p∗2 V) ⊕ HomΓ (p∗1 V, p∗2 V) ∗ ⊕ ⊕ 0→ −−−−→ HomΓ (OX×X , p2 V) −“−−→ − → O∆ → 0, ” t ( a′ s′ ) b′ OX×X ⊕ OX×X s′′ HomΓ (p∗1 V, OX×X ) where



   0 C s′ (λ) = λi2  , s′′  I  = Ji1 0 J   η t = tr(η|∆ ) − λ|∆ . µ

Here (·)|∆ is the restriction to ∆, and we have used the isomorphism p∗1 V|∆ ∼ = p∗2 V|∆ to define the trace tr(η|∆ ). In fact, (4.11) is a modified version of the exact sequence L ∗in [50,∗ 3.6]. It gives a resolution of the diagonal whose higher terms are in the form p1 E ⊗ p2 F for some E and F . Using the argument similar to that in Chapter 2, Kronheimer and the author gave a description of holomorphic vector bundle on X [50]. Moreover, (4.11) gives an interesting interpretation of the McKay correspondence. Irreducible summands Vk of the tautological vector bundle V have following properties: (i) The restriction of Vk to π −1 (C2 \ {0}/Γ ∼ = C2 \ {0}/Γ comes from the representation Rk ([50, 2.2]). (ii) {c1 (Vk )}k=1,...,n is the dual basis of the basis given by irreducible components of π −1 (0) ([66, 5.8,5.10]). In this picture, the correspondence between irreducible representations of Γ (except the trivial representation) and irreducible components of the exceptional set becomes concrete. It is realized by the tautological bundles Vk ’s. In [66, 5.8], we have shown the correspondence respects the multiplicative structures, one given by the tensor product and one given by the cup product. In fact, using (4.11), we can show that two matrices  Z  − c1 (Vk ) ∪ c1 (Vl ) X

and k,l=1,...,n

(2δkl − akl )k,l=1,...,n (where Q ⊗ Rk =

M

akl Rl )

are inverse to each other. The proof is free from the classification of simple singularities. Question 4.12. Is it possible to give an analogue of the McKay correspondence between irreducible representations of the symmetric group Sn and cycles in the Hilbert [n] scheme (C2 ) ? It seems likely that the tautological bundle V plays a fundamental role.

4.3. TAUTOLOGICAL VECTOR BUNDLES

51

How the multiplicative structures relate under the correspondence ? Note that the di[n] mension of H∗ ((C2 ) is the number of partitions of n which is equal to the number of irreducible representations of Sn by Corollary 5.9. Although there are several higher dimensional analogue of the McKay correspondence, the study of multiplicative structures is missing.

CHAPTER 5

Poincar´ e polynomials of the Hilbert schemes (1) [n]

In this chapter we shall calculate the Poincar´e polynomial of (C2 ) . This was first accomplished by Ellingsrud and Strømme [14]. They have used the Bialynicki-Birula decompo[n] sition associated with the natural torus action on (C2 ) , and then compute the Poincar´e polynomial using the Weil conjecture. Our approach is essentially the same, but we use Morse theory instead of the Weil conjecture. For a later purpose (Chapter 7), we shall explain the perfectness of the Morse function given by the moment map of a torus action on a general symplectic manifold. However, [n] when the fixed points of a torus action are all isolated, such as the case of (C2 ) , the perfectness follows easily from the Morse inequality since they all have even indices. The [n] reader who has interests only in (C2 ) could skip §5.1. 5.1. Perfectness of the Morse function arising from the moment map Let (X, ω) be a compact symplectic manifold and T a compact torus. We suppose that there exists a T -action on X preserving ω with the corresponding moment map µ : X → t∗ . As explained in Chapter 3, µ : X → t∗ is called a moment map if it satisfies (5.1)

dhµ, ξi = iξ∗ ω

for ξ ∈ t,

where t is the Lie algebra of T and ξ ∗ is the vector field on X generated by ξ. We take a non-zero element ξ ∈ t and use f = hµ, ξi : X → R as a Morse function. Since ω is non-degenerate, x ∈ X is a critical point of f if and only if ξx∗ = 0. This is equivalent to the condition that g · x = x for any g ∈ exp Rξ. If we choose a generic element ξ, we have exp Rξ = T . Therefore in such a case, the critical point is the same as the fixed point of the torus action. In the following, we assume ξ is generic, and hence Crit(f ) = X T . Now we fix a Riemannian metric g which is invariant under the T -action. The symplectic form ω together with the Riemannian metric g gives an almost complex structure I defined by ω(v, x) = g(Iv, w). With this almost complex ` structure, we regard the tangent space Tx X as a complex vector space. Let X T = ν Cν be the decomposition into the connected components. For each x ∈ Cν , we have the weight decomposition M Tx X = V (λ), λ∈Hom(T,U(1))

52

5.1. PERFECTNESS OF THE MORSE FUNCTION ARISING FROM THE MOMENT MAP

53

where V (λ) = {v ∈ Tx X | t · v = λ(t)v for any t ∈ T }. We define X X V (λ), Nx− = V (λ). Nx+ = √ h −1dλ,ξi>0

√ h −1dλ,ξi<0

Since ξ is generic, we have Tx X = Nx+ ⊕ V (0) ⊕ Nx− . The exponential map gives a T -equivariant isomorphism between a neighborhood of 0 ∈ Tx X and a neighborhood of x ∈ X. This shows that Cν is a submanifold of X and that the tangent space at x is given by Tx Cν = V (0). Moreover, f is approximated around x by the map X 1X √ Tx X ∋ v = vλ 7→ h −1dλ, ξikvλk2 , 2 λ λ where vλ is the V (λ)-component of v. Hence we have, 1X √ h −1dλ, ξikvλk2 . Hess f (v, v) = 2 λ This shows that the Hessian of f is positive definite (resp. negative definite) on Nx+ (resp. Nx− ). Therefore f is non-degenerate in the sense of Bott, i.e. the set of critical points is a disjoint union of submanifolds of X, and the Hessian is non-degenerate in the normal direction at any critical point. We put dν = dimR Nx− = 2 dimC Nx− which is the index of f at the critical manifold Cν . Note that the index dν is always even in this case. Let us denote by Wν+ (resp. by Wν− ) the stable manifold of Cν (resp. the unstable manifold of Cν ). These are defined by def.

Wν+ = {x ∈ X | lim φt (x) ∈ Cν }, t→−∞

def.

Wν− = {x ∈ X | lim φt (x) ∈ Cν }, t→+∞

where φt is a gradient flow of f with respect to the T -invariant metric g on X. The stable manifoldS Wν+ (resp. the unstable manifold Wν− ) is diffeomorphic to the positive normal S bundle x∈Cν Nx+ → Cν (resp. the negative normal bundle x∈Cν Nx− → Cν ), and in the following we assume these identifications. Notice that Wν− is an orientable real vector bundle of rank dν on Cν since Nx− consists of non-zero weight spaces for the T -action. It is well known (see e.g., [1, p.537]) that there exists a partial ordering < on the index set of the critical manifolds with the property S (1) Wν+ ⊂ ν≤µ Wµ+ , (2) µ ≤ ν implies f (Cµ ) ≤ f (Cν ).

Suppose that for c ∈ R there exists only one critical manifold Cν with f (Cν ) = c. This assumption is just for saving notations, and the argument for the general case is essentially

54

´ POLYNOMIALS OF THE HILBERT SCHEMES (1) 5. POINCARE

S S the same. We define Xc,− = µ<ν Wµ+ and Xc,+ = µ≤ν Wµ+ = Xc,− ∪ Wν+ . Then the cohomology exact sequence for the pair (Xc,+ , Xc,− ) gives (5.2)

jq+1

jq

· · · → H q (Xc,+ , Xc,− ) − → H q (Xc,+ ) → H q (Xc,− ) → H q+1 (Xc,+ , Xc,− ) −−→ · · · .

We split this sequence into the following short exact sequences. (

0 → im jq → H q (Xc,+ ) → H q (Xc,− ) → ker jq+1 → 0 0 → ker jq → H q (Xc,+ , Xc,− ) → im jq → 0.

Note that Xc,+ is an open submanifold of X and that Wν+ is a closed submanifold of Xc,+ . Let Nν be the normal bundle of Wν+ in Xc,+ . Then the restriction of Nν to Cν is the unstable manifold Wν− and the inclusion (Wν− , Cν ) ֒→ (Nν , Wν+ ) becomes a homotopy equivalence. Since Wν− is orientable, so is Nν , and we have the following identification given by the Thom isomorphism H q (Xc,+ , Xc,−) ∼ = H q (Nν , Nν \ Wν+ ) ∼ = H q (Wν− , Wν− \ Cν ) ∼ = H q−dν (Cν ). Therefore we have

( bq (Xc,+ ) = bq (Xc,− ) − aq+1 + cq bq−dν (Cν ) = aq + cq ,

where aq = rank(ker jq ), cq = rank(im jq ) and bq ’s are the q-th Betti numbers. From these equalities, we have Pt (Xc,+ ) = Pt (Xc,− ) + tdν Pt (Fν ) − (1 + t)Rν (t),

P where Rν (t) = q∈Z aq+1 tq . Summing over the critical values c ∈ R, we have the wellknown Morse inequality Pt (X) =

X ν

tdν Pt (Fν ) − (1 + t)R(t),

P where R(t) = ν Rν (t). This is the standard argument for the Morse inequality. If Cν is a point, or more generally H odd (Cν ) = 0, the cohomology long exact sequence (5.2) splits into short exact sequences and we have R(t) = 0, i.e. the Morse function is perfect. However, we can show the perfectness of the Morse function in our situation without any condition on Cν . We shall follow the argument in [47]. The above argument shows that the perfectness of the Morse function is equivalent to the condition that H q (Xc,+ , Xc,− ) → H q (Xc,+ ) is injective for all q ∈ Z and for all the critical values c ∈ R. Let us consider the

5.1. PERFECTNESS OF THE MORSE FUNCTION ARISING FROM THE MOMENT MAP

55

following diagram. H q (Xc,+ , Xc,− ) x  

H q (Nν , Nν \ Wν+ ) x  

H q (Wν− , Wν− \ Cν ) x  

jq

−−−→ H q (Xc,+ )  ∗ yi H q (Nν )  ∗ yi

H q (Wν− )  ∗ yi

H q−dν (Cν ) H q (Cν ) The left vertical line is the Thom isomorphism, and the right one is given by the restriction maps. From the definition of the Thom isomorphism, we know that the composite map ψ : H q−dν (Cν ) → H q (Cν ) is given by the multiplication by the Euler class of Wν− → Cν . If ψ : H ∗ (Cν ) → H ∗ (Cν ) is injective, then jq must also be injective, and therefore we can prove the perfectness of the Morse function. Unfortunately, this argument does not work since ψ is not injective. The main idea to save the argument is to use the equivariant cohomology instead of the usual cohomology. Let ET → BT be the universal T -bundle. This is given by a product of S ∞ → CP ∞ which is the inductive limit of the Hopf bundle S 2n+1 → CP n , where we regard S 2n+1 as a unit sphere in Cn+1 . For a topological space M with a T -action, the equivariant cohomology HT∗ (M) is defined by HT∗ (M) = H ∗ (MT ), where MT = ET ×T M. Notice that MT → BT is a fiber bundle with fiber M. Hence using the equivariant cohomology means that we replace M with a family of M’s parameterized by BT . This shows that the above argument also works for the equivariant cohomology. Therefore we have the Morse inequality also for the equivariant cohomology. X PtT (X) = tdν PtT (Cν ) − (1 + t)RT (t), ν

T

where P is the Poincare polynomial for the equivariant cohomology. The same argument shows that RT (t) = 0 if ψ T : HT∗ (Cν ) → HT∗ (Cν ) is injective. Since T acts trivially on Cν , we have (Cν )T = BT × Cν . Hence we have HT∗ (Cν ) = H ∗ (BT ) ⊗ H ∗ (Cν ). If we write the equivariant Euler class of Wν− as M eT (Wν− ) = eν ⊗ 1 + λ ∈ H ∗(BT ) ⊗ H 0 (Cν ) ⊕ H ∗ (BT ) ⊗ H q (Cν ), q>0

where eν ∈ H dν (BT ) is given by the Euler class of ET ×T Nx− → BT for x ∈ Cν . Since λ is nilpotent, ψ T is injective if eν 6= 0. Moreover, Nx− consists of the non-zero weight spaces, hence we always have eν 6= 0. This shows that f is equivariantly perfect. Therefore we have X X PtT (X) = tdν PtT (Cν ) = Pt (BT ) tdν Pt (Cν ), ν

ν

56

´ POLYNOMIALS OF THE HILBERT SCHEMES (1) 5. POINCARE

where Pt (BT ) = Pt ((P∞ )r ) = (1 + t2 + t4 + · · · )r = (1 − t2 )−r with r = rank T . To conclude the perfectness of the Morse function, we consider the Leray-Serre spectral sequence for the fibration XT → BT . The E2 -term is given by E2p,q = H p (BT, H q (X)) = H p (BT ) ⊗ H q (X) since π1 (BT ) = 0. Because the Er -term is given by the cohomology of p,q the Er−1 -term, we have rank Erp,q ≤ rank Er−1 . Hence we have X X X p,q bn (XT ) = rank E∞ ≤ rank E2p,q = bp (BT )bq (X). p+q=n

p+q=n

p+q=n

This shows the next inequality. X Pt (BT ) tdν Pt (Cν ) = PtT (X) ≤ Pt (BT )Pt (X), ν

where the use is made of the fact that f is equivariantly perfect. Hence we have X tdν Pt (Cν ). Pt (X) ≥ ν

Since the usual Morse inequality gives the opposite inequality, we have X Pt (X) = tdν Pt (Cν ). ν

This completes the proof of the perfectness of our Morse function. Notice that our argument also gives the proof of the perfectness of the Morse function in the case of a noncompact symplectic manifold if the appropriate conditions on f are satisfied. For example, the condition that f −1 ((−∞, c]) is compact for all c ∈ R is sufficient, and this is the case for (C2 )[n] as will be shown later. When X is a K¨ahler manifold, the stable and the unstable manifolds can be expressed purely in terms of the group action. Notice that an T -action on X extends uniquely to a holomorphic T C -action on X. For our Morse function f = hµ, ξi, we have df = ω(ξ ∗, ) = g(Iξ ∗, ), where ξ ∗ is the infinitesimal action of ξ ∈ t and I is the complex structure on X. This shows that the gradient vector field of f is given by grad f = Iξ ∗, and that the gradient √ flow φt is expressed as φt (x) = e −1tξ · x. Therefore the stable manifold Wν+ and the unstable manifold Wν− are given by √

Wν+ = {x ∈ X | lim e

−1tξ

· x ∈ Cν }

Wν− = {x ∈ X | lim e

−1tξ

· x ∈ Cν }.

t→−∞ t→+∞

√

The unstable manifold Wν− becomes important in Chapter 7 when we study a holomorphic symplectic manifold.

5.2. POINCARE POLYNOMIAL OF (C2 )

[n]

5.2. Poincare polynomial of (C2 )

57

[n]

Let us apply the above result to our situation. Recall that we have two different [n] descriptions of (C2 ) . The first one, given in Theorem 1.14, is simple, but does not give a K¨ahler metric. The second one, given in√Theorem 3.23, defines a K¨ahler metric, but is complex because of the equation µ1 = −1dχ. We use both descriptions properly according to situations. We define an action of 2-dimensional compact torus T 2 = U(1) × U(1) on C2 by C2 ∋ (z1 , z2 ) 7→ (t1 z1 , t2 z2 ) for (t1 , t2 ) ∈ T 2 . [n]

This induces the action on (C2 ) . In terms of the ADHM data, this action is given by [n]

[(B1 , B2 , i)] 7→ [(t1 B1 , t2 B2 , i)] for (t1 , t2 ) ∈ T 2 and [(B1 , B2 , i)] ∈ (C2 ) , [n]

and the corresponding moment map µ : (C2 ) → (t2 )∗ is given by √  √ −1 −1 2 2 kB1 k , kB2 k . µ([(B1 , B2 , i)]) = 2 2 Note that we are using the description 3.23, otherwise kBα k2 are not√well-defined. Now we apply the general argument in §5.1. We choose ξ = −2 −1(1, ε) ∈ t2 . Then our Morse function f is given by def.

f ([(B1 , B2 , i)]) = hµ([(B1 , B2 , i)]), ξi = kB1 k2 + εkB2 k2 . For generic ε (0 < ε ≪ 1), the critical points of f coincide with the fixed points of T√2 -action. Moreover f −1 ((−∞, c]) is compact, since f ≤ c together with the equation √ −1 ([B1 , B1† ]+ [B2 , B2† ]+ ii† ) = −1dχ implies a bound on kB1 k2 + kB2 k2 + kik2 , Therefore 2 we can apply Morse theory to our situation. [n] First, we shall identify the fixed point set. By definition, [(B1 , B2 , i)] ∈ (C2 ) is a fixed point if and only if there exists a homomorphism λ : T 2 −→ U(V ) satisfying the following conditions: t1 B1 = λ(t)−1 B1 λ(t), (5.3)

t2 B2 = λ(t)−1 B2 λ(t), i = λ(t)−1 i.

Suppose [(B1 , B2 , i)] is a fixed point. Then we have the weight decomposition of V with respect to λ(t), i.e. M V = V (k, l), k,l

where

V (k, l) = {v ∈ V | λ(t) · v = tk1 tl2 v}.

´ POLYNOMIALS OF THE HILBERT SCHEMES (1) 5. POINCARE

58

From the conditions (5.3), the only components of B1 , B2 and i which might survive are B1 : V (k, l) −→ V (k − 1, l),

B2 : V (k, l) −→ V (k, l − 1), i : W −→ V (0, 0).

