Typical singularities of differential 1-forms and Pfaffian equations (Translations of Mathematical Monographs)

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Typical Singularities of Differential 1-Forms and Pfaffian Equations

Recent Titles in This Series Micbail Zbitomirskil, Typical singularities of differential 1-forms and Pfaffian equations, 1992 112 S. A. Lomov, Introduction to the general theory of singular perturbations, 1992 111 Simoa Gindikin, Tube domains and the Cauchy problem, 1992 110 B. V. Shabat, Introduction to complex analysis Part II. Functions of several variables, 113

1992

Isao Miyadera, Nonlinear semigroups, 1992 108 Takeo Yokoatuna, Tensor spaces and exterior algebra, 1992 107 B. M. Makarev, M. G. Golazina, A. A. Lodkin, and A. N. Podkorytov, Selected problems in real analysis, 1992 106 G.-C. Wen, Conformal mappings and boundary value problems, 1992 105 D. R. Yafaev, Mathematical scattering theory: General theory, 1992 104 R. L DobrushLy R. Koteckfi, and S. Shlosman, Wulff construction: A global shape from local interaction, 1992 103 A. K. Tsikh, Multidimensional residues and their applications, 1992 102 A. M. Il'in, Matching of asymptotic expansions of solutions of boundary value problems, 1992 101 Zhang Zhi-fen, Ding Tong-fen, Haang Wen-zoo, and Doug Zhen-xi, Qualitative theory of differential equations, 1992 100 V. L. Popov, Groups, generators, syzygies, and orbits in invariant theory, 1992 99 Norio Shimakara, Partial differential operators of elliptic type, 1992 98 V. A. Vassillev, Complements of discriminants of smooth maps: Topology and applications, 1992 97 Itiro Tamura, Topology of foliations: An introduction, 1992 96 A. I. Markuskevich, Introduction to the classical theory of Abelian functions, 1992 95 Guangchang Dong, Nonlinear partial differential equations of second order, 1991 94 Yu. S. Il'yasbeako, Finiteness theorems for limit cycles, 1991 93 A. T. Fomenko and A. A. Tnzbllin, Elements of the geometry and topology of minimal surfaces in three-dimensional space, 1991 92 E. M. Nikiabin and V. N. Sorokin, Rational approximations and orthogonality, 1991 91 Matttorn Mimura and Hirosi Toda, Topology of Lie groups, I and II, 1991 90 S. L. Sobolev, Some applications of functional analysis in mathematical physics, third edition, 1991 89 Valeril V. Kozlov and Dmitrh V. Treshchgv, Billiards: A genetic introduction to the dynamics of systems with impacts, 1991 88 A. G. Kbovanskil, Fewnomials, 1991 87 Aleksandr Robertovich Keener, Ideals of identities of associative algebras, 1991 86 V. M. Kadets and M. I. Kadets, Rearrangements of series in Banach spaces. 1991 85 Mikio Ise and Masaru Takeuchi. Lie groups 1, II, 1991 84 Dio Trong Thi and A. T. Fomenko, Minimal surfaces, stratified multivarifolds, and the Plateau problem, 1991 83 N. 1. Porteako, Generalized diffusion processes, 1990 82 Yasutaka Siboya, Linear differential equations in the complex domain: Problems of analytic continuation, 1990 81 I. M. Gelfand and S. G. Gindikin, Editors, Mathematical problems of tomography, 1990 80 Junjiro Noguchi and Takashlro Ochial, Geometric function theory in several complex variables, 1990 79 N. I. Akhiezer, Elements of the theory of elliptic functions, 1990 109

(Continued in the back of this publication)

Translations of

MATHEMATICAL MONOGRAPHS Volume 113

Typical Singularities of Differential 1-Forms and Pfaffian Equations Michail Zhitomirskii

American Mathematical Society, Providence. Rhode Island in cooperation with

MHXAHJI )KHTOMHPCKHft

THIIHLIHbIE OCOEEHHOCTH EPEHIJHAJIbHbIX T

1- DOPM H YPABHEHH1 HOAbA The present translation is published under an agreement between MIR Publishers and the American Mathematical Society. 1991 Mathematics Subject Classification. Primary 58-02; Secondary 35F20, 53C15, 58A10, 58A17, 58A30, 58F36, 93B52. ABSTRACT. This book is devoted to the problems formulated by J. Pfaff at the start of the 19th century: to what simplest form can a differential form be reduced by a change of coordinates (local classification of differential 1-forms) or by a change of coordinates and multiplication by a function (local classification of Pfaffian equations)? Answers to these classification problems are applied to basic questions related to the geometry of singularities, stable and finitely determined germs, and normal forms. Modern applications of these classification results include contact geometry, theory of differential equations, control theory, nonholonomic dynamics, and variational problems. Some of these applications are given in the first three appendices. In the other appendices, analytic and topological classification, the classification of distributions and modules of vector fields, and the classification of closed differential 2-forms are discussed. This book contains many fundamental concepts, techniques, and methods for the study of singularities that appear in any classification problem. Library of Congress Cataloging-lu-Publication Data

Zhitomirskii, Michail. [Tipichnye osobennosti differentsiaYnykh 1-forms i uravnenii Pfaffa. English] Typical singularities of differential 1-forms and Pfaffian equations/Michail Zhitomirskii. p. cm.-(Translations of mathematical monographs, ISSN 0065-9282; v. 113) Includes bibliographical references and index. ISBN 0-8218-4567-5 1. Differential forms. 2. Singularities (Mathematics) 3. Pfafrs problem. I. Title. 11. Series. QA381.Z4513

1992

92-24410 CIP

515'.36-dc2O

Copyright ©1992 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America Information on Copying and Reprinting can be found at the back of this volume. The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. This publication was typeset using AA S-TEX, the American Mathematical Society's TEX macro system.

10987654321

979695949392

To the Memory of Professor Jean Martinet

Contents Introduction Chapter I. Main Results § 1. Stable and finitely determined germs; normal forms §2. Geometry of singularities Chapter II. Basic Notions, Definitions, Notation, and Constructions §3. Differential 1-forms and Pfaffian equations §4. Singularities and their characteristics §5. The homotopy method and its modifications §6. The infinitesimal equation, functional moduli, and "wild" jets §7. Classsification of submanifolds of a contact manifold §8. Solvability of equations with respect to germs of flat functions §9. Commentary Chapter III. Classification of Germs of Differential Forms §10. The class of a germ; preliminary normal form; Darboux theorem § 11. Singularities and their adjacencies § 12. Classification of coclass 1 singularities § 13. Classification of point singularities § 14. Basic results and corollaries; tables of singularities; list of normal forms; examples § 15. Commentary

Chapter IV. Classification of Germs of Odd-Dimensional Pfaffian Equations § 16. Class of Pfaffian equations; classification of 1 -jets; preliminary normal form § 17. Singularities

§ 18. Classification of germs at points of second degeneration manifolds § 19. Point singularities of 3-dimensional Pfaffian equations §20. Degenerations of codimension > 4 §21. Point singularities of Pfaffian equations in R2k+1

ix 1 1

5

9 9 14 17

20 25 26 28 31

31

35 41

47 53 57 59 59 61 71

82 91

99

viii

CONTENTS

§22. Basic results and corollaries; table of singularities; list of normal forms; examples §23. Commentary Chapter V. Classification of Germs of Even-Dimensional Pfaffian Equations §24. Singularities associated with the decrease of germ class; preliminary normal form §25. Other singularities (of the class n - 3 } §26. Classification of first occurring singularities of Pfaffian

equations in R', n = 2k > 6 §27. Degenerations of codimension > 4 §28. Normal forms of Pfaffian equations in R4 §29. Point singularities §30. Basic results and corollaries; table of singularities; list of normal forms §31. Commentary

113 116 119

119 120 128 137 140 147 147 148

Appendix A. Local Classification of First-Order Partial Differential Equations

153

Appendix B. Classification of Submanifolds of a Contact Manifold

155

Appendix C. Feedback Equivalence of Control Systems

157

Appendix D. Analytic Classification of Differential Forms and Pfaffian Equations 159

Appendix E. Distributions and Differential Systems

162

Appendix F. Topological Classification of Distributions

164

Appendix G. Degenerations of Closed 2-Forms in R2k

165

References

167

Author Index

171

Subject Index

173

List of Symbols

175

Introduction Ordinarily in any local classification problem interest focuses on the generic case: classification of germs of a generic object on a manifold (generic function, vector field, differential form, etc.). Usually the orbit of the germ of a generic object at a generic point is an open and everywhere dense set in the space of germs. Nongeneric points (for which this property is violated) are said to be singular. When classifying germs of a generic object at singular points, we study typical singularities. Roughly speaking typical singularities are irremovable under a small perturbation while nontypical singularities may be eliminated by a suitable small perturbation (they decompose into typical ones). In this monograph we deal with typical singularities of differential 1-forms

and Pfaffian equations. Pfaffian equations in modern terms are modules of differential 1-forms generated by one differential 1-form. Local equivalence of differential 1-forms corresponds to the action of the group of local diffeomorphisms (reversible coordinate transformations). The classification of Pfaffian equations may be considered to be the classification of differential 1-forms: in addition to a change of coordinates we can multiply a 1-form by a nonvanishing function. The problem of classifying differential 1-forms and Pfaffian equations was formulated by Pfaff at the start of the 19th century (in terms of reduction to "simplest" forms). The first basic step in this problem was made by Darboux, whose theorem can be formulated as follows: for a generic differential 1 form (generic Pfaffan equation) on a manifold M there exists an open everywhere dense subset k c M such that all germs of the 1 form (Pfaffian equation) at points of M are equivalent to a standard germ. Martinet was the first to study singularities (classification of germs at points

of M) systematically. His results are collected in [Mar] which is both the starting point and a guide for other studies including the present one. Unfortunately I knew Professor Martinet only by correspondence. My wish to meet him personally will never come true. Singularities and the classification of 1-forms and Pfaffian equations are

interesting not only as a classical problem but also (and perhaps mainly) is

x

INTRODUCTION

because of their applications (in contact geometry, the theory of partial differential equations, control theory, nonholonomic dynamics, and variational problems). Some important applications have appeared in the last 10-15 years. Most applications in contact geometry are due to the relative Darboux theorem, which was proved by Givental in 1982. This theorem states that two submanifolds of a contact manifold are contactly locally equivalent (i.e., their germs lie in the same orbit of the group of diffeomorphisms that preserve the germ of the contact structure) if and only if the Pfaffian equations obtained by the restriction of the contact structure to the submanifolds are locally equivalent (with respect to the action of the complete group of diffeomorphisms).

Classification results for Pfaffian equations can be reformulated as those for submanifolds of a contact manifold. The classification of Pfaffian equations also leads to the classification of first-order partial differential equations since the latter may be considered as hypersurfaces in a contact manifold of 1 -jets.

An application to control theory is associated with the fact that a Pfaffian equation generated by a 1-form w defines a module of vector fields v such

that w(v) = 0. On the other hand, a differential 1-form w defines an affine module of vector fields v such that w (v) = 1 . Such modules may be interpreted as control systems, linear with respect to the control. In this monograph we collect results on the geometry of singularities and classification of differential forms and Pfaffian equations. We also present applications and closely related classification problems. All the results are given with proofs. In the proofs we use the technique of jets on a manifold, the homotopy method and its modifications, the transversality theorem, the necessary and sufficient conditions for a germ's stability and finite determinacy, the relative Darboux theorem and related results, and theorems by Belitskii and Roussarie on the solvability of equations with respect to germs of flat functions. In Chapter II we collect the relevant material and the basics of singularity theory. In Chapters III-V we discuss differential 1-forms, and odd-dimensional and even-dimensional Pfaffian equations. At the end of each chapter we summarize the main results, tabulate the singularities, and list the normal forms. The main results of the book are also collected in Chapter I. In Appendices A, B, and C we apply the results respectively to the classification of the first-order partial differential equations, to the study of the geometry of submanifolds of a contact manifold, and to some problems of control theory. Our main results hold relative to the C°°-equivalence or C'-equivalence for arbitrary r < oc. Nevertheless, we also dwell (on the level of formulation, conjecture, and brief discussion) on analytic and topological classification problems (Appendices D and F). We present some classification results for distributions, differential systems (modules of vector fields), and closed differential 2-forms (Appendices E and G).

INTRODUCTION

u

Some of the results in the book are due to Martinet (degenerations of codimension 1: first occurring singularities of 1-forms and odd-dimensional Pfaffian equations [Mar]), some to Lychagin (point singularities in the evendimensional case associated with the vanishing of 1-forms; such singularities correspond to first occurring singularities of first-order partial differential equations [L1]), other results were obtained by the author (degenerations of codimension > 2, in particular, first occurring singularities of evendimensional Pfaffian equations [Z3, Z4, Z8, Z9]). Some of the results are published for the first time (not taking into account the preprint [Z14]). The results in this monograph give complete answers to the principal questions of the problems of local classification. I would like to express my profound gratitude to Professor V. Arnold, to my teacher Professor G. Belitskii, to Professors Yu. Il'yashenko and V. Lychagin, with whom I discussed both some of the concrete results and the book as a whole. I am very thankful to Professor D. Leites who published the preliminary text [Z14] in the transactions of his Seminar on Supermanifolds and called my attention to related classification questions in superanalysis in his introduction to the text.

CHAPTER I

Main Results Our basic result is a complete list of stable germs of differential 1-forms and Pfaffian equations and a complete list of finitely determined typical singularities. Recall that a germ of an object is said to be stable if a nearby object has the same (up to equivalence) germ at a point close to the source of the initial germ; finite determinacy means that the orbit of a germ in the set of all germs with a fixed source has finite codimension (a germ is k-determined if its k -jet defines its orbit). Darboux established a classification for the generic case (germs of generic 1-forms and Pfaffian equations at generic points), Martinet [Mar] gave a classification for degeneracies of codimension I (first occurring singularities of 1-forms and odd-dimensional Pfaffian equations), and Lychagin [L1] produced a classification of point singularities of even-dimensional 1-forms and the corresponding Pfaffian equations. Other classifications can be found in

recent papers by Zhitomirskii [Z3, Z4, Z8, Z9, Z14]. In this chapter we do not discuss normal forms with functional parameters corresponding to singularities that are not finitely determined. J I. Stable and finitely determined germs; normal forms

THEOREM 1.1 (on stable germs of Pfaffian equations). (1) A germ of a generic Pfaffian equation at a generic point of an n-dimensional manifold is

stable. Nongeneric points form a set of codimension >_ 4 if n > 5 and of codimension > 3 if n = 3 or n = 4. (2) Any stable germ of a Pfaffian equation on an n-dimensional manifold is equivalent to one and only one germ on the following list:(' )

n = 3:

dz+xdy=0,

(1.1)

dy+x2dz = 0;

(1.2)

dy, +x2dy2 = 0;

(1.3)

n-4: (')For a Pfaffian equation E (a module of 1 -forms generated by a single 1-form) we use the

notation w = 0, where w is one of the generators of E. 1

1. MAIN RESULTS

2

n=2k+1>5: =0,

(1.4)

(1.5)

=0, =0;

dy, dy,

n=2k>6:

=0,

dyi

(1.6) (1.7) (1.8)

dy,+x2dy2+...+xk_1dyk_I+x1yk(1+A+x2)dxk

+Xlxkdyk =O,

dy +x2 dy

(1.9)

A E (0, 1),

dy,k-1 + (x xk(2A+x)-x yk )dx, 2 (1.10) A E (0, x). +(XIxk +xlyk(2A+x2))dyk = 0, REMARK. The normal forms (1.1), (1.3), (1.4), and (1.8) are the classical Darboux models. One of the coordinates does not occur in the normal forms (1.3) and (1.8). The normal forms (1.2) and (1.5) are equivalent to those obtained by Martinet [Mar]. The normal forms (1.6), (1.7), (1.9) and (1.10) 1

2

9-1

1

1

were obtained in [Z8, Z9, Z14]. The existence of a parameter A in the normal forms (1.9) and (1.10) is not in conflict with stability (because the corresponding singularities are not isolated). THEOREM 1.2 (on stable germs of differential 1-forms). (1) A germ of a generic differential 1 -form at a generic point of an n-dimensional manifold is stable. Nongeneric points form a set of codimension _> 2. (2) Any stable germ of a Pfaffian equation on an n-dimensional manifold is equivalent to one and only one germ on the following list:

n=2k+1:

(1.11)

(1 +xi)dyl

(1.12)

n = 2k: (1.13) (1.14)

REMARK. The normal forms (1.11) and (1.13) are classical Darboux models. The normal forms (1.12) and (1.14) were obtained by Martinet [Mar]. THEOREM 1.3 (on finitely determined germs of Pfaffian equations). (1) Any

germ of a generic Pfafan equation on a 3-dimensional manifold is finitely determined (2) Let n > 5 (respectively, n = 4 ). In the set of Pfaffian equations on an n-dimensional manifold M there exists an open subset A satisfying the following property. for each Pfaffian equation E E A there exists a submanifold

U. STABLE AND FINITELY DETERMINED GERMS

3

M C_M of codimension 4 (respectively, 3) such that at none of the points a E M is the germ of E finitely determined. (3) Let M(E) be the set of points at which the germ of a Pfafran equation E: {w = 0} on a manifold M is unstable but finitely determined and let M0(E) be the set of points at which the 1 -form w vanishes. Then for a generic Pfgftan equation E (3.1) M(E) consists of isolated points; (3.2) M0(E) C R(E); (3.3) if the dimension of M is even, then M(E) = M0(E) . (4) Any finitely determined germ of a generic Pfgftan equation on an ndimensional manifold is either stable (in which case it is equivalent to one and

only one of the germs (1.1)-(1.10)) or unstable and equivalent to one and only one germ in the following list

n = 3:

zdz+2(xdy-ydx)+dG,(x, y)=0,

(1.15)

dy+(xy+x2z+bx3z2)dz=0, (xy+x3/3+xz2 +bx3z2)dz = 0; dy+

(1.16)

(1.17)

n=2k>4: k E (xrdy,-y,dx,)+dG.. ,µk= 0;

(1.18)

r=1

n=2k+1>5: k

zdz+2(xidyi -y1dxi)+dG,,

,

µk

=0,

(1.19)

r=1

dy,+Ex.dy1+{xl[yl+2E x1dy1-Gµz....,µk+x1z+bxiz2]}dz=0, i=2

i=2

(1.20)

dy1+F,xtdyr+{xl[yl+2>xrdy1-GA'....,µk+3x2 +z2 1=2

=2

+b ( S xi + 3x1 z2 + z4) ] } dz = 0. \\

///

(1.21)

Here Gµ - Gµ (xl , yl , ... , xk , yk) is the normal form of a µk µk quadratic Hamiltonian in the symplectic space (R2k,

dx1 Ady1 +..-+dxk Adyk) with invariants ±u1 , ... , fpk (eigenvalues of the corresponding Hamiltonian vector field); µ=,...,µk = G µ = ...., µk (x2 , y2 , ... , xk, yk) and G,(x, y)

I. MAIN RESULTS

4

are similar notations of normal forms of quadratic Hamiltonians in the symplectic spaces (It2k-2 , dx2 A dye + - + dxk A dyk) and (R2, dx A d y) , respectively. A list of normal forms of quadratic Hamiltonians can be found in [A2]. Parameters b, R , µI , ... , U k are the moduli of normal forms. REMARK. The normal form (1.18) was obtained (in other terms) by Lychagin [L1]. The normal forms (1.16) and (1.17) can be found in [Z8]. The normal forms (1.15), (1.19)-(1.21) are given here for the first time. THEOREM 1.4 (a complete classification of germs of generic Pfaffian equations on a 3-dimensional manifold). Any germ of a generic Pfaffian equation on a 3-dimensional manifold is equivalent to one and only one of the germs

(1.1), (1.2), (1.15), (1.16), (1.17). THEOREM 1.5 (on finitely determined germs of 1-forms). (1) In the set of all differential 1 forms on an n-dimensional manifold M there exists an open subset A satisfying the following property: for each 1 form to c A there exists a submanifold M c M of codimension_2 such that the germ of to is finitely determined at none of the points a E M . (2) The set of points at which the germ of a generic differential 1 form to is unstable but finitely determined coincides with the set of points at which w vanishes.

(3) Any finitely determined germ of a generic 1 -form on an n-dimensional manifold is either stable (in which case it is equivalent to one and only one of germs (1.11)-(1.14)) or unstable and equivalent to one and only one of the germs:

n=2k: k

Z1: (x,dy,-y;dx1)+dG# 1

(1.22)

.....vk;

i=I

n=2k+1: fzdz+

k

F'(x;dyr-y;dx;)+dGM.

=I

µk

(1.23)

(the notation is the same as in Theorem 1.3.) All the above classifications of differential 1-forms and odd-dimensional Pfaffian equations hold with respect to Co equivalence; for even-dimensional Pfaffian equations the given results are true, generally speaking, with respect

to C' equivalence for any 6 < r < oc (the C°° classification is different). We do not discuss the analytic classification (see Appendix D) or the two-dimensional case for Pfaffian equations because a local classification of Pfaffian equations in the plane immediately leads to a local classification of vector fields in the plane with respect to orbital equivalence. Indeed, it is easy to check that the two Pfaffian equations a d x+ b d y= 0 and a d x+ b d y=

§2. GEOMETRY OF SINGULARITIES

5

0 are equivalent if and only if the vector fields a(8/8y) - b(8/8x) and a(8/8y) - b(810x) are orbitally equivalent. Results on local orbital equivalence of vector fields in the plane can be found in [Boll. §2. Geometry of singularities

2.1. Singularities of a generic Pfaffian equation E: w = 0 on a manifold M of dimension 2k + I > 5. At a generic point a c M we have (a) w A (dw)k1R : 0 (here and below conditions on w are valid or violated simultaneously for all generators of E). The germ of E at a point a satisfying (a) is stable, 1-determined, and reducible to the model (1.4). Degeneration (i.e., the violation of (a)) takes place on a stratified submanifold M1 c M of codimension 1. For a generic point a E M, the following conditions are valid:

(b) locally, near a, the set MI is a smooth hypersurface; (c) w n (dw)k-1IQ 54 0;

(d) the space Kerw A (dw)k- IIQ c TM is transversal to MI . The germ of E at a point a E satisfying (b), (c), and (d) is stable, 2-determined, and reducible to the model (1.5). We can show that under condition (c) dim Kerw n (dw)k-1 = 2. This implies that the second degeneration takes place on a stratified submanifold MI

M2 C Mi of codimension 3 in M (w does not vanish but one of the conditions (b), (c), (d) is violated) and at isolated points forming a set Mo C MI (w vanishes); Mo n M2 = 0. For a generic point a E M2 the following conditions are valid:

(e) locally, near a, the sets Ml and M2 are smooth submanifolds; (f)

wA(dw)k-II"_j 0;

(g) Kerw A (dw)k-I1R C T Ml ; but (dw)k-'IR

(h) Kerw n ®T M2 =T M ; (i) Kerwi« is transversal to M1 . ' The germ of E at a point a E M2 satisfying conditions (e)-(i) is stable, 1

3-determined, and reducible either to the model (1.6) or to the model (1.7). The third degeneration takes place on a submanifold M3 c M2 of codimension 4 in M ((g) and (i) are valid, one of conditions (e), (f), (h) is violated) and at isolated points forming a set M4 ((i) is violated); M4nM3 = 0; M4 n Mo = 0. The submanifold M3 divides M2 into two parts corresponding to models (1.6) and (1.7). In M3 there are no points where the germ of E is finitely determined. At each point a E M4 the germ of E is unstable but 5-determined and reducible to normal forms (1.20) or (1.21). At each point a E Mo the germ of E is unstable but 1-determined and reducible to (1.19).

1. MAIN RESULTS

6

2.2. Singularities of a generic Pfaffian equation E: w = 0 on a 3-dimensional

manifold M. At a generic point a E M we have (a) w n dwI. 0. The germ of E at a point a satisfying (a) is stable, 1-determined, and reducible to the model (1.1). Degeneration (i.e., the violation of (a)) takes place on a smooth surface MI . For a generic point a c MI the following conditions are valid: (b) the space Ker w Io c T,M is transversal to MI ; (c) wI',:)6 0.

The germ of E at a point a E MI satisfying (b) and (c) is stable, 2determined, and reducible to the model (1.2). The following degenerations take place at isolated points of the manifold MI . Let M2 be the set of isolated points a E MI at which w1,, 0 but (b) is violated, and let MO be the set of points at which w vanishes. At each point a E MO the germ of E is unstable but 1-determined and reducible to

the normal form (1.15). At each point a c M2 the germ of E is unstable but 5-determined and reducible either to the normal form (1.16) or to normal form (1.17).

2.3. Singularities of a generic Pfaffian equation E: w = 0 on a manifold M of dimension 2k > 4. At a generic point a E M we have (a) w A (dw)k-IIQ # 0. The germ of E at a point a satisfying (a) is stable, 1-determined, and reducible to the model (1.8). Degeneration (i.e., the violation of (a)) takes place on a closed stratified

submanifold MI c M of codimension 3 (w does not vanish but (a) is violated) and at isolated points forming a set MO (w vanishes). This structure of singular points was obtained by Martinet [Mar]. At each point a E MO the germ of w = 0 is unstable but 1-determined and reducible to the normal form (1.18).

In the case k = 2 the germ of E is finitely determined at none of the points of the curve MI . In the case k > 3 the germ of E at a generic point a E MI is stable and reducible either to the normal form (1.9) or to (1.10). To define conditions of general position for a point a E MI let us introduce a volume form u on M and define a vector field X by the following equation µ(X ,

YI ,

...

,

Y2k_1) = w n

(dw)k-i(Y1

, ...

, Y2k_I )

(for arbitrary vector fields Y1 , ... Y2k-1). The field X is defined within multiplication by a nonvanishing function. Each point a e MI is a singular

point of the field X. Moreover, (2k - 3) eigenvalues of X at a E MI are necessarily zero; the other three eigenvalues Al , A2 , k3 satisfy the condition Al + A2 + A3 = 0. For a generic point a e MI we have (b) ).1).2).3 # 0;

(c) ).1, A2 , and A3 are distinct. At a point a E MI satisfying (b), (c),

§2. GEOMETRY OF SINGULARITIES

7

and one more condition of general position (see Chapter V) the germ of E is stable and reducible to the normal form (1.9) (if Im,l1 = 0, i = 1, 2, 3) or to (1.10) (if, within the numeration, Im),1 = -Im )'2 0) . The following degeneration is associated with violating (b) or (c). It takes place on a codimension 4 submanifold. At points of this submanifold the germ of the Pfaffian equation is not finitely determined. 2.4. Singularities of a generic differential 1-form w on a 2k-dimensional

manifold M. At a generic point a c M we have (a) (dw)kI. 96 0; (b) w1. 0 0 . The germ of w at a point a satisfying (a) and (b) is stable, I-determined, and reducible to the model (1.13). Degenerations take place on a stratified submanifold M1 ((a) is violated) and at isolated points forming a set Mo ((b) is violated); M1 n Mo = 0. For a generic point a E M1 the following conditions are valid: (c) w n (dw)k-11, 0 and consequently dim Kerw A (dw)k-1IR = 1 ; (d) locally, near c f, the set M1 is a smooth hypersurface; (e) the direction Kerw A (dw)k-II" C TM is transversal to M1 . The germ of w at a point a E M1 satisfying (c), (d), and, (e) is stable, 2-determined, and reducible to the normal form (1.14). The following degeneration (violation of one condition in (c)-(e)) takes

place on a submanifold M2 c M1 of codimension 2 in M. It divides the manifold M1 into two parts corresponding to the signs + and - in the normal form (1.14). At each point a c M2 the germ of w is not finitely determined.

At each point a E Mo the germ of w is unstable but 1-determined and reducible to normal form (1.22).

2.5. Singularities of a generic differential 1-form w on a manifold M of dimension 2k + 1 . At a generic point a c M we have (a) w n (dw)kIQ 76 0.

The germ of w at a point a satisfying (a) is stable, 1-determined, and reducible to the model (1.11). Degeneration takes place on a stratified manifold M1 , codim M1 = 1. For a generic point a E M1 the following conditions are valid: (b) wIR 96 0;

(c) (dw )k IR V- 0 and consequently dim Ker dw I, = 1 ;

(d) locally, near a, the set M1 is a smooth hypersurface; (e) the direction Kerdwlo C T M is transversal to M1 . The germ of w at a point a E M1 satisfying (b)-(e) is stable, 2-determined, and reducible to normal form (1.12). The following degenerations take place on a submanifold M2 C M1 of codimension 2 in M (violation of one of conditions (c)-(e)) and at isolated

S

1. MAIN RESULTS

points forming a set MO ((b) is violated); MO n M2 = 0. The manifold M2 divides Ml into two parts corresponding to the signs + and - in the normal form (1.12). At no point a E M2 is the germ of in finitely determined. At each point a E MO the germ of w is unstable but 1-determined and reducible to the normal form (1.23).

CHAPTER II

Basic Notions, Definitions, Notation, and Constructions All objects considered below (differential forms, vector fields, diffeomorphisms) and families of objects will be assumed to be smooth (Co), °unless there is an explicit mention to the contrary. We use M to denote a basic n-dimensional manifold on which we consider differential 1-forms and Pfaffian equations. 1-forms defined on M will usually be denoted by the Latin letter w and their germs by the Greek letter cc. Basic constructions in this chapter include: (1) the transversality theorem; (2) the homotopy method and its modification necessary to prove that a jet is sufficient or that a germ is stable, and to obtain preliminary and invariant normal forms; (3) the method to prove the absence of the sufficient jet and stability property or to prove the existence of functional moduli and the method of their presentation; (4) the relative Darboux theorem, normal forms of hypersurfaces, and functions in contact space; (5) the solvability of partial differential equations with respect to germs of flat functions. In this chapter we give a number of examples. Some of these examples are not obvious, and the statements in the examples follow from the results obtained in Chapters III-V.

§3. Differential 1-forms and Pfafan equations In this section we give examples of Pfaffian equations as the restrictions of a contact structure on submanifolds, and trace the relation between Pfaffian equations and distributions. We introduce notation for the spaces of 1-forms and Pfaffian equations, and their germs and jets at a point. We introduce the notion of a jet on a manifold, which will be used many times for studying nonisolated singularities, and we introduce the notion of the equivalence of germs and their jets at a point and on a manifold. We discuss the topology in the spaces of 1-forms (Pfaffian equations) and their germs.

U. BASIC NOTIONS. DEFINITIONS, NOTATION, AND CONSTRUCTIONS

10

3.1. Pfaftiian equations. Let AI(M) be the space of external differential 1-forms (referred to below as 1-forms) on M. A Pfaffian equation can be defined as a module E C A' (M) over the ring of smooth functions generated by a single 1-form w E A' (M) . We say that this form represents the Pfaffian equation. A Pfaffian equation is represented by 1-forms that are equal when multiplied by a function that is nonvanishing on M. We will use the notation {w = 0} for a Pfaffian equation E, where w is one of the forms representing E. If E: {w = 0} is a Pfaffian equation, where w is nonvanishing on M, then E can be considered a distribution of codimension 1 on M (a field of hyperplanes), since in the tangent space TaM (a e M) the hyperplane consisting of tangent vectors annihilated by the form wla is defined. For example, a contact structure, i.e., the field of kernels of a contact 1form on an odd-dimensional manifold is a Pfaffian equation. A variety of Pfaffian equations can be obtained by the restriction of a contact structure on odd- and even-dimensional manifolds. EXAMPLE 3.1. Consider the contact structure {dz + x dy + u dv = 0} in 1R5 . Its restriction to the three-dimensional manifold {u = v = 0} coincides with the Pfaffian equation {dz + x dy = 0} , i.e., with a contact structure in

R. 3 EXAMPLE 3.2. The contact structure {d z + x dy + u dv = 0} in IR5 restricted to the 4-manifold {u = 0} is a Pfaffian equation in R4 (with coordinates z, X, y, V). Such an equation is called a quasicontact structure. EXAMPLE 3.3. The restriction of the above contact structure to the submanifold {y = 0, u = x2} is the distribution {w = dz +x2 dv = 0} in R3 , but it is not a contact structure since w A dw vanishes in the plane {x = 01 . EXAMPLE 3.4. The restriction of the above contact structure to the fourdimensional submanifold { z - f (x , y, u, v) = 0} is a distribution if and

only if grad f j4 0 at each point (x, y, u, v) . NOTATION. The set of all Pfaffian equations on a manifold M will be denoted by PAI (M) .

3.2. Germs of 1-forms and Pfaffian equations. Jets at a point and on a The set of germs of 1-forms at a point a E M is denoted by W (M) and the set of germs of Pfaffian equations by PW (M) . The point

manifold.

a is called the source of a germ. When studying germs we can often assume that M = R" and a = 0 E R". The set of all germs of 1-forms and Pfaffian equations at 0 E R" will be denoted W(n) and PW(n) respectively. There exist natural bijections ZrQ: W (M) --. W(n) and Ira: PW (M) PW (n) . Sometimes we shall identify (using these bijections) W (M) with

W(n) and PW (M) with PW(n). In addition we use ;ra to denote the maps AI(M)

- W(n) and

PAI(M) - PW(n)

§3. DIFFERENTIAL I-FORMS AND PFAFFIAN EQUATIONS

II

such that IraE = 7rae,

7raw = Iraw,

where w e At (M) , E E PAl (M) , to is the germ of w , and e is the germ

of E at the point a. A germ e c PW(n) is a submodule of W(n) over the ring of the germs of smooth functions generated by co E W(n). For germs in PW(n) we use the notation {w = 0). If H is the germ of a function at 0 E R", H(0) # 0, then {co = 0} and {Hw = 0} are the same equations. Next we define the k -jet of a 1-form (Pfaffian equation, function, vector field) on the germ of a manifold S C R" at the point 0 E R". The notation below refers to all these objects simultaneously.

Let 9JIs be the ideal of germs at 0 E R" of functions that vanish at k to denote the

each point of S and 9)12. be its kth power. We also use following spaces of germs of vector fields and 1-forms:(')

f,,v1, f E9Jts,

9RS=

911 Efw,,j.E9Jts,wiEW(n) Elements of the factor-spaces

Jk

C°° n

Jk

9Jtk+1

k+t W(n)/9t,

Jk

Vect n 9Rk+l

are called the k -jets of functions, 1-forms and vector fields on the germ of the manifold S respectively. We use is to denote the map sending an object to its equivalence class from JS . In the case S = {0} the elements of the spaces mentioned above are simply called k -jets. In this case the notation is k

k

`m(01 = 9R

k

k ,

J(o) = J

,

k 1{0}

k J

Given coordinates x,, ... , x,, in a neighborhood of 0 E R', the k -jet of a function f(x, , ... , x") is identified with the respective polynomial

f

jkJ = IaIGk

a... a x"

It , XI

a

(0)xi ... xn"

e

where a=(at,... ,a"), Ian=al+...+a",a!=a,!...a"! If a manifold S E R" is given by the equations {xt = 0, ... , xl =

01, then the k -jet of the function f (x, , ... , x") on the germ of S at 0 can be considered as a polynomial in the variables x, , ... , x1 whose coefficients ()Below CO°(n) is the set of function-germs at 0 E R" ; Vect( n ) is the set of germs of vector fields at 0 c R"

.

12

11. BASIC NOTIONS, DEFINITIONS, NOTATION, AND CONSTRUCTIONS

are function-germs of the variables x1+1 , a191f

Jsf =

11

BI <_k

...

, x,, :

i (0, ... '0'X l+l , ... axl ... ax,

,

X,91 ,

where 6 _ ($1, ... , $r) . The relations

jsk (Ea,(x)dx) = > js(a1(x))dx1,

js (1: bl(x)ax

Ejs(bi(x)) 0 ,

are representatives of k -jets for 1-forms and vector fields in the case of fixed coordinates.

The set of Pfaffian equation k -jets on a germ of a manifold S is also denoted JS . It is the factorization of PW(n) by the following equivalence relation: two germs are equivalent if there exist representatives of them in W (n) with coinciding k -jets on S. A typical representative of the equivalence class for a germ {co = 0} is the germ {jsco = 0} . EXAMPLE 3.5. Consider the germ co = e" l"3dx3 + x2ex,dx4 E W(4) and the manifold S given by {x1 =x2 = 01. Then

jlw=dx3+x2dx4, 2w = (1 + x1x3)dx3 + x2(l + x3)dx4 ,

jsw = (1 +xlx3)dx3 +x2e"'dx4, jsw = (1 +XIx3+XIx3i2)dx3 +x2ex'dx4. The germ {dx3+x2(1 +x3)dx4 = 0} is a representative of the jet j2{w = 01 (as well as the germ {.2w=0}). In the case of fixed coordinates x1 , ... , xR we denote by fits the set of all function-germs (germs of vector fields or differential forms) which coincide

with their i-jet on S and belong to fits Let fits be the space of the germs at 0 of flat functions on a manifold S C R" (a function is flat on S if it and all its derivatives vanish on S). NOTE. Let S JO}. Then fits ;E n, 9)1s . EXAMPLE 3.6. Consider the smooth functions ;(v) =

j 0,

IYI <_ I/i,

l y.(y), Iyl > I/i, where 0 < y,(y) < I , and the sequence yr > 0 tends to zero sufficiently S : {x = 01. Then f E nr<m fills > fast. Let f = f (x, y) _ E We also use fits to denote the set of germs of vector fields that are flat on S: v = > f,,v, , j E Ms, vi E Vect(n) and a similar set of differential 1-forms flat on S.

§3. DIFFERENTIAL I-FORMS AND PFAFFIAN EQUATIONS

13

3.3. Local and jet equivalence. Let Diff(n) be a group of diffeomorphism

germs at 0 E R" with fixed point 0. The group Diff(n) acts naturally in W(n). The Diff(n)-action in PW(n) is defined by the relation

{w=0}={40'w=0}

(l EDiff(n),wE W(n))

(one can easily see that the result does not depend on the choice of the representative).

Germs wl E W (M), w2 E W,(M) (germs el E PW (M), e2 E PW,(M) respectively) are said to be equivalent if the germs rr,, col , rrpw2 (rrne, , 7r,e2 respectively) lie on the same orbit of the Diff(n)-action.

We willl also need the notion of C'-equivalence, r > 1 . Germs cal E W (M) , (02 E WP(M) are called C'-equivalent if there exists a germ of a C'-diffeomorphism with fixed point 0 that transforms 7r.6w, into The equations {cot = 0} E PW (M) and {w2 = 0} E PW,(M) are said to be C'-equivalent if for some C'-function H, H(0) 94 0, the germs Hw1 E W (M) and w2 E W,(M) are C'-equivalent. The Diff(n)-action is also defined in the space of k -jets of 1-forms and Pfaffian equations, namely, _ 1k 3 (3.1) 4)

where 40 E Diff(n), is a k -jet, E W(n) (or E PW(n)), jk4 = . Consider the subgroup Diffs(n) c Diff(n) of diffeomorphisms preserving

the germ of a manifold S. The group Diffs(n) acts in the space of k -jets on the germ of S at 0 E S, viz.,

-by =jkd'µ,

(3.2)

where u is a k jet on S, K E W(n) (PW(n)), jsC = a. An easy check shows that the actions (3.1) and (3.2) are correctly defined (i.e., the result is independent of the choice of or k ). 3.4. Topology. Generic forms and Pfaffian equations. In order to define a generic 1-form (Pfaffian equation) we need the notion of the structure of a topological space in A' (M) and PAS (M) . We use the Whitney topology. Consider the space of the k -jets of all 1-forms or Pfaffian equations on a manifold at all points of M. Denote this space by (M, J k ). Its elements

are pairs (a,c), The basis of neighborhoods in the Whitney topology consists of nonempty

sets parametrized by pairs (k, U), where k > 0 and U is an open set in the space (M, Jk) . The pair (k, U) defines the set of 1-forms {w E A'(M)I(a, .krrnw) E U, a E M} or the set of Pfaffian equations

{E E PA'(M)I(a, jkJr,E) E U, a E M}.

11. BASIC NOTIONS, DEFINITIONS, NOTATION, AND CONSTRUCTIONS

14

If the manifold M is compact, then the Whitney topology is equivalent to the topology of uniform convergence.

We say that a set is thick if it is an everywhere dense intersection of a countable number of open sets. By saying that some property is valid for a generic form (generic Pfaffian equation) we mean that it is valid for some thick set in A'(M) (PA'(M)) relative to the Whitney topology. Now we define the topology in the space W(n). The basis of neighbor-

hoods consists of sets parametrized by pairs (k, U), where k > 0 and U is an open set in Jk . A pair (k, U) defines the neighborhood {w E W(n)Ijkw E U}.

The topology in the space PW(n) is defined similarly. Note that this is a topology in the space of formal series corresponding to 1-forms and Pfaffian equations. §4. Singularities and their characteristics

We need some terminology used in singularity theory. For singularities of functions this terminology is contained in [AVG, AVGL]. In this section we introduce similar notions for 1-forms. They cover Pfaffian equations and finitely-smooth equivalence of forms and Pfaffian equations almost entirely. Note that in contrast to singularities of functions, singularities of 1-forms and Pfaffian equations are nonisolated and the classification of germs with unfixed sources differs from the classification of germs in W(n) or PW(n) (with a fixed source 0 E 1R" ).

Also, in this section we present a necessary variant of the transversality theorem.

4.1. Singularity classes. According to our definition, a singularity is the orbit of a germ to c- W(n) .(2) A set S C W(n) is called a singularity class if it is closed relative to the Diff (n)-action. We usually need singularity classes which are defined by some condition on an 1-jet

S = {w E W(n)ljtw E S),

(4.1)

where S is some submanifold (or stratified submanifold with a finite number of strata [AVG]) in the space J1 of 1 -jets. In the case of a stratified submanifold S each stratum defines its singularity subclass as well. By saying that some property is valid for a generic germ in W(n), we mean that it is valid for some subset of germs satisfying (4.1), where S is an open and everywhere dense set in J1. (2)lt is convenient to include the case of an open orbit in W(n). This case is however excluded in the phrases like the first occuring singularity is ...

.

S4, SINGULARITIES AND THEIR CHARACTERISTICS

{

15

Consider the singularity class (4.1). Let 1 < k < oc and let Sk = E JkI j ! E &}. The validity of some property of generic germs in the

set S like (4.1) means that it is valid for some subset Q c S such that for I < k < oc (i) jk Q C Sk , (ii) jkQ is open in Sk , and (iii) j k Q is everywhere dense on the union of strata having the minimum codimension. The codimension of the singularity class S given by (4.1) in W(n) is the codimension of S in J1 (the minimum codimension of all strata). EXAMPLE 4.1. A generic germ co E W(2k + 1) satisfies the condition co A (dw)k 1o 34 0 (this is a relation on the 1 -jet of a germ). EXAMPLE 4.2. For a generic germ {w = 0} of a Pfaffian equation in i 6 the condition to A (d w)210 0 is valid. Consider the class S of singularities violating this relation. One can show (see §24) that codim S = 3. EXAMPLE 4.3. Let S be the previous singularity class. For a generic germ

0 is valid. In spite of this there {w = 0} E S the relation co A dwlo exist germs /< E S such that for each germ µ E S close to A we have

µ A dµjo = 0. We may take for p a germ {w = 0), where to E W(6), wlo = 0, rankdwlo = 6. The point is that the relation to n dwlo = 0 is violated for 1 -jets of the set { C J' I j° = 0} of codimension 6 which is one of the strata of S C J I . Next we introduce notation connected with a singularity class S, a 1-form

w EA'(M) andagerm wE W (M). We use S(w) to denote the set of points a E M for which the germ of w at a belongs to S after the translation Jr,, . EXAMPLE 4.4. Let S = {w E W (4)lw n dwlo = 01, w - dy, + x2y2 dx, E A' (R4) . Then S(w) is a plane given by {x2 = yZ = 01.