Thus we have a commutative diagram, x   (5.4)

B

x  

−−−→ V (k, l − 1) −−−1→ V (k − 1, l − 1) −−−→ x x B B ,  2  2 −−−→

V (k, l) x  

B

−−−1→

V (k − 1, l) x  

−−−→

where the commutativity follows from the condition [B1 , B2 ] = 0. This diagram has the following properties. Proposition 5.5. (1) If k > 0 or l > 0, then V (k, l) = 0. (2) For every k, l ∈ Z, we have dim V (k, l) ≤ 1 (3) If k ≤ 0 and l ≤ 0, then dim V (k, l) ≥ dim V (k, l − 1) and dim V (k, l) ≥ dim V (k − 1, l). (4) Maps between non-zero vector spaces in (5.4) are non-zero. Proof. The stability condition is equivalent to the condition that V = Cn is spanned by elements of the form B1p B2q · i(1) (p, q ∈ Z≥0 ). From this, Assertion (1) follows. We also have Assertion (2) for k or l = 0, and dim V (0, l) ≥ dim V (0, l − 1), dim V (k, 0) ≥ dim V (k − 1, 0), (a part of Assertion (3)). Assertion (4) holds for maps between weightspace in 0-th row, or 0-th column. The stability condition and the commutativity of (5.4) exclude the following situations: B1 V (k, l − 1) ∼ = C −−−→ V (k − 1, l − 1) 6= 0 x x  B ,  2 

V (k, l) ∼ =C 0 x  

−−−→

0

−−−→ V (k − 1, l − 1) 6= 0 x B  2 , B

1 V (k, l) ∼ = C −−−→

V (k − 1, l) ∼ =C

where V (k, l), V (k, l − 1) (resp. V (k, l), V (k − 1, l)) are supposed 1-dimensional in the diagram above (resp. below). Moreover, if V (k, l), V (k −1, l), V (k, l−1) are 1-dimensional,

5.2. POINCARE POLYNOMIAL OF (C2 )

[n]

59

then V (k − 1, l − 1) is 1-dimensional or 0-dimensional by the commutativity and stability: B1 V (k, l − 1) ∼ = C −−−→ V (k − 1, l − 1) x x B B ,  2  2

V (k, l) ∼ =C

Now all assertions follow by induction.

B

−−−1→ V (k − 1, l) ∼ =C

Now we switch to the descriptionQin Theorem 1.14. We can normalize all non-zero maps in (5.4) to 1 by the action of k,l GL(V (k, l)). In particular, the critical point is uniquely determined from the diagram (5.4). From the above proposition, we can give a bijective correspondence between a critical point and a Young diagram of weight n as follows. Suppose a critical point is given. Then we take the weight space decomposition of V and get a diagram (5.4). The above proposition shows that we get a Young diagram of weight n by putting a box when dim V (k, l) = 1. Conversely, if we are given a Young diagram of weight n, we can associate a diagram (5.4) satisfying conditions in Proposition 5.5. For instance, suppose the following diagram corresponds to a critical point. (We suppose all V (k, l)’s bellow are non-zero.) 0 x  

0 −−−→ V (0, −2) −−−→ x  

−−−→

0 x  

0 x  

0 −−−→ V (0, −1) −−−→ V (−1, −1) −−−→ V (−2, −1) −−−→ 0. x x x       0 −−−→ V (0, 0) −−−→ V (−1, 0) −−−→ V (−2, 0) −−−→ 0 x x x      

0 0 Then the corresponding Young diagram is

0

. For a Young diagram D of weight n, we denote by νi (resp. νj′ ) the number of boxes in the i-th column (resp. in the j-th row). For example, we have ν1 = 3, ν2 = 2, ν3 = 2 and ν1′ = 3, ν2′ = 3, ν3′ = 1 for the above diagram. (Note our convention differs from one used in [54]. Rotate 90◦ in anti-clockwise if the reader prefer one used in [54].) In this way, a Young diagram corresponds to a partition ν = (ν1 , ν2 , . . . ) and its conjugate partition

60

´ POLYNOMIALS OF THE HILBERT SCHEMES (1) 5. POINCARE

ν ′ = (ν1′ , ν2′ , . . . ). In terms of the critical point they are given by X X νi = dim V (k, 1 − i), νj′ = dim V (1 − j, l). k

l

[n]

Next, we shall calculate the character of TZ (C2 ) as T 2 -module for Z ∈ Crit(f ) = [n] 2 ((C2 ) )T . We define l(s) = νi − j and a(s) = νj′ − i for each box s ∈ D which sits at the i-th row and j-th column. Hence l(s) (resp. a(s)) is the number of boxes sitting on the right of s (resp. above s) in D. (See Figure 5.6.)

(5.6)

♥ ♥ s ♠♠

l(s) = number of ♠ a(s) = number of ♥

Let R(T 2 ) = Z[T1 , T2 ] be the representation ring of T 2 , where Ti denotes the one dimensional representation given by Ti : (t1 , t2 ) 7→ ti , i = 1, 2. Then the weight decomposition of [n] TZ (C2 ) is given by the next proposition. Proposition 5.7. Let D be the Young diagram corresponding to the fixed point Z ∈ [n] Fix(T 2 ). Then the character of TZ (C2 ) is given by X l(s)+1 −a(s) [n] −l(s) a(s)+1 TZ (C2 ) = (T1 T2 + T1 T2 ). s∈D

[n]

Proof. In order to calculate the weight decomposition of TZ (C2 ) , we adopt the [n] first description of (C2 ) . Let (B1 , B2 , i, 0) be the data corresponding to Z given by Theorem 1.14. Consider a complex

(5.8)

Hom(V, Q ⊗ V ) ⊕ V a b Hom(W, V ) Hom(V, V ) − → − →Hom(V, V ) ⊗ 2 Q, ⊕ V Hom(V, 2 Q ⊗ W )

where Q is 2-dimensional T 2 -module, and a and b are defined by     ξB1 − B1 ξ C1 ξB2 − B2 ξ  C2   , b   = [B1 , C2 ] + [C1 , B2 ] + iJ. a(ξ) =    I ξi 0 J

Since a is the differential of GLn (C)-action and b is the differential of µC in (3.17), the [n] tangent space of (C2 ) at Z can be identified with Ker b/ Im a. (This is the restriction of the tautological complex (see §4.3) to the diagonal.) Recall that when Z = (B1 , B2 , i, 0) is a critical point, there exists a homomorphism λ : T 2 −→ U(V ) which satisfies (5.3) and

5.2. POINCARE POLYNOMIAL OF (C2 )

[n]

61

that we regard V as a T 2 -module. We put the one dimensional representation make the complex T 2 -equivariant. Since a is injective and b surjective, we have TZ (C2 )

[n]

= Hom(V, V ) ⊗ (Q − = V ∗ ⊗ V ⊗ (Q −

2 ^

2 ^

Q − 1) + Hom(W, V ) + Hom(V, W ) ⊗ 2 ^

Q − 1) + V + V ∗ ⊗

Q.

2 ^

V2

Q to

Q

P Pνj′ −j+1 −i+1 PN Pνi We put ν1 = M and ν1′ = N. Since V = M T2 = i=1 j=1 T1−j+1T2−i+1 i=1 T1 j=1 V2 and Q − Q − 1 = T1 + T2 − T1 T2 − 1 = (T1 − 1)(1 − T2 ), we have V ⊗ (Q −

2 ^

Q − 1) = =

M X j=1

M X j=1

ν′

T1−j+1(T1 − 1)

j X

i=1

T2−i+1 (1 − T2 )

−νj′ +1

T1−j+1(T1 − 1)(T2

− T2 ).

Hence we have ∗

V ⊗ V ⊗ (Q − =

νi N X X

2 ^

′ T1j −1 T2i−1

i=1 j ′ =1

=

N X M X i=1 j=1

Q − 1)

M X j=1

T1−j {

νi X

′

−νj′ +1

T1−j+1(T1 − 1)(T2 ′

i−νj′

(T1j +1 − T1j )}(T2

j ′ =1

− T2 )

− T2i )

N X M X i−ν ′ = (T1−j+νi+1 − T1−j+1 )(T2 j − T2i ) i=1 j=1

N X M X i−ν ′ i−ν ′ = {(T1−j+νi+1 T2 j − T1−j+1 T2i) − T1−j+1(T2 j − T2i ) − (T1−j+νi+1 − T1−j+1)T2i }. i=1 j=1

Moreover, we have N X M X i=1 j=1

i−ν ′ T1−j+1(T2 j

− T2i ) =

M X

ν′

j X −j+1

T1

j=1

=V −

i=1

(T2−i+1 − T2N −i+1 )

νj′

M X X j=1 i=1

T1−j+1T2N −i+1 .

62

´ POLYNOMIALS OF THE HILBERT SCHEMES (1) 5. POINCARE

Similarly, we have N X M X

(T1−j+νi+1

i=1 j=1

T1−j+1)T2i

−

=V∗⊗

[n]

2 ^

ν′

Q−

j M X X

T1−M +j T2i .

j=1 i=1

Thus, if we put R = TZ (C2 ) , we have 2 ^

R =V ∗ ⊗ V ⊗ (Q − =

N X M X

Q − 1) + V + V ∗ ⊗ i−νj′

(T1−j+νi+1 T2

i=1 j=1

− T1−j+1 T2i )

2 ^

Q

ν′

+

j M X X

(T1−j+1T2N −i+1 + T1−M +j T2i ).

j=1 i=1

P

We write R = R(T1 , T2 ) = (k,l)∈Z2 ck,l T1k T2l , and define R>0 = P k l k≤0 ck,l T1 T2 . From the above expression, we have R>0 =

νi N X X

P

k l k>0 ck,l T1 T2

and R≤0 =

i−νj′

T1−j+νi+1 T2

i=1 j=1

=

X

l(s)+1

T1

−a(s)

T2

.

s∈D

V Since R = R∗ ⊗ 2 Q, we have R(T1 , T2 ) = R(T1−1 , T2−1 )T1 T2 . Hence we have another expression of R as N X M X −i+ν ′ +1 R= (T1j−νi T2 j − T1j T2−i+1 ) i=1 j=1

ν′

+

j M X X

(T1j T2−N +i + T1M −j+1T2−i+1 ).

j=1 i=1

From this expression, we have

R≤0 = =

νi N X X

−i+νj′ +1

T1j−νi T2

i=1 j=1

X

−l(s)

T1

a(s)+1

T2

.

s∈D

Therefore we have R = R>0 + R≤0 =

X s∈D

This completes the proof of proposition.

l(s)+1

(T1

−a(s)

T2

−l(s)

+ T1

a(s)+1

T2

).

5.2. POINCARE POLYNOMIAL OF (C2 )

[n]

63

[n]

Corollary 5.9. The Poincar´e polynomial of (C2 ) is given by   X [n] Pt (C2 ) = t2(n−l(ν)) , ν

where ν runs over all partitions of n and l(ν) is the length of ν, i.e. the number of nonzero entries in ν. Proof. We only need to calculate the index of the critical point Z. By the choice of our Morse function, this is equal to the sum of dimensions of the weight spaces which satisfy either of the following conditions, (i) the weight of t1 is negative, (ii) the weight of t1 is 0 and the weight of t2 is negative. By the above proposition, this is equal to the twice of the number of boxes in D which satisfies l(s) > 0. Hence the index of f at Z is 2(n − N) = 2(n − l(ν)).   [n] If we introduce the generating function of Pt (C2 ) , the corollary can be expressed as ∞ ∞   Y X 1 n 2 [n] q Pt (C ) = . 2m−2 m) (1 − t q n=0 m=1

This is G¨ottsche’s formula for the case of C2 . The general case is the subject of Chapter 6. [n] Let π : (C2 ) → S n (C2 ) be the Hilbert-Chow morphism. In Chapter 1, we have shown that dimC π −1 (n · [0]) ≤ n − 1, and there exists at least one (n − 1)-dimensional component. [n] [n] By the corollary, we have Hk ((C2 ) ) = 0 for k ≥ 2n − 1 and rank H2n−2 ((C2 ) ) = 1 which is generated by the closure of the unstable manifold Lν corresponding to the partition ν = (n). Therefore we have the next theorem. Theorem 5.10. There is exactly one (n − 1)-dimensional irreducible component in π (n · [0]). −1

In fact, much stronger result is known. Theorem 5.11 (Brian¸con). π −1 (n · [0]) is irreducible. For our purpose (in particular, in Chapter 6), Theorem 5.10 is enough. Hence we do not give the proof of Brian¸con’s theorem. Remark 5.12. As we shall see in Chapter 7, fixed points of the torus action correspond to ideals generated by monomials. In fact, the result in this section can be translated to the language of ideals. Finally, we shall calculate the critical values of our Morse function f for a later purpose √ √ −1 (Chapter 7). If we write −1dχ = 2 c · id for some c ∈ R>0 , the condition for the real moment map becomes [B1 , B1† ] + [B2 , B2† ] + ii† = c · id .

64

´ POLYNOMIALS OF THE HILBERT SCHEMES (1) 5. POINCARE [n]

Proposition 5.13. Let Z = [(B1 , B2 , i)] ∈ (C2 ) be a critical point corresponding to a partition ν and its conjugate partition ν ′ . Then the critical value of Z is given by X X f (Z) = c{ (j − 1)νj + ε (i − 1)νi′ }. j

L

i

Proof. Let V = k,l V (k, l) be the corresponding weight decomposition of V and L Vi = k V (k, 1 − i). If we write B1,i = B1 |Vi : Vi → Vi and B2,i = B2 |Vi : Vi → Vi+1 , we have † † † [B1,i , B1,i ] + B2,i−1 B2,i−1 − B2,i B2,i = c · idVi , for i = 2, 3, . . . from the above equation. Taking the trace of the both hand sides, we have kB2,i−1 k2 − kB2,i k2 = c dimC Vi = c · νl .

Hence we have

kB2 k2 = 2

Similarly, we have kB1 k = c

P

X i

kB2,i k2 = c

X (i − 1)νi′ . i

− 1)νj . Therefore we have X X f (Z) = c{ (j − 1)νj + ε (i − 1)νi′ }. j (j

j

i

CHAPTER 6

Poincar´ e polynomials of Hilbert schemes (2) In this chapter, we shall prove the following formula for Poincar´e polynomial of Hilbert scheme X [n] of n-points on a quasi-projective nonsingular surface X: Theorem 6.1. The generating function of the Poincar´e polynomials of the Hilbert scheme X [n] , parameterizing n-points in X, is given by ∞ ∞ X Y (1 + t2m−1 q m )b1 (X) (1 + t2m+1 q m )b3 (X) n [n] q Pt (X ) = (6.2) , (1 − t2m−2 q m )b0 (X) (1 − t2m q m )b2 (X) (1 − t2m+2 q m )b4 (X) n=0 m=1 where bi (X) is the i-th Betti number of X.

The formula (6.2) for projective surfaces was proved by G¨ottsche [28] using the Weil conjecture. Our proof given here is due to G¨ottsche-Soergel [31]. They used BorhoMacPherson’s formula [10] for the direct image of the intersection cohomology complex by a projective semismall morphism. Borho-MacPherson’s formula, in turn, is a direct application of Beilinson-Bernstein-Deligne-Gabber’s decomposition theorem [7]. The proof of the theorem is out of reach of this lecture. Hence we just recall results on the intersection cohomology in §6.1 and apply them to our situation. It is very suggestive to compare (6.2) with Macdonald’s formula for the symmetric product [53] (6.3)

∞ X n=0

q n Pt (S n X) =

(1 + tq)b1 (X) (1 + t3 q)b3 (X) . (1 − q)b0 (X) (1 − t2 q)b2 (X) (1 − t4 q)b4 (X)

One sees that this picks the term m = 1 in (6.2). Exercise 6.4. Prove Macdonald’s formula (6.3). Hint: H ∗ (S n X, Q) is Sn -invariant part of H ∗ (X n , Q), where the action is given by the permutation of factors (twisted by a sign for odd degrees). 6.1. Results on intersection cohomology For an algebraic variety X, we consider the derived category of complexes of sheaves on X of Q-vector spaces with bounded constructible cohomology. We denote by QX the constant sheaf on X, regarded as a complex concentrated in degree 0. For irreducible X and a local system L defined on a Zariski open set, we denote by IC(X, L) the intersection cohomology complex with coefficients in L (see [32, 7]). When L is trivial, we simply denote by IC(X). If X is nonsingular, we have IC(X) ∼ = QX [dim X]. 65

66

´ POLYNOMIALS OF HILBERT SCHEMES (2) 6. POINCARE

Lemma 6.5 ([10, Proposition 1.4],[31, Proposition 3]). Let G be a finite group acting on a nonsingular irreducible variety X. Then IC(X/G) = QX/G [dim X]. This lemma is equivalent to saying that X/G is a rational homology manifold. It holds because we have the Poincar´e duality isomorphism for the cohomology groups with rational coefficients on X/G. Lemma 6.6 ([31, Lemma 1]). Let κ : X → Y be a finite birational morphism between irreducible varieties. Then κ∗ IC(X) = IC(Y ). Let π : Z → Y be a proper projective morphism between algebraic varieties. Suppose that YS decomposes into a finite number of irreducible nonsingular subvarities, called strata: Y = y Oy . Here y denotes a distinguished point in Oy . We further assume the following condition: For each stratum Oy , the restriction of π to π −1 (Oy ) is a topological fiber bundle with base Oy and fiber π −1 (y). Definition 6.7. The map π is called semismall , if it satisfies 2 dim π −1 (y) ≤ codim Oy

for any stratum Oy . We say a stratum Oy is relevant for π, if the equality holds. −1

Consider the top degree cohomology group H 2 dim π (y) (π −1 (y), Q) of the fiber π −1 (y), and denote it by Vy . It has a basis dual to the irreducible components of π −1 (y). The fundamental group π1 (Oy ) of the stratum Oy acts on Vy by monodromy. The action is simply the permutation of irreducible components. Let M Vy = V(y,φ) ⊗ Rφ be the irreducible decomposition of Vy as a π1 (Oy )-module. Here Rφ is an irreducible module. We denote by Lφ the local system on Oy given by the representation φ of π1 (Oy ). Then we have the following formula:

Theorem 6.8 (Decomposition theorem for a semismall morphism [10, Theorem 1.7]). Let π : Z → Y be a semismall projective morphism. Assume that Z is nonsingular. Then M π∗ QZ [dim Z] = IC(Oy , Lφ ) ⊗ V(y,φ) , (y,φ)

where the summation runs through all relevant strata Oy and irreducible representations of π1 (Oy ). (Here “ ” denotes the closure. )

This result is deduced from the decomposition theorem of Beilinson, Bernstein, Deligne and Gabber [7], which has been proved via the ℓ-adic intersection cohomology of varieties over fields of positive characteristic. Another proof was given by M. Saito’s mixed Hodge modules [75], which is also very deep. For the application of Theorem 6.8 to the Springer representation of the Weyl group, we refer to [10]. Another interesting application to the representation theory was given in [12].

6.2. PROOF OF THE FORMULA

67

6.2. Proof of the formula Let X be a quasi-projective nonsingular surface and X [n] the Hilbert scheme of n-points on X. Recall that the n-th symmetric product S n X has a stratification [ Sνn X S nX = ν

indexed by partitions ν of n (1.6). We need a study of each stratum Sνn . A partition ν = (ν1 ≥ · · · ≥ νk ) of n has another presentation written by ν = α1 α2 (1 2 · · · nαn ) where αi = #{l | νl = i}. In this notation, we allow αi = 0. Then, we define S ν X = S α1 X × S α2 X × · · · × S αn X.

Let κ = κν : S ν X → Sνn X be a morphism defined by n X κ(C1 , . . . , Cn ) = iCi for Ci ∈ S αi X. i=1

Then it is easy to see the following

Lemma 6.9. κ is finite and birational. It induces an isomorphism between   α1 αn S(1,...,1) X × · · · × S(1,...,1) X \ D

and Sνn X. Here

D = {(C1 , . . . , Cn ) | Supp Ci ∩ Supp Cj 6= ∅ for some i 6= j}. In particular, we have dim Sνn X = 2 where l(ν) =

P

i

X

αi = 2l(ν),

i

αi is the length of the partition ν.

[n] n Lemma 6.10. (1) The Hilbert-Chow S n morphism π : X → S X is semismall with ren spect to the stratification S X = ν Sν X (1.6). (2) The fiber π −1 (C) is irreducible for any C ∈ S n X.

Proof. (1) Let C be a 0-cycle contained in Sνn X for ν = (ν1 ≥ ν2 ≥ · · · ≥ νk ), i.e. C=

k X i=1

Then (6.11)

νi [xi ]

(xi 6= xj for i 6= j).

π −1 (C) ∼ = π −1 (ν1 [x1 ]) × · · · × π −1 (νk [xk ]),

where π −1 (νi [xi ]) in the right hand side is the inverse image of νi [xi ] ∈ S νi X under the Hilbert-Chow morphism π : X [νi ] → S νi X. (We should write πνi in order to emphasize

68

´ POLYNOMIALS OF HILBERT SCHEMES (2) 6. POINCARE

maps π appeared in the factors of the right hand side are different. But we simply denote them by π for simplicity.) More generally,   νk ν1 −1 n −1 −1 ∼ π (Sν X) = π (S(ν1 ) X) × · · · × π (S(νk ) X) \ D ′

where (νi ) is the partition of νi consisting of the single integer νi , and D ′ is the subvariety {(Z1 , . . . , Zk ) | Supp Zi = Supp Zj for some i 6= j}. Since the dimension of π −1 (νi [xi ]) is νi − 1 by Theorem 5.11, the dimension of π −1 (C) is equal to −1

dim π (C) =

k X i=1

(νi − 1) = n − k = n − l(ν).

On the other hand, the codimension of the stratum Sνn X is codim Sνn X = 2n − 2k = 2(n − l(ν)). Hence we have 2 dim π −1 (C) = codim Sνn X. Moreover, it is clear that π|π−1 (Sνn X) : π −1 (Sνn X) → Sνn X is a topological fiber bundle −1 n n n X) : π because the corresponding assertion holds for π|π−1 (S(n) (S(n) )X → S(n) X∼ = X. n −1 (2) We already know S(n) X is irreducible (see Theorem 5.11). Then π (C) for general C is irreducible by (6.11). Now we can apply Theorem 6.8. We have M π∗ QX [n] [dim X [n] ] = Vν,φ ⊗ IC(Sνn X, Lφ ). (ν,φ)

But we know that the fiber π −1 (C) is irreducible. Hence Vν,φ is nontrivial only when φ is trivial. In other words, we have Vν = Vν,trivial . We also have IC(Sνn X) = κ∗ IC(S ν X) by Lemma 6.6. On the other hand, since S ν X has only quotient singularities by finite groups, we have IC(S ν X) = QS ν X [dim S ν X] = ⊠ QS αi X [dim S αi X] i

by Lemma 6.5. Combining all these formulae, we finally get M (6.12) H i+2n (X [n], Q) = H i+2l(ν) (S ν X, Q). ν

6.2. PROOF OF THE FORMULA

69

Now let us compute the generating function of the Poincar´e polynomials. Let an (t) be the coefficients of q n in (6.3). That is the Poincar´e polynomial of S n X. Then, we get ∞ X X X q n Pt (X [n] ) = qn t2(n−l(ν)) Pt (S ν X) n=0

n

=

XX n

= = Thus we have proved (6.2).

ν

XX n ∞ Y

ν

ν

q α1 +2α2 +···+nαn t2(α2 +2α3 +···+(n−1)αn ) aα1 (t) · · · aαn (t) aα1 (t)q α1 aα2 (t)(t2 q 2 )α2 · · · aαn (t)(t2(n−1) q n )αn

(1 + t2m−1 q m )b1 (X) (1 + t2m+1 q m )b3 (X) . (1 − t2m−2 q m )b0 (X) (1 − t2m q m )b2 (X) (1 − t2m+2 q m )b4 (X) m=1

Remark 6.13. Using M. Saito’s mixed Hodge modules [75] instead of intersection cohomology complex, one can compute the generating function of Hodge numbers. (See [31] for detail.)

CHAPTER 7

Hilbert scheme on the cotangent bundle of a Riemann surface In this chapter, we study the Hilbert scheme of points on the cotangent bundle of a Riemann surface. In this case, there exists a natural holomorphic symplectic structure and a C∗ action on the Hilbert scheme, and this gives us an alternative method to investigate the topology of the Hilbert scheme. Using this strategy, we rederive G¨ottsche’s formula for the Poincar´e polynomials in this case. Our method is essentially the same as the one given by Morse theory used in Chapter 5. The main difference is that the fixed points are not necessarily isolated. Moreover, each component of the fixed point set of C∗ -action defines a Lagrangian variety which is essentially the closure of the unstable manifold. This plays an important role in Chapter 9. There exists close analogy between the Hilbert scheme of points on the cotangent bundle of a Riemann surface and the moduli space of Higgs bundle on a Riemann surface. This analogy and the actual relation recently found by Hurtubise([40]) is explained in the end of this chapter. 7.1. Morse theory on holomorphic symplectic manifolds In Chapter 5, we have studied Morse theory on a symplectic manifold X given by an action of a compact torus T . As noted there, when X is a K¨ahler manifold, the gradient flow is given by the associated holomorphic action of the complexification T C of T . Hence, the stable and the unstable manifolds can be expressed purely in terms of the group action. The Hilbert scheme of points on the cotangent bundle of a Riemann surface has a natural holomorphic symplectic structure together with a natural C∗ -action. In this case, the unstable manifold is very important since it becomes a Lagrangian submanifold. The same kind of situation appears in many cases, for example when one studies the moduli space of Higgs bundle or the quiver varieties [62], and it is worth explaining this point before studying the specific example. Let X be a K¨ahler manifold with a holomorphic symplectic form ωC . Suppose there exists a C∗ -action on X with the property that ψt∗ ωC = tωC for t ∈ C∗ , where we denote the C∗ -action on X by ψt : X → X. Let Cν be a connected component of the fixed point set of the C∗ -action, and consider the subset Lν defined by Lν = {x ∈ X | a limit lim ψt (x) exists and contained in Cν }. t→∞

This is the unstable manifold Wν− for the appropriate choice of ξ ∈ t in Chapter 5. 1 2

Proposition 7.1. The manifold Lν is Lagrangian, i.e. ωC |Lν = 0 and dimC Lν = dimC X. In particular, the index at Cν is equal to dimC X − 2 dimC Cν . 70

7.2. HILBERT SCHEME OF T ∗ Σ

71

Proof. For x ∈ Cν , the tangent space Tx X is a C∗ -module. Let M Tx X = H(m) m∈Z

be the weight decomposition. Since

tωC (u, v) = ψt∗ ωC (u, v), H(m) and H(1 − m) is the dual space to each other. Hence we have X X 1 dimC H(m) = dimC X. dimC H(m) = 2 m>0 m≤0

Since X is a K¨ahler manifold, the exponential map gives a local C∗ -equivariant isomorphism Lbetween open neighborhoods V of 0 ∈ Tx X and U of x ∈ X. Let x ∈ Cν and Tx X = the weight decomposition. By the above isomorphism, Lν ∩ U m∈Z H(m) beL can be identified with m≤0 H(m) ∩ V . It follows that Lν ∩ U is a smooth submani1 fold of dimension 2 dimC X, and that for y ∈ Lν ∩ U and u ∈ Ty Lν there exists a limit limt→∞ (ψt )∗ u. Since t(ωC )y (u, v) = (ψt∗ ωC )y (u, v) = (ωC )ψt (y) ((ψt )∗ u, (ψt )∗ v), there exists a limit limt→∞ t(ωC )y (u, v) for y ∈ Lν ∩ U and u, v ∈ Ty Lν . This is possible only when (ωC )y (u, v) = 0. Therefore we have ωC |Lν ∩U = 0. This shows that Lν is a Lagrangian submanifold in a neighborhood of Cν . Let y be an arbitrary point in Lν . Since there exists a limit limt→∞ ψt (y) ∈ Cν , ψt maps C∗ -equivariantly a neighborhood U ′ of y into a neighborhood U considered above for sufficiently large t ∈ R. This shows that Lν ∩ U ′ is also a smooth submanifold of dimension 1 dimC X. Moreover, since ψt∗ ωC = tωC , it follows that Lν is a Lagrangian submanifold 2 also near y. Remark 7.2. In the above argument, we assumed that the C∗ -action on a holomorphic symplectic K¨ahler manifold X satisfies ψt∗ ωC = tωC for t ∈ C∗ . This is possible only when X is non-compact. The reason is as follows. If X is compact there exists a critical manifold corresponding to the maximum of the Morse function for which the unstable manifold is open submanifold of X, but this contradicts the propostion which asserts that every unstable manifold is Lagrangian. 7.2. Hilbert scheme of T ∗ Σ Now we shall study the Hilbert scheme of points on the cotangent bundle of a Riemann surface. Let Σ be a Riemann surface and T ∗ Σ its cotangent bundle. There exists a natural holomorphic symplectic form ωC on T ∗ Σ. The multiplication by a complex number on each fiber gives a natural C∗ -action on T ∗ Σ, and with respect to this action we have ψt∗ ωC = tωC for t ∈ C∗ , where we denote the action of t by ψt : T ∗ Σ → T ∗ Σ. As explained in Theorem 1.10, the Hilbert scheme T ∗ Σ[n] inherits a holomorphic symplectic form and a C∗ -action, denoted by ΩC and Ψt respectively. Since the holomorphic symplectic form on T ∗ Σ[n] is essentially the one on S n (T ∗ Σ) pulled back by the Hilbert-Chow morphism π : T ∗ Σ[n] → S n (T ∗ Σ), it follows that Ψ∗t ΩC = tΩC for t ∈ C∗ .

72

7. HILBERT SCHEME ON THE COTANGENT BUNDLE OF A RIEMANN SURFACE

Now we shall determine the fixed point set in T ∗ Σ[n] . Since T ∗ Σ is locally isomorphic to T ∗ C, it is sufficient to consider the case when Σ = C. Under the identification T ∗ C ∼ = C2 , we have   I is an ideal, ∗ [n] ∼ (T C) = I ⊂ C[z, ξ] , dimC C[z, ξ]/I = n

on which C∗ acts by ψt (z, ξ) = (z, tξ) for t ∈ C∗ . Let D be a Young diagram of weight n. We denote by νi (resp. νj′ ) the number of boxes in the i-th column (resp. in the j-th row), so that for the diagram of Figure 7.1, we have ν1 = 4, ν2 = 3, ν3 = 1 and ν1′ = 3, ν2′ = 2, ν3′ = 2, ν4′ = 1. (Note that our convention differs from one used in [54]. Our diagram is rotated by π/2 from one used in [54]. ) In this way, a Young diagram corresponds to a partition ν = (ν1 , ν2 , . . . ) and its conjugate partition ν ′ = (ν1′ , ν2′ , . . . ). ξ4 zξ 3 z2 ξ 2 z2 ξ z3 Figure 7.1. Young diagram and ideal Definition 7.3. For a Young diagram D and z0 ∈ C, we define the ideal ID,z0 generated by the elements (z − z0 )j−1 ξ νj , j ∈ Z>0 , i.e., where k = ν1′ .

ID,z0 = (ξ ν1 , (z − z0 )ξ ν2 , . . . , (z − z0 )k−1 ξ νk , (z − z0 )k ),

For example, ID,0 = (z 3 , z 2 ξ, z 2 ξ 2, zξ 3 , ξ 4) for the Young diagram in Figure 7.1. It is ∗ easy to see that ID,z0 ∈ ((T ∗ C)[n] )C . Conversely we have the following proposition. Proposition 7.4. Each element I ∈ ((T ∗ C)[n] )C can be uniquely expressed as ∗

I = ID1 ,z1 ∩ · · · ∩ IDk ,zk for some distinct z1P , . . . , zk ∈ C and for some Young diagrams D1 , . . . , Dk of weight n1 , . . . , nk satisfying kj=1 nj = n. Proof. Let I ∈ (T ∗ C)[n] be a fixed point of the C∗ -action. There exist distinct points z1 , z2 , . . . , zk ∈ C such that I can be expressed as I = I1 ∩ · · · ∩ Ik ,

7.2. HILBERT SCHEME OF T ∗ Σ

73

where each Ij , j = 1, 2, . . . , k is generated by elements of the form (z − zj )m ξ n with m, n ∈ P Z≥0 . If we write nj = dimC C[z, ξ]/Ij , we have n = kj=1 nj , since C[z, ξ]/I ∼ =

k M

C[z, ξ]/Ij .

j=1

Let us write a box at (m, n) ∈ R2 if a monomial (z − zj )m ξ n is not an element of Ij . Since Ij is an ideal, the number of boxes must decrese in low and in column. Hence we get a Young diagram Dj of weight nj , and Ij must be IDj ,zj . Therefore we have I = ID1 ,z1 ∩ ID2 ,z2 ∩ · · · ∩ IDk ,zk . By the proposition, we can express an element of the fixed point set as in Figure 7.2: D1

z1

D2

z2

D3

z3

Σ = C ⊂ T ∗C

Figure 7.2. Young diagrams on a curve Suppose I = ID1 ,z1 ∩ ID2 ,z2 is an element of ((T ∗ C)[n] )C . What happens when z1 and ′ z2 collide ? Since IDj ,zj , j = 1, 2 is generated by the elements (z − zj )(νj )l ξ l−1 for l ∈ Z>0 , ′ ′ I is generated by the elements (z − z1 )(ν1 )l (z − z2 )(ν2 )l ξ l−1, which converge to the elements ′ ′ (z − z1 )(ν1 )l +(ν2 )l ξ l−1 when z1 collides with z2 . Therefore if we denote by D1 ∪ D2 the Young diagram which has (ν1 )′l + (ν2 )′l boxes in the l-th row, we have limz2 →z1 ID1 ,z1 ∩ ID2 ,z2 = ID1 ∪D2 ,z1 . (See Figure 7.3.) This shows that the connected component is determined by D = D1 ∪ · · · ∪ Dk . For fixed D of weight n, the open strata consists of elements of the form I = ID1 ,z1 ∩ · · · ∩ IDk ,zk where Dj , j = 1, . . . , k has only one column and D1 ∪ · · · ∪ Dk = D, as depicted in Figure 7.4. Notice that if D has only one column with n boxes, we have ∗

ID,z0 = ((z − z0 ), (z − z0 )ξ, . . . , (z − z0 )ξ n−1, ξ n ) = ((z − z0 ), ξ n )

= {f ∈ C[z, ξ] | f (z0 , 0) = ∂ξ f (z0 , 0) = · · · = ∂ξn−1 f (z0 , 0) = 0}.

Hence ID,z0 corresponds to the points at (z0 , 0) vanishing in the fiber direction with multiplicity n. If we regard (z, ξ) as a local coordinate in T ∗ Σ, we can define an ideal ID,p of OT ∗ Σ for a Young diagram D and p ∈ Σ, i.e. ID,p = (ξ ν1 , zξ ν2 , . . . , z k−1 ξ νk , z k ),

74

7. HILBERT SCHEME ON THE COTANGENT BUNDLE OF A RIEMANN SURFACE

D1 ∪ D2

D2 D1

z2 → z1 z1

z2

z1

Figure 7.3. Collision of z1 and z2 Dk D2

D3

D1

z1

···

z2

z3

···

zk

Figure 7.4. open stratum where z is a local coordinate around p such that z(p) = 0, and ξ is the associated coordinate on Tp∗ Σ. This ideal does not depend on the choice of a local coordinate (z, ξ). Let ν be the partition of n corresponding to the Young diagram D. We write ν = (1α1 2α2 . . . nαn ), where αi denotes the number of times the integer i occurs as a part. Then the above argument shows that an element of the component of the fixed point set corresponding to D consists of α1 single-points, α2 double points, . . . , and αn multiple points of multiplicity n all sitting at the zero section Σ and vanishing in the fiber direction. A typical example is shown in Figure 7.5. We have shown the following proposition. Proposition 7.5. The fixed point set of C∗ action on T ∗ Σ[n] is given by a ∗ (T ∗ Σ[n] )C = S ν Σ, ν: partition of n

where S ν Σ = S α1 Σ × S α2 Σ × · · · × S αn Σ for the partition ν = (1α1 2α2 . . . nαn ). Now we shall calculate the Poincare polynomial. In order to apply the general argument of Morse theory, we need to find an appropriate S 1 -invariant K¨ahler metric. Though we believe that there exists a natural K¨ahler metric on T ∗ Σ[n] , we do not know how to construct

7.2. HILBERT SCHEME OF T ∗ Σ

75

.. . ···

··· | {z }

.. .

k boxes

> > > > > > > > ;

···

| {z }

α1 times

···

9 > > > > > > > > =

| {z }

α2 times

αk times

Figure 7.5. a point in S ν Σ for ν = (1α1 2α2 . . . nαn )

it at present moment. (We could produce a hyper-K¨ahler metric on (T ∗ P1 )[n] using the ADHM description on the ALE space T ∗ P1 .) Therefore we shall take another route. Suppose first that Σ = C. In this case there exists a natural K¨ahler metric on (T ∗ C)[n] since it is obtained by the hyper-K¨ahler quotient construction. In the notation of the 2 ADHM description, we may take f ([(BP 1 , B2 , i)]) = ||B2 || As shown in Proposition 5.13, ν the critical value is given by f (S C) = i (i − 1)νi . Now we shall consider the general case. As remarked before, we may define “the stable manifold” Wν+ and “the unstable manifold” Wν− by using the C∗ -action, i.e. Wν+ = {Z ∈ T ∗ Σ[n] | there exists a limit lim ψt (Z) ∈ S ν Σ} t→0

Wν−

∗

[n]

= {Z ∈ T Σ

| there exists a limit lim ψt (Z) ∈ S ν Σ}. t→∞

P Let U1 , . . . , Uk be disjoint open sets in Σ, and n1 , . . . , nk ∈ N satisfying i ni = n. Then S n1 (T ∗ U1 ) × · · · × S nk (T ∗ Uk ) ⊂ S n (T ∗ Σ), and there exists an isomorphism π −1 (S n1 (T ∗ U1 ) × · · · × S nk (T ∗ Uk )) ∼ = (T ∗ U1 )[n1 ] × · · · × (T ∗ Uk )[nk ] ,

preserving C∗ -action and holomorphic symplectic structures. It follows that T ∗ Σ[n] is locally a K¨ahler manifold, and that Wν− is a Lagrangian submanifold. Motivated by the special case of Σ = C, we define “the critical value at S ν Σ” to be P aν = i (i − 1)νi . The same argument as in §5.1 shows the perfectness of the stratification S S X = ν Wν+ if one can show Wν+ \ Wν+ ⊂ aν
where l(ν) =

P

ν

αi is the length of the partition ν = (1α1 2α2 . . . nαn ).