For the germ co of a form w E A' (M) at a point a E M we denote by S(w) the germ of S(w) at a. Clearly, the definition of S(w) is independent of the chosen w E A1(M). Typical and isolated singularities. A singularity class S c W(n) is called typical if there exist an open set U c A' (M) and a thick set A C A' (M) such that S(w) # 0 for any form w E A f1 U. If in this definition the condition S(w) / 0 is replaced by the condition that S(w) consists of isolated points, then singularities belonging to S are called isolated or point singularities (more exactly, S is a point singularity class).

Adjacencies. Let k, p < oc, I = max (k, p). Then we say the class Si = {w E W(n)I jk(d E S, } is adjacent to the class S2 = {w E W(n)I j°w C S2) if

the closure of SZ

!

in J1 contains the entire set Sl

ESi},

1

. Here

s2.!_{'EJ'II"'ES2}.

11. BASIC NOTIONS. DEFINITIONS, NOTATION, AND CONSTRUCTIONS

16

EXAMPLE 4.5. The class S1 = {w E W(6)I(dw)21o = 0} is adjacent to the class S2 = {w C W(6)Iw A (d(0)Z1o = 0, who 01.

EXAMPLE 4.6. The singularity class S1 = {w E W(4)fwIo = 0} (of codimension 4) is not adjacent to the class S2 = {w E W(4)I(dw)2Io = 0, wl0 0} (of codimension 1). We use the notation S2 ~ S1 to denote the adjacency of S1 to S24.2. The transversality theorem. For l-forms and Pfaffian equations we have the exact analog of the Thom transversality theorem (see [AVG, Mar, Math]). We use the following version of this result. THEOREM 4.1 (see [Mar]). Let S be a stratified submanifold in the space

(M' Jk) For a generic 1-form w E A' (M) its k -jet extension (i.e., the map M - (M, Jk) transforming a point a E M into the pair (a, jk7rQw)) is .

transversal to k REMARK. Under some assumptions on S the transversality condition takes place not only for a thick set but for some open everywhere dense set of 1forms as well. In particular, this is true if codimS > dim M [Mar] when S is a closed smooth manifold [AVG]. Let S c J1 be a stratified manifold with strata Si . Consider the singularity class S given by (4.1). Theorem 4.1 implies the following corollaries.

COROLLARY 4.1. For a given generic 1 form w E A' (M) the set S(w) is

a stratified manifold: S(w) = US,(w), where S,(w) is a singularity class corresponding to the stratum S,

.

COROLLARY 4.2. A singularity class S is typical if and only if codim S < n

and is isolated if and only if codim S = n. EXAMPLE 4.7. Consider the following singularity classes of 1-forms (a) {w E W(nflwlo = 0} ; (b) {w E W(n)1m1o = 0, dwlo = 01; (c) {w E W(n)I(m n dw)Io = 0, wlo :A 0). The singularity class (a) is typical and isolated for an arbitrary n ; the singularity class (b) is not typical; the singularity class (c) is typical for n < 4 and

is not typical for n > 5 (see § 11.1). The transversality theorem and its corollaries extend literally to the case of Pfaffian equations. 4.3. Singularity characteristics. Finite determinacy. A germ co E W(n)

is said to be k-determined if the relation jkw = jkU(,U E W(n)) implies the equivalence of the germs to and 4. A germ is called finitely determined if it is k-determined for some k < oc. The smallest k for which a germ is k-determined is called the index of finite determinacy. If a germ is kdetermined, then its k -jet is called sufficient (it defines the entire orbit of the germ). A germ to E W (M) is called k-determined if the germ 7r.m is k-determined.

15. THE HOMOTOPY METHOD AND ITS MODIFICATIONS

17

EXAMPLE 4.8. By the Darboux theorem (see § 10.4) an arbitrary germ to E

W(2k+1) whose 1 jet satisfies the relation wn(dco)klo $ 0 is 1-determined. Normal forms, modality. Consider a singularity class S. A normal form in S is a subset R C S such that any germ T E S is equivalent to a germ i E R. A normal form is called invariant if no two of its germs are equivalent. In the opposite case a normal form is called preliminary. Suppose that all the germs belonging to S are k-determined. Then we can choose a normal form that contains the germs whose coefficients, in suitable coordinates, are k-degree polynomials.

The modality of a germ to E W(n) is defined as the minimum of the natural numbers m such that a sufficiently small neighborhood of co in W(n) may be covered by a finite number of m-parametric families of orbits. If a germ is k-determined, then its modality is the least number m such

that a sufficiently small neighborhood of jkw in Jk may be covered by a finite number of m-parametric families of the orbits of k -jets. The modality of a k-determined germ belonging to the given singularity class S in the general case differs from the number of parameters of an invariant normal form in S (if these parameters depend smoothly on the kjet of the germ, they are said to be the internal moduli (in S )). The moduli in S distinguish close but not equivalent germs from S. If a germ is not finitely determined, its modality equals 0C. The converse is false (a finitely determined germ can have infinite modality as well (see § 14).

Stability. The germ of w E A' (M) at a point a E M is called stable if for an arbitrary small neighborhood V of a there exists a neighborhood U of w in A' (M) such that an arbitrary 1-form w E U has a germ at some point fi E V, which is equivalent to the germ of w at the point a. One can verify that stability is a local property (i.e., depends only on the germ). Notice that there exist stable germs with nonzero modality. In §26 we show

that stable but unimodal germs (i.e., germs whose modality is 1) correspond to the first occurring singularities of even-dimensional Pfaffian equations (this is only true for C'-equivalence for any finite r < oc ). §5. The homotopy method and its modifications

The idea of the homotopy method is given in detail in [AVG] where it is applied to the proof of sufficiency for some jet of a function relative to R-equivalence. The homotopy method is applicable to the study of other object singularities, in particular, 1-forms and Pfaffian equations. It is the basic method for proving the sufficiency of a given jet. In §5 we describe the homotopy method for 1-forms and Pfaffian equations and give its modification enforcing its power (it enables us to introduce a normal form for any germ with a given jet and to normalize a singularity jet at a point or on a manifold).

11. BASIC NOTIONS, DEFINITIONS, NOTATION, AND CONSTRUCTIONS

18

5.1. More about the equivalence of Pfaffian equations. Let Q(n) be the set

of pairs (H, D) , where cl E Diff(n) and H is a function-germ at 0 E R" H(O) # 0. We define a group structure on Q(n) by

(Hi , '1)(H, fi) - (H1H(c1), One can easily see that the map

((H,4)),w)-(H,cp)co=HVto defines the Q(n)-action on W(n). It is obvious that the Pfaffian equations {w1 = 0} and {cot = 0} are equivalent if and only if the germs w1 and w2 belong to the same orbit of the Q(n)-action. 5.2. I-parametric families of functions, vector fields, and diffeomorphisms. Let Vecto(n) be the space of vector fields from Vect (n) belonging to 9R (i.e.,

with the singular point 0 E R"). Denote by L0(n) the set of pairs (h, v), where v E Vecto(n) and h is an arbitrary function-germ. Let v, be a family of vector fields in Vecto(n) and let (h,, v1) be a family belonging to Lo(n), t E [0, 1 ] . Define the families b, E Diff(n) and (H,, ID,) E Q(n) by d dd(x) = v,(4),(x)),

4)0(x)

dH,(x) = H,(x)h,(4',(x)), aF;

= x,

Ho(x) = 1,

x E R" (inclusions cfi, E Diff(n) and (H,, 4),) E Q(n) are easily checked). LEMMA 5.1. Let to, E W (n) (t E [0, 11) be a family of germs of 1 forms. Then

dtfv,,dcol+d(v,,(0,)+dd`J,

(5.1)

dt((H1,),)w1)=H,II [v,dw +d(v,W,)+ h1w, + dtll.

(5.2)

We can prove both relations using coordinates. Take

w,=>a,(t,x)dx,,

v,b,(t,x)

a

,

19x,

h,=h(t,x),

4',(x) = (c01(t , X), ... , c"(1, x)). Introduce

A(t, x) = (a1(t, x), ... , a"(t, x)),

B(t, x) _ (b1(t, x), ... b"(t, x)). ,

Let

(H,, 0,)w, _

a,(t, x)dx,,

A(t, x) _ (a1(t, x), ...

,

a"(t, x)).

15. THE HOMOTOPY METHOD AND ITS MODIFICATIONS

19

Then A(t, x) = H(t, x)+p(t, x)A(t, 4),(x)), where +p(t, x) is the transTaking into account the last relation we can easily position of check (5.1) and (5.2). 5.3. The homotopy method for the proof of the equivalence of two germs of 1-forms or Pfaffian equations. Let co, w E W(n) and co, = w + t(uh - (0). PROPOSITION 5.1.

(1) If the equation

v,,dw,+d(v,j(0,)=to -w

(5.3)

can be solved with respect to the family v, E Vecto(n). t e [0, 1], then the germs w and iw are equivalent. (2) If the equation

v,Jdw,+d(v,,Cod +h,w, =w-m is solvable with respect to the f a m i l y (h, , v,) E Lo(n) , t E [0, Pfaffian equations {w = 0} and {w = 0} are equivalent.

(5.4) 1] ,

then the

PROOF. We prove the second part (the first part is proved similarly). Let (h, , V,) be the solution of equation (5.4) and let the family (H,, (D,) E Q(n)

be defined as in §5.2. By Lemma 5.1 d(H,, 'D,)w,/dt = 0. But

(H0, '0)wo = co,

since H o - 1, 4y0 = id, too = w . Hence (H, , fi1)wl = (HI, fi1)w = to. The last relation implies the second part of Proposition 5.1. 5.4. Modification of the homotopy method. Let P be a linear operator in the space W(n) and let Ker P be its kernel. Retaining the notation of §§5.2 and 5.3 we can formulate the following statement. PROPOSITION 5.2.

(1) Let the equation

P[v,,dco, +d(v,-iw,)] = P(w -w) (5.5) have a solution v, such that for any p c Ker P and t E [0, 1] we have 0,,u E Ker P . Then the l form & is equivalent to a form co + , with C KerP. (2) Let the equation (5.6)

have a solution (h,, v,) E L0(n) such that for any p E KerP and t E [0, 1] we have (H,, 0,)A E Ker P. Then the Pfaffian equation {ii = 0} is equivalent to an equation {w + = 01, with E KerP. PROOF. As before we will concentrate on the proof of the second statement. By Lemma 5.1 we have dt[P(H,, d',)w,] = P(H,, 0,)y,

20

U. BASIC NOTIONS, DEFINITIONS, NOTATION. AND CONSTRUCTIONS

where y E KerP. Then our assumptions imply that dP(H,, 4),)w,/dt - 0 and, in particular, P(Ho, 00)wo = P(HI, 01)w1

i.e., Pwo = P(H1 , (I )wl , which concludes the proof of the second statement.

COROLLARY 5.1 (normalization of a k -jet). Let equation (5.5) (equation (5.6) respectively) have a solution v, E Vect0(n) (a solution (h,, v,) E L0(n) respectively) for t E [0, 1] and let the projector P: P(co) = jk cu. Then k -jets of the 1 forms to, iv (Pfaffian equations {w = 0} , {io = 0) respectively) are equivalent.

PROOF. Indeed, the set of germs with a vanishing k -jet is invariant relative

to the Q(n)-action. COROLLARY 5.2 (normalization of a k-jet on a manifold). Let S be a germ at 0 of a submanifold in R" . Let equation (5.5) (equation (5.6) respectively) have a solution v, E 911s c Vecto(n) (a solution (h,, v1) , v, E M. respectively

(see §3.2)) for t E [0, 1] and let the projector P: P(w) = jsw. Then the kjets on S of the germs wl , w2 (the germs {w, = 0} , {cot = 0} respectively) are equivalent.

PROOF. For any t E [0, 1] 0, preserves S. So, the condition js w = 0 implies jS(H,, 0,)w = 0. Q.E.D. EXAMPLE 5.1. Let z, X , y (i = 1 , ... , k) be coordinates in R 2k+1 Using Proposition 5.1, one can prove that the 1 -jet of the 1-form dz + t

k

x; dy1 is sufficient (the Darboux theorem). EXAMPLE 5.2. Let z, xi, y,. (i = 1, ... , k) be coordinates in R2k+I

Using Proposition 5.2, we can prove that an arbitrary germ {w = 0} with the 1 jet jiw I x; dy, + dz is equivalent to the germ k-1

dz +>2 xi dy, + fdyk = 01, I

J

where f is some function-germ. To prove this we must introduce the projector P : k

k

k

k-I

=adz+>a1dx;+>F,dyi

P I

1

1

and prove the solvability of equation (5.6) with respect to the family (h, , v,) 1] . for which vi,dyk = 0 , Propositions 5.1 and 5.2 will be used for proofs of further results.

§6. The infinitesimal equation, functional moduli, and "wild" jets

Results obtained in §5 give sufficient conditions for the finite determinacy of a germ; they can also be used for the proof of its stability. In this section

16. THE INFINITESIMAL EQUATION, FUNCIIONAL MODULI, AND "wILD" JETS

21

we adduce the necessary conditions of finite determinacy and stability. They are based on the check of the solvability of the infinitesimal equation. These necessary conditions will help us prove the existence of functional moduli in the problem of classification of (typical) singularities.

6.1. Infinitesimal equation and finite determinacy. The equation

v,dw+d(v,to) = r

(6.1)

with respect to the unknown vector field v c Vect( n) is called the infinitesimal equation for a germ co E W (n) . The equation

v,dw+d(v,w)+hw= r

(6.2)

with respect to the unknown function-germ h and vector field germ v E Vect( n) is called the infinitesimal equation for the germ {w = 0} of a Pfaffian equation.

The right-hand side of (6.1) and (6.2) is a given germ of 1-form. The left-hand sides of these equations are obtained from (5.3) and (5.4) by the substitution t = 0. The left-hand side of (6.1) is the Lie derivative of the form w along the field v , i.e., 1f=0(151 w , where fit is the flow correspond

ing to v. We also need infinitesimal equations in jets, i.e.,

j1(v.dco+d(v,to)) = j1r, j1(v,dw+d(v.4w)+hw)=JT.

(6.3) (6.4)

Introduce the subgroups Diff 1(n) C Diff (n) and Q1(n) C Q(n) , viz., Diff1(n) _ {fi E Diff(n)I4)'(0) = id}, Q, (n) = {(H, 0) E Q(n)Ifi E Diff1(n), H(0) = 1}. Transformations in these subgroups are called transformations with unit linear part.

Let W E W(n). We use T,(w) to denote the orbit of the jet jtw in the space J1 relative to the Diff (n)-action and we use T,(to) to denote the orbit of j1 co in J1 relative to the Diff (n)-action. Let T1({w = 0}) be the orbit of j'w relative to the Q(n)-action and let T1({w = 0}) be the orbit relative to the Q1(n)-action (i.e., T1({w = 0}) is the orbit of the 1 -jet of the Pfaffian equation in the space of 1 -jets at 0 of the Pfaffian equation). We need the following proposition which is valid for a variety of objects and transformations and follows from the general Belitskii's theory on the classification of formal series. We formulate this result for the Diff 1(n)- and Q1(n)-action in the space W(n). 1

II. BASIC NOTIONS, DEFINITIONS. NOTATION, AND CONSTRUCTIONS

22

PROPOSITION 6.1 (see [Bel4]). The following statements are equivalent : (i) the sequence of codim T,(w) (codim T({w = 0)) respectively) is

bounded as 1-- oo ; (ii) there exists k < oo such that the infinitesimal equation (6.3) (equation (6.4) respectively) has a solution v E Vect°(n) , j 1 v = 0 (solution (h , v) E L°(n), j v= j°h = 0 respectively) for any 1 and a germ r with vanishing I

k -jet;

(iii) the germ co is k-determined in the formal category relative to the Diff1(n)-action (the Q, (n)-action respectively).

The general construction which proves this proposition is proved in the monograph [Be14]. An analog of this proposition for germs of vector fields is proved separately in [Ic 1 ].

PROPOSITION 6.2. For any k < oo let there exist an I < oo and a germ r with a vanishing k -jet such that equation (6.3) (equation (6.4) respectively) is unsolvable relative to v E Vect°(n) (relative to (h, v) E L°(n) respectively).

Then the germ co E W(n) (the germ {co = 0) respectively) is not finitely determined (relative to the complete group of diffeomorphism germs).

PROOF. By Proposition 6.1 codimT?(w) -+ oo (codimT,({(O = 0})

oo respectively) as I oc. Let d 1(d2 respectively) be the dimension of the group Diff(n)/Diff1(n) (the group Q(n)/Q, (n) respectively). Then dim Tt(w) < d1 + dim Tt(w) , codim T,(co) > codim T,(w) - d, , and codim T!({w = 0}) > codim T,({w = 0}) - d2. Therefore, codim T,(w) -+ oc and codimTl({w = 0}) - oo as I oc. This implies that the germ w({w = 0} respectively) is not finitely determined even in formal series. In Chapters III-V we show that starting from some codimension the germ jets become "wild", i.e., their arbitrary extension is not sufficient. A singularity class (4.1) is called wild if the 1 -jet of each germ belonging to this class is wild. In other words a singularity class is wild if none of its singularities is finitely determined. In principle it is possible that a singularity class adjacent to a wild class is not itself wild. EXAMPLE 6.1. Consider the set of 1 jets of vector fields

S-

a 1.

t21 xi

ex + A2x2 ax + 23x3 ax l

2

3

Let S1 be the set of jets belonging to S such that A, > 0, 22 > 0, 13 < 0 and AI /'13 , 22/23 are rational numbers. Introduce the set S2 of jets belonging to S such that Al , 22 > 0, 2.3 < 0 and the 3-tuple A, , 22 , 23 is not

resonant.(3) Let S, be a class of germs X such that i1 X E §i, i = 1, 2. (3)i.e., A, # m1A1 + m222 + m3A3 for arbitrary integers m1 , mz , m3 > 0, F m1 > 2,

and iE{1,2,3}.

§6. THE INFINITESIMAL EQUATION, FUNCTIONAL MODULI, AND "WILD" JETS

23

Then S, - S2, the class S, is wild (this follows from [Icl], [Ic2]), but each germ belonging to S2 is 1-determined (this follows from the Poincare-Dulac theorem (see [Al]) in the formal case, and from the Chen theorem ([Chen], see also [H]) in the smooth case). If we want to preserve the wildness property of a singularity class relative to ajacency, we should require that the infinitesimal equation be unsolvable uniformly relative to all singularities of the given class.

PROPOSITION 6.3. For any k < oc let there exist I = 1(k) such that for any germ w({w = 0} respectively) belonging to the given class E c W(n) (E c PW(n) respectively) there exists a germ r with a vanishing k -jet for which equation (6.3) (equation (6.4) respectively) is unsolvable. Then the given class and all ajacent singularity classes are wild.

PROOF. The wildness of the singularity class E C W(n) follows from Proposition 6.2. Let El be a class ajacent to E. Assume that w E E, is a finitely determined germ. By Proposition 6.2 there exists k < oo such that equation (6.3) is solvable for co = ii for all 1 < oo and for any germ r with vanishing k -jet. Let 10 = 10(k) be the number mentioned in the formulation

of Proposition 6.3. Then the germ w has a neighborhood U c W(n) such that equation (6.3) is solvable for I = 10, j'` r = 0, and for w = p for any germ l1 E U. Since E - E, we can take UL E U r 1. This contradiction proves Proposition 6.3 for 1-forms. In the case of Pfaffian equations the arguments are the same. 6.2. A necessary condition for germ stability. It is obvious that if a germ

of a form w E A'(M) at a point a E M is stable and represents a class of isolated singularities E c W(n) (i.e., the germ of E(w) at the point a only consists of one point a), then necessarily codim T,(R,,w) = 0 for all I. A similar remark is true for Pfaffian equations. However, for Pfaffian equations the following interesting effect takes place. The condition codim T (Ruw) = 0 is not necessary for nonisolated singularities of a germ with nonzero modality to be stable (see §26).

be a germ at a point a of a 1 -form or a Pfaffian is stable, then for any I

PROPOSITION 6.4. Let

equation. If

codim T,(;r.i;) < n.

(6.5)

PROOF. Let i; be a germ of a 1-form and let w e A' (M) be a representative of the germ c . Consider the singularity class

E, = {w E W(n)ljrw E TT(r w)}. If for some l condition (6.5) is violated, then by the transversality theorem (see the statements of §4.2) there exists a 1-form w arbitrarily close to w and such that L,(tu) = 0. Therefore, for any point 6 E M the germ is not equivalent to the germ Ruw . This contradicts to the stability of the germ

11. BASIC NOTIONS, DEFINITIONS, NOTATION, AND CONSTRUCTIONS

24

w and hence condition (6.5) is valid for all 1. The proposition is proved (for Pfaffian equations the argument is repeated literally).