76

7. HILBERT SCHEME ON THE COTANGENT BUNDLE OF A RIEMANN SURFACE

Proof. By the argument above, we only need to calculate the index of the fixed point set S ν Σ. Since S ν Σ is a Lagrangian submanifold, we have   1 ν ∗ [n] ν ind(S Σ) = 2 dimC T Σ − dimC S Σ 2 X = 2(n − αk ) = 2(n − l(ν)). k

Using the above lemma, we can deduce G¨ottsche’s formula in the case of T ∗ Σ. Corollary 7.7. ∞ X

∗

[n]

n

Pt (T Σ )q =

n=0

∞ Y

(1 + t2d−1 q d )b1 (Σ) . 2d−2 q d )b0 (Σ) (1 − t2d q d )b2 (Σ) (1 − t d=1

Remark 7.8. In order to compute the Poincare polynomial, we do not actually need the holomorphic symplectic form on T ∗ Σ. The fact that it is locally isomorphic to T ∗ C is enough. Hence the above argument holds and shows G¨ottsche’s formula also for the case of the total space of a holomorphic line bundle over Σ, not necessarily T ∗ Σ. The only difference is that “the unstable manifold” Wν− becomes Lagrangian in the case of T ∗ Σ. As remarked above, we may define “the unstable manifold” Wν− of the fixed point set S ν Σ for the total space of a holomorphic line bundle over Σ. More generally, let X be a surface containing a curve Σ. By identifying a tubular neighborhood of Σ with the total space of the normal bundle of Σ in X, we can define “the unstable manifold” Wν− . In fact, this can be defined intrinsically as follows. Let π : X [n] → S n X be the Hilbert-Chow morphism. We define L∗ Σ =

∞ a

{J ∈ X [n] | Supp(O/J ) ⊂ Σ}.

n=0

Then we have the stratification L∗ Σ =

[

Wν− .

ν:partition

The irreducible component of L∗ Σ is obtained by taking the closure of Wν− . Namely, we define (7.9)

Lν Σ = {J ∈ X [n] |π(J ) ∈ Sνn Σ},

P where Sνn Σ = { i νi [xi ] ∈ S n X|xi ∈ Σ, and xi ’s are distinct }. The subvariety Lν Σ was first considered by Grojnowski [33]. It defines a middle dimensional homology class [Lν Σ] ∈ H2n (X [n] ) and plays an important role in Chapter 9.

7.3. ANALOGY WITH THE MODULI SPACE OF HIGGS BUNDLES

77

7.3. Analogy with the moduli space of Higgs bundles The Hilbert scheme T ∗ Σ[n] is analogous in many ways with the moduli space of Higgs bundle over a Riemann surface introduced by Hitchin [36]. Let Σ be a compact Riemann surface. The moduli space NΣ of Higgs bundle over Σ is defined by   E : holomorphic vector bundle over Σ,    NΣ = (E, Φ) Φ ∈ H 0 (Σ, End(E) ⊗ KΣ ) isomorphism .     stability condition We shall not explain what the stability condition means. The interested reader may consult the original papers [36, 37] for the precise definition and the properties of NΣ mentioned in the following. Now we shall explain how much T ∗ Σ[n] looks like NΣ . In both cases, there exists a holomorphic symplectic form ωC and a C∗ -action which satisfies ψt∗ ωC = tωC for t ∈ C∗ . The holomorphic symplectic form on NΣ is given as follows. For (E, Φ) ∈ NΣ , the tangent space of NΣ at (E, Φ) is given by T(E,Φ) NΣ = H 1 (Σ, End(E)) ⊕ H 0 (Σ, End(E) ⊗ KΣ ).

By the Serre duality, there exists a natural non-degenerate pairing between H 1 (Σ, End(E)) and H 0 (Σ, End(E) ⊗ KΣ ), and this defines a holomorphic symplectic form on NΣ . The C∗ -action on NΣ is given by ψt (E, Φ) = (E, tΦ) for t ∈ C∗ .

From the definition, it follows that ψt∗ ωC = tωC . Let MΣ be the moduli space of stable vector bundles over Σ. For E ∈ MΣ , the tangent space of MΣ at E is given by TE MΣ = H 1 (Σ, End(E)).

As noted above, the cotangent space is given by TE∗ MΣ = H 0 (Σ, End(E) ⊗ KΣ ). This shows that NΣ looks roughly like T ∗ MΣ . These are not exactly the same since E may not be stable for (E, Φ) ∈ NΣ . On the other hand we have shown that S n Σ ⊂ T ∗ Σ[n] . Since S n Σ is Lagrangian, the normal bundle is identified with T ∗ (S n Σ). Moreover T ∗ (S n Σ) is dense in T ∗ Σ[n] since it is identified with the stable manifold of S n Σ. Hence we may say that T ∗ Σ[n] looks like S n (T ∗ Σ) but not exactly. Now let us consider the fixed point set of the C∗ -action. In the case of NΣ , the obvious component of the fixed point set is MΣ which corresponds to the set of a point of the form (E, 0) ∈ NΣ . There exist other components, and this shows that NΣ is not exactly the same as T ∗ MΣ . These components are described as follows. By definition (E, Φ) ∈ NΣ is a fixed point if and only if there exists an isomorphism between (E, Φ) and (E, tΦ). This

78

7. HILBERT SCHEME ON THE COTANGENT BUNDLE OF A RIEMANN SURFACE

is possible even when Φ 6= 0 if E decomposes as

E = E 1 ⊕ · · · ⊕ Ek ,

since the automorphism of E = E1 ⊕ · · · ⊕ Ek given by t idE1 ⊕ · · · ⊕ tk idEk defines the isomorphism between (E, Φ) and (E, tΦ) for Φ of the form   0 ∗ 0    ..   . 0 ∗ Φ= .  .  .. ... 0   0 ∗ 0 The situation is very similar in the case of T ∗ Σ[n] . There exists an obvious component Σ[n] = S n Σ of the fixed point set which corresponds to the partition ν = (1, 1, . . . , 1). | {z } n times

Besides this component, there also exist other components S ν Σ corresponding to other partitions of n, as shown in Proposition 7.5. Now the importance of NΣ lies in the fact that it is related with an integrable system. Hitchin shows that there exists a fibration rank ME p : NΣ → H 0 (Σ, KΣ⊗i ) Lr

i=1

r−i given by p(E, Φ) = in det(λ idE −Φ)) with r = rank E, and i=1 ( the coefficient of λ that this gives an integrable system on NΣL . This means that if we express p as p = E (p1 , . . . , pN ) with respect to a linear basis of rank H 0 (Σ, KΣ⊗i ), pi ’s satisfy the following i=1 conditions. ( {pi , pj } = 0, for i, j = 1, . . . , N dp1 ∧ · · · ∧ dpN is nowhere vanishing on NΣ ,

where { , } is the Poisson bracket on the space of holomorphic functions on NΣ defined by the holomorphic symplectic form. Roughly speaking this follows from the fact that pi ’s depend only on Φ which is just “the half of the canonical coordinate”. The function pi is called Hitchin’s Hamiltonian and plays an important role for an approarch to the geometric Langlands conjecture. The D-modules on the moduli stack of vector bundles given by the geometric Langlands (=Hecke eigensheaf) are supposed to have characteristic varieties contained in p−1 (0) (=Laumon’s global nilpotent cone). (Interested reader should glance at Juten Lecture Note no.12 (in Japanese).) In our situation, p−1 (0) corresponds to L∗ Σ. Now it seems to be natural to ask whether T ∗ Σ[n] is related with an integrable system. In fact, the relation between NΣ and T ∗ Σ[n] was recently found by Hurtubise [40]. He shows that there exists a symplectic equivalence over an open set of NΣ and T ∗ Σ[n] . We shall only sketch his construction and the reader may consult the original papers [40, 37] for the precise statement.

7.3. ANALOGY WITH THE MODULI SPACE OF HIGGS BUNDLES

79

Lr ⊗i 0 Let π : T ∗ Σ → Σ be the projection. For a = (ai ) ∈ i=1 H (Σ, KΣ ), we define a section sa ∈ H 0 (T ∗ Σ, π ∗ KΣ⊗r ) by sa = λr + a1 (x)λr−1 + · L · · + ar (x), where λ is the ∗ ⊗r tautological section of π KΣ . Let us take an open subset V of ri=1 H 0 (Σ, KΣ⊗i ) such that s−1 a (0) is non-singular and reduced. We consider the family of “spectral curves” S → V given by S = {(p, a) ∈ T ∗ Σ × V | sa (p) = 0}. The fiber of S is a smooth curve of genus g ′ = r 2 (g−1)+1 by the adjunction formula, where g is the genus of Σ. Notice that g ′ is also equal to the dimension of V . Now consider the fiber′ wise symmetric product S g S → V . For each fiber Sa = Sa ⊂ T ∗ Σ, there exists a natural in′ ′] clusion S n (Sa ) ֒→ T ∗ Σ[n] . This defines a map π : S g S → T ∗ Σ[g L . Suppose p1 , . . . , pg′ ∈ T ∗ Σ r ⊗i 0 be distinct points, then generically there exists unique a ∈ i=1 H (Σ, KΣ ) such that sa (pi ) = 0 for i = 1, . . . , g ′. This shows that φ is an isomorphism on an open dense set. other hand, there exists a nice description of a generic fiber of p : NΣ → Lr On the ⊗i 0 i=1 H (Σ, KΣ ) due to Hitchin. The idea is as follows. Suppose p(E, Φ) = a, then sa = det(λ idE −Φ) by definition. Hence Sa can be considered as a set of eigenvalues of Φ, and this is the reason why Sa is called the spectral curve. From this, one sees that the fiber p−1 (a) consists of a pair (E, Φ) such that the eigenvalues of Φ is Sa . It is also natural to consider the eigenspaces of Φ. According to [40], we consider the cokernel of λ idE −Φ instead of the kernel. For a ∈ V , this induces a line bundle L over Sa . Conversely, if we have a line bundle L over Sa for a ∈ V . we can associate a vector bundle E = π∗ L over Σ. Moreover, the multiplication by each element p ∈ Sa ⊂ T ∗ Σ defines Φ : E → E ⊗ KΣ . The relation between the degrees of E and L is given by deg(L) = deg(E) + (r 2 − r)(g − 1) + 1. This shows that the generic fiber of p is the Jacobian of a spectral curve Sa . Let J → V be the family of Jacobians associated with a family S → V . Then by the above argument there exists an inclusion φ : J → NΣ which preserves each fiber. Suppose we have a section of J → V which specifies the base point of a fiber Ja = Ja . ′ Then we have the Abel-Jacobi map I : S g S → J which is birational. Now such a section exists at least locally, and we have the following maps. ′

φ

′

I

ψ

T ∗ Σ[g ] ←− S g S −→ J −→ NΣ .

′

This gives an isomorphism between dense open sets of T ∗ Σ[g ] and NΣ , at least locally. Moreover, it is shown in [40] that this isomorphism preserves symplectic structures.

CHAPTER 8

Homology group of the Hilbert schemes and the Heisenberg algebra In this chapter, we shall review the author’s work on the homology group of the Hilbert scheme [68]. It started from an observation that G¨ottsche’s formula (6.2) for the Poincar´e polynomial is the same as the character formula for the representation of a product of the Heisenberg algebra and the Clifford algebra. (The author learned this observation in Vafa-Witten’s paper [79].) It motivated the author to construct a representation of the Heisenberg/Clifford algebra in a geometric term. The similar result was also obtained by Grojnowski [33]. 8.1. Heisenberg algebra and Clifford algebra First we have a brief review on Heisenberg algebra and Clifford algebra and their representations. The general reference is [43, §9.13] The infinite dimensional Heisenberg algebra s is a Lie algebra generated by p[m] (m ∈ Z \ {0}) and K with the following relations     (8.1) p[m], p[n] = mδm+n,0 K, p[m], K = 0. For every a ∈ C∗ , the Lie algebra s has an irreducible representations on the space R = C[p1 , p2 , . . . ] of polynomials in infinitely many indeterminates pi defined by ( am ∂p∂m (m > 0) p[m] 7→ , K 7→ a id . p−m (m < 0)

This representation has a highest weight vector 1, and R is spanned by elements pj11 pj22 · · · pjnn = p[−1]j1 p[−2]j2 · · · p[−n]jn · 1. We extend s by a derivation d0 defined by   d0 , p[m] = −mp[m].

The above representation R can be extended by d0 7→

X

mpm

m

80

∂ . ∂pm

8.2. CORRESPONDENCES

81

Then it is easy to check the character formula ∞ X Y d0 def. i q dim{v ∈ R | d0 v = iv} = (8.2) trR q =

1 . 1 − qj j=1

i

Next let us review on the infinite dimensional Clifford algebra Cl. (See [19].) It is generated by p[m](m ∈ Z \ {0}) and K with the following relation: {p[m], p[n]} = p[m]p[n] + p[n]p[m] = |m|δm+n,0 K. V This algebra has a representation on the exterior algebra F = ∗ of an infinite dimensional vector space V = Cdp1 ⊕ Cdp2 ⊕ · · · defined by ( (m > 0) am ∂p∂m , K 7→ a id, p[m] 7→ dp−m ∧ (m < 0) (8.3)

where a ∈ C∗ and denotes the interior product. This has the highest weight vector 1 and spanned by elements dpi1 ∧ · · · ∧ dpin = p[−i1 ] · · · p[−in ] · 1,

We extend Cl by d0 defined by It acts on F by

  d0 , p[m] = −mp[m]

d0 (dpi1 ∧ · · · ∧ dpin ) = The character is given by (8.4)

(i1 > i2 > · · · > in ).

trF q

d0

=

X  ik dpi1 ∧ · · · ∧ dpin

∞ Y

(1 + q j ).

j=1

Remark 8.5. In the literature, the defining relation (8.3) for the Clifford algebra is {p[m], p[n]} = δm+n,0 K.

Our convention is adapted for the later construction. In fact, the most natural language is the super-Lie algebra. 8.2. Correspondences Our method to construct a representation of the Heisenberg/Clifford algebra is to use correspondences in the product of the Hilbert schemes. We first briefly recall the theory of correspondences in general manifolds. For more detail and the formulation in Chow rings, the reader should consult Fulton’s book, Chapter 16, [22] or Chriss and Ginzburg’s book [12]. (In this chapter, any homology or cohomology group is assumed to have rational coefficients.) For a locally compact topological space X, let H∗lf (X) denote the homology group of possibly infinite singular chains with locally finite support (Borel-Moore homology). The

82

8. HEISENBERG ALGEBRA

usual homology group of finite singular chains will be denoted by H∗ (X). If X = X ∪ {∞} is the one point compactification of X, we have H∗lf (X) is isomorphic to the relative homology group H∗ (X, {∞}). If X is an n-dimensional oriented manifold, we have the Poincar´e duality isomorphism (8.6) H lf (X) ∼ = H n−i (X), Hi (X) ∼ = H n−i (X), c

i

where H ∗ and Hc∗ denote the ordinary cohomology group and the cohomology group with compact support respectively. Let M1 , M2 be oriented smooth (possibly non-compact) manifolds of dimensions d1 , d2 respectively. Take a submanifold Z in M1 × M2 such that (8.7)

the projection Z → M2 is proper.

Then the fundamental class [Z] defines an element in M (8.8) [Z] ∈ HdimR Z (M1 × M2 , M1 × {∞}) =

i+j=dimR Z

Hi (M1 ) ⊗ Hjlf (M2 ),

where M2 = M2 ∪ ∞ is the one point compactification of M2 and we have used the K¨ unneth formula. More generally, if [Z] is a cycle whose support Z satisfies (8.7), the same construction works. Then by identifying Hilf (M2 )Lwith (Hd2 −i (M2 ))∗ by the Poincar´e duality (8.6), [Z] can be regarded as an element of j Hom(Hd2 −dim Z+j (M2 ), Hj (M1 )). More explicitly, [Z](c) = p1∗ (p∗2 c ∩ [Z]) for c ∈ H∗ (M2 )

where pi : M1 × M2 → Mi (i = 1, 2) are i-th projections, and p∗2 c ∩ means the cap product of the pull-back of the Poincar´e dual of c by p∗2 . When we regard [Z] as an operator on homology groups, we call it a correspondence. When we have a morphism ϕ : M2 → M1 , we can consider its graph graph(ϕ) ⊂ M1 ×M2 as a correspondence. Obviously, the composite of [graph ϕ] and [graph ψ] for ψ : M3 → M2 is [graph(ϕ◦ψ)]. In this sense, correspondences are generalization of morphisms. Moreover, instead of homology groups, if we have appropriate theory which has operators ∩, ( )∗ , ( )∗ , then correspondences can be defined. For example, we can replace homology groups by Chow groups or K-groups. Another example, which should be kept in mind, is the convolution product: if we have a function Z(x, y) on M1 × M2 , we have an operator from functions on M2 to functions on M1 by Z ∞ C (M2 ) ∋ f 7−→ Z(x, y)f (y)dy ∈ C ∞ (M1 ). M2

The cap product, (resp. pushforward ( )∗ ) is changed to the multiplication (resp. integration). The composition of correspondences can be described as follows. Suppose Z ⊂ M1 ×M2 and Z ′ ⊂ M2 × M3 such that the projection Z → M2 and Z ′ → M3 are proper.