COROLLARY 6.1. Let a germ to E W(n) (a germ {w = 0} E PW(n) respectively) satisfy the assumptions of Proposition 6.2. Then the germ to (the germ {w = 0} respectively) is unstable.

Indeed, Proposition 6.1 implies that codim T}(co) -+ oo

(codim T!({w = 0}) -+ oc respectively)

oc. When proving Proposition 6.2 we obtained that if this is true, then codim Tl(w) -+ oc, (codim Ti({w = 0} -i oc respectively). Proposition 6.4 now gives the necessary result.

as l

6.3. Moduli in jets and functional moduli. In this subsection we explain the construction which gives functional invariants for a class of nonisolated singularities in the case where we have sufficiently many moduli for the classification of finite jets.(4) Consider a singularity class E C W(n). Assume that in the classification problem for 1-jets of germs belonging to E there exist p moduli, i.e., there exists a locally surjective map y: jIE - R" , transforming an orbit from jIE into a point and different orbits into different points. Let w E AI(M) , and a singularity of the class E occur at a point a: a E

1(w). Consider a map from the germ of the stratified manifold E(w) at a to RP : $ -i y(jI2r,w) . Denote the germ of the image of this map at the point y(jI naw) by D,,,(a). One can easily see that the following statement is valid.

PROPOSITION 6.5. Let the germ at a E M of a form w1 E A'(M) be equivalent to the germ at Q E M of a form w2 C AI (M) . Then D. (a) _ D,,, (fl)

Let to E W(n). Denote by D(co) the germ D,,,(0) , where w is a 1-form with the germ to at 0. By Proposition 6.5 D(to) is the invariant of to. In the case where singularities belonging to E are nonisolated (codim E <

n) and p > dim E = n

codim E the invariant D(w) is functional. In this

case for a generic germ to E 1, D(w) is a germ of a submanifold in RP whose codimension is positive but less than p . The set of all germs of the manifolds

{D(w), co E E} in this case cannot be described by a finite number of parameters. In particular, under given assumptions on codim E , n, and p the germ co is not finitely determined (i.e., the class E is wild). Indeed, by adding to co a germ u with a vanishing k -jet of a given arbitrarily high order k we cannot change the k-jet at 0, but this addition can "move" the (4)This construction (which was suggested to the author by Givental) is given for 1-forms, although it can literally be expanded to Pfaffian equations and other objects.

§7. CLASSIFICATION OF SUBMANIFOLDS OF A CONTACT MANIFOLD

25

manifold D(cv) in the class of manifolds which contact it with the order k at the point y(jrm) . In some cases (see, for example, §28) the functional moduli corresponding to wild jets can be represented more explicitly, i.e., one can obtain an invariant normal form with function-germs as parameters. In other cases (see, for example, §§ 12.3 and 12.4) it is difficult to represent functional moduli explicitly, but we are able to prove that the classification problem for the considered objects (forms, Pfaffian equations) contains (is harder than) the classification problem for other objects with deliberately wild jets (for example, the classification problem for germs of k-dimensional distributions in R" for n > 5, 2 < k < n - 2, see Refs. [JP2], [VG2], [ZI 1 ]).

§7. Classification of submanifolds of a contact manifold

Let S, and S2 be the germs at 0 C

R2k+I

of equal dimensional submanifolds in a contact space (R2k+t , w). They are called equivalent if one of them can be transformed into another by a contact diffeomorphism

0 E Diff (2k + 1), i.e., by a diffeomorphism 0 satisfying the condition

Vm = Hw, where to is the germ at 0 of a 1-form w and H is some function-germ, H(0) 0. The definition of equivalence of functions (other objects) in a contact space is similar. THEOREM 7.1. Let wr = cwis, i = 1 , 2. The germs S, , S2 are equivalent if and only if the germs {mI = 01 and {w2 = 01 are equivalent relative to Diff(2k).

This theorem was proved by Givental (see [Al], [AG]). It is called the relative Darboux theorem. Let us point out some corollaries for hypersurfaces in a contact space.

We say that a germ at a point a of a hypersurface in a contact space ) is generic if the hyperplane of the form w at the point a (i.e., its kernel at a) is transversal to S. This condition is equivalent to the following one: the form w Is does not vanish at the point a. R2k+, W)

(

THEOREM 7.2. Generic germs of hypersurfaces in a contact space are equivalent.

This result goes back to Sophus Lie. It can be obtained as a corollary of Theorem 7.1 and the classical Darboux theorem for even-dimensional Pfaffian equations (see §24). Theorem 7.2 has the following stronger variant. We shall say that a contact

diffeomorphism fi is exact if cYw = w. THEOREM 7.3 (see Lychagin [L 1 ]). Let SI and S2 be generic germs of hyPersurfaces in a contact space, that are transversal to Ker who and to Ker d wi0 . Then there exists an exact contact diffeomorphism fi transforming Sl to S2 .

II. BASIC NOTIONS, DEFINITIONS, NOTATION, AND CONSTRUCTIONS

26

The following statements are corollaries of this theorem and will be used in Chapters III-V. THEOREM 7.4. Let co = dz + x1 dy, + + xk dyk . A generic germ at 0 of a hypersurface in the contact space (R2k+1 , w) can be reduced to the form {x, = 0}({y1 = 0}) by a contact diffeomorphism.

THEOREM 7.5. Let co = d z+x1 dyI + +xk dyk , let f be a function-germ at 0 and let [df n d z] I0 $ 0 , a (0) $ 0 . Then there exists a diffeomorphism cp reducing the pair (co, f) to the pair (co, H(x1 - z) + f(O)), where H is a function-germ, H(0) $ 0.

One also can prove a closely related result about contact equivalence of functions. THEOREM 7.6. All function-germs f : I[t" --, RI , f(0) = 0 satisfying the condition (df A co)lo : 0 are equivalent in the contact space (R" , to).

As a corollary, we obtain THEOREM 7.7. Let

coo =dz+xldy1+...+xkdy ,

f(Z,x1,Y1,... ,x/.,Yk)

be a function-germ with df n 60010 = df n dz1o 0. Then there exists a diffeomorphism c1 reducing the pair (wo , f) to (Hwo, f (0) + y,), where H is a function-germ, H(0) $ 0. We also announce the following result. THEOREM 7.8. Let co be a germ of 1-form in

R2k

,

R2k

let S c

be a germ

of a hypersurface. If the pair (co, S) is generic, i.e., (dw)k10 0 and S is transversal to the kernel of co n (dw)k-1 , then (co, S) is reducible to

{x, =0}). §8. Solvability of equations with respect to germs of flat functions In this section we formulate some results obtained by Belitskii [Bell- Bel6]

and Roussarie [Ro] on the solvability of functional equations and partial differential equations. We point out some corollaries of these results which are used in Chapters III-V.

Let S c R" be a submanifold of codimension p, 0 E S. Let V be a germ of a vector field vanishing on S: Vas = 0. Then the spectrum of V-linearization at the point 0 E R" is of the form Consider the equation

(V +v,)(u,)+g,(x)u, =f(x),

AP , 0, ... , 0) . (8.1)

where yr is a family of germs of vector fields in R" , g,, and f are families of function-germs, and u, = u,(x) is the unknown family of function-germs,

tE[0, 1].

§8. EQUATIONS WITH RESPECT TO GERMS OF FIAT FUNCTIONS

THEOREM 8.1. Let Re A,

0

27

(i = 1 , ... , p),(5) V, E 9Jls f E 9Jts ,

(t E [0, 1]). Then equation (8.1) is solvable.

THEOREM 8.2. Let Re. $ 0 (i = I , ... , p), r < oc. Then there exists 1 = l (r) < oe such that for every v, E 9Jls , f, E 9JtS equation (8.1) has a C'-solution u,(x) . THEOREM 8.3. Let S = J01, n = 2, yr E 9R°° , f E 9Jt°° and let V be a germ of a vector field If V is finitely determined in the formal category, then equation (8.1) is solvable. THEOREM 8.4. Let V

2

=xiax -x2ax +(xI +x2)ax I

2

3

be a germ of a vector field in R' (xI , ... , X"), S: {xl = x2=01, V, E 9RS , g, E 9JiS , f, E 9Jis . Then equation (8.1) is solvable. Theorem 8.1 is a corollary of the following result. THEOREM 8.5. Let ReAj $ 0 ( i = 1 , ... , p) , and let H,(x, u) be a f a m i l y o f germs o f vector functions, x E III" , u E lit'. t E [0, 1 ] . If H, (x, 0) E TT

,

then the equation

V(ut) = H,(x, u,) has a solution u, = u,(x) = (u, I (x), ... , ut ,(X)).

(8.2)

Theorem 8.5 implies, in particular, the equivalence of the vector fields V and V + v, and, therefore, the statement of Theorem 8.1. The proof of Theorem 8.5 is based on the replacement of equation (8.2) by the functional equation

u,(F(x)) = ft, (x , u,) ,

(8.3)

where F is a diffeomorphism hyperbolic on S and H,(x, 0) E 9J1; [Be16, Bell]. The solvability of equation (8.3) is shown in [Bell, Bel3]. Similarly, Theorem 8.2 is a corollary of the following statement. THEOREM 8.6. Let F be a diffeornorphism hyperbolic on S, r < 00. There

exists an I = 1(r) < oo such that for any Cr family H,(x, u) satisfying the condition H,(x, 0) E 9Jlts and for any vector field family V, E 9Jlts equation

(8.3) has a C'-solution u,(x)

.

The proof of Theorem 8.6 can be found in [Be12]. Theorem 8.3 is proved in [Be16].

The proof of Theorem 8.4 for the case g,(x) = 0 can be found in [Ro]. In the case g,(x) 0 0 we can seek u,(x) in the form u,(x) = e9'(X)u,(x) , where (5)This condition is equivalent to the hyperbolicity condition of the field V on the manifold S, see [Bc14], [AI].

U. BASIC NOTIONS, DEFINITIONS, NOTATION, AND CONSTRUCTIONS

28

the germ ip,(x) satisfies the equation (V + v,)(rp,) = -g,. Then equation (8.1) will be transformed to (V + v,)(u,) = e 9'(")f,(x) . This equation is solvable (as an equation for u,) by [Ro]. Now let us consider equation (8.1), where g,(x) and f,(x) are families of functions (not germs) defined in a domain U c R" (which may be unbounded). Assume that the field V +v, has the flow FF(x) = exp(s(V +v,)) in the domain U for Isl <_ so. Let r > 1 . Among these equations we distinguish equations satisfying the following conditions for each t c- [0, 1 ] , 0 < s < so, a E U (1) P(Fr (x), S) <_ P(x, S)[1 -C(s)pr(x, S)], (2) Ig,(x)I
We denote the class of such equations by H(U, S, r). REMARK. If U is bounded, then condition (3) is equivalent to the condition f E 93ts . THEOREM 8.7 (see [Be14]). Let equation (8.1) belong to the class H(U, S, r)

and for a sufficiently small s > 0 the condition

II(F')'(x)II <- 1 +E(s)pr(x, S) (8.4) hold true. Then there exists a family u, (x) satisfying equation (8.1) in the domain U and satisfying the estimates

Ilu;j'(x)II
0
for some Rid . §9. Commentary §3. A k -jet on a manifold S of a function-germ can also be defined as an

equivalence class of C°°(n) by the following equivalence: two germs f, g

are equivalent if f(a) - g(a) = 0(pk(a, S)), where a E R" and

is

some metric in R" . One can consider a k -jet on a manifold as a germ of a map from this manifold to the space of k-degree polynomials. Example 3.6 shows that working with jets on a manifold, one must be more careful than working with simple jets. For instance, two germs can have the same k -jets. on a manifold for an arbitrary k < oo, but their difference may not be a germ of a function flat on this manifold. We shall deal with these difficulties in § 18 and §21 and §26 and §28. But, in principle, the technique of jets on a manifold for nonisolated singularities seems to be very useful. §4.

The existence of stable and at the same time unimodal germs is

possible because, according to our definition, modality is a property of germs with a fixed source and stability is a property of germs whose source is not

§9. COMMENTARY

29

fixed. Nevertheless, we do not know examples of stable germs with nonzero modality, except the singularities considered in §26. §5. The homotopy method allows us to use algebraic arguments instead of geometric ones. In the survey [Mar] the classical Darboux theorem for closed 2-forms is proved by the homotopy method; this proof is the shortest of numerous proofs. The homotopy method for problems concerning differential forms or some objects in a contact space is used in various works (for

example, [Mos], [Ro], [A3], [Ko]). As to the modification in §5.4, it is quite simple. Nevertheless, it is used frequently in our work, as well as its corollaries. §6.

Proposition 6.5 and its corollaries show the validity of the following

heuristic principle: nonisolated singularities are either stable or wild. Of course, it is not an exact formulation, but after some specifications in each concrete case this statement (or some statement like this) is valid. §7. One can formulate the relative Darboux theorem as follows: the internal geometry defines the external one [AG]. The internal geometry defines the external geometry not only for submanifolds of a contact manifold, but also for some submanifolds with singularities (stratified manifolds) [A3].

§8.

Results of §8 allow us to conclude in a number of cases that the

smooth equivalence of germs follows from their formal equivalence or from

the equivalence on the level of k -jets on a manifold. First results in this direction were obtained by Sternberg [St2] and Chen [Chen] (see also [H]). Theorem 8.1 for the case S = 10} was obtained by Lychagin [L2]. References [Be12], [Sa] are devoted to reduction to normal forms by finitely smooth transformations and to the solvability of equations in a finite smoothness class.

We do not deal here with numerous works devoted to C°°- and C'normalization of vector fields and diffeomorphisms in the plane. Notice that the most complete list of C°°-orbital normal forms of germs of vector fields in the plane can be found in works by Bogdanov (see [Bol], [Bo2]). The most general results on the solvability of equations (8.2) and (8.3) belong to Belitskii. Their immediate corollaries give various sufficient con-

ditions of w-determinacy (we use the notation of [Be14]): a germ is wdetermined if its C-equivalence to another germ follows from their formal equivalence (in other words, to-determinacy means that the problem of C°°classification is reduced to the problem of formal classification). Results on w-determinacy and related questions can be found in [Be14] and in works by Belitskii mentioned in §8. Notice that in these works Theorems 8.5, 8.6, and 8.7 were proved for the case when the parameter t E [0, 1] is absent, but the presence of the parameter belonging to a compact set does not change the proofs in the mentioned works. Belitskii's theory for normalization of formal series includes the construc-

30

II. BASIC NOTIONS, DEFINITIONS, NOTATION, AND CONSTRUCTIONS

Lion of invariant normal forms for formal series corresponding to various objects [Be14], [Be15]. It is based on a generalization of the following

LEMMA (Takens, [T]). Let X and Y be vector fields, j' X = j' Y = 0, jk[X, Y] = 0. Then

jk+l(X+[X, Y]).

CHAPTER III

Classification of Germs of Differential Forms In Chapter III we stratify the space of germs of differential 1-forms into singularity classes. We show that the generic germs and the germs representing the first occurring singularities are stable (Darboux and Martinet theorems). Isolated even-dimensional singularities due to the vanishing of 1-forms at a point are unstable but 1-determined (Lychagin theorem). The same is true for isolated odd-dimensional singularities. We show that all other singularity classes are wild. We give their preliminary normal forms with functional parameters.

§10. The class of a germ; preliminary normal form; Darboux theorem

In §10 we classify 1 -jets of 1-forms. In the case of nonvanishing 0 -jet the only invariant of the 1 jet is a positive integer p called the class of the 1-form. If this class is defined by the maximum possible value p = n, the 1 -jet is sufficient (the Darboux theorem), i.e., all germs having a 1 -jet of the maximum class are equivalent. In the case p < n we give a preliminary normal form to which we can reduce any germ with a 1 jet of the given class p. The preliminary normal form can be used to study singularities of positive codimension. 10.1. Class of it germ of 1-form. Given W E A' (R") we can consider its differential, i.e., a closed 2-form dw . Recall that the rank of dw at a point cr (rank dw la ) is the number 2p where

(dw)'I,, j4 0,

(dw)°+'IQ = 0.

Let rankdwla = 2p. DEFINITION. If w 0 then the class of w at a is the number 2p + 1 if w A (dw)°la # 0 and the number 2p if w A (dw)'l, = 0. For who, = 0 the class of w at a is defined to be 0 (note that if wla # 0, the class of w at a is no less then 1 and does not exceed n ). The class of a form at a

characterizes only its germ at a and, moreover, only its 1 -jet. NOTATION. The class of a form w at a point cr will be denoted claw . Suppose that co E W(n), w E A' (It"), co = now . The class of the germ Co is defined as clow and will be denoted cl co. 77

III. CLASSIFICATION OF GERMS OF DIFFERENTIAL FORMS

32

EXAMPLE 10.1. The class of a contact 1-form at any point of (2k + I)dimensional contact manifold is maximum and equals (2k + 1) . equals 3 at EXAMPLE 10.2. The class of the form dx + xz dy E A' the generic points of R3 (x # 0) , equals 2 at the generic points of the plane {x = 01, and equals 1 at the points of the line {x = z = 01, 10.2. Classification of 1-jets. The following theorem shows that all 1 -jets

of the same positive class are equivalent. For germs of the zero class (i.e., representing isolated singularities) the situation is quite different (see § 13). THEOREM 10.1 .

(1) All 1 jets of class (2p + 1) are equivalent to the jet (10.1)

(for coordinates z, x., y,, (i= 1, ...,p), ul (j=l,...,n-2p-1)). (2) All 1 jets of class 2p, p > l , are equivalent to the jet

(l+XI)dy1+x2dy2+ +xpdyp

(10.2)

(for coordinates x., y. (i = 1, ...,p), uj (j = 1, ...,n-2p) The coordinates u do not occur in the normal form. PROOF. The proof is based on the algebraic Darboux theorem for 2-forms ([Stl], [Mar]) which can be formulated as follows: the 0-jet of the germ at 0 E ]R of a closed 2-form with rank 2p (at 0) can be reduced to dx1 Adyl +

+dxpndyp. First consider an odd class. Let cl w = 2p + 1 . Then rank dwIo = 2p and by the algebraic Darboux theorem the [Jet is reduced to (10.3)

XI dyI

where h is a function of the variables

X = (XI, ...

,

xp)

,

y = (Y1 , ... , yp) ,

u = (uI , ... , un-2D-1) ,

Z.

The condition co n (dw)Dl0 96 0 means that az(0) 0 0 or a (0) # 0 for some j. We can assume that aZ (0) # 0. Then a transformation of z enables us to get the normal form (10.1) from (10.3).

Now suppose that clw = 2p. Then the 1 -jet is reducible to (10.3) and j1h has the form alxl + ply1 + + apxp + flpyp. Since p >- 1 we have dhI0 # 0. Without loss of generality we can assume that fl, 96 0 (for aI A 0 the arguments are similar). By changing the coordinate y1 to yI = h we get the 1 -jet D

(1 + LI } dyl - R, xI dxI +

X'

Al xI ! dy`

Rl x1

dx'J

S 10. THE CLASS OF A GERM; PRELIMINARY NORMAL FORM; DARBOUX THEOREM

Next introduce the coordinates z, = xi - (f 1/f 1)xI (i > 2). Then co =

where

/

(1 +

L') 2

2fil

y, = y, + aix1 /f

1

P

-dQ,

dy'I

alxI Q=

,

33

2) 2aIxI5c7I + a, fl, xI

+ =2

By changing the coordinate yI to yl = yI - Q, and the coordinate xI to XI = x1 /f we obtain (with the obvious change of notation) the normal 1

form (10.2). The proof is complete. 10.3. Class calculation in local coordinates. Suppose that in some local coordinates x = (x1 , ... , x,,) the 1 -jet of a form w at 0 is (10.4) (c + Ax, dx),

where cERR, c#0, A is n x n matrix and dx=(dx1,... ,dx,,), (.,.) denotes a scalar product in RR . LEMMA 10.1. The class of a 1 form at 0 can be calculated as follows

Glow = dim{u E RRIu = (A - A')v + ac, v ERR , a E RI}

(10.5)

or Glow = rank B, where B = (A - A', c). PROOF. Let F = (FI, ... , FR) . In new coordinates x = 1'(y) the form

(F(x), dx) looks like ((3')'F(O), dy). Thus, the pair (c, A) transforms into the pair (T'c, T'AT + B), where B = B', det T 9 0. Taking this fact into account we can easily show that the right-hand side of (10.5) does not depend on the choice of local coordinates. In order to prove Lemma 10.1 it remains to note that by formula (10.5) for normal forms (10.1) and (10.2) the class is correctly defined. 10.4. Normal form of a germ with a given class. The following theorem resembles the theorem on the normal form of a function-germ at a degenerate

critical point [AVG], ["GL]. THEOREM 10.2. (1) Let w E W (n) , cl co = 2p + I. Then the germ w is equivalent to a-2p-I

E ft(z,x,y,u)dut

(10.6)

j=1

(for coordinates

x = (x1, ... , XP), y = (Y1, ... , yp), Z' U = (U],---

u n- 2p-1)).

(2) Let co c W(n), cl co = 2p > 2. Then the germ co is equivalent to a-2p (1

E f1(x, y, u) du, j=I

(10.7)

.... . .oo.rat Al Iun Ur I,CKMJ OF DIFFERENTIAL FORMS

(/or coordinates x = (xt , ... 'XP), y = (Y], ...

, yp) .

u = (u1 , .. .

un_2p) ).

Here the f are functional parameters with vanishing 1 jet. REMARK. The number of parameters of the preliminary normal forms is equal to the coclass of the germ w, i.e., n - cl a). Therefore, as a corollary of Theorem 10.2 we obtain the classical Darboux theorem. DARBOUx THEOREM (on 1-forms). All germs belonging to W(2p + 1)

(W(2p) respectively) of the maximum class 2p + I (2p respectively) are equivalent to the germ given by (10.1) ((10.2) respectively).