8.3. MAIN CONSTRUCTION

83

Hence [Z] and [Z ′ ] are defined, and their composition [Z][Z ′ ] makes sense. Let pij : M1 × M2 × M3 → Mi × Mj be the projection to the product of the i-th and the j-th factors. Since [Z] ∈ H∗ (M1 ) ⊗ H∗lf (M2 ) and [Z ′ ] ∈ H∗ (M2 ) ⊗ H∗lf (M3 ), the intersection p∗12 [Z] ∩ p∗23 [Z ′ ] is an element in H∗ (M1 ) ⊗ H∗ (M2 ) ⊗ H∗lf (M3 ). Taking the push-forward by p13 , we get p13∗ (p∗12 [Z] ∩ p∗23 [Z ′ ]) ∈ H∗ (M1 ) ⊗ H∗lf (M3 ). We can also check that the projection −1 ′ ′ p13 (p−1 12 Z ∩ p23 Z ) → M3 is proper. Then the composition [Z][Z ] is equal to this class: [Z][Z ′ ] = p13∗ (p∗12 [Z] ∩ p∗23 [Z ′ ]).

lf By the intersection product, we have a paring between Hj (Mi ) and Hn−j (Mi ). (Note that this pairing is non-degenerate by the Poincar´e duality.) Then the adjoint [Z]∗ : lf Hdlf1 −j (M1 ) → Hdim Z−j (M2 ) of [Z] is given by

[Z]∗ (c) = p2∗ (p∗1 c ∩ [Z]) for c ∈ H∗lf (M2 ) This is nothing but the homomorphism defined by identifying (Hj (M1 ))∗ with Hdlf1 −j (M1 ) in (8.8). 8.3. Main construction We apply the construction in the previous section introducing certain correspondences in the product of the Hilbert schemes. [n] Let X be a quasi-projective surface, and X Hilbert scheme parametrising n-points ` the [n−i] in X. For i > 0, we introduce cycles P [i] ⊂ n X × X [n] by (8.9)

def.

P [i] =

a n

(J1, J2 ) ∈ X [n−i] × X [n] | J1 ⊃ J2 , Supp(J1 /J2) = {x} for some x ∈ X .

Similarly, we define P [i] for i < 0, exchanging J1 and J2 . Let us define a morphism Π : P [i] → X by (8.10)

Π(J1 , J2 ) = x for (J1 , J2 ) ∈ P [i],

where x is the only one element of Supp(J1 /J2 ). The dimensions of P [i] ∩ (X [n−i] × X [n] ) are given by (8.11)

dimC P [i] ∩ (X [n−i] × X [n] ) = 2n − i + 1.

This is calculated as follows. Suppose i > 0. First note that J1 runs over X [n−i] whose complex dimension is 2(n−i). if we fix J1 , then the set of J2 ∈ X [n] which satisfies J1 ⊃ J2 i X), where and Supp(J1 /J2 ) = {x} for some x ∈ X\Supp(OX /J1) is identified with π −1 (S(i) −1 i X) is π is the Hilbert-Chow morphism. Since π is semismall (Lemma 6.10), dimC π (S(i) equal to i + 1. The variety with x ∈ Supp(OX /J1 ) has smaller dimension (See the proof of Lemma 8.31.) Therefore we have dimC P [i] ∩ (X [n−i] × X [n] ) = 2(n − i) + i + 1. The case i < 0 is similar.

84

8. HEISENBERG ALGEBRA

For α ∈ H∗lf (X) and β ∈ H∗ (X), we define homology classes Pα [i], Pβ [−i] (i > 0) by def.

(8.12)

Pα [i] = Π∗ α ∩ [P [i]] , def.

Pβ [−i] = Π∗ β ∩ [P [−i]] ,

Q where Pα [i], Pβ [−i] ∈ n H∗ (X [n∓i] ) ⊗ H∗lf (X [n] ). Here we have a slight inaccuracy in the definition of Pβ [−i], since P [−i] → X [n] may not be proper. The precise definition is as follows: Consider P [−i] as a cycle in X [n+i] ×Q X [n] × X. Then, the projection to [n] X × X is proper. Hence it defines a class [P [−i]] in Q n H∗ (X [n+i] ) ⊗ H∗lf (X [n] ) ⊗ H∗lf (X). [n] If β ∈ H∗ (X), the cap product Π∗ β ∩ [P [−i]] is in n H∗ (X [n+i] ) ⊗ H∗lf (X ) ⊗ H∗ (X). Q [n+i] [n] Taking a pushforward by the projection onto X ×X , we get a class in n H∗ (X [n+i] )⊗ H∗lf (X [n] ). We denote the resulting class by Π∗ β ∩ [P [−i]] for simplicity, hoping it makes no confusion. L L [n] [n∓i] These cycles Pα [i], Pβ [−i] define correspondences H (X ) → ) by ∗ n n H∗ (X the procedure in §8.2. By the dimension formula (8.11), Pα [i] maps the degree (2n + k)part in H∗ (X [n] ) to the degree (2(n − i) + k + deg α − 2)-part in H∗ (X [n−i] ). Note that the middle degree space (i.e. k = L 0) is preserved when deg α = 2. [n] Next theorem asserts that ∞ n=0 H∗ (X ) becomes a representation space of the Heisenberg/Clifford algebra by Pα [i] and Pβ [−i]. Theorem 8.13 (Nakajima [68], Grojnowski [33]). The following relations hold,   (8.14a) Pα [i], Pβ [j] = (−1)i−1 iδi+j,0 hα, βi id if (−1)deg α·deg β = 1,  (8.14b) Pα [i], Pβ [j] = (−1)i−1 iδi+j,0 hα, βi id otherwise,

where hα, βi = p∗ (α ∩ β) with p : X → pt being the unique morphism from X to the point pt. Remark 8.15. (1) In [68], we could not determine the constant (−1)i−1 i in (8.14). The coefficients (−1)i−1 i was calculated later by Ellingsrud and Strømme [16]. We shall give another proof in Chapter 9. Moreover our definition is slightly different from the original one. The author learned the formulation given here from A. King. (2) Our proof given below works for Chow groups instead of homology groups. Comparing character formula (8.2), (8.4) with G¨ottsche’s formula (6.2), we find L [n] Corollary 8.16. The homology group ∞ n=0 H∗ (X ) is an irreducible highest weight representation as a representation of the Heisenberg/Clifford algebra, where the highest weight vector is the generator of H0 (X [0] ) ∼ = Q. Before proving the theorem, let us consider a basic example and check the relation above. Example 8.17. Fix a point x ∈ X and set i = 1, j = −1, α = [X] and β = [x]. Take distinct points x1 , . . . , xn ∈ X \ {x}, the ideal J1 of functions vanishing at x1 , . . . , xn . We identify J1 with the set of its vanishing points, namely, J1 ∼ = {x1 , . . . , xn }.

8.3. MAIN CONSTRUCTION

85

  Let us calculate Pα [i], Pβ [j] [J1 ]. First P[x] [−1][J1 ] is [{x}∪J1 ]. Then P[X] [1]P[x] [−1][J1 ] is the sum of cycles of the form [({x} ∪ J1) \ {a point}], i.e. P[X] [1]P[x] [−1][J1 ] = [{x1 , . . . , xn }] + = [J1 ] +

n X i=1

n X i=1

[{x} ∪ {x1 , . . . , xbi , . . . , xn }]

[{x} ∪ {x1 , . . . , xbi , . . . , xn }],

where xbi means the element xi is removed. Considering similarly, we have ! n X P[x] [−1]P[X] [1][J1 ] = P[x] [−1] [{x1 , . . . , xbi , . . . , xn }] i=1

=

n X i=1

[{x} ∪ {x1 , . . . , xbi , . . . , xn }].

Thus we conclude  
Pα [i], Pβ [j] [J1 ] = P[X] [1]P[x] [−1] − P[x] [−1]P[X] [1] [J1 ] = [J1 ]. (See Figure 8.1.)

x

P[X] [1]

P[x] [−1] x1 x2 x3 x4

P[X] [1] P[x] [−1]

  Figure 8.1. The proof of P[X] [1], P[x] [−1] [J1 ] = [J1 ] Remark 8.18. When i, j = ±1, the relations hold for arbitrary dimensional X, if we consider the symmetric products of X. Comparing with Macdonald’s formula (6.3),

86

8. HEISENBERG ALGEBRA

L n 0 one finds that n H∗ (S X) is generated by Pα [±1] from 1 ∈ H0 (S X). That is, Macdonald’s formula is the same as the character formula for the product of 1-dimensional Heisenberg/Clifford algebras. This observation was informed to us by A. King. 8.4. Proof of Theorem 8.13 The relations (8.14) will be checked for each summand H∗ (X [n] ). Hence we fix n during the proof. We study the case (a) i, j > 0 or i, j < 0 and the case (b) i > 0, j < 0 or i < 0, j > 0 separately. (a). The case i > 0, j > 0. Let pab : X [n1 ] × X [n2] × X [n3 ] → X [na ] × X [nb ] be the projection to the product of the a-th and the b-th factors. Then Pα [i]Pβ [j] is given by (8.19)

Pα [i]Pβ [j] = p13∗ (p∗12 [P [i]] ∩ Π∗1 α ∩ p∗23 [P [j]] ∩ Π∗2 β),

where we denote by Π1 (resp. Π2 ) the morphism Π in (8.10) for P [i](resp. P [j]). In order to calculate the intersection product, we need to study the set theoretical intersection
def. −1 [n−i−j] P [i, j] = p−1 × X [n−j] × X [n] 12 (P [i]) ∩ p23 (P [j]) ∩ X   Supp(J1 /J2 ) = {x} for some x ∈ X = (J1 , J2 , J3 ) J1 ⊃ J2 ⊃ J3 , . Supp(J2 /J3 ) = {y} for some y ∈ X

−1 We separate P [i, j] into the part with x 6= y and x = y. First, p−1 12 (P [i]) and p23 (P [j]) intersect transversely along P [i, j] ∩ {x 6= y} (see the proof of Claim (1) below), and the dimension is given by

dimC P [i, j] ∩ {x 6= y} = 2(n − i − j) + (i + 1) + (j + 1) = 2n − i − j + 2, def.

which is the expected dimension. Let P [i, j] = P [i, j] ∩ {x 6= y}. Then we have (8.20)

p∗12 [P [i]] ∩ p∗23 [P [j]] = [P [i, j]] + ι∗ R

for some R ∈ H∗lf (P [i, j] ∩ {x = y}), where ι : P [i, j] ∩ {x = y} → P [i, j] is the inclusion. This R comes from the contribution of the intersection along P [i, j] ∩ {x = y}. Similarly, we have (8.21)

Pβ [i]Pα [j] = p13∗ (p∗12 [P [i]] ∩ Π∗1 β ∩ p∗23 [P [j]] ∩ Π∗2 α)

and (8.22)

p∗12 [P [j]] ∩ p∗23 [P [i]] = [P [j, i]] + ι′∗ R′ ,

where R′ ∈ H∗lf (P [j, i] ∩ {x = y}) and ι′ : P [j, i] ∩ {x = y} → P [j, i] is the inclusion. Claim. (1) There exists an isomorphism (8.23)

∼ =

P [i, j] ∩ {x 6= y} − → P [j, i] ∩ {x 6= y},

8.4. PROOF OF THEOREM 8.13

87

which exchanges Π1 and Π2 . (2) We have (8.24)

p13∗ (Π∗1 α ∩ Π∗2 β ∩ ι∗ R) = 0, p13∗ (Π∗1 β ∩ Π∗2 α ∩ ι′∗ R′ ) = 0.

Once this claim is proved, we can see that P [i, j] and P [j, i] cancel out as follows: (8.25)

p13∗ ([P [j, i]] ∩ Π∗1 β ∩ Π∗2 α) = p13∗ ([P [i, j]] ∩ Π∗2 β ∩ Π∗1 α)

= (−1)deg α·deg β p13∗ ([P [i, j]] ∩ Π∗1 α ∩ Π∗2 β).

On the other hand, the terms ι∗ R, ι′∗ R′ in (8.20), (8.22) do not contribute thanks to Claim (2). Hence we have the assertion for the case i, j > 0. The case i, j < 0 can be checked similarly. Proof of Claim. (1) There is a correspondence between (J1 , J2, J3 ) ∈ P [i, j] ∩ {x 6= y} and (J1 , J2′ , J3 ) ∈ P [j, i] ∩ {x 6= y} given by J1 /J2′ = J2 /J3,

J2′ /J3 = J1 /J2.

See Figure 8.2 where the supports of above sheaves are drawn with shaded circles and white circles respectively, and support of OX /J1 is drawn in black circles. (The number of circles represents the multiplicity.) This correspondence gives rise the required isomorphism under which Π1 and Π2 are exchanged.

J2

x

y

J1

J3

J2′ Figure 8.2. Correspondence between J1 ⊃ J2 ⊃ J3 and J1 ⊃ J2′ ⊃ J3

88

8. HEISENBERG ALGEBRA

(2) First, let us consider the following commutative diagram. Π′

P [i, j] ∩ {x = y} −−−→   ιy

(8.26)

P [i, j]

Π ×Π

X   ∆y

2 −−1−−→ X × X,

where ∆ is the diagonal embedding and Π′ is the projection given by def.

Π′ (J1 , J2, J3 ) = Π1 (J1 , J2) = Π2 (J2 , J3 ) Using this diagram, we have Π∗1 α ∩ Π∗2 β ∩ ι∗ R = ι∗ (ι∗ (Π∗1 α ∩ Π∗2 β) ∩ R) (8.27)

= ι∗ (ι∗ (Π1 × Π2 )∗ (α × β) ∩ R) = ι∗ (Π′∗ ∆∗ (α × β) ∩ R)

(projection formula) (commutativity of (8.26))

= ι∗ (Π′∗ (α ∩ β) ∩ R) ,

where the intersection α ∩ β is performed in X. To calculate push-forward by p13∗ , we introduce the following another commutative diagram P [i, j] ∩ {x = y}   p13 ◦ιy

Π′

−−−→ X

Π′′

P [i + j] = {(J1 , J3 )| J1 ⊃ J3 , Supp(J1 /J3 ) = {x}} −−−→ X, where Π′′ is the map defined as in (8.10). Then we have (8.28)

p13∗ ι∗ (Π′∗ (α ∩ β) ∩ R) = Π′′∗ (α ∩ β) ∩ p13∗ ι∗ (R).

On the other hand, the dimension of P [i + j] is given by dimC P [i + j] = 2n − i − j + 1. This dimension is less than the expected dimension. So we have p13∗ ι∗ (R) = 0. From (8.28), it follows that p13∗ (Π∗1 α ∩ Π∗2 β ∩ ι∗ R) = 0. Similarly, we also have p13∗ (Π∗1 β ∩ Π∗2 α ∩ ι′∗ R′ ) = 0. (b). The case i > 0, j < 0. The product Pα [i]Pβ [j] is also given as (8.19). Fixing n, we introduce
def. −1 [n−i−j] Q[i, j] = p−1 × X [n−j] × X [n] 12 (P [i]) ∩ p23 (P [j]) ∩ X   Supp(J1 /J2 ) = {x} for some x ∈ X = (J1 , J2, J3 ) J1 ⊃ J2 ⊂ J3 , , Supp(J3 /J2 ) = {y} for some y ∈ X

8.4. PROOF OF THEOREM 8.13

89


def. −1 [n−i−j] Q′ [j, i] = p−1 × X [n−i] × X [n] 12 (P [j]) ∩ p23 (P [i]) ∩ X ( ) ′ Supp(J /J ) = {y} for some y ∈ X 1 2 = (J1 , J2, J3 ) J1 ⊂ J2′ ⊃ J3 , . Supp(J2′ /J3 ) = {x} for some x ∈ X

Then we have an isomorphism similar to (8.23): (8.29) Q[i, j] ∩ {x 6= y} ∼ = Q′ [j, i] ∩ {x 6= y}.

The correspondence between (J1, J2 , J3 ) ∈ Q[i, j] ∩ {x 6= y} and (J1 , J2′ , J3) ∈ Q′ [j, i] ∩ {x 6= y} is given by J2′ /J1 = J3 /J2, J2′ /J3 = J1 /J2. −1 −1 −1 (See Figure 8.3.) Note that p−1 12 (P [i]) and p23 (P [j]) (resp. p12 (P [j]) and p23 (P [i])) in-

x

y

J2

J3

J1

J2′ Figure 8.3. Correspondence between J1 ⊃ J2 ⊂ J3 and J1 ⊂ J2′ ⊃ J3 tersects transversely along Q[i, j] ∩ {x 6= y} (resp. Q′ [j, i] ∩ {x 6= y}). In particular, Q[i, j] ∩ {x 6= y} and Q′ [j, i] ∩ {x 6= y} have the expected dimension 2n − i − j + 2. Let ′ Q[i, j] = Q[i, j] ∩ {x 6= y} and Q [i, j] = Q′ [j, i] ∩ {x 6= y}. Then there exist decompositions similar to (8.20), p∗12 [P [i]] ∩ p∗23 [P [j]] = [Q[i, j]] + ι′′∗ R′′ , ′

′′′ p∗12 [P [j]] ∩ p∗23 [P [i]] = [Q [i, j]] + ι′′′ ∗ R ,

for some R′′ ∈ H∗lf (Q[i, j] ∩ {x = y}) and R′′′ ∈ H∗lf (Q′ [j, i] ∩ {x = y}), where ι′′ : Q[i, j] ∩ {x = y} → Q[i, j] and ι′′′ : Q′ [j, i] ∩ {x = y} → Q′ [j, i] are inclusions. Then Q[i, j] and ′ Q [i, j] cancel out by exactly the same argument in (8.25).