PROOF OF THEOREM 10.2. We start with the case cl w = 2p + 1 . Let + xp dyp. By Theorem 10.1 the germ w can be coo = dz + xI dy1 + expressed in the coordinates Z,

X=(XI,... , p), y=(y1,... ,yp), u=(ul,... , un-2p-l)

as wo + p , where j1 p = 0. Denote by wt the family wt = wo + t p ,

t C-

[0, 1 ] . Introduce the projector P transforming the germ A d z + > (B1 d xi + Ci d y ) + r_ Ri du J into the germ A d z + F ,(B, d x, + Ci d yi) . In order to obtain the necessary result it suffices to prove the solvability of the equation

P[vt,dw, +d(v,,wt)] _ -Pp

(10.8)

with respect to the family of germs v, E Vect(n) satisfying the conditions

(a) j'vt = 0 (t E [0, 1]),

(b)v,,du1=0 (tE[0, 11, j=1,... ,n-2p-1).

(10.9)

Indeed, if equation (10.8) is solvable in the mentioned class of families, then

by Proposition 5.2 the germ w is reducible to coo + , where P4 = 0, i.e., to (10.6). Moreover, the reducing diffeomorphism has a unit linear part,

whence j' f = 0. We are to prove now the solvability of (10.8) in the class of families (10.9).

Let R, = vt , w, and

vt=y,az+Epi 5 We

+wi tya

denote by a.,fi,r the function-germs a/axi,p,a/ayi,u and a/az,p

respectively. Equation (10.8) can be rewritten as a system

aRt

,p,

ax, aRt

aRt

az

+...=t,

Rt=y,+...

§1I. SINGUTARTr1ES AND THEIR ADJACENCIES

35

is a linear combination of the form E(ar9',,+brw; t)+c', , where a,, b, , c are some germs of functions depending on the parameter t E [0, 11 and vanishing at the point 0 E R" for all 1. System (10.10) can be reduced to a single equation OR ='Z where

8Zj + Q(R

1

)+BR 1

1

t'

where Q, is the germ of a vector field, B, , it are function-germs, and j°Qt = j' it = 0 for all t E [0 , 1] since j I a1 = j',8, = j I T = 0. The family of germs 8/Bz+Q, can be "straightened", i.e., we can find a family of diffeomorphisms 9t E Diff(n), with

(az

OZ,

Therefore equation (10.11) can be transformed into the equation for Rt = R,((P,), viz., OR

(10.12)

+ B,(q't)R,

Equation (10.12) has a solution R, and this solution can be choosen to

satisfy the relation j2Rt = 0. Then j2Rt = 0, and by (10.10) we have i' Vi., = j I wi t = j 2 yt = 0. Thus, j'v, = 0 and the solution v, satisfies conditions (10.9). In the even-dimensional case we can use Proposition 5.2 similarly and the proof will be reduced to the solvability of the equation (10.13) OR I + Rt + Qt(RI) = it ,

8xi

where Q, is a family belonging to Vect(n), j°Qt = jITt = 0 and R, is the sought-for family satisfying j2Rt = 0. The solvability of (10.13) in the mentioned class of families can be proved exactly as for (10.11). The proof is complete. §11. Singularities and their adjacencies

In § 11 we divide the set of germs W(n) into singularity classes, calculate their codimensions, and give all adjacencies.

11.1. Singularities associated with the decrease of the germ class. Consider the set WJ C W(n) consisting of class j germs, j < n . Obviously, for j > 1 W' is a singularity class defined by a relation on a 1-jet; for j = 0 by a relation on a 0-jet. The set of 1 -jets of the j-class is a submanifold in it ; its closure (i.e., the set of 1 jets of class < j) is a stratified manifold consisting of the strata j1" W' and j' W"j ° \ ' W'" , jI W° fljI W'" , i _<j. EXAMPLE 11.1. For n = 2 the space JI can be considered as the space of pairs c = (c1 , C2) and A = (a,) , I, j = 1 , 2. The stratum j I W2 is defined 1.

W21 is defined by by the condition a12 A a21, c # (0, 0) . The stratum j 1 a12 = a21 , c 0 (0, 0) ; the stratum j' W ° by c = (0 , 0) . The complete

stratification of J' is J1

=l 'WZ 2I U 'W'uj (' W°/'WZ')u('W21 j j n j 'u'°) z 2

EXAMPLE 11.2. For n = 3 the space J' can be treated as the space of pairs c = (c1, c2, c3) and A = (a1f), i, j = 1, 2, 3. The stratum j1 WW is defined by the condition (A - A) 11- 0, rank (A - A', c) = 3 (see Lemma 10.1). The stratum j1 W 3 is defined by the condition (A-A,) 0 0, rank (A( 0 , 0,0); the A` , c) = 2. The stratum j' W 3 ' is given by A - A` = 0,c stratum j°W3 by c = (0, 0, 0). REMARK. One can show that the closure of the manifold gularities only at the points of the closure of j' W,:-2 .

has sin-

THEOREM 11.1. [Mar] For j > 1 the codimension of W' is equal to

(n - j) x (n - j + 1)/2. The codimension of W is equal to n. First we prove the following statement.

LEMMA 11.1. The set of antisymmetric n x n matrices of rank 2p has in the space of all antisymmetric matrices the codimension equal to the dimension

of the space of all antisymmetric (n - 2p) x (n - 2p) matrices, i.e., equal to

(n - 2p)(n - 2p - 1)/2. PROOF. All antisymmetric matrices R with rank 2p are equivalent rel-

ative to the group of transformations R - T'RT, det T 41- 0. Thus, the codimension is equal to the codimension of the orbit of the standard Darboux matrix

0

-E 0 0

1

where E is a unit p x p matrix. The tangent space to the orbit consists of all antisymmetric matrices B 0

This implies the formulated result. PROOF OF THEOREM 11.1. Let j 1 w = (c + Ax, dx). The pair (c, R), R = A - A' entirely defines the class of the 1 jet, so that the codimension of each class can be found as the codimension of the corresponding set of pairs

§11. SINGULARITIES AND THEIR ADJACENCIES

37

(c, R) in the space of all pairs (c, R). Consider the following four cases and use Lemmas 10.1, 11.1.

(1) n = 2k + l , j = 2p + 1 . The corresponding set of pairs consists of (c, R) satisfying the relation rank R = 2p , rank (R, c) = 2p + 1 , i.e., the set consists of generic pairs with rank R = 2p . By Lemma 11.1 codim Wi

(n - 2p)(n - 2p - l)/2 = (n - j)(n - j + 1)/2. (2) n = 2k + 1 , j = 2p > 2. The corresponding set of pairs consists of (c, R) : rank R = 2p, rank (R, c) = 2p. Let Sp be the set of pairs (c, R) : rank R = 2p . The codimension of the given set in Sp is equal to 2k + 1 - 2p (as one can easily calculate). Thus, codim WR = 2k + 1 - 2p + codim Sp. By

Lemma 11.1 we have codim K = (n - j)(n - j + 1)/2. (3) n = 2k, j = 2p. The corresponding set consists of generic pairs (c, R) E Sp. Thus, codim WJ = codimSp = (n - j)(n - j + 1)/2. (4) n = 2k, j = 2p. The corresponding set consists of (c, R) E Sp with rank (R, c) = 2p. The codimension of this set in Sp is equal to 2k - 2p. Thus, codim W ' = 2k - 2p + codim Sp = (n - j) (n - j + l )/2 . The proof is complete (the second statement of Theorem 11.1 is obvious). COROLLARY 11.1. The class of an arbitrary germ of a generic 1 form on

an n-dimensional manifold is not less than n - [ 2n -+1 /4 - 1/2] or equal

to 0. EXAMPLE 11.3. A generic 1-form in R5 can have class 4 singularities (on a hypersurface), class 3 singularities (on a 2-dimensional submanifold), and class 0 singularities (at isolated points). These statements are corollaries of Theorem 11.1 and results from §3.4.

11.2. Other singularities. Singularities belonging to W1,"_ are the first we encountered; for a generic f o r m they occur on a hypersurface. In § 1 1.2 the class W" will be divided into subclasses of singularities. I

We start with the case n = 2k + 1 > 3. To each singularity co E W k+1

the following objects are invariantly related: (a) the germ W k+i (CO) (recall that according to the notation of §4 this is the germ of the set of points realizing singularities E Wzk+1 for w E AI(R") with the germ co at 0);

(b) the germ of a field of directions on WZ +I(w) : at each point a E Wk+1 (w) near 0, a direction in TR" is defined that consists of vectors annulated by the form dwI , . We use X. to denote the field of kernels of dw. For a generic germ w E Wyk+1 the following conditions are valid:

(1) the set Wzk+1(w) has at 0 E R" the structure of a codimension I submanifold for each germ CO E WZk+I close to w ; (2) thedirection Xj,, istransversal (at OER" ) to the manifold W2 1(w)

.

Ill. CLASSIFICATION OF GERMS OF DIFFERENTIAL FORMS

38

REMARK. In the general case condition (1) does not follow from the fact

that W2+I (w) has the structure of a manifold at 0. For example for w = (1 + x) dy + dz3 E W(3) the set W2(W) is given by the equation {z = 0} and for w = co - e d(zx2) the set W Z ( i u ) is given by {3z2 - ex2 = 0} .

The set of germs for which conditions (1) and (2) are valid is denoted by WZk ° . The set of germs for which condition (1) is violated is denoted t'I sk+I , and the set of germs for which (1) is satisfied and (2) is violated is denoted Wzk+II

THEOREM 11.2. The set W2k+II has codimension 2 in W(2k + 1). The set W k+I has codimension 2k + 2 in W(2k + 1). PROOF. Each germ w E Wzk+l can be represented in local coordinates as

xldyl+...+xkdyk+dH(x,Y, z), x= (XI,... ,xk), Y=(YI,... ,Yk)

since rank dw = 2k (we can use the Darboux theorem on closed 2-forms of maximum rank)('). Moreover, a -(0) = 0. The set Wzk+l(w) is given in R2k+' by the condition a = 0. Condition (1) is satisfied if and only if d (aH o 0. This implies that codim W2k+,j = 2k + I (in Wzk+I ). Hence, codim Wzk+J = 2k + 2 (in W(2k + 1) ). Assume now that (1) is satisfied. The direction X( at 0 is generated by the vector 8/8z . That is why the transversality condition (2) leads to a4(0) 0 0. Thus, codim Wzk+i - 1 (in WZ+I)and codim WZk+' = 2 (in W(2k + 1)). Q.E.D. Consider now the case n = 2k. The following objects are invariantly related to each singularity w E zk (a) the germ WZk-' 2k (w} (b) the germ X. of the field of directions on WZk -' (o)), i.e., at each point a c WZk -' (w) a direction in T Rzk is defined consisting of vectors annulated by the form dwI0 and by the form col.. The correctness of the definition of X. follows, for example, from the normal form (10.6) with n = 2k, p = k - 1 (we can assume that a = 0, then in the corresponding coordinates X 0 is generated by 8/8u1 ). For a generic germ CO E Wzk -I the conditions (1), (2) are valid (with the substitution of WZk-' for WZk+I ). The set of germs for which these I

Wzk-'

conditions are valid is denoted Wk-'' ° , the set of germs for which condition and the set of germs for which condition (1) is violated is denoted Wzk-', zk (1) is valid and condition (2) is violated is denoted Wk2k-1, ' (')Note, that jkH is defined by jku, and is not defined by jk-iu,.

§ 11. SINGULARITIES AND THEIR ADJACENCIES

39

THEOREM 11.3. The set WWk-1' 1 has codimension 2 in W(2k) and the set W2k Zk-1, has codimension (2k + 1 ) in W(2k) '

PROOF. Using the normal form (10.6), we represent the germ a) E in local coordinates as dx1 + x2 dy2 + ' ' + xk dyk + f(x, y) dy1 , X, y E -I(W) _ {(x, y) = 0} and therefore the IItk . In these coordinates Wik WZk-'

is equivalent to the condition d (}

inclusion w E WZk -1

o

= 0. This

implies the second statement of Theorem 11.3. The direction X at 0 is generated by a/ay1 . Therefore co E W -1' 1 if and only if a e f (0) = 0. This implies the first statement of Theorem 11.3. Q.E.D.

REMARK. The proofs of Theorems 11.2. and 11.3 show that the singularity 1,o

classes WR

n-11

,

Wn-1, are defined in W(n) by a condition on

the 2 -jet.

11.3. Typical singularities. Adjacencies.

THEOREM 11.4. An arbitrary germ of a generic [-form on an n-manifold belongs to one of the singularity classes given on the diagram below. The diagram contains all the adjacencies of typical singularity classes.

n=1

n=2k+1>3 W(n) 4--

WR-1'o

Wn-1 ,

1

R

_--

4-- W`

WR-2

R

R

I Won

n=2k>2

W(n) WW1 ._ Wn n

n

R

z

r

...

W'n

T

Wo n

2 n+4-1.

n-

11

L

PROOF. In view of the results of §§ 1 1.1 and 11.2 it suffices to prove the following statements: (a)

W. n-"' - WR (n = 2k + 1 > 3);

(b) there are no adjacencies

Wn-1.1

W (n = 2k + 1 > 3) ;

-.-..,..+..

A-A11UN OF CitKMN OF DIFFERENTIAL FORMS

(c) there are no adjacencies Wn-l '0 ~- W5° (n = 2k) ; WnWr-z (n=2k+1> 3); Wn-'' Wnn-2 (n = 2k > 4). (e) n Let us prove these statements. (a) A generic germ w E W k+I satisfies the condition rank dcwI0 = 2k . By the Darboux theorem on 2-forms such a germ can be reduced to xl dyl + . + xk d yk + d H (x , y, z), d H I 0 = 0. For an arbitrary small e j4 0 the germ w + e dy1 belongs to Wn-' . Therefore, the closure of Wn-' (and, hence of W,,"-1 0) contains W,° . (b) Let w E W k+I be a generic germ. We can assume that w = x1 dyl + +xk dyk +d H(x , y, z). The manifold W n-' (w) is given by the equation {aHlaz = 0} . It is transversal to the kernel of dwJ0 (i.e., to the direction 8/19z) viz., as (0) & 0. The transversality condition is also valid for germs from W n - t close to w . Therefore, there is no adjacency W :-I ' t 4-- W (c) For n = 2k the condition clw = n -l signifies that rank d wl0 = n-2. At the same time rank d wI0 = n for a generic germ co E Wo . 2k-1 By Theorem 10.2 we can assume that w = dz + (d) Let w E Wzk+I . d X1 d YI y, "' +x_ Yk-1 +fdx k+ gdy,k where ' = 1' k depend on the variables x1 , y1, ... , Xk , Yk , z . Consider the family wt = w + (e xk + e2yk) d z . It can be reduced to (d)

1f j,0+f, g

1

I

1

+h1dxk+h2dyk

(1

by the transformation(2) Xk

f(

h1 =

h2 =

1

E

,

x' 5y'

Xk

,

where

e2yk

Yk ,

Xk

{g {

ele2yk

Z)

, A, Z)

=(xl,yl,...

- ele2h1

,Xk-1,Yk-1)'

0, The manifold Wz

The germ wE belongs to Wzk for e1 1

1(w) is given

by the equation R = 0, where jI R = ahl/ayk - 8hz/axk . The direction XW at 0 is generated by the vector a/ayk . Therefore, in order to prove statement (d) it suffices to show that for arbitrary generic function-germs f and g with vanishing 1 -jets there exist arbitrarily small el , C2 for which z Al

ayk

(0)

z

0Xkd Yk

(2)Here and below the old and new coordinates are denoted by the same letters; in the given case we use the coordinate Xk (instead of xk ) where xk = (x'k - CZyk )lc, and then change

zk to xk .

§ 12. CLASSIFICATION OF COCLASS I SINGULARITIES

41

It is easy to show that this condition is equivalent to

ayk(0)-ax2ayk(o)+el $(o)-a kayk(0)1 =0. La

The last relation is satisfied for suitable arbitrarily small c1 , c2 . (e) Let w E , k > 2. By Theorem 10.2 we can assume that WWk-2

x2dy2+..+xk_1dyk_1+fdxk+gdyk,

w=(l+x1)dy1+

JIf=.%Ig=0. By considerations similar to those in the proof of statement (d) we can show that there exist arbitrarily small el , e2 such that the germ w+el dxk +c2 d yk belongs to WZk -1 ' . Q.E.D. 1

§12. Classification of coclass 1 singularities In this section we prove Martinet's theorem on germs belonging to and show that the codimension 2 singularity class W,"-1' 1 is wild, i.e., it contains no finitely determined germs. Notice that the stability and 1-determinacy of generic germs (i.e., belonging WW"-1,0

to W°) follows from the Darboux theorem; for fixed n all such germs are equivalent.

12.1. Odd-dimensional singularities from W"-1 ' o THEOREM 12.1 [Mar]. Any germ w E WZk+l is equivalent to one of the germs

dyk±zdz.

(12.1)

PROOF. When proving Theorem 11.2 we have shown that co has normal form

x1dy1+ +xkdyk+dH(x, y, z),

(12.2)

where ate (0) = 0, as (0)

0. The Morse theorem with parameters (see [AVG], [AVGL]) implies that there exists a nondegenerate transformation z -+ W (x , y, z) reducing the germ (12.2) to xldyl+...+xkdyk+d AO(x,Y)± 21

,

(12.3)

where Ao(x, y) is a function-germ. Since dA0I0 j4 0, the germ x1 dy1 + +xkdyk+dAo(x , y) belongs to Wk . According to the Darboux theorem there exists a germ 0 E Diff(2k) reducing this germ to (1 +x1) d y1 +x2 d y2+ + xkdyk and the germ (12.3) to (12. 1). Q.E.D. We need the following result.

a,i. tu+.3lr1C:ATION OF (ilKMS OF DIFFERENTIAL FORMS

LEMMA 12.1. Let 1 E Diff(2k + 1) be a germ preserving the germ of the

2form dxl Adyl+ +dxkAdyk in R2k+I(XI , yl , ... , Xk l yk l Z). Then 0

is of the form (x, y, z) r(fix,spy,rpz),where apps/az0, a9ylaz=0, and the transformation (x, y) -+ (co co) is symplectic.

PROOF. It suffices to prove that aqx/az - 0, aqy/az = 0. By equating the coefficients of the terms dx1 A dxj ; dy1 A dyl in the forms dx1 A d y1 + + dxk A dyk) we can show that the - + dxk A dyk and 1 (dx1 A dy1 + matrix a(qx , (py)/a(x , y) is nondegenerate at 0. By equating the coefficients of d xi A d z and d y, A d z we can find a system of linear equations for the

vector (aqx/az, aqy/oz) with r nondegenerate matrix and zero right-hand side. This proves the lemma.

COROLLARY 12.1 . The germs co' and w- of the form (12.1) are not equivalent.

PROOF. Assume that (D (w ) = w+ . By Lemma 12.1 the transformation

D is of the form x - 01(x , y) , y - CD2 (x , y) , z --, R(x, y, z) , where the transformation x --' c1 , y - 02 preserves the standard Darboux 2-form. Such a transformation reduces the germ co- to the germ

where M(x, y) is a function-germ. But the equation M(x, y)-R2(x , y, z)= z2 is unsolvable with respect to the germs M(x, y) and R(x, y, z), MI (0) 0. We obtain a contradiction. Q.E.D. 12.2. Even-dimensional singularities from W.

n_1

'°

THEOREM 12.2 [Mar]. Any germ w E W I'° is equivalent to one of the germs w*=(,±X2 (12.4) ) dy, + x2 dy2 + + xk dyk-

At first we prove the following LEMMA 12.2. Any germ w E Wzk -I is reducible to the normal form

H(x, y)[dy1

(12.5)

PROOF. By Theorem 10.2 a germ CO E Wzk 1 can be reduced to the nor-

+ xk dyk + f dx1, jl f = 0. The germ mal form co = dy1 + x2 dy2 + (1 + x1)w belongs to WZk , therefore the germ V [(1 + x1)w] is of the form (1+x1)dy1+x2dy2+-'-+xkdyk for some diffeomorphism (D E Diff(2k). This implies that the germ co is equivalent to the germ

H(x,y)[dyl+1+x

dye+...+1+x dyk], 1

1

§12. CLASSIFICATION OF COCLASS I SINGULARITIES

43

where H(0) # 0. The normal form (12.5) can be obtained now by the coordinate transformation x, -' x1(1 + x1) , i > 2. Q.E.D. PROOF OF THEOREM 12.2. We can assume that the germ co is of form (12.5). The condition rank dwI0 = 2k - 2 signifies that aH (0) = 0 and the transversality condition signifies that 'H (0) # 0. According to the Morse lemma with parameters (see [AVG]) there exists a transformation reducing the germ (12.5) to

[A(y1,x2,Y2,...

,xk,Yk)+x?][dyl+x2dy2+...+xkdyk]

(12.6)

Notice that A(0) 4 0 and the germ A[dyl +x2 d y2 +' +xk d yk] belongs 2k-I to WZk_I . By the Darboux theorem there exists a transformation 0 of the coordinates yl , x2 , y2 , ... , xk , Yk, reducing this germ to dyI +x2 d y2 +'- + xk dyk. This transformation reduces (12.6) to X2'

1+

A(0)

[dyl +x2dy2 + ... +xk dyk ].

In order to obtain the normal form (12.4) it suffices to introduce the coordinate x'1 =x1/ fj A(CD)I. Q.E.D.