90

8. HEISENBERG ALGEBRA

′′′ Thus our remaining task is to compute the contribution of ι′′∗ R′′ and ι′′′ ∗ R . We have the following commutative diagram similar to (8.26) Π′

Q[i, j] ∩ {x = y} −−−→   ιy Q[i, j]

Π ×Π

X   ∆y

2 −−1−−→ X × X,

where Π′ is defined as in (8.26). (We use the same symbol for brevity.) Then as (8.27), we have Π∗1 α ∩ Π∗2 β ∩ ι′′∗ R′′ = ι′′∗ (Π′∗ (α ∩ β) ∩ R′′ ) .

(8.30)

Similar formula holds for R′′′ . In order to calculate the push-forward of the above under p13∗ , we need to know the image of Q[i, j]∩{x = y} and Q′ [j, i]∩{x = y} under p13 . It will be studied in the following lemma. Lemma 8.31. Let L be   J1 = J3 outside x def. [m] [n] . L = (J1 , J3) ∈ X × X π(J1 ) = π(J3 ) + (m − n)[x] for some x ∈ X

Then we have the following. (1) If m > n, then L has one irreducible component with dimC L = m + n + 1. The other irreducible component are lower dimensional. (2) If m = n, then there are n + 1 irreducible components L0 , L1 , . . . Ln with dimC Li = 2n. Proof. (1) Suppose Supp J3 does not contain x. Then we can decompose J1 as J1 = J3 ∩ J1′ ,

where J1′ ∈ X [m−n] satisfies π(J1′ ) = (m − n)[x]. If we fix J2 , J1′ (therefore J1 ) runs over m−n π −1 (S(m−n) X) whose dimension is (m − n) + 1. Since J3 runs over a 2n-dimensional set, the total dimension is (m − n) + 1 + 2n = m + n + 1. Suppose Supp J3 contains x. Then we can decompose J3 as J3 =

J3′

∩

J3′′

so that

Accordingly J1 is decomposed as J1 = J3′′ ∩ J1′′

J3′ ∈ X [k],

J3′′ ∈ X [n−k] ,

so that J1′′ ∈ X [m−n+k] ,

Supp(OX /J3′ ) = {x}

Supp(OX /J3′′ ) ∩ {x} = ∅ Supp(OX /J1′′ ) = {x}.

.

If we fix x ∈ X, J2′′ runs over a 2(n − k)-dimensional set. And J1′′ and J2′ run over a (m − n + k − 1)-dimensional and a (k − 1)-dimensional set respectively. The point x can be moved in X. Hence the total dimension is 2(n − k) + (m − n + k − 1) + (k − 1) + 2 = m + n. This is less than m + n + 1. Hence this is a lower dimensional component.

8.4. PROOF OF THEOREM 8.13

91

(2) If J1 ∼ = J3 on whole X, then (J1 , J3) ∈ ∆X [n] and this is 2n-dimensional. Set L0 = ∆X [n] . Suppose J1 6∼ = J3 . Then we can decompose J1 as J1 =

J1′

∩

J1′′

so that

J1′ ∈ X [k],

Supp(OX /J1 ) = {x}

J1′′ ∈ X [n−k] ,

Supp(OX /J1′′ ) ∩ {x} = ∅

.

Similar decomposition exists for J3 , J3 = J3′ ∩ J3′′ . Then J1′′ must be isomorphic to J3′′ . Hence J1′′ ∼ = J3′′ runs over X [n−k] whose dimension is 2(n − k). For fixed x, {J1′ } and {J3′ } are (k − 1)-dimensional spaces. Moving x in X, we find that the total dimension is 2(n − k) + 2(k − 1) + 2 = 2n. This is Lk (k = 1, . . . , n). Now we begin to calculate the push-forward of (8.30). Note that the image of Q[i, j]∩{x = y} and Q′ [j, i] ∩ {x = y} under p13 is contained in L in Lemma 8.31 with m = n − i − j. We shall study the case i + j 6= 0 and i + j = 0 separately. (c). The case i + j 6= 0. Our argument for this case is very similar to that for the case i, j > 0. We assume i + j < 0. The case i + j > 0 can be treated by exchanging J1 and J3 . Let us consider the following commutative diagram. Π′

(8.32)

Q[i, j] ∩ {x = y} −−−→   p13 ◦ι′′ y L

Π′′

X



−−−→ X,

where L is as in Lemma 8.31 with m = n − i − j > n, and Π′′ (J1 , J3) = x is the unique point outside which J1 and J3 are isomorphic. Using the commutativity of the above diagram (compare (8.28)), we have p13∗ ι′′∗ (Π′∗ (α ∩ β) ∩ R′′ ) = Π′′ (α ∩ β) ∩ p13∗ ι′′∗ R′′ Applying Lemma 8.31 (1), we have dim L = 2n − i − j + 1 which is less than the expected dimension. Thus we conclude p13∗ ι′′∗ R′′ = 0. Similar argument proves p13∗ ι′′′ R′′′ = 0. Thus we have finished the proof in the case i + j 6= 0. (d). The case i + j = 0. This is only the case that the right hand sides of (8.14) are nontrivial. The identity operators in (8.14) come from the diagonal ∆X [n] in X [n] × [n] X [n] . Hence we first study that the restriction to p−1 × X [n] \ ∆X [n] ), then study the 13 (X contribution of the diagonal later.

92

8. HEISENBERG ALGEBRA

Outside the diagonal, we have the following commutative diagram: Π′

[n] Q[i, j] ∩ {x = y} ∩ p−1 × X [n] \ ∆X [n] ) −−−→ 13 (X   p13 ◦ι′′ y

(8.33)

Π′′

X



L \ ∆X [n] = L1 ∪ · · · ∪ Ln −−−→ X, where L is as in Lemma 8.31 with m = n. Note that the irreducible component L0 = ∆X [n] was extracted, and hence there is a well-defined morphism Π′′ : L \ ∆X [n] → X which map (J1 , J3 ) to the unique point x outside which J1 and J3 are isomorphic. Let [n] j ′ : Q[i, j] ∩ {x = y} ∩ p−1 × X [n] \ ∆X [n] ) → Q[i, j] ∩ {x = y} 13 (X

j ′′ : L \ ∆X [n] → L

be the inclusion. They induce homomorphisms [n] j ′∗ : H∗lf (Q[i, j] ∩ {x = y}) → H∗lf (Q[i, j] ∩ {x = y} ∩ p−1 × X [n] \ ∆X [n] )) 13 (X

j ′′∗ : H∗lf (L) → H∗lf (L \ ∆X [n] )

.

They are defined by identifying locally finite homology groups with relative cohomology groups. For example, the lower homomorphism is induced from H 8n−∗ (X [n] × X [n] , X [n] × X [n] \ L) → H 8n−∗ (X [n] × X [n] , X [n] × X [n] \ (L \ ∆X [n] )).

Then we have

j ′′∗ p13∗ ι′′∗ (Π′∗ (α ∩ β) ∩ R′′ ) = p13∗ ι′′∗ j ′∗ (Π′∗ (α ∩ β) ∩ R′′ ) ′′∗

= Π (α ∩ β) ∩

p13∗ ι′′∗ (j ′∗ R′′ )

(base change) (commutativity of (8.33)).

By Lemma 8.31, L \ ∆X [n] is 2n-dimensional, hence we have p13∗ ι′′∗ (j ′∗ R′′ ) = 0. Thus the above equation must be 0. Then p13∗ ι′′∗ (Π′∗ (α ∩ β) ∩ R′′ ) must be contained in the image of H∗ (∆X [n] ) → H∗ (X [n] ) ⊗ H∗lf (X [n] ), since its pull-back by j ′′∗ vanishes. The same thing ′∗ ′′′ holds for p13∗ ι′′′ ∗ (Π (α ∩ β) ∩ R ). If deg α + deg β < 4, then α ∩ β = 0. If deg α + deg β > 4, we have deg (p13∗ ι′′∗ (Π′∗ (α ∩ β) ∩ R′′ )) = 4n + deg α + deg β − 4 > 4n = dimR ∆X [n] .

Thus in the case deg α + deg β 6= 4, we have

p13∗ ι′′∗ (Π′∗ (α ∩ β) ∩ R′′ ) = 0,

′∗ ′′′ p13∗ ι′′′ ∗ (Π (α ∩ β) ∩ R ) = 0.

The proof is completed in this case. Let us consider the remaining case deg α + deg β = 4. In this case, the degree is equal ′∗ ′′′ to dim ∆X [n] , henceP p13∗ ι′′∗ (Π′∗ (α ∩ β) ∩ R′′ ) and p13∗ ι′′′ ∗ (Π (α ∩ β) ∩ R ) are multiples of [∆X [n] ]. If α ∩ β = nk [xk ] for [xk ] ∈ H0 (X), then X p13∗ ι′′∗ (Π′∗ (α ∩ β) ∩ R′′ ) = nk p13∗ ι′′∗ (Π′∗ [xk ] ∩ R′′ ).

8.4. PROOF OF THEOREM 8.13

93

(When X is connected, we could take a single point y and write α ∩ β = n[x]. However, it may be Pimpossible when we work with Chow groups. This is the reason why we write α ∩ β = [xk ].) In order to determine the multiple, we take a particular J3 and compute the intersection product with {J3 }×X [n] . Take distinct points {x1 , . . . , xn } different from xk ’s and consider the corresponding ideal J3. We want to compute p13∗ (p∗12 [P [i]] ∩ p∗23 [P [j]] ∩ Π∗2 [xk ])[J3 ] − p13∗ (p∗12 [P [j]] ∩ p∗23 [P [i]] ∩ Π∗1 [xk ])[J3 ].

Example 8.17 shows this is equal to [J3 ] when i = 1 and j = −1. Hence we get X (Pα [1]Pβ [−1] − (−1)deg α·deg β Pβ [1]Pα [−1])[J3 ] = nk [J3 ] = hα, βi[J3]. When i ≥ 2, the following situation can not happen:

J3 ⊂ J2′ , Supp J2′ /J3 = {a point}, where J3 ∈ X [n] , J2′ ∈ X [n+i] .

Then we have

p13∗ (p∗12 [P [−i]] ∩ p∗23 [P [i]] ∩ Π∗ [xk ]) ∩ [X [n] × {J3 }] = 0.

On the other hand,

i p13∗ (p∗12 [P [i]] ∩ p∗23 [P [−i]] ∩ Π∗ [xk ]) ∩ [X [n] × {J3 }] = h[π −1 (S(i) )], [π −1 (i[xk ])]i · [J3 ].

i Since h[π −1 (S(i) )], [π −1 (i[xk ])]i is determined by a neighborhood of xk , it is a universal constant independent of X. We shall compute it in the next chapter. The answer is (−1)i−1 i. This proves (8.14).

CHAPTER 9

Symmetric products of an embedded curve, symmetric functions and vertex operators This chapter is continuation of the previous chapter. We shall study the operator P[Σ] [i] associated with an embedded curve Σ in X. There are also distinguished homology classes [Lν Σ] in H∗ (X [n] ) (n = |ν|) introduced in Chapter 7. The operator P[Σ] [i] preserves these classes. We shall give the precise formulas for the action. We shall find they are exactly the same as relations in symmetric functions when we replace P[Σ] [i] by the power sum pi , [Lν Σ] by the orbit sum mν . In particular, the generating function of symmetric products [S n Σ] is the same as the vertex operator ([9],[20, §8.4],[43, §14.8]). In §9.3, we briefly recall the definition of the vertex algebra, and give its interpretation from our point of view in §9.4. def. i As an application, we shall determine the coefficients ci = h[π −1 (S(i) X)], [π −1 (i[x])]i, which was left in the previous section. As we have remarked before, these are calculated by Ellingsrud and Strømme [16]. Our approach is totally different. The idea to use symmetric products of an embedded curve is due to Grojnowski [33], although the relation to symmetric functions seems to new. The relation to vertex algebras is also due to him. The reader should keep results in Chapter 7 in mind. 9.1. Symmetric functions First we briefly recall the theory of symmetric functions which will be needed in the next section. See [54] for detail. Let ΛN be the right of symmetric functions with rational coefficients ΛN = Q[x1 , . . . , xN ]SN , where the symmetric group SN acts by the permutation of the variables. It is a graded ring: M ΛN = ΛnN , n≥0

where ΛnN consists of the homogeneous symmetric functions of degree n. It is more relevant for us to consider symmetric functions in “infinitely many variables” formulated as follows: let M ≥ N and consider the homomorphism Q[x1 , . . . , xM ] → Q[x1 , . . . , xN ] 94

9.1. SYMMETRIC FUNCTIONS

95

which sends xN +1 , . . . , xM to 0. We have induced homomorphisms ρnM,N : ΛnM → ΛnN ,

which is surjective for any M ≥ N, and bijective for M ≥ N ≥ n. Let def.

n Λn = ← lim − ΛN . Then the ring of symmetric functions in infinitely many variables is defined by M def. Λn . Λ = n

In the relationship between symmetric functions and Hilbert schemes, the degree n corresponds the number of points, while the number of variables N are irrelevant. This is the only reason why we use the different notation from [54]. There are several distinguished classes of symmetric functions. The first class is the orbit sum mν . Let ν be a partition with l(ν) ≤ N. Let def.

mν (x1 , . . . , xN ) =

X

α∈SN ·ν

xα1 1 · · · xαNN =

X νσ(1) 1 ν x1 · · · xNσ(N) , #{σ ∈ SN | σ · ν = ν} σ∈S N

where α = (α1 , . . . , αN ) is a permutation of ν = (ν1 , . . . , νN ). (We allow νk = 0 in this notation.) If l(ν) ≤ N, we have ρnM,N mν (x1 , . . . , xM ) = mν (x1 , . . . , xN ).

Hence mν defines an element in Λ, which is also denoted by mν . Then {mν }ν is a basis for Λ. For each nonnegative integer n, let us define the n-th elementary symmetric function en by X def. en = xi1 xi2 . . . xin = m(1n ) . i1
Their generating funtion is (9.1)

def.

E(z) =

∞ X n=0

∞ Y en z = (1 + xi z). n

i=1

If we define eν = eν1 eν2 · · · for a partition ν = (ν1 , ν2 , . . . ), {eν }ν gives another basis for Λ. The n-th complete symmetric function hn is X def. hn = mν . |ν|=n

Since 1 = 1 + xi z + x2i z 2 + · · · , 1 − xi z

96

9. SYMMETRIC PRODUCTS, SYMMETRIC FUNCTIONS AND VERTEX OPERATORS

the generating function for hn is def.

H(z) =

∞ X

n

hn z =

n=0

In particular, we have

∞ Y i=1

1 . 1 − xi z

E(z)H(−z) = 1. If we define hν = hν1 hν2 · · · for a partition ν = (ν1 , ν2 , . . . ), {hν }ν gives another basis for Λ. Finally, the n-th power sum is X def. pn = xni = m(n) . Its generating function is def.

P (z) =

∞ X n=1

Hence we have (9.2)

pn z

n−1

=

∞ X ∞ X

xni z n−1

i=1 n=1

H(z) = exp

Z

∞ X 1 d d = log = log H(z). dz 1 − xi z dz i=1

P (z)dz = exp

∞ X pn z n n=1

n

We also have (9.3)

E(z) = H(−z)−1 = exp

∞ X n=1

pn z n (−1)n−1 n

!

.

!

.

These formulas will be used later. For a partition ν = (ν1 , ν2 , . . . ), let pν = pν1 pν2 · · · . Then {pν }ν is a basis for Λ. Since {mν }ν , {eν }ν and {pν }ν are additive bases for Λ, there exist transition matrices between them. We need the following elementary lemma whose proof is left to the reader. Lemma 9.4. If we multiply the power sum pi to the orbit sum mν , we get X pi mν = aνµ mµ , µ

where the summation is over partitions µ of |ν| + i which is obtained as follows: (a) add i to a term in ν, say νk (possibly 0), and then (b) arrange it in descending order. The coefficient aνµ is #{l | µl = νk + i}. For example, p1 m(4,3,2) = m(4,3,2,1) + 2m(4,4,2) + m(5,3,2) + 2m(4,3,3) p3 m(4,3,2) = 2m(4,3,3,2) + m(5,4,3) + m(6,4,2) + m(7,3,2) .