LEMMA 12.3. Let x = (x1, ... , xk), y = (Y1, yk) be coordinates in ,(x, y) preserving the Pfaffian R2k . A transformation xi -4 Ip; (x , Y), y1 equation dyI +x2dy2+ +xk dyk = 0 satisfies the condition a9x /axl = 0.

any /axl = 0 (i > 1). The proof is similar to that of Lemma 12.1.

COROLLARY 12.2. The germs co' and w- of the form (12.4) are not equivalent.

PROOF. The germs w are equivalent to the germs iv'* _ (1 ± x, )[dyI +

x2 dyZ+...+xkdyk]. Let Vis =W+ and r=(y1,x2,Y2,... ,xk,Yk) By Lemma 12.3 the transformation CD is of the form r - i yi(r), x1 R(x1, r), a RR (0) 0 0. Then the equation [I - RZ(x1 , r)]M(r) = l + x2 for the germs R(xl , r), M(r) has a solution (R, M), a (0) ;A 0. But this cannot be true, hence the germs w+ and w are not equivalent. Q.E.D. Thus, the germs belonging to W"-1.0(n > 2) are reducible to simple normal forms (without parameters). This immediately implies their stability and 2-determinacy. So the following result is proved. PROPOSITION. Any germ w E

W"-I.0

is stable, 2-determined, and equiv-

alent to the invariant normal form (12.1) for n = 2k + 1 or (12.4) for

n=2k.

111. CLASSIFICATION OF GERMS OF DIFFERENTIAL FORMS

44

12.3. Classification of germs from

Wit+,1.

THEOREM 12.3. For k > 1 a generic germ w E WZk 11 can be reduced to the normal form

where A(x , y) (y1,... yk).

y)z+z3], (12.7) is a function-germ, A(0 0) 0 0, x = (x, , ... , xk) , y = ,

can be reduced to the normal form x, dy, +dH(x, y, z), where dHjo -A 0, 02 H (0) = 0 , (0) = 0. For a generic germ a 4(0) 0 0. Suppose that H PROOF. As it was shown above, every germ w E

WZk+,I

is a function of one variable z with the parameters x, y. Then H can be reduced to the form Bo(x, y) + B,(x, y)z + z3 by a transformation z - 4)(x, y, z) , a (0) 0 0 (since the function H has a versal deformation to + C, z + z3 relative to R-equivalence, see [AVG]). This transformation reduces co to x, d y, + . . . +xkdyk + d [Bo + B1 z + z 3] . Arguing as in the proof of Theorem 12.1 we can show that the latter germ is equivalent to the germ

(1+x1)dy,+xZdyZ+-..+xkdyk+d[B,(x,y)z+z3].

(12.8)

Notice that B1(0)=0. For a

Let

generic germ the condition aB (0) 0 0 holds. This condition means that the

kernel of w0 A (d(oo)k-1 is transversal to the manifold {(x, y)1B1(x, y) _ 01. By Theorem 7.8 there exists a transformation 'P of the xy-plane such that 'Ywo = w0, B1('P) = x1A(x, y) , where A is a function-germ, A(0) 0 0. This transformation reduces the germ (12.8) to the normal form (12.7). Q.E.D.

THEOREM 12.4. For any function-germ A, A(O) # 0, the germ (12.7) is not finitely determined.

PROOF. Assume that the germs w0 + df1 (x, y, z) and w0 + df2(x , y, z) are equivalent, where wo = (1 + x1) d y, + x2 d y2 + +xkdyk , z3 +x1zA1(x, y). Lemma 12.1 implies that their equivalence is realized

j=

by a transformation 'I': (x, y, z) -p (O(x, y), R(x, y, z)), where c (971 , 972)

,

Vdw0 = dw0 . Let Vw0 = coo +

+f1('P)=f2

y). Then the equality (12.9)

holds.

We treat this relation as the equality of two functions of one variable z with the parameters x, y. Assume that A, (0) >0. Let E: {(x, y) l x1 < 01. For every (x , y) E E the function f2 has two critical points z = f -x1 AZ 3 The corresponding critical values are ±4(-x1A2)312/ close to 0.

30. At the same time the function

+ f ('F) has the critical values

§12. CLASSIFICATION OF COCIASS 1 SINGULARITIES

45

y) for every (x, y) E E (the domain E is necessarily invariant under the diffeomorphism (b). Then by equality (12.9)

(x, y) _- 0 for all (x, y) E E and the germs g1 = -x1A1 and g2 = -x1A2 are R-equivalent in the domain E. Moreover, g1(c) = g2 at every point (x, y) E E close to 0. Since = 0 (in E), cYwo = wo (in E). This implies that in the formal category g1(1') = 92 , I coo = too . Therefore, the problem of classifying the germs (12.7) includes the problem of classifying

the functions of 2k variables x, y under the transformations preserving the form wo . Since the generic germ of a l-form in R2k is equivalent to wo , this problem is equivalent to the classification of generic pairs, consisting of the germ of a [-form and a function-germ (under the whole group of diffeomorphisms). Let Nk be the set of k -jets of the mentioned pairs and let Qk be the set of k-jets of diffeomorphisms belonging to Diff(2k). The Diff(2k)-action on the jets from Nk is reduced to the action of the jets belonging to Qk+1 . But when k is sufficiently large dim Qk+1 < dim Nk . This proves the statement of Theorem 12.4. The case A.(0) < 0 or the case A1(0) > 0, A2(0) < 0 can be proved in a similar manner. Q.E.D. COROLLARY 12.3. The singularity classes W+11 and WZk+1 (1 <_ 2k - 1) are wild (i.e., in these classes there are no finitely determined germs). The germs of these classes are unstable. 11

be a class of singularities, satisfying the conditions of "general position" used in the proof of Theorem 12.3. The proof of this theorem shows that the conditions of Proposition 6.3 are valid. At the same time, all classes mentioned in the formulation of the corollary are adjacent to the class WZk+i' . Consequently, the classes are wild. The instability of the germs of these classes follows from Proposition 6.4 (see Corollary 6.1). PROOF. Let WWk

COROLLARY 12.4. Let µ(1) be the number of the moduli in l jets appearing in the classification ofgenericgerms from WZk 11 Then u(l) is asymptotically

equivalent to the number y(l, 2k) of the coefficients of a degree l polynomial in 2k variables (1 -+ oo).

PROOF. Theorem 12.3 implies that µ(1) tends to infinity no faster than y(1,2k). At the same time dim Q1+1-2ky(1,2k), dim N, -(2k+l)y(l, 2k) (see the notation at the end of the proof of Theorem 12.4). Therefore U(1) tends to infinity no slower than y(1, 2k). Q.E.D. 12.4. Classification of germs from WZk -1 ' 1 THEOREM 12.5. A generic germ w E W2k -I ' I (k > 2) can be reduced to the normal form (12.10)

-.. --

..,... avi, yr ,caamJ VY UIJ

K1 NIIAL FUKMS

where r = (y1 , x2, y2, ... , xk , Yk) , A(O) # 0, xz = x2 - Y,

-

PROOF. As was shown in the proof of Theorem 12.2, any germ co c WZk -''' can be reduced to the normal form H(x1 , r)wo , where coo =

H(O) j4 0, a L(0)= az (0)=0. Forageneric germ we have a 4(0) # 0. The change x, -+ R(x,l , r) of the coordinates reduces the germ to the normal form (Bo(r) + B, (r)x, + xi )w0 . Arguing as in the proof of Theorem 12.2, we can show that the normal form (1 + B(r)x, + xi )w0 (12.1 1) can be obtained by a transformation of the coordinates in the space (y1 , x2 ,

Y2,... ,xkIYk) Notice that B(0) = 0. By Theorem 7.5 there exists a coordinate transformation r -p 'P(r) such that 'P' w0 = w0 , B('P) = A(r)z2 , A(0) 96 0 (under the conditions dB A d y,10 710, a B (0) ; 0 that are valid for generic germs). This transformation reduces the germ (12.11) to the normal form (12.10). Q.E.D.

THEOREM 12.6. None of the germs (12.10) is finitely determined.

PROOF. Suppose that two germs f1w0 and f2w0 are equivalent and f = 1 + xi + x1z2A,(r). Lemma 12.3 implies that the equivalence is realized by the transformation Y': (r, x,) - (4)(r), 'P(xl , r)). Besides, we have 0*w0 = Hw0 , where H = H(r) is a function-germ. This implies the equality

Hf1('P) = f2 (12.12) We treat (12.12) as the equality of two functions of one variable x1 with parameters r. Assume that A1(0) > 0 and let E: {rjz2 < 0}. When r E

E the function f2 has two critical points x1 = f -z2A2(r)/3 with the respective critical values

If

a

(-X2A2) 3/2. The function H f, ('P) has the critical values H(r) [1 ± 343((-x2A1)(4)))3h2] for every r c- E. This implies 3

3

that H(r) - 1 (in E) and Vwo = wo (in E). At the same time the germs g1 = -22A1 and g2 = -z2A2 are equivalent, i.e., g, (1) = g2 (in E ; the

domain E is invariant under the diffeomorphism (b). The final arguments are the same as at the end of the proof of Theorem 12.4. Q.E.D.

-' '

COROLLARY 12.5. The singularity classes Wk_"'

(k > 2) and WZk

(1 < j < 2k - 2) are wild and the germs of these classes are unstable.

COROLLARY 12.6. Let p(l) be the number of moduli appearing in the clas-

sifcationofthegeneric germs belonging to WZk ''. Then u(1) - y(l, 2k-1) (see the notation in Corollary 12.4). These corollaries are proved in the manner as Corollaries 12.3. and 12.4 respectively.

§13. CLASSIFICATION OF POINT SINGULARITIES

47 1

THEOREM 12.7. A generic germ belonging to the singularity class WZ'' can be reduced to the normalform (1 +A(y)x+x3) dy, A (0) = 0. There are no finitely determined and stable germs belonging to the class W.''' . PROOF. When proving Theorem 12.5 we obtained the normal form (12.11)

without using the condition k > 2. For k = 1 the normal form (12.11) coincides with that given in Theorem 12.7. For generic germs A(y) = ay + ... , a 0 0. We can assume that a > 0. The proof of the absence of finite determinacy is similar to the arguments in Theorem 12.6 (we should take

the domain {(x, y)Iy < 0} instead of the domain E = {rjx2 < 0} ). The instability of the germs follows from Proposition 6.4.

Q.E.D.

§13. Classification of point singularities

We show that a generic germ w E W,0 is 1-determined. Consequently, the classification of the generic germs belonging to H is reduced to the classification of their 1 -jets. The classification of 1 -jets is reduced, in its turn, to the classification of quadratic Hamiltonians. Results of §§13.1 and 13.2 belong to Lychagin [L1], but we prove them in another way. 13.1. Classification of the 1 jets of the germs w E WWk . The germ of a vector field X. defined by the equation X., d w = co is invariantly related to every germ w E Wk satisfying the condition rank dwlo = 2k . It is clear that R2k is a singular point for XG, . Nevertheless, the map w E Wk - X. 0E to the set of germs of vector fields with singular point 0 is not surjective.

THEOREM 13.1. The field X. is the sum of a Hamiltonian field (corresponding to the symplectic structure (R2k , dx n dy)) and of the Euler field

1 r aaxi k

Xi

i=l

k

+

a

i=l

yi ayj

PROOF. By the Darboux theorem the germ w can be reduced to the normal

form w = k 1 x, dyi + dH(x, y), d H10 = 0. In these coordinates the field X. assumes the form k

xw _ H=H+.IrkIx,y,,

Let

Xco

2

k

i=l

r I

x, + ay,) a ,,

ax! a)

then

a\

a x'ax, +yiay, ] +

I aH a_ aH a i=l

This implies the statement of the theorem.

[ayi axi

Q.E.D.

ax, ay,]

48

111. CLASSIFICATION OF GERMS OF DIFFERENTIAL FORMS

COROLLARY 13.1. The spectrum of the linearization of X. at 0 E IIB2k i = 1 , ... , k, Rep, > 0 . ; '4k+i = . - pi ,

consists o f the eigenvalues iii = 2 + µi

Consider the set of germs w E W? satisfying the conditions (a) rank dwI0 = 2k; (b) 'li#)'i (c) 01AI +

(i#.1);

+ a2kA2k 0 1 for any integer ai > 0, aI +

+ a2k > 3.

Let W2k be a set of such germs.

THEOREM 13.2. Let w be the germ of a generic 1 form at a point where this form vanishes. Then w E W k

PROOF. Let LI C Wk be the singularity class consisting of the germs for which condition (a) is violated, Eii C W k be the set of germs for which condition (b) is violated for fixed i, j, and Etr c W°k be the set of germs for which condition (c) is violated for a fixed multi-index a = (al , ... , a2k) . Any of these classes is defined by a condition on the 1 jet and the codimension

of each class is 2k + I . There is a countable number of these classes, and they can be denoted EI , E2 , E3 , .... According to §4.2 there exist open everywhere dense sets Ai C A' (M) such that for any form w E Ai we have

Ei(w) = 0. Suppose that A= fi Ai. Then A is a "thick" set of 1-forms for which the statement of the theorem holds. Q.E.D. The invariants u, for a germ w E W k (see Corollary 13.1) are divided into three groups: real positive #1 , ... , s1; pure imaginary pl+i/ = iOl

, ...

, Al+$ = i03 , and pairs u,+s+1 = al + if l

µ1+s+2 = a2 + ifl2' µl+s+p+2 = a2 - tf2'

,

u,+s+p+, - al - 1fl ,

µl+s+p - ap + ifp ,

'

ap-iflp. Here O >0, aj >0, fl>0, k=l+s+2p.

p1+s+2p

THEOREM 13.3. The 1 jet of a germ belonging to Wk can be reduced to the invariant normal form 2

[ix1a'- yi i

((

r

d l

1

r

r

i

1

r

l+i

+ y) !+i +

1

r 1

(13.1)

where Qi = ai(x1+s+iyl+s+r + xl+s+p+iyl+s+p+!) + flifxl+s+iyl+s+p+i - x1+s+p+iyl+s+i)

(with arbitrary combinations of signs in Ei

PROOF. In suitable coordinates (x, y) the l jet of a germ w E W k is k

j'w = 2 E(xi dyi - yi dxi) +dH(x, y), 1

(13.2)

§13. CLASSIFICATION OF POINT SINGULARITIES

49

where H(x, y) is a quadratic polynomial. Linear transformations preserving the normal form (13.2) are necessarily symplectic (for the standard symplectic structure d X A d y ). Every linear symplectic transformation preserves not only the 2-form dxndy but also the 1-form >i (x; dy, -y; dx;) since it can be written as (J(y) , d (y)) , where J is a standard Darboux matrix, and the

linear transformation with matrix T carries the matrix J into the matrix

T`JT. Thus, the classification of 1-jets (13.2) can be reduced to the classification of quadratic Hamiltonians (i.e., two normal forms (13.2) with H = Hl and H = H2 equivalent if and only if the Hamiltonians Hl and Hz are equivalent). Let XH be the Hamiltonian vector field corresponding to H. It is easy to check that

Xro=XH+2E (x +yjay`) '

.

Thus t,ui are the eigenvalues of the XH-linearization at 0. Now, using the well-known classification of quadratic Hamiltonians (see [A2]) we obtain the invariant normal form (13.1). Q.E.D.

It is convenient to use complex coordinates in order to write the germs (13.1) in a more compact form. Such coordinates are connected with the coordinates x1+1 , ... , Xk , Y1+1 , ... , Yk by the symplectic complex transformation (below i = i.e., J8jx1+j(1 - i)/2 - (1 + i)y,+j12

j = 1 , ... , s,

Bj,

y1+j _ V/"ojx1+j(1 - i )/2 + ( 1 + i)y1+ /2vj ,

j1 ,

x1+s+j = - iXi+s+j + xi+s+p+j, xl+s+P+j ix,+S+j + xl+s+P+j ,

J = 1 , ... , P,

yl+s+j = (Iy1+s+j + yi+s+P+j )/2 , yl+s+p+j

= (-iY,+s+j +Y1+s+P+j)l2,

j = 1 , ... , p.

We omit the tilda and use the same notation for the new coordinates as for the old ones. The normal form (13.1) becomes k

k

>(xj dyi - yj dx1) + d > µ;x;y; or k 1

(.Ui

1

- 2) Yj dx, + ( "I + 2) Xjdyi.

(13.3)

50

III. CLASSIFICATION OF GERMS OF DIFFERENTIAL FORMS

13.2. Sufficiency of the 1-jet of a germ co E Wk

THEOREM 13.4. A germ co Ek is 1-determined. COROLLARY 13.2. A germ w E Wk is reducible to the invariant normal form (13.1) or (in complex coordinates) to the invariant normal form (13.3). Let

be the germ (13.1).

LEMMA 13.1. The homological equation (13.4)

T

with respect to the vector field X has a solution X E f'3f t) for any right-hand

Side TE!)31>,I>2. PROOF OF THE LEMMA. Let T = )1 (T1i dx1 + TZi dy1) in the complex coordinates of the normal form (13.3) and let k

X

k

a

ax.

a

+ W1

y,

in these coordinates. Suppose that u = X, where

is the germ (13.3).

Then equation (13.4) can be rewritten as 1P, +

8u

__

au

FT k

u=>

1

(y;-2

yiVi+Epi+21 xiVi1

This system is reduced to one equation for the germ u, viz., k

k

u-E(2-k;) y,au-E(2+p') x'8xi yj ,

=y,

(13.5)

where y is a germ belonging to Ml(t+I) Let y

E

6=(f1,...,fk).

I"I+101=t+I

Then equation (13.5) has the solution u = Z ra IDy" Ox"y0 , where r" 0 = 1 - E a, (1 /2 +,u,) - E fi (1 /2 - a,) (it follows from condition (c) (see § 13.1) that for I a I + Ifl I = I + 1 > 3 we have r" 0 # 0 ). Thus, the vector field from

JI' of the form r, (i2;

- ay) a + i (-T1i + ax) ay is the solution of

the homological equation. Q.E.D. We also need the following statement.

LEMMA 13.2. Let w E Wk . Then the field X, is hyperbolic at 0 (i.e., the spectrum of its linearization at 0 does not interesect with the imaginary axis).

§I3. CLASSIFICATION OF POINT SINGULARITIES

51

PROOF. If there exists an eigenvalue ,l, = ri, r E R, then the spectrum also contains .12 = -ri, A3 = 1 - ri. But in this case condition (c) of § 13.1 23 = 1 holds true. Q.E.D. fails, since the equality 2A, + PROOF OF THEOREM 13.4. We can assume that co = 4 +.u, where 4 is the

germ (13.1), j' k = 0, Suppose that we have proved the equivalence of the

(1 - 1) jets jr-'w and jr-'4. Let us show that the 1 -jets j!w and jI 4 are also equivalent. We can assume that jiw = 4 + T, T E 93T«l . According to Corollary 5.1, it suffices to verify the solvability of the equation

jI[Xr,d(4+tT)+d(Xr,(4+tT))] = -T.

(13.6)

We seek the solution X, in the form Xr = X E Jli<< . Then equation (13.6) will be reduced to the homological equation (13.4), which is solvable by Lemma 13.1. Thus, for an arbitrary finite number I the 1-j ets jtw and jI are equivalent. Note that a transformation that "kills" the additional term r E 9A(1 in the /Jet of 4 + T, differs from the identity by terms of degree

> 1. This implies that the germs w and 4 are equivalent not only at the level of arbitrary finite jets, but also in the formal category. According to the Borel theorem, there exists a smooth diffeomorphism 0 such that the formal expansions of the germs caw and 4 at 0 coincide. Then P co = 4+0 , where 0 E JJIO° , i.e., 0 is the germ of a flat 1-form. Now the proof is reduced to the proof that the germs 4 and 4 + 0 are equivalent. The Darboux theorem

implies that we can assume without loss of generality that d9 = 0. We shall use the homotopy method (Proposition 5.1). It suffices to prove the solvability of the equation

Xr,d(4+t0)+d[Xr,(4+t9)] _ -0 with respect to the family of vector fields Xr, t E [0, 1 ] . Let

0=>2(0,,dx;+62,dyi),

Xr=E 91,ta r+witayi

ur=Xr,(4+te). The given equation can be written as a system eayi u _ 8u _ ezr - wit + axi = ett) qtr +

(13.7)

ur=Xr'(4+10).

After eliminating cir , wI: system (13.7) is reduced to the equation (13.8)

U, = YY(ur)+Or,

with respect to the family of germs ur , where Yr = Xc+re is a family of vector fields, 0, is a family of germs of flat functions. The family Y, has the form X, + Yr , where Yr is a family of germs of flat vector fields. By Lemma 13.2 the field Y, is hyperbolic at 0 (t E [0, 1]). Now we can apply Theorem 8.1 by which equation (13.8) has a solution u, E J31°° .

Q.E.D.

S2

III. CLASSIFICATION OF GERMS OF DIFFERENTIAL FORMS

13.3. Classification of the germs w E Wk+I

We begin with the simple

.

case k = 0. A germ w E WI0 is of the form df (x) , f (O) = f (0) = 0. A singularity is typical if and only if f'(0) 0. It is clear that in this case the germs are reduced to the normal form -Lx dx. The germs x dx and -x dx are not equivalent. Consider now the case k > 1 . It was shown in § 11.3 that there is an W k+l. For a generic germ w E WZk+I the following adjacency W2 conditions hold: (a) rankdwjo = 2k ; (b) the set WZk+I (p) is a smooth manifold near 0 for all germs p E that are close to w (3); (c) the kernel of the 1-form dwI0 is transversal to the manifold WZkt(w)

+l '

Wk+I

at 0. Under condition (a) the germ can be reduced to the normal form E xi d y, +d H(x , y, z) . Conditions (b) and (c) taken together imply that as (0) 0.

Then a transformation z

O(x, y, z) reduces the germ to the form k

w=±zdz+xidy,+dA(x,y),

(13.9)

where dAlo = 0. Let & be a restriction of co on the manifold WZk+I (w) . Then w E W W.k In the coordinates of the normal form (13.9) the manifold W2k+I(w) and the 1-form w become k

W. k+I(w) = {(x, Y, z)Iz = 0} ;

w=

x, dy, + dA(x, y).

Thus, the following result holds. THEOREM 13.5. The generic germs wI , w2 E Wk+I are equivalent if and only if their restrictions on the corresponding sets of the first degeneration Wik+I (wt) , i = 1, 2, are equivalent.

Here by the generic germs we mean those satisfying conditions (a), (b), and (c). We introduce a new condition of "general position" is a germ belonging to Wk (d) the restriction of w to Let Wk+I be the set of germs that satisfy conditions (a), (b), (c), and (d). The next result follows from § 13.2 and Theorem 13.5. THEOREM 13.6. (1) There exists a thick set A of 1 forms w E A' (M) such

that if w E A and w 1" = 0 the germ of w at the point a belongs to Wk+I ()Note that 0 0 W22,..,(w) , but 0 E W24}1(w)

.

§I4. BASIC RESULTS AND COROLLARIES

53

(2) A germ w E Wk+[ is 1-determined and can be reduced to the invariant normal form

fzdz+E ( (generally speaking, in complex coordinates, see § 13.1) where, p, are moduli_

REMARKS. (1) For fixed p, the germs (13.10) with terms z d z and -z d z are not equivalent. This fact is proved in the same manner as Corollary 12.1. (2) In contrast to the case w E WZk+I the 1 -jet of H in the normal form I x; dy;+dH(x, y, z) ofa germ to E WZk+[ is defined by jr-`w . So, the singularity class W +I is distinguished from W(2k + 1) by a condition on the 1 -jet.

§14. Basic results and corollaries; table of singularities; list of normal forms; examples

In what follows (1), (2), ... signify the numbers of the normal forms in the list of normal forms in Table 14.2 (see page 56). In all formulations except Theorem 14.6 we assume that n > 2. THEOREM 14.1. Any stable germ ofa 1 form on an n-manifold is equivalent to one and only one of the normal forms

(1), (3) for n = 2k + 1 ;

(2), (4) for n = 2k.

THEOREM 14.2. Any unstable but finitely determined germ of a generic 1form on an n-manifold is equivalent to the normal form (10) for n = 2k + 1

or (ll) for n = 2k. THEOREM 14.3. For a generic 1 form w on a manifold M there exists a

subset j4 c M of codimension > 2 such that the germ of w at any point a E M \ M is stable. THEOREM 14.4. There exists an open set A C AI(M) such that for any 1form w E A there exists a codimension 2 subset M C M with the following property : the germ of w at any point a E M is not finitely determined.

THEOREM 14.5. Let S be a set where a generic 1 form w vanishes. Then S consists of isolated points, at which the germ of w is unstable but finitely

determined. At the points of M \ S the germ of w is either stable or not finitely determined. THEOREM 14.6. The germ ofa generic 1 form given on a line or on a circle is stable at each point and equivalent to one of the three germs dx, x dx, -x dx.

We give below a table of typical singularity classes and a list of normal forms. Note that we use the partition into singularity classes for which all

III. CLASSIFICATION OF GERMS OF DIFFERENTIAL FORMS

54

the given characteristics depend only on the singularity class (they do not depend on the germ belonging to a fixed class). It is also important that the number of moduli that distinguish close nonequivalent germs from the given class can in the general case differ from the modality. For the singularities given in the table this number (m) equals the modality, except for the singularities from W k w2k+1 Notation in Table 14.1 of singularity classes.

codim codimension of a singularity class I minimum order of a jet which enables us to determine the given p

singularity class index of finite determinacy of the germs from the given class (p = oc when there is no sufficient jet)

m number of moduli that distinguish close nonequivalent germs from the given class st

sign of stability: + (the germs from the given class are stable), -

*

(the germs from the given class are unstable) takes place only for the generic germs from the given singularity class preliminary (not invariant) normal form

**

Comments on the list of normal forms In Table 14.2.

Coordinates

in (2), (4), (6), (11) xl , YI , ...

Xk , Yk

,

in (1), (3), (5), (10) z , xl , YI , ... , Xk , Yk in (9) xI , Y1 , ... , xp , Yo , U1, - , uR-2p in (8) z, xl , Y1 , ... , Xp , Yp , U1 , ... , Un-2p-l -

in (7) x, y Numerical moduli

in (10), (1) pf E C (generally speaking), Rep J > 0

Functional parameters in (5)-(9) A, j, jlf = 0 EXAMPLE 14.1(4). At the generic points of a 2-manifold the germ of a generic 1-form is stable and equivalent to the normal form (I + x) d y . De-

generations take place on a curve L and on a set S of isolated points, S n L = 0. At the generic points of the curve the germ remains stable and equivalent either to the normal form (1 + x2) d y or to (I - x2) d y . At isolated points of the curve the germ is reducible to the normal form (1 + A(y)x + x3) d y and is not finitely determined. At the points of S the germ is unstable but 1-determined and equivalent to the normal form (p + 1/2) y dx + (p - 1/2) x dy for real or purely imaginary values of the invariant u. (4)In Examples 14.1-14.5 we assume that for the considered 1-form 11 E AI (M) all typical singularity classes E are realized, i.e., 1(w) j4 0.

§ 14. BASIC RESULTS AND COROLLARIES

55

Table 14.1. Typical singularity classes Singularity class

codim

I

p

m

st

Nor-

Remarks

mal

form 1

0

+

(1)

1

1

0

+

(2)

2

2

0

+

(3)

2

2

0

+

(4)

2

2

00

00

2

2

ao

oo

2

2

00

00

1

00

00

W2k+l'

0

W2k`

0 1

1

W2k 0

W22k-t'0 2k.1 W2k+l W2k_t't

W2 '1

1

Wnj

-

(5)

(6) (7)

*, .

( g)

j#

j=2p+1
k > 1, , :.

0,

codim
n

2

1

00

00

-

(

k

-

(11)

k

-

(10)

9)

j=2p
j

0,

codim
Wk

2k

1

1

W2kl

2k + I

1

1

k> 1

EXAMPLE 14.2. At the generic points of a 3-manifold the germ of a generic

1-form is stable and equivalent to the normal form dz + x dy . The first degeneration takes place on a hypersurface M1 . At the generic points of M1 the germ remains stable; it is equivalent to the normal form (1 +x) dy±z d z .

The next degenerations hold on a curve L c M1 and on a set S c M1 of isolated points, S n L = 0. At the points of L the germ is not finitely determined, at the generic points of L it is reducible to the normal form (1 +x) dy+d(xA(x, y)z+z3) . At the points of S the germ is not stable but is 1-determined and equivalent to the normal form ±z dz + (p + 1/2) y dx + (A - 1/2) x d y , where p is a real or purely imaginary modulus. EXAMPLE 14.3. At the generic points of a 4-manifold the germ of a generic

1-form is stable and equivalent to the normal form (1 + x1) d y1 + x2 d y2 . Degenerations hold on a hypersurface M1 and on a set S of isolated points, S fl Ml = 0. At the generic points of M1 the germ is stable and equivalent to the normal form (1 ± xi) dy1 + x2 d y2 . The nongeneric points form a 2manifold M2 c M1 . At the points of M2 the germ is not finitely determined;

at the generic points of M2 the germ is reducible to the normal form (6) (k = 2). At the points of S the germ is unstable but 1-determined. It is equivalent to the normal form (11), where k = 2, Re p1 > 0, Re p2 > 0 and either p1 = ±J2 , l22 = fP2 , or 91 = Q2 .

,u. I.Lw3Mr,4.n, LON Or U KMS OF DIFFERENTIAL FORMS

Table 14.2. List of normal forms No

Normal form

(1) (2)

(1+x1)dy1+x2dy2+x3dy3+- +xkdyk

(3) (4) k

(5)

(1+x1)dy1+Lxidyi+d[z3+A(x1,... ,Xk,y1,...

Yk)X1Z]

,=2

(6)

3

[ l + x l + x1 A(yl , x2 , y2 , ... , Xk , Yk)(XZ - Y1 }]

x[dyl +x2dy2 +...+xkdyk] (7)

(1 + A(y)x + x3) dy,,

(8)

dz+xldy1 +

(9)

n-2p-1

A(O)=O

4(Z' X1 , y1 , ... , Xp , Yp

141 ,

.. , un-2p-1) dui

(1 +x1)dy1 +x2dy2+- +xpdyp n-2p

+ > f (, X1, Y1, ...

, XP , Yp , u1 , ... , u17-2P) d ui

i=l

(10)

k

k

j=1 1

i=1

fzdz+> (p,-7)y,dxi+ k

(11) i=1

(,ui+ )x, dy1

k

(yi- i)yidxi+> (/i+ lf)x,dyi i=1

EXAMPLE 14.4. At the generic points of a 5-manifold the germ of a generic

form is stable and equivalent to the normal form d z + x1 dy1 + x2 d y2 . A degeneration holds on a hypersurface M1 At the generic points of M1 the .

germ is stable and reducible to the normal form ±z d z + (1 +x1) d y 1 +x2 d Y2 .

The next degeneration holds on a 3-manifold M2 c M1 (at the points of M2 the germ is not finitely determined) and at isolated points outside M2 . At these points the germ is unstable but 1-determined; it is reducible to

the normal form (10), where k = 2, Re u, > 0, Re z2 > 0, and either Jul = tP1 , P2 = +f[2 or u, = #2 EXAMPLE 14.5. The set of first degeneration for a generic 1-form, defined

on a manifold of dimension 6 < n < 14, is a stratified manifold whose singular points form a smooth codimension 6 manifold. For n > 15 the set of singular points is a stratified manifold (the set of its singular points is of codimension 15).

S 15. COMMENTARY

57

§15. Commentary §10. There are various ways to define a germ's class. We use the definition in [F], [Mar]. Theorem 10.1 is a corollary of the following Darboux theorem (see [Stl]): all 1 forms having a constant class 2p+ 1 (2p respectively) in a neighborhood of 0 E IIt° are locally equivalent to the 1 form (10.1) (to (10.2) respectively). Class calculation in local coordinates and related questions were consid. ered in [Z2]. One can compare Theorem 10.2 with the Poincare-Dulac theorem giving

the normal form of the germ of a vector field at a singular point which depends only on the 1 -jet of the vector field at this point (see [Al]). Notice, that Theorem 10.2 holds in the smooth category (as well as in the analytic one).

§11, 12.1, 12.2. The material is based on the results of Martinet [Mar]. In [Mar] the class W11-t'0 is defined in a similar way.

§§12.3, 12.4. The fact that all finitely determined germs co, wlo 54 0 are

reducible to Darboux and Martinet models was announced in [Z3]. The construction of functional moduli for the case W21' I can be found in [AG], [Z3]; in the work [JRe] a close problem of classification of singularities of control systems in the plane is considered and a complete list of normal forms (including degeneration of an arbitrary codimension) is obtained. Corollaries 12.4 and 12.6 show that the normal forms (12.7) and (12.10) are invariant in the asymptotic sense. Such normal forms for distributions can be found in [Z 11 ]. §13. The results in §§13.1-13.2 are due to Lychagin [L1]. He obtained them from the classification of partial differential equations. We essentially simplify the proof of Theorem 13.3 given in [Ll] by completely reducing the corresponding algebraic problem to a well-known problem. The existence of moduli in the classification problem for germs belonging to W0 was also shown in [GT fl. The results of §13.3 were announced in [Z4]. [Z7] describes the use of complex coordinates in various classification problems for real germs. The resonance case in the classification of point singularities was considered in [Z1], [Z5]. Classification of germs from W0 (versal deformations for arbitrary (nontypical) singularities) can be found in [Ko]. Lychagin also proved an analog of Sternberg's theorem: i f c o, w E W , w - w E fl , and X. is a hyperbolic field, then the germs Co and w are equivalent [L2].

CHAPTER IV

Classification of Germs of Odd-Dimensional Pfaffian Equations In this Chapter we classify singularities of Pfaffian equations on a mani-

fold M of dimension n = 2k + 1 > 3. The stratification into singularity classes and classification results are essentially different from those for the case of 1-forms. In particular, the number of nonequivalent stable germs increases.

The cases n - 3 and n > 5 for Pfaffian equations are essentially different and should be considered separately beginning with codimension 3. We show that a germ of a generic Pfaffian equation on a 3-manifold is stable outside a set of isolated points and that the germ at each point is finitely determined. For n = 2k + I > 5 the germ is stable outside a codimension-4 set of points.

We use the notation and results of Chapters II and III. In addition, we introduce the following special notation:

x=(x1,... ,xk), Y=(Yl,... ,yk), r = (y1 , x2 , y2 , ... )xk , Yk) , wo=dy1+x2dy2+x3dy3+...+xkdyk.

§ 16. Class of Pfaffian equations; classification of 1 -jets; preliminary normal form

As well as for 1-forms, the only invariant of the 1 -jet of a Pfaffian equation at a point is the class of the Pfaffian equation at this point (if we do not consider isolated singularities connected with the vanishing 0 -jet). Classification of 1-jets of Pfaffian equations and preliminary normal forms can be obtained from the corresponding results for 1-forms.

Let {w = 0} E PA`(M), a E M,

16.1. Class of a Pfaffian equation. WI" 34 0.

DEFINITION. The class of a Pfaffian equation {w = 0} at a point a is the number (2p + 1), where W n (dw)°ln j4 0,

wA 59

(dw)°+'I,,

= 0.

(16.1)

60

IV. CLASSIFICATION OF GERMS OF ODD-DIMENSIONAL PFAFFIAN EQUATIONS

REMARK. The definition is independent of the choice of the 1-form rep-

resenting a Pfaffian equation, since if wI = Hw we have w1 A (dw1)' _ H'+'w A (dw)' . Thus, the class of a Pfaffian equation {w = 0} at the points where the form w does not vanish is odd. The maximum value of the class is equal to n = 2k + 1, the minimum value is equal to 1. We denote the class of Pfaffian equation {w = 0} at a point a by clo {w = 0}. We agree that for wI0 = 0 c1Q{w = 0} = 0. Obviously, clQ{w = 0} is a characteristic of the 1 -jet of the Pfaffian equation at the point a . By the class of a germ e E PW(n) we mean the class

of the equation E E PA' (R") at 0 E R", where e is the germ of E at 0. We use cl{w = 0} to denote the class of the germ {w = 01. THEOREM 16.1. Let w E PW (n) , n = 2k + I, cl{w = 0} = 2p + 1, and

w be an arbitrary germ representing the equation {w = 01. Then

(1) 2p+l
(2) for p < k there are germs w1 , w2 E W(n) representing the equation {w = 0} such that cl wi =2p+ 1, cl w2 = 2p+2. PROOF. The first statement follows directly from relation (16.1). We prove

the second part. Assume that (dw)" '10 = 0. We can choose a functiongerm H such that dH A w A (dw)pjo :yt 0, H(0) :yt 0. One can easily see that [d (Hw)f +'to :yt 0, and hence cl (Hw) - 2p + 2. Thus, the existence of the form w having the class 2p + 2 and representing the Pfaffian equation {w = 0} is proved. By Theorem 10.1 the form j'w in suitable coordinates

is (l+xi)dyl+x2dy2+--.+x pdyp+xp+Idyp+I. Thentheform (l+xi)-'w represents the Pfaffian equation and its class is 2p + 1 As a corollary to Lemma 16.1 we obtain

.

Q.E.D.

LEMMA 16.1 (class calculation in local coordinates). Let j'w = (C + Ax, dx). Then for C V lm (A - A') we have cl{w = 0} = rank (A - A`) + 1

andfor CEIm(A-A') we have cl{w=0}=rank (A-A')-1. EXAMPLE 16.1. Let w E A' (R5), w = dx + xy d z + yv du. At generic

points cl{w = 0} = 5; at generic points of the hypersurfaces {x = 0} and {y = 0} we have cl{w = 0} = 3, and at the points of the 2-plane x = y = v = 0 we have cl{w=0}=1. 16.2. Classification of 1 -jets.

THEOREM 16.2. Any 1 -jet of a Pfaffian equation of class (2p + 1) can be reduced to

PROOF. By Theorem 16.1 it is possible to choose a 1-form w that represents the Pfaffian equation and has class 2p + 1 . By Theorem 10.1 this germ can be reduced to the above form. Q.E.D.

§ 17. SINGULARITIES

61

16.3. Preliminary normal form.

THEOREM 16.3. Any germ of a Pfaffian equation of class 2p + 1 can be reduced to the normal form

w=dz+x,dy,+ -+x dyp+f,(x,y,z)dyp+, k

+ E [fj(x, y, z) dx,,

(16.2)

J-p+2

in the coordinates x = (xl , ... , xk) , y = (y, , .. , yk) , Z. where f,(x, y, z) and g1(x, y, z) are functional parameters, j1 j = j'gi = 0. PROOF. For p = k there are no functional parameters and we can fulfil the reduction to (16.2) by the Darboux theorem for 1-forms. Assume now that p < k. By Theorem 16.1 for the germ of the Pfaffian equation we can choose the germ w of the 1-form that represents the germ of the Pfaffian equation and has class 2p + 2. By Theorem 10.2 this germ can be reduced to the form & + x, dz. Thus, the Pfaffian equation can be represented by the form (&+ x, d z) l (1 +x,). The normal form (16.2) can now be obtained by the coordinate substitution xi -' i, = x,) . Q.E.D. As a corollary of this theorem we obtain the classical Darboux theorem for contact structures: all contact structures (M, w), dimM = 2k + I , are locally equivalent. EXAMPLE 16.2. The germ of a Pfaffian equation in R2k+1 of a minimum

nonzero coclass 2 can be reduced to a normal form with one functional parameter, viz.,

{wo+f(x, y, z)dz=0}(1),

j'f=0.

(16.3)

However, the normal form (16.3) is only preliminary. In §17.2 we show that a generic germ of coclass 2 can be reduced to a stable normal form. § 17. Singularities

In § 17 the singularities of Pfaflian equations associated with the decrease of a germ's class are described. The first singularities we meet have coclass 2. The corresponding singularity class will be divided into subclasses. It follows from Theorem 12.1 that a generic germ of coclass 2 can be reduced to a stable normal form. The scheme of adjacencies for singularity classes and (preliminary) normal forms for the germs of coclass 2 with additional degenerations are also given. 17.1. Singularities associated with the decrease of a germ's class. The set of germs belonging to PW(n) and having class 2p + 1 is a singularity class. We denote it PW2°+' (')The notation wo is given in the Introduction to Chapter IV.

62

IV. CLASSIFICATION OF GERMS OF ODD-DIMENSIONAL PFAFFIAN EQUATIONS

THEOREM 17.1 [Mar]. codimPW2p+1 = (n - 2p - 1)(n - 2p - 2)/2.

= 0 and the statePROOF. For 2p + 1 = n we have codim Let 2p + 1 < n, Then by Theorem 16.1 ment of the theorem is valid. PW2°+'

codim PW2p+I = codim W2P+2 (in W(n) ). Now the statement follows from Theorem 11.1. Q.E.D. REMARK. It is clear that the codimension of the class consisting of the

germs {w = 01, m10 = 0 is equal to n. COROLLARY 17.1. The class of a germ of a generic Pfaffian equation on an

n-manifold is greater than or equal to n - [ 1/2 +

2n -+1 /4] or is equal to

0

EXAMPLE 17.1. The class of a germ of a generic Pfaffian equation in ilt equals 5 at generic points, equals 0 at isolated points, and equals 3 at all other points (which form either a set of codimension 2 or the empty set). The classes j I PW' , i = n, n - 2, n - 4, ..., 1 stratify the space J' of I -jets of Pfaffian equations. The closure of the stratum j I PW' consists of the strata j I PW$ , s < i. The stratum j I P W' is a smooth manifold at every point except for the points of the closure of the stratum j' PWn-2 .

17.2. Generic germs of coclass 2. By Theorem 17.1 the class codimension 1.

PW"-2

has

LEMMA 17.1. Let e E , e: {w = 01. Then the (n - 2) form w A (dw)k-I 1° has a kernel of dimension 2. PW,"-2

PROOF. We can use the local coordinates from Example 16.2. In these coordinates Kerw A = (a/ax1 , 49/49z). At the same time the kernel of w A is defined invariantly (it is independent of the choice of the 1-form representing the Pfaffian equation). Q.E.D. Each germ e E PW"-2 is invariantly related not only with KerwA(dw)k-' but also with the set PWn-2(e) (see the notation in §4). We call this set the set of the first degeneration and denote it SI or S, (e) . For a generic germ e E PW"-2 the following conditions are valid: that (a) S, (e) is a germ of a hypersurface for all the germs e E are close to e. (b) Ker w A is transversal to Si (e) at 0. Let PWR -2' ° be a set of germs for which conditions (a) and (b) are (dw)k_'I0

(dw)k-I

PWn-2

(dw)k-I10

satisfied.

THEOREM 17.2. Any germ e E PWn-2.0 is equivalent to the germ

w0+x2 dz. The proof is based on the following lemma.

(17.1)

§17. SINGULARITIES

LEMMA 17.2. Let e: {w = 0} E PWR -Z

.

63

For each vector A E Kerw A

(d w)k- I there exists a germ H such that

H(0)

0,

dim Kerd(Hw)10 = 1 ,

A E Kerd(Hw)10.