9.2. SYMMETRIC PRODUCTS OF AN EMBEDDED CURVE

97

Although it is not related to the rest of this chapter, it is worth while to remark that the relationship between Chern classes/characters and symmetric functions. If E is a complex vector bundle of rank N, we can define its i-th Chern classes ci . The total Chern class is its generating function: c(E) =

N X

cd (E)z d .

d=0

If it is factored as (9.5)

c(E) =

N Y

(1 + xi z),

i=1

xi is called Chern roots of E. The splitting principle says that there exists a continuous e → X such that map f : X e is injective, (i) f ∗ : H ∗ (X) → H ∗ (X) ∗ (ii) f c(E) is factored. A symmetric function in xi , which is a polynomial in cd (E)’s, defines a well-defined cohomology class in H ∗ (X). Though c(E) is not necessarily factored in H ∗ (X), we could think as if xi ’s are well-defined classes when we want to prove a universal formula involving Chern classes. This follows from the splitting principle. For example, we could define the Chern character by ch(E) =

N X

1 exp(xi ) = N + (x1 + · · · + xN ) + (x21 + · · · + x2N ) + · · · . 2 d=1

Since cn (E) corresponds to the n-th elementary symmetric function en by (9.1) and (9.5), the above symmetric function in xi can be written in terms of en ’s. In fact, H 2n -part of Chern character corresponds to the n-th power sum pn divided by n!. Moreover, hn is the n-th Segre class of the dual vector bundle E ∨ (see [22, Chapter 3]). [n]

Question 9.6. Let V be the tautological bundle over the Hilbert scheme (C2 ) of 2 C . How the correspondence P[pt] [i] acts on Chern classes ck (V) ? This question should relate to Question 4.12. 9.2. Symmetric products of an embedded curve Let Σ ⊂ X be an embedded curve. We assume that Σ is non-singular throughout in thisLchapter. InLChapter 8, we define correspondences Pα [−i] = Π∗ α ∩ [P [−i]] ∈ Hom( n H∗ (X [n] ), n H∗ (X [n+i] )) for i > 0 and α ∈ H∗ (X) (8.12). (The case i < 0 will be discussed later.) We want to study the case α = [Σ] in detail. Let us denote Pα [−i] = P[Σ] [−i] by PΣ [−i] for brevity. It can be represented by a subvariety a (9.7) (J1 , J2) ∈ X [n+i] × X [n] | J1 ⊂ J2 , Supp(J2 /J1 ) = {x} for some x ∈ Σ . n

98

9. SYMMETRIC PRODUCTS, SYMMETRIC FUNCTIONS AND VERTEX OPERATORS

The difference to the definition of P [−i] (8.9) lies in the last condition, whether x can be moved in X or Σ. In this section, we use the same notation PΣ [−i] for the homology class and the above representative. L Note that PΣ [−i] preserves the middle degree part of H2n (X [n] ). It also preserves classes whose support are contained in a def. L∗ Σ = {J ∈ X [n] | Supp(OX /J ) ⊂ Σ}. n

Recall that (see Chapter 7) the irreducible components of L∗ Σ are indexed by partitions ν and denoted by Lν Σ (7.9). They are defined by (see (7.9)) P

Lν Σ = π −1 (Sνn Σ),

where Sνn Σ = { i νi [xi ] | xi ∈ Σ, xi 6= xj for i 6= j} and π is the Hilbert-Chow morphism. It is clear that Lν Σ is an open subvariety of L∗ Σ∩X [|ν|] , and has dimension |ν| since the fiber of π has dimension |ν|−l(ν) (Lemma 6.10) and the base has dim Sνn Σ = l(ν). Furthermore, Lν Σ is irreducible since the L fiber of π is irreducible (Theorem P 5.11). Hence the homology ν [n] ∗ classes [L Σ] form a basis of H2n (X ∩L Σ). In particular, q n dim H2n (X [n] ∩L∗ Σ) is equal to the character of the Heisenberg algebra the following homomorphism L (8.2).[n]Hence, ∗ from the ring Λ of symmetric functions to H2n (X ∩ L Σ) is an isomorphism:

(9.8)

pν 7−→ PΣ [−ν1 ]PΣ [−ν2 ] · · · PΣ [−νk ] · 1

for ν = (ν1 , ν2 , . . . , νk ).

In other words, we can define an isomophism so that the action of PΣ [−i] corresponds to the multiplication of pi in Λ. The following lemma determines the action of PΣ [i] on the basis given by irreducible components [Lν Σ]. Theorem 9.9. For i > 0, we have PΣ [−i][Lν Σ] =

X

aνµ [Lµ Σ],

µ

where the set over which the summation runs and the coefficient aνµ are same as in Lemma 9.4. By the induction on the length and the dominance order of partitions, we immediately get the following L H2n (X [n] ∩L∗ Σ) given by (9.8), the Corollary 9.10. Under the isomorphism Λ ∼ = class [Lν Σ] corresponds to the orbit sum mν .

In particular, we can write down [Lν Σ] in terms of PΣ [i]’s. For example, for |ν| = 3, we find   1 1 1 (1,1,1) 3 3 [L Σ] = [S Σ] = PΣ [−3] − PΣ [−2]PΣ [−1] + PΣ [−1] · 1 3 2 6 [L(2,1) Σ] = (−PΣ [−3] + PΣ [−2]PΣ [−1]) · 1 [L(3) Σ] = PΣ [−3] · 1.

9.2. SYMMETRIC PRODUCTS OF AN EMBEDDED CURVE

99

By (9.3),(9.2), the generating function for symmetric products S n Σ (n ≥ 0) and Lν Σ have the following nice expressions ! ∞ ∞ i X X z P [−i] Σ (9.11) z n [S n Σ] = exp · 1, i−1 i (−1) n=0 i=1 ! ∞ ∞ i X X X X z P [−i] Σ z |ν| [Lν Σ] = zn [Lν Σ] = exp (9.12) · 1. i n=0 ν: partition i=1 ν: partition of n Rather surprisingly, these expressions were appeared in the vertex operator [9],[20, §8.4],[43, §14.8]. We shall explain this analogy in detail later in §9.4. Proof of Theorem 9.9. Let pa : X [n] × X [n] → X [n] be the projection onto the a-th factor (a = 1, 2). As in (9.7), we represent PΣ [−i] as a subvariety. Then its set theoretical ν intersection with p−1 2 (L Σ) is (9.13)

{(J1, J2 ) | J2 ∈ Lν Σ, J1 ⊂ J2 , Supp(J2 /J1 ) = {x} for some x ∈ Σ}.

Let µ = (µ1 , µ2 , . . . , µN ) be a partition of |ν| + i which does not necessarily satisfy the condition in Lemma 9.4. In order to determine the coefficients of [Lµ Σ] in PΣ [−i][Lν Σ], we can take arbitary point Jµ in Lµ Σ and restrict cycles to a neighborhood of Jµ . We choose the point Jµ in Lµ Σ so that the support of OX /Jµ consists of N distinct points x1 , . . . , xN with length ((OX /Jµ )xk ) = µk . Hence Jµ = Jµ;1 ∩ · · · ∩ Jµ;N such that Supp(OX /Jµ;k ) = {xk }. Furthermore, we assume Jµ;k = (ξ µk , z) for some coordinate (z, ξ) around xk such that Σ = {ξ = 0}. Suppose this point Jµ is contained in the image of (9.13) under the projection p1 , i.e., there exists J2 such that (Jµ , J2 ) is a point in (9.13). Then the point x must be one of xk ’s, and ′ J2 = Jµ;1 ∩ . . . Jµ;k−1 ∩ Jµ;k ∩ Jµ;k+1 ∩ · · · ∩ Jµ;N

′ where Jµ;k = (ξ µk −i , z).

Since J2 must be a point in Lν Σ, µ must be as in Lemma 9.4, that is µ is obtained by (a) adding i to νk , and then (b) arranging in descending order. Moreover, if aµν is as in Lemma 9.4, there are exactly aµν choices of xk ’s. This explains the coefficient aµν in the formula. Thus the only remaining thing to check is that each choice of (J1 , J2 ) contributes to PΣ [−i][Lν Σ] by [Lµ Σ]. This will be shown by checking

ν (9.14.1) PΣ [−i] and p−1 2 (L Σ) intersect transversally, (9.14.2) the intersection (9.13) is isomorphic to Lν Σ under the first projection p1 ,

in a neighborhood of (J1, J2 ).

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9. SYMMETRIC PRODUCTS, SYMMETRIC FUNCTIONS AND VERTEX OPERATORS

Since J1 and J2 are isomorphic outside xk , we can restrict our concern to Jµ;k and ′ We take the following coordinate neighborhood around (Jµ;k , Jµ;k ) in X [µk ] × X [µk −i] :  µ ((ξ k + f1 (ξ), z + g1 (ξ)), (ξ µk −i + f2 (ξ), z + g2 (ξ))) | f1 , g1 , f2 , g2 as follows

′ Jµ;k .

f1 (ξ) = a1 ξ µk −1 + a2 ξ µk −2 + · · · + aµk ,

g1 (ξ) = b1 + b2 ξ + · · · + bµk −i ξ µk −i−1

+ (bµk −i+1 + bµk −i+2 ξ + · · · + bµk ξ i−1 )(ξ µk −i + f2 (ξ))

f2 (ξ) = a′1 ξ µk −i−1 + a′2 ξ µk −i−2 + · · · + a′µk −i ,

g2 (ξ) = b′1 + b′2 ξ + · · · + b′µk −i ξ µk −i−1

where (a1 , . . . , aµk , b1 , . . . , bµk ) (resp. (a′1 , . . . , a′µk −i , b′1 , . . . , b′µk −i )) is in a neighborhood of 0 in C2µk (resp. C2(µk −i) ). Then the above ideal is contained in PΣ [−i] if and only if the followings hold (i) ξ µk + f1 (ξ) = ξ i (ξ µk −i + f2 (ξ)), (ii) g1 (ξ) − g2 (ξ) is divisible by ξ µk −i + f2 (ξ).

Namely, the defining equation for PΣ [−i] is

a1 = a′1 , a2 = a′2 , . . . , aµk −i = a′µk −i , aµk −i+1 = · · · = aµk = 0, b1 = b′1 , b2 = b′2 , . . . , bµk −i = b′µk −i .

ν On the other hand, the defining equation for p−1 2 (L Σ) is

a′1 = a′2 = · · · = a′µk −i = 0. Now our assertions (9.14) are immediate. As we promiced, we determine the constant left in Chapter 8. def.

i Theorem 9.15. The constant ci = h[π −1 (S(i) X)], [π −1 (i[x])]i is (−1)i−1 i.

The remainder of this section is devoted to the proof of this theorem. Since the constant ci does not depend on the point x or the surface X, we may assume X = P2 . We take lines Σ, Σ′ . As we have proved in Chapter 8, the commutator   two different ′ PΣ′ [−i], PΣ [−i] = hΣ , Σici id. Since Σ and Σ′ intersect at one point transversely, the intersection number hΣ′ , Σi = 1. Note that [Σ] = [Σ′ ] and PΣ′ [−i] is equal to PΣ [−i] as elements of the homology group. Lemma 9.16. (9.17)

PΣ′ [i][S n Σ] = [S n−i Σ].

9.2. SYMMETRIC PRODUCTS OF AN EMBEDDED CURVE

101

Proof. Let pa : X [n] × X [n] → X [n] be the projection onto the a-th factor (a = 1, 2). As in (9.7), we represent PΣ′ [i] as a subvariety a (J1 , J2) ∈ X [n−i] × X [n] | J1 ⊃ J2 , Supp(J1 /J2) = {x} for some x ∈ Σ′ . n

n Then its set theoretical intersection with p−1 2 (S Σ) is

{(J1, J2 ) | J2 ∈ S n Σ, J1 ⊃ J2 , Supp(J1 /J2 ) = {x} for some x ∈ Σ′ }. If p denotes the intersection Σ ∩ Σ′ , the above point x must be p. Moreover, we have J1 ∈ S n−iΣ and J2 = J1 + i[p],

where we have used additive expression for S n Σ. Exactly as in the proof of Theorem 9.9, n we can check that the PΣ′ [i] and p−1 2 (S Σ) intersect transversely, and the set theoretin−i cal intersection is isomorphic to S Σ under the projection p1 . Therefore we have the assertion. Since S n Σ can be expressed by PΣ [−i]’s by (9.11), we can extract the commutator relation of PΣ [−i] and PΣ′ [j] from Lemma 9.16 as follows. Let def.

Σ(z)− = Using (9.11), we get PΣ′ [i]

∞ X

!

z n [S n Σ]

n=0



∞ X z n PΣ [−n] n=1

(−1)n−1n

.

= PΣ′ [i] exp Σ(z)− · 1 

= exp Σ(z)− exp (− ad Σ(z)− ) P [i] · 1   1 2 = exp Σ(z)− PΣ′ [i] − ad Σ(z)− PΣ′ [i] + (ad Σ(z)− ) PΣ′ [i] + · · · · 1. 2! Σ′

where we have used exp(−A)B exp(A) = exp(− ad A)B in the second equality. Since we have z i ci id, (−1)i−1 i z i ci (ad Σ(z)− )2 PΣ′ [i] = − ad Σ(z)− id = 0, (−1)i−1 i ad Σ(z)− PΣ′ [i] = −

we get PΣ′ [i]

∞ X n=0

!

z n [S n Σ]

z i ci z i ci = exp Σ(z) · 1 = − (−1)i−1 i (−1)i−1 i

∞ X n=0

!

z n [S n Σ] .

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9. SYMMETRIC PRODUCTS, SYMMETRIC FUNCTIONS AND VERTEX OPERATORS

Comparing with (9.17), we find ci = (−1)i−1 i. Exercise 9.18. Suppose two curves Σ, Σ′ are embedded in a surface X. Show that ∞ X ′ z n h[S n Σ], [S n Σ′ ]i = (1 + z)h[Σ],[Σ ]i . n=0

This can be shown by using (9.11), but the direct calculation is possible when h[Σ], [Σ′ ]i ≥ 0. There are other important classes of symmetric functions, such as Schur functions, Jack polynomials, Macdonald polynomials, etc (see [54]). In [70], we shall discuss a geometric interpretation of Jack polynomials. 9.3. Vertex algebra We shall review the definition of the vertex algebra briefly. We do not explain the original motivation from the string theory, or the monster moonshine. The interested reader should consult [9, 20, 23]. In the next section, we shall see that many formulas presented below have geometric interpretation. However, our understanding stays in a superficial level and we still lack an answer to a fundamental question “why homology groups of Hilbert schemes (more precisely, moduli spaces of rank 1 torsion-free sheaves) form a vertex algebra ?” It seems that the string theory is hidden behind the Hilbert schemes. L Definition 9.19. A vertex algebra is a Z-graded vector space V = n∈Z V(n) equipped with bilinear operations indexed by n ∈ Z V ⊗ V −→ V u ⊗ v 7−→ un v,

and two distinguished elements, called vaccum vector and conformal vector 1 ∈ V(0) and ω ∈ V(2)

satisfying the followings. (9.19.1) For any u, v ∈ V , there exists an integer N = N(u, v) such that un v = 0 for n ≥ N. (9.19.2) For any u, v, w ∈ V and k, l, m ∈ Z, we have ∞   X k i=0

i

  m (um+i v)k+l−i w = (−1) {uk+m−i(vl+i w) − (−1)m vl+m−i (uk+iw)} , i i=0

where

∞ X

i

  n def. n(n − 1) · · · (n − i + 1) = i! i

(9.19.3) u−11 = u and un 1 = 0 for n ≥ 0.

  n for i > 0, and = 1. 0

9.3. VERTEX ALGEBRA

103

(9.19.4) ω0 u = u−21 for any u ∈ V , ωn ω = 0 for n ≥ 4, c ω3 ω = 1 for some c ∈ C, 2 ω2 ω = 0, ω1 ω = 2ω. def.

(9.19.5) If we set L0 = ω1 , we have L0 v = nv for all v ∈ V(n) . In other words, the gradation of V is given by the eigenvalues of L0 . We consider un as an endormophism of V , and introduce a generating function Y (u, z) for un by X def. (9.20) Y (u, z) = un z −n−1 ∈ End(V )[[z, z −1 ]]. n∈Z

Since (9.19.2,3) implies 1−1 x = x and 1n x = 0 for n 6= −1, we have Y (1, z) = id .

Let us define L−1 : V → V by L−1 u = u−2 1 = ω2 . This is a derivation, i.e., L−1 (un v) = (L−1 u)n v + un (L−1 v). Moreover, we have (9.21)

Y (L−1 u, z) = ∂Y (u, z),

where ∂ is the differentiation with respect to z. More generally, if we define operators Ln on V by ωn+1 , it satisfies n3 − n δn+m,0 c id . 12 This defining relation of the Virasoro algebra follows from (9.19.4). By (9.19.5), if u ∈ V(k) , the operator un maps V(i) into V(i+k−n−1) , i.e. it shifts the def. L ∗ grading by k − n − 1. Hence if φ ∈ V and ψ ∈ V ′ = n∈Z V(n) (the restricted dual of V ), X hψ, Y (v, z)φi = hψ, un φiz −n−1 [Ln , Lm ] = (n − m)Ln+m +

n∈Z

is a finite sum. From now, when we say an “operator A”, we mean that hψ, Aφi is defined for all φ ∈ V and ψ ∈ V ′ . It may not mean that A is an endomorphism on V . (See [20] for detail.) Then (9.19.2) is equivalent to saying that the following three series which converges in the indicated regions are analytically continued to a common operator-valued rational function (in the above sense): Y (u, z)Y (v, w) Y (v, w)Y (u, z) Y (Y (u, z − w)v, w)

in |z| > |w|, in |w| > |z|,

in |w| > |z − w|.

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9. SYMMETRIC PRODUCTS, SYMMETRIC FUNCTIONS AND VERTEX OPERATORS

The equality “Y (u, z)Y (v, w) = Y (v, w)Y (u, z)” is called commutativity, and “Y (u, z) Y (v, w) = Y (Y (u, z − w), w)” associativity. These two equalities are equivalent to (9.19.2). Moreover, it is also known that the associativity follows from the commutativity. In some literature, the commutativity is used in the definition of the vertex algebra instead of (9.19.2). Substituting (9.20) into Y (u, z)Y (v, w) = Y (Y (u, z − w)v, w), we get X Y (un v, w) Y (u, z)Y (v, w) = . (z − w)n+1 n We often write this in the so-called operator product expansion ∞ X Y (un v, w) Y (u, z)Y (v, w) ∼ , n+1 (z − w) n=0

where ∼ means that the difference has no pole at z = w. Note that the right hand side is finite sum by (9.19.1). P −n−2 For example, the energy momentum tensor T (z) = z Ln = Y (ω, z) has the following operator product expansion: T (z)T (w) ∼

c/2 2 ∂T (w) + T (w) + . 4 2 (z − w) (z − w) z−w

Definition 9.22. A vertex operator algebra is a vertex algebra V satisfying (9.22.1) the eigenspaces of L0 = ω1 are finite dimensional, (9.22.2) for sufficiently small integers, the corresponding eigenspace is 0. Vertex algebra from a lattice. It is known that a vertex algebra is construted from an even lattice L ([9, 20]). We briefly recall the construction. Let us denote by h·|·i the bilinear form of the lattice. Let e hL be the Heisenberg Lie algebra associated with L, that is a Lie algebra generated by α(n) (α ∈ L, n ∈ Z \ {0}) and K with relation (9.23)

(aα + bβ)(n) = aα(n) + bβ(n)

[α(m), β(n)] = hα|βimδm+n,0K,

[K, α(m)] = 0.

e<0 Subalgebra generated by α(n) (n > 0) (resp. n < 0) is denoted by e h>0 L (resp. hL ). The Fock representation of e hL is the induced representation e

L Indehh>0 Q, ⊕QK L

where e h>0 L ⊕ QK acts on Q by

α(n)1 = 0 for n > 0,

K1 = 1.

It is spanned by elements of the form α1 (n1 )α2 (n2 ) · · · αk (nk )1,

αi ∈ L, ni ∈ Z<0 .