First we show how Theorem 17.2 follows from Lemma 17.2. According to condition (b) we can choose a vector A E Ker co A (dw)k- 110 transversal

to the manifold S1(e). By Lemma 17.2 it is possible to choose a form & representing the Pfaffian equation and such that dimKerdw10 = 1 , A E Kerdw10 . The form w satisfies the condition of Theorem 12.1 and can be reduced to co} = +z d z + (1 + x1) dy1 + x2 dy2 + + xk d yk . It is easy to note that the germs {w} = 0} are equivalent (we can multiply the germ co- by (-1) and change y, to (-y,) ). Therefore, all the germs belonging to PWR Z' ° are equivalent. But the germ (17.1) belongs to PW"-Z' ° (since for it SI : {x1 = 01, Ker w A (d w)k-' _ (a /axI , a /a z)) and this gives the statement of Theorem 17.2. PROOF OF LEMMA 17.2. By Theorem 16.1 we can assume that rankdwl0 = 2k. Then we can also assume that j 1 co = (1 + x1) dyl + x2 dy2 + + xk dyk . In these coordinates Ker W A (d w) k-' 10 = (0/ax1 , a/az). Let A=

a(a/ax1)+f(a/az) . Let us choose a and b so as to satisfy a(l+a)+fb = 0, (1 +a )2 +b2 O .Then the germ H = ax, + bz + 1 satisfies the assertion given in Lemma 17.2. Q.E.D. REMARK. We can formulate Lemma 17.2 as follows: the set of directions {Kerdw10}

,

where iv represents a Pfaffian equation {w = 01, is the space

Kerw A (dw)k10.

17.3. Partition of the class into subclasses. Let PWn-z' * be a PW"_2' I be a set of germs for which condition (a) of § 17.2 is violated and set of germs for which condition (a) is valid but condition (b) is violated. PW"-2

THEOREM 17.3. codim PWR-Z' = n + 1 , codim PWR -2' = 3. I

PROOF. Every germ e E PWR -2 can be reduced to the normal form (16.3) of Example 16.2. In these coordinates the set S1 (e) is given by the equation

a f/ax1 = 0 while Kerw A (dw)k-1I0 = (a/ax1 , a/az). Condition (a) is violated if d (a f/ax1)10 = 0 . Condition (b) is violated if

ox,

(0) =

o

(0) = 0. This gives the state-

ment of the theorem, since codim PW"-2 = 1 . Q.E.D. Thus, the singularity class PR is nontypical. REMARK. From the proof of Theorem 17.3 it is clear that the singularity classes PW"-"0 , P RV,.

PWR Z'' are distinguished from PW(n) by

conditions on the 2 -jet.

64

1V. CLASSIFICATION OF GERMS OF ODD-DIMENSIONAL PFAFFIAN EQUATIONS

Consider the singularity class PWR -2' . The set PWR -2' 1(e) is invari-2,1 . We call this set the set of the antly related with each germ e E PWn second degeneration and denote it S2(e). Let n = 2k + 1 > 5. For a generic germ e: {w = 0) E PWn-2' the following conditions are satisfied: (a) the plane of the form co at 0 (Ker wl0 c TOR") is transversal to S1 (e) ; W,-2,1 (b) for germs e' E P close to e the set S2(e) at 0 has the structure of a smooth manifold of codimension 3; (O)k(c) S2(e) is transversal to Ker w A (d I l0 in S,(e) j e., 1

1

T0S2(e) ® Kerw A (dw)k-1 10 = T0S1.

Now we can introduce some subclasses of the class PWn-2, 1 ,0

W.

,+'

PW"-2, I' I

(a), (b) hold, (c) fails;

, PW"-2, 1

R

viz.,

(b) fails;

n R

I

(a), (b), (c) hold;

I

PW"-2, 1

PW"-2'

,2

(a) fails.

I

REMARK. In coordinates the equation for S2(e) can be found using the following lemma. PW"-2, I. LEMMA 17.3. Let e: {w = 0} E The set S2(e) consists of the points where the (n - 2) form co n (dw)k-1IS1(e) vanishes.

PROOF. Let p = w A

(dw)k-1,

f[ = uIS1(e) , a E S2(e). Then the form

1a has a 2-dimensional kernel. Since 1o is an (n - 2)-form in T Rn-1 we have 0.

Given p: k1.=0 we assume that there exists a vector Y T S1 , pla(Y)= 0. Since every vector Z E T R" can be represented in the form Z = aY+X , X E T S1 , our assumption implies that 1uI.(Z) = 0, i.e., ul. - 0. But this contradicts the condition e E PWn-2'2,1 (if ILI,, = 0 then cl(e) < n - 2). Therefore, for every Y ¢ T,S1 we have pIQ(Y) # 0. Then Kerpl, C T S1 and hence a E S2(e). Q.E.D. 17.4. Normalizing the 3-jet of germs w E PWR

-z, I

1 .2 can be reduced to THEOREM 17.4. (1) The 2 -jet of a germ e E P the normal form {wo + XLy1 d z = 0}. PW"-2' I\PWR-2, 1.2 , n > 5, can be reduced (2) The 3 -jet of a germ e E W"-2.

to the normal form {w0 + (XI y2 + f) d z = 0} , where f is a homogeneous polynomial of degree 3, a fla y2 = 0' f Ixi-0 = 0 . First we prove an auxiliary statement.

§17. SINGULARITIES

65

LEMMA 17.4. Let t be a germ of a 1 form from 0(" , I > 2. Then the 1 -jet of the germ {wo +,r = 0} can be reduced to the form {w0 + x1 f d z = 0}, f E by a transformation that differs from identity by terms of degree > 1.

PROOF. Let PI be a projector which transforms a function C(x, y, z) j1_1 into a function C + (j(1)C)Ixr=0 . Consider a projector P which sends a

1-form EAidxi+EBidy1+Cdz into a form j(EA,dx1+r_ Bidyi)+ P1 C dz. The kernel of P consists of the forms whose /-jets are x1 f d z ,

f

E(1- `)

.

Let wt = w0 + tt. By Proposition 5.2 in order to prove the theorem it suffices to show the solvability of the equation P[X,,dwl+d(X,, wt)+h1wt+ T] = 0 with respect to the family (ht , X,), t E [0, 1 ] , where j 1 _ I X1 = 0,

X, dx1=X1,dz=0. Set

ht=hE9RX,=rPiax +EY/, 1>2

T=

1>1

u=X,wO, yi

(r1;dx;+t21dy1)+t3dx1+t4dy1+T5dz. i>2

We can write this as a system 49U

49U

-Y/1+ax +T1i=Vi+ay +r21=0

(i>2),

r

ay +h+T4= 8u

au'

+t3=0,

u=+V1+x11//i,

+T5Lx,_o=0.

az

It is obvious that this system has a solution u E M?(1+1) Wr E A'i1(i > 2), EM(I+1),hE9311. W1 Q.E.D. PROOF OF THEOREM 17.4. We can use the normal form for germs be-

longing to PWn-z By Lemma 17.4 the corresponding normal form for 2 -jets is reducible to the form {w0 + x1gdz = 0} , where g is a homogeneous linear function. It was shown that for germs from PWn-2.1 we have 3;x (0) = box (0) = 0. The condition that a germ belongs to the class .

PWn-2.1,2

signifies in these coordinates that g = ay1 , where a # 0 (for 0 the set of the first degeneration is not a manifold or, more precisely, condition (b) of § 17.3 is violated). By changing the coordinate x1 for the coordinate x1 = ax1 we reduce the 2 -jet to the form {w0 + x1y1 dz = 01. The first statement of the theorem is proved.

In order to prove the second statement we note that for germs from I , 2 the kernel of the form dy1 is transverse to the

PW"-2' 1

IV. CLASSIFICATION OF GERMS OF ODD-DIMENSIONAL PFAFFIAN EQUATIONS

66

surface {g = 0} . Since g = g(r) , there exists a transformation 7 of the coordinates r such that cI wo = Hwo , g(cV) = y2 , H(O) j 0. This follows from Theorem 7.7. The transformation 7 reduces the 2 -jet to the form {wo + H - I (r)x, y2 d z = 0}. The transformation x, -* x, -H(r) finally normalizes the 2 -jet, i.e., j2 e: {wo + x, y2 d z = 01. We can now use Lemma 17.4 and show that in suitable coordinates a germ from PW"-2' I\PWn-z' 1,2 has a 3-j et of the form {wo + (x,y2 + f) dz =

0}, where f is a homogeneous polynomial of degree 3, f _0 = 0. The condition 8j/8Y2 = 0 is provided by a transformation x, -. x, + ip, where is a suitable homogeneous polynomial of degree 2. Q.E.D. REMARK. The proof shows that the class PWn-2' 1.2 is distinguished from PW (n) by a condition on the 2-jet and has codimension n , i.e., for a generic equation E E PA' (M) the singularities of the class pw,-2 ' I' 2 are realized at isolated points. ip

REMARK. For n = 3 the class PWn-2' I ' 2 coincides with the class pW,,-2,1 . The preceding remark also holds for n = 3. Let n > 5, e E PWn-2' I\PW"-z' I'2 . Assume that the 3 -jet of the germ e coincides with the normal form given in Theorem 17.4. In the appropriate coordinates the manifold SI (e) is given by the equation y2+8 jf/8x, +r = 0, where jet = 0. Now using Lemma 17.3 we shall find the equation of the set of the second degeneration. Let e: {w = 0) . The restriction w = wjs,(e) coincides with the restriction of w on the manifold {y2 + a flax, = 0) up to a 1-form with a vanishing 2 -jet. We can assume that {x1 , yI , x2 , x, , y,- , i > 3} is a coordinate system

on SI(e). Then J2w

dy, -x2d (OX

+x3dy3+...+xkdyk I

and z

&A(dw)k-' = [2Lf dx, + BX,az dz Ady,Adx2ndx3ndy3A. .ndxkndyk 2 I

up to a numerical coefficient. It follows from Lemma 17.3 that the set S2(e) is given by the equation {F = G = 0), where j1 F = 82 f/8xi , j I G = (d6)k-Il0 = (0/ax, , 8/az) we conclude that elf/(8x18z) . Since Kerw A

e E PWn-2' 1'' if and only if d

(ax ) Io n d

\iX18z }

= 0; 0

(17.2)

§ 17. SINGUTARITIES

e E PWn-2'

1

'

1

67

if and only if a

det

3

f

ax

a3f

(17.3)

0.

ax;az axlaz2 In particular, the following statement holds true. THEOREM 17.5. The classes PWrt-2' 1'' and are distinguished from PW(n) by a condition on the 3 -jet and have codimension n + 1 and 4 PWn-2' 1' 1

respectively.

Indeed, it follows from relations (17.2) and (17.3) that codim PW"-2' 1 = 1 ) and codim PWn-2' 1 , , = n - 2 ( in PW"-2' ( In PW"-2 " n ), since 1 ,

1

1

of/aye - 0. At the same time codim PW"-2.1 = 3. 17.5. Invariant field of directions. Let e: {w = 0} E PWn 2 and assume that S1(e) is a smooth manifold. At the points a E SI (e)\S2(e) the direction in T=S1(e) is invariantly defined. It is the intersection of the 2-dimensional kernel of the form w n and T SI(e) . We denote this direction (dw)k-1IR

d,,(e)

. PWn-2.1

. We define the field of directions at all points of Now let e E SI (e) including the points of S2(e) . Assume that co e W(n) is an arbitrary

germ representing the germ e and let p be an arbitrary germ at 0 of a nondegenerate volume form on SI (e) . We define a vector field X = Xe on S1(e) by the equality X,p = [Co A (des)k-1]IS

(e)'

(17.4)

THEOREM 17.6. (1) The germ of the vector field Xe is uniquely defined up

to orbital equivalence, i.e., the set of germs {HXe, H E C°°(S1(e))} is an invariant of the germ e. (2) If a E SI(e)\S2(e) then X;1. E da(e).

(3) Each point a E S2 (e) is a singular point of the field Xe , i.e., Xe Io = 0. (4) Let 'il (a), ... , A"-1 (a) be the spectrum of the linearization of the field Xe at a point a E S2(e) Then up to a numerical coefficient (depending on a) and the order of numeration only one of the following alternatives is possible. (a) (A1(c), ..., .,_1(a)) = (1, -1, 0, ..., 0); .

(b)

0),1=V---l;

(c) (A,(a), ..., A'-1(a)) _ (0, 0, ..., 0). (5) Let n > 5. At the pointsa E PW. -2' 1'0 the spectrum has the form PW,n-2.1.1 (a) or (b) and at the points of all the eigenvalues are equal to 0.

PROOF. (1) The freedom in the definition of the field X is connected with the choice of the 1-form representing the Pfaffian equation and with

68

IV_ CLASSIFICATION OF GERMS OF ODD-DIMENSIONAL PFAFFIAN EQUATIONS

the choice of the volume form u. But if in (17.4) we replace co by HI w , HI (0) # 0 and the germ u 1by the germ H2,u , H2(0) 0, then the field Xe is transformed into the field (Hi H2 I)Xe . (2) if a E SI\S2 , then there exists a vector Z E T lR"\T SI from the kernel of co n (d w)k-11 It follows from (17.4) that co n (dw)k-IIR(XejQ, YI

, ..., Y"-3) = 0

for any vectors Y, E TS,(e) . Since every vector from T ]R" can be represented in the form Y + tZ , Y E T SI (e) , t E JR1 , we have Xelo E Ker Co n (dw)k-1 In

.

(3) By Lemma 17.3 the right-hand part of (17.4) vanishes at the points a E S2(e), hence the field Xe also vanishes at these points. (4) The form wn(dw)k-I IS,(e) vanishes at the points a E S2(e) . Therefore

the form (dw)kls (e also vanishes at these points. Hence, it follows from

(17.4) that d(X .,u)I,, = 0. Then .11(a) + ... + "_I(a) = 0. Since the manifold S2(e) has codimension 2 in S1 (e) , some (n-3) of the eigenvalues

are necessarily equal to 0 (moreover this is true if a c PW"-2' 1,*(e) and S2(e) is not smooth). Therefore, there can be only one of the alternatives mentioned above for the spectrum (up to a coefficient and numeration). (5) We shall describe the field Xe in the local coordinates of the normal form of the 3 -jet given in Theorem 17.4. In § 17.4 we have found how the normalization of 1-jet of the form wn (dw)k-I , w = (01S, looks like in these coordinates. Let u = dx1 A dx2 A dy1 A dx3 A dy3 A A dxk n d yk . Then

IX _ a2f a - ax1aZ axI

a

2

fa

ax; (9Z,

(17.5)

This implies that all the eigenvalues of the linearization of Xe at 0 are equal

to 0 if and only if condition (17.3) holds, i.e., if e E PWn-2' 1.1 or if e E PW"-2 °

1

.

Q.E.D.

PW"-2,1,0 It follows from Theorem 17.6 that we can divide the class PWn-2,1 'e consisting of the germs e for into subclasses 1'h and which the spectrum of Xe at 0 respectively has the form (a)-(b) up to a coefficient (see the formulation of Theorem 17.6). We say that the germs in the class PWn-2,1.h are hyperbolic and the germs in the class are elliptic. The germs in the class PWrt-2' I' 1 can be called parabolic. The set of parabolic germs belongs to the closure of the set of hyperbolic germs and to the closure of the set of elliptic germs simultaneously. PW1"-2, 1,e

17.6. Singularities from the classes PW31' and PWn-2.1'2. The parti2,1 into the subclasses given in § 17.5 is only possible tion of the class PW,"-2' 1

§17. SINGUTARiTIES

for n > 5.

69

For the germs from PW31 ' the kernel of a representing 1form is tangent to the first degeneration manifold at 0. For e E the field Xe is defined on a 2-dimensional surface and has a singular point 0 E R3 . The spectrum of the linearization of Xe at 0 looks like {± 1 } , {±i} 1

PW31.1

or {0; 0} . With respect to these alternatives the singularity class PW3I' I can be divided into the subclasses PW31 ' I'h (of hyperbolic germs), PW3I' 1.e (of

elliptic germs) and PW3I'''D (of parabolic germs). It is not hard to show that these classes are distinguished from PW(3) by conditions on the 3-jet and have codimension 3, 3, and 4 respectively. Indeed, we can argue as in the proof of the first statement of Theorem 17.4 and show that in suitable I coordinates x, y, z E lR the 2 -jet of a germ belonging to PW31 ' looks like I {dy + xy d z = 0). By Lemma 17.4 the 3 -jet of a germ e E is reducible to the form {w = d y + (xy + f) d z = 0} where f is a homogeneous polynomial of degree 3, f Io = 0. By the substitution x - x + ip, ip E M? 1

PW31'

we can also obtain the condition a flay = 0. The restriction &' of the form w on the manifold of the first degeneration S1: {y - -of/ax} has a 1 jet -d ((9f/ax) (we can assume that x and z are coordinates in S1 (e) ). Therefore the field X has a 1 -jet 492f/axe a/az - a2f/axaz a/ax (up to a multiplier). All the eigenvalues of the X linearization are zero if and only if the Hamiltonian O flax in the symplectic manifold (S1(e), dx rid z) is degenerated, i.e., in the case a

det

3

f

a

3

f

ax2oz

a3 ax2az

= 0.

(17.6)

axaz2

Condition (17.6) is imposed on a germ's 3 -jet.

Using similar arguments for the singularities from PW°-2'',2 (n > 5) we obtain the normal form of the 3 -jet

j3e: {wo+(xly1+f)dz=0} with f a homogeneous polynomial of degree 3, fs

= 0, of/ay, = 0.

Now it is easy to show that in these coordinates the field Xe has 1 -jet 2f 49

f1X

Hence, the class

s

a

49x2 az 2'

1 1, 2

a2f a - ax10z 49x1

can also be divided into 3 subclasses

PWn-2.1.2.h, PWrt -2.1 , 2,e , and PWi-2.1.2'D . All of them are distinguished by conditions on 3 -jets. But the singularities of the classes PW"-2.1.2'° and PW31'' 'D

are nontypical, i.e., they do not occur for generic

Pfaffian equations.

IV. CLASSIFICATION OF GERMS OF ODD-DIMENSIONAL PFAFFIAN EQUATIONS

70

17.7. Scheme of adjacencies.

THEOREM 17.7. At any point of a generic Pfaffian equation on an n-manifold the germ belongs to one of the singularity classes given in the scheme below. All the adjacencies of the typical singularity classes are shown in this scheme-

n=3:

PW(3) , PW , PW31 '0

-

PW3I'

I .h

PW31,I,e

P

3

n=2k+l>5:

PWn-2, I,2,h

n PWn2.1.h

PWn-2.U

PW(n) I- PWn F-

PWn-2,1,1

PWn-2'

PWn-2,1,2,e

PW° n

n

PWn-4 n T

T PWn-2i

n

2n+1/4

i=

41, +

PROOF. Taking into account the results of §§17.3-17.6, it suffices to prove the statements

(1) PWR-24-PW°; (2) there is no adjacency PWn-2' I - P 3 (3) PWn-2,1.h n

-

+

(4) there is no adjacency PWn 4

(5) p2'1'' n

;

PWn-2,I,e

PWn-2.1,2,h

n PW"-2.1.1.

PWry -2' I'2

PWn-2,1,2,e. n

n

We prove these statements one by one. (1) A generic germ e E PW° is reducible to the form j o) = xI dy1 +

+

xk dyk + dH(x, y, z)), dHJ0 = 0. The germ {w + e dxl = 0} belongs to PWn-2,

I .

(2) By Theorem 13.6 a generic germ e E PW° can be reduced to {f z d z+ W1 = 0} , where o), is a germ of a 1-form in R2k , x and y are coordinates

§ 18. GERMS AT POINTS OF SECOND DEGENERATION MANIFOLDS

71

in R2k . The closure of the set PWn-2(e) is a manifold {z = 01. The

restriction of the form ±z d z+wi on the manifold {z = 0} is the germ wi . The germ dwi has maximum rank at 0, i.e., (dw1)kJo -A0. Let e: f@=O) be a germ from PWn-2 close to e. Then we have [d(rvISi(e))]kIo 0. But for any germ e: {co = 0} belonging to PWn-2' I Lemma 17.3 implies that {wIS1(e) A [d(wISi(e))]k-1}1o = 0, so [d(wlSi(e))]k1O = 0. Therefore PWn-2. ¢ PWn-2.1.2 is reducible to the normal form {wo+ (3) A generic germ e E

(x1y1+f)dz=0}, f E9313. The germ e: {wo+(xly1+exly2+f)dz=0} can be reduced to the form {wo + (x1y2 + f) dz = 0) (see the proof of Theorem 17.4). We can see that j3fl r=o - j3 f (6x1)Ir=o , where S E R1 , 0. This implies that the spectra of the linearizations at 0 of the vector S fields Xe and Xc coincide up to a nonzero real coefficient. Consequently the given adjacencies do exist. (4) For a generic germ e E PWn -2, 1 , 2 the spectrum of the linearization of the field Xe at 0 does not vanish entirely. This is also true for any germ PWn-2, 1

in

close to e.

(5) A complete proof of this adjacency is very cumbersome and we only give its scheme.

By Theorem 16.3 a generic germ e E PWn-4 has the form {w = dy1 + x2 dy2+' . +xk_1 dyk_I+Adxk+Bdyk+Cd z = 0} (in suitable coordinates), where j 1 A= j 1 B = j i C = 0. Consider the deformation

e: {w+(exk+61xI +52z)d(eyk+suxI) =0}. Note that E PWrt-2 for sufficiently small C, SI , 62 , u, e # 0. The PWn-2.1 , condition e E is equivalent to a system of three algebraic equations for the unknowns e , S1 , 62 , u (the coefficients of these equations are defined by the 3-jets of the germs A, B, C