9.3. VERTEX ALGEBRA

105

Since e h<0 L is commutative, the Fock representation is isomorphic to the symmetric algebra S(e h<0 L ). From now, we drop “1” in the above element and simply denote it by α1 (n1 )α2 (n2 ) · · · αk (nk ). Operators α(m) with m < 0 (resp. m > 0) are called creation operators, (resp. annihilation operators). Let us define the Fock space associated with the lattice L by (9.24)

e def. VL = IndehhL>0 ⊕QK Q ⊗ Q[L] = S(e h<0 L ) ⊗ Q[L], L

e<0 where Q[L] is the group algebra of L, and S(e h<0 L ) is the symmetric algebra of hL . The α element in Q[L] correpsonding to α ∈ L is denoted by e as usual. In order to have a correct commutation relation, we need to twist the action of eα as follows: choose a multiplicative 2-cocycle ε(·, ·) : L × L → {±1} such that ε(α, β)ε(α + β, γ) = ε(β, γ)ε(α, β + γ) ε(α, β)ε(β, α) = (−1)hα|βi . def.

Then we define the twisted product by eα eβ = ε(α, β)eα+β . From now, we only use the twisted product, so we do not introduce a new symbol. Each element α(m) (α ∈ L, m ∈ Z \ {0}) acts on VL by α(m) ⊗ 1. We introduce the zero mode operator by (9.25)

def.

α(0)(v ⊗ eβ ) = hα|βi v ⊗ eβ .

It commutes with the action of the Heisenberg algebra e hL . Hence (9.23) holds even for m or n = 0.

Remark 9.26. In some literature, α(0) is included in e hL . In some sense, it is more e natural, since hL becomes the central extension of the loop algebra L ⊗ Q[t, t−1 ] ⊕ QK,i.e. an affine Lie algebra. But from the view point of Hilbert schemes, the role of α(0) and other α(n)’s are totally different. This is the reason why we do not include α(0) in e hL . We −1 denote L ⊗ Q[t, t ] ⊕ QK by b hL . Let z be a formal parameter z. Define a formal expression X z −n def. φα (z) = α(n) + α(0) log z + α. −n n∈Z\{0}

Only formal power series make sense in vertex algebras, hence the above does not make sense. But its differential X ∂φα (z) = α(n)z −n−1 n∈Z

does make sense. (When we differentiate the above formal expression, we regard the last term α as a constant.)

106

9. SYMMETRIC PRODUCTS, SYMMETRIC FUNCTIONS AND VERTEX OPERATORS

Consider the exponential of φα (z). This does not make sense yet since we have an infinite sum in coefficients of the z-expansion. In order to remedy the difficulty, we need to introduce so-called normal ordering, which defines an order on operators α(n), eα ’s. Definition 9.27. (1) Let S(b hL ) be the symmetric algebra of the affine Lie algebra −1 b hL = L ⊗ Q[t, t ] ⊕ QK. Identifying α ⊗ tn with α(n), we have a decomposition S(b hL ) = ≶0 <0 >0 0 0 e b e e b S(hL )⊗S(hL)⊗S(hL ), where hL is as above and hL is the algebra generated by α(0) and b0 e≶0 b0 K. Since e h≶0 L and hL is commutative, we can identify S(hL ) and S(hL ) with the universal b0 enveloping algebra U(e h≶0 L ) and U(hL ) respectively. The multiplication in the universal b0 e>0 b enveloping algebra U(b hL ) defines a linear map U(e h<0 L ) ⊗ U(hL ) ⊗ U(hL ) → U(hL ) given by <0 0 >0 <0 0 >0 x ⊗ x ⊗ x 7→ x x x . Composing these maps, we get a linear map S(b hL) → U(b hL ) and denote it by : · :. It simply means that we move the negative part to left, positive part to right, and zero part to middle, and then multiply them in that order. (2) We extend the definition to operators including elements of Q[L] by

:eα1 · · · eαk : def. = eα1 +···+αk ,

:eβ α(0): = :α(0)eβ : def. = eβ α(0).

Since eβ , α(0) commute with elements in U(e h≶0 L ), the above definition is enough.

Remark 9.28. Our normal ordering is slightly different from that in [20]. Ours satisfies :hh′ : = :h′ h:. For α ∈ L, we introduce the associated vertex operator def.

Y (eα , z) = :eφα (z) :

! ∞ X α(−m) m α α(0) = exp z e z exp m m=1

(9.29)

= exp(φα (z)− )eα z α(0) exp(φα (z)+ ),

∞ X

α(m) −m − z m m=1

!

where φα (z)±

∞ X α(±m) ∓m = z . ∓m m=1

def.

The general vertex operator is defined by ∂ n1 φα1 (z) ∂ nk φαk (z) (9.30) Y (v, z) = : ··· Y (eβ , z): for v = α1 (−n1 ) · · · αk (−nk ) ⊗ eβ . (n1 − 1)! (nk − 1)! def.

By definition, we have (9.31)

Y (uv, z) = :Y (u, z)Y (v, z):.

9.3. VERTEX ALGEBRA

107

Finally, we consider the element r 1 X ab ω = g αa (−1)αb (−1) ⊗ 1, 2 a,b=1 def.

where {αa }ra=1 is a basis of L, and (g ab )a,b is the inverse matrix of (gab )a,b = (hαa |αb i)a,b . Theorem 9.32 ([9, 20]). VL together with bilinear operations given by Y (v, z), the vacuum vector 1 = 1 ⊗ e0 , and the conformal vector ω is a vertex algebra. If in addition L is positive definite, it is a vertex operator algebra. We check only parts of conditions for vertex algebras. For the proofs of the full statement, see [20]. Moreover, the proof below will not be used in the next section. First let us study the conformal vector ω. The associated vertex operator is given by r 1 X ab g :∂φαa (z)∂φαb (z): Y (ω, z) = 2 a,b=1

=

r 1 X ab X g :αa(n − m)αb (m):z −n−2 . 2 a,b=1 m∈Z

In particular, L−1 is given by

L−1 v = =

r r ∞ X X 1 X ab X g ab αa (−1 − m)αb (m)v g :αa (−1 − m)αb (m):v = 2 a,b=1 m=0 m∈Z a,b=1

k X p=1

np αa1 (−n1 ) · · · αap−1 (−np−1 )αap (−np − 1)αap+1 (−np+1 ) · · · αak (−nk ) ⊗ eβ + αa1 (−n1 ) · · · αak (−nk )β(−1) ⊗ eβ

for v = αa1 (−n1 ) · · · αak (−nk ) ⊗ eβ . Hence (9.33)

Y (L−1 v, z) = ∂ :

∂ n1 φα1 (z) ∂ nk φαk (z) ··· Y (eβ , z): = ∂Y (v, z). (n1 − 1)! (nk − 1)!

We get L−1 = ω0 . We leave the commutation relation of Ln ’s for the exercise.

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9. SYMMETRIC PRODUCTS, SYMMETRIC FUNCTIONS AND VERTEX OPERATORS

We have r 1 X ab X L0 v = g :αa (−m)αb (m):v 2 a,b=1 m∈Z

(9.34)

r X

!

∞ X

1 αa (0)αb (0) + αa (−m)αb (m) v 2 m=1 a,b=1 ! k X 1 for v = αa1 (−n1 ) · · · αak (−nk ) ⊗ eβ . = np + hβ|βi v 2 p=1

=

g ab

We also remark that

Y (v, z) =

∞ X (z − w)n n=0

(9.35)

=Y

n!

∂ n Y (v, w)

∞ X (z − w)n n=0

n!

Ln−1 v, w

(Taylor expansion) !

(by (9.33))

= Y (exp ((z − w)L−1 ) v, w)

where we assume 0 < |z − w| < |w| to ensure that Y (v, z) is holomorphic. Let us study the composition of two vertex operators of (9.29). We need the following elementary lemma Lemma 9.36. Suppose [A, B] = C commutes with A and B. Then exp A exp B = exp C exp B exp A. Proof. The assertion follows from exp A· B · exp(−A) = exp(ad A)(B) = B + [A, B] = B + C. From the above lemma, we get

! ∞ X 1 wm exp(φα (z)+ ) exp(φβ (w)− ) = exp −hα|βi exp(φβ (w)− ) exp(φα (z)+ ) m m z m=1   w  = exp hα|βi log 1 − exp(φβ (w)− ) exp(φα (z)+ ) z  w hα|βi = 1− exp(φβ (w)− ) exp(φα (z)+ ), z where we have assumed |z| > |w| in the second equality. Hence Y (eα , z)Y (eβ , w)

= exp(φα (z)− )eα z α(0) exp(φα (z)+ ) exp(φβ (w)− )eβ w β(0) exp(φβ (w)+ )

= ε(α, β)(z − w)hα|βi exp(φα (z)− ) exp(φβ (w)− )eα+β z α(0) w β(0) exp(φα (z)+ ) exp(φβ (w)+ ) = ε(α, β)(z − w)hα|βi :Y (eα , z)Y (eβ , w):.

9.3. VERTEX ALGEBRA

109

Similarly, when |w| > |z|, we have Y (eβ , w)Y (eα , z) = ε(β, α)(w − z)hα|βi :Y (eβ , w)Y (eα , z):. Hence Y (eα , z)Y (eβ , w) and Y (eβ , w)Y (eα , z) define the same rational function. More generally, we have

:

k Y

αi

Y (e , zi ): :

i=1

(9.37) =

Y i,j

l Y

Y (eβj , wj ):

j=1

ε(αi , βj )(zi − wj )

hαi |βj i

:

k Y

αi

Y (e , zi )

i=1

l Y

Y (eβj , wj ):

j=1

when |zi | > |wj |. Now we begin to study general vertex operators (9.30). We consider an element u=

k Y i=1

exp(φαi (zi )− )eαi ∈ VL [[z1 , . . . , zk ]].

In fact, the coefficients of this function span V as αi and k vary. Hence it is enough to check the commutativity and associativity for vectors of this form. Setting v = eαi , z = zi in (9.35), we have exp(φαi (zi )− )eαi = Y (eαi , zi )1

= Y (exp((zi − w)L−1 )eαi , w)1 = exp(zi L−1 )eαi ,

where we set w = 0 in the last equality. Remark that we can use the Taylor expansion (9.35) at w = 0 since Y (u, w)1 is regular there. Hence Y (u, z) = Y (

k Y i=1

(9.38)

=:

k Y i=1

=:

k Y

exp(zi L−1 )eαi , z)

Y (exp(zi L−1 )eαi , z):

(by (9.31))

Y (eαi , z + zi ):

(by (9.35)).

i=1

Let us take another vector v=

l Y j=1

exp(φβj (wj )− )eβj .

110

9. SYMMETRIC PRODUCTS, SYMMETRIC FUNCTIONS AND VERTEX OPERATORS

By (9.37), it follows that both Y (u, z)Y (v, w) and Y (v, w)Y (u, z) are analytically continued to the following rational function Y

(9.39)

i,j

ε(αi , βj )(z + zi − w − wj )

hαi ,βj i

:

k Y

αi

Y (e , z + zi )

i=1

l Y

Y (eβj , w + wj ):.

j=1

Now we check the associativity of vertex operators. We have Y (u, z − w)v = lim Y (u, z − w)Y (v, z ′ )1 ′ =:

k Y i=1

=

Y i,j

Hence

z →0

αi

Y (e , z − w + zi ): :

l Y

Y (eβj , wj ):1

j=1

Y (Y (u, z − w)v, w) =

Y i,j

=

Y i,j

k Y

ε(αi , βj )(z − w + zi − wj )hαi ,βj i :

ε(αi , βj )(z − w + zi − wj )

hαi ,βj i

ε(αi , βj )(z − w + zi − wj )

hαi ,βj i

Y (:

i=1

k Y i=1

:

k Y i=1

Y (eαi , z − w + zi )

αi

Y (e , z − w + zi ) αi

Y (e , z + zi )

l Y

l Y

Y (eβj , wj ):1.

j=1

l Y

Y (eβj , wj ):1, w)

j=1

Y (eβj , w + wj ):,

j=1

Q where we have used (9.38) in the second equality replacing u by ki=1 exp(φαi (z − w + Q zi )− )eαi lj=1 exp(φβj (wj )− )eβj . This is the same rational function as (9.39). 9.4. Moduli space of rank 1 sheaves

As we explained in the previous section, we have a vertex algebra associated with a lattice L. On the other hand, as we saw in Chapter 8, the Fock representation of the Heisenberg algebra can be naturally understood as the homology group of the Hilbert scheme. In this section, we go a little bit further, the Fock space VL associated with L can be also understood as a homology group of certain moduli spaces. For this purpose, we take a lattice which is smaller than the full homology group of X, i.e., the N´eron-Severi lattice of X. In this section, We assume X is simply-connected for simplicity. Let NS(X) be the N´eron-Severi group of X. By the assumption this is a finitely generated free abelian group. The intersection form defines a non-degenerate symmetric bilinear form, which we denote by h·, ·i. The Hodge index theorem (see e.g., [5]) says that its index is (1, n). For c1 ∈ L and ch2 ∈ 1/2Z (if the intersection form is even we may assume ch2 ∈ Z), R let M(c1 , ch2 ) be the moduli space of rank 1 torsion-free sheaves with c1 (E) = c1 , ch(E) = ch2 , where c1 (E) is the first Chern class of E, and ch(E) the Chern character X of E. If E ∈ M(c1 , ch2 ), its double dual E ∨∨ is a line bundle. Then E ⊗ (E ∨∨ )∨ is a rank

9.4. MODULI SPACE OF RANK 1 SHEAVES

111

R 1 torsion-free sheaf with c1 (E ⊗ (E ∨∨ )∨ ) = 0, X ch(E ⊗ (E ∨∨ )∨ ) = ch2 − 12 hc1 , c1 i. The double dual of E ⊗ (E ∨∨ )∨ is the trivial sheaf OX , hence E ⊗ (E ∨∨ )∨ can be considered as 1 an element of X [−ch2 + 2 hc1 ,c1 i] . Conversely, if L is a line bundle with c1 (L) = c1 (such an L is unique by the Rassumption) and J ∈ X [n] , then L ⊗ J is a rank 1 torsion-free sheaf with c1 (L ⊗ J ) = c1 , X ch(L ⊗ J ) = −n + 21 hc1 , c1 i. Hence we have the natural identification M(c1 , ch2 ) ∼ = X [n] ,

1 where n = −ch2 + hc1 , c1 i. 2

For cohomology groups, we have M M H∗ (M(c1 , ch2 )) ∼ H∗ (X [n] ) ⊗ Q[NS(X)], = c1 ∈L,ch2

n

where Q[NS(X)] is the group algebra of NS(X) with coefficients in Q. By results in L Chapter 8, n H∗ (X [n] ) forms a representation of the Heisenberg/Clifford algebra modelled L on H∗ (X). We consider the subspace n HNS (X [n] ) generated by elements Pα [i] · 1 for α ∈ NS(X), i ∈ Z<0 , where 1 is the generator of H0 (X [0] ) = H0 (pt). It forms a representation of the Heisenberg algebra modelled on NS(X). The corresponding subspace in H∗ (M(c1 , ch2 )) is denote by HNS (M(c1 , ch2 )). We shall study M M def. VNS(X) = HNS (X [n] ) ⊗ Q[NS(X)] ∼ HNS (M(c1 , ch2 )), = n

c1 ∈NS(X),ch2

where the notation VNS(X) is consistent with (9.24). L [n] According to the discussion in the previous section §9.3, the space n HNS (X ) ⊗ Q[NS(X)] has a structure of the vertex algebra if the intersection form of NS(X) is even. Our aim is to explain a geometric interpretation of the vertex operator. First, an element α ∈ NS(X) can be identified with the generator of

1 H0 (M(α, hα, αi)) ∼ = H0 (pt), 2 where the relevant moduli space consists of a single point corresponding to the unique line bundle L = Lα with c1 (L) = α. The corresponding operator eα on VNS(X) , which maps eβ to eα+β and commutes with the action of the Heisenberg algebra, corresponds to an operator given by the tensor product with Lα : 1 M(c1 , ch2 ) ∋ E 7→ E ⊗ Lα ∈ M(c1 + α, ch2 + hc1 , αi + hα, αi) 2 where the integral of the Chern character was calculated by Z

Z

ch(E ⊗ Lα ) = ch(E) ∪ ch(Lα ) X Z Z Z 1 1 = ch(E) + c1 (E) ∪ c1 (Lα ) + c1 (Lα ) ∪ c1 (Lα ) = ch2 + hc1 , αi + hα, αi. 2 X 2 X X X

112

9. SYMMETRIC PRODUCTS, SYMMETRIC FUNCTIONS AND VERTEX OPERATORS

The operator α(i) in the Heisenberg algebra, of course, corresponds to the operator Pα [i] constructed in Chapter 8. But our commutator relation (8.14) differs from the standard one, we need to modify operators. In fact, it is more natural to change also the sign of the bilinear form. Hence we define def.

h·|·i = −h·, ·i = (−1) × (intersection form), ( −Pα [i] (i > 0), α(i) = i−1 (−1) Pα [i] (i < 0). For i = 0, the operator α(0) in (9.25) corresponds to an operator measuring the first Chern class, i.e., hα|c1 i Id

on HNS (M(c1 , ch2 )).

The degree given by the eigenvalue of L0 is given by the integral of the Chern character multiplied by −1, i.e., L(0) = −ch2 id

on HNS (M(c1 , ch2 )).

Now (a part of) vertex operator corresponds is given by ! ! ∞ ∞ X X z i Pα [−i] α(−i) i z = exp exp φα (z)− = exp i (−1)i−1 i i=1 i=1

If the line bundle Lα has a section s with Zero(s) = Σ, the above expression is the same as the generating function of the symmetric products Σ. The correspondence is summarized in the following table: Vertex algebra Geometric Objects lattice L the N´eron-Severi group NS(X) bilinear form h·|·i (−1) × (intersection form) L V = H VL = S(h<0 ) ⊗ Q[L] NS (M(c1 , ch2 )) NS(X) L c1 ,ch2 α∈L the line bundle Lα with c1 (Lα ) = α ∈ NS(X) eα the operator induced by E 7→ E ⊗ Lα α(i) (i 6= 0) ±Pα [i] in (8.12) α(0) hα|c1 i Id on HNS (M(c1 , ch2 )) −ch2 Id on HNS (M(c1 , ch2 )) L0 exp φα (z)− the generating of the symmetric P∞function i i products i=0 [S Σ]z , where [Σ] = α.

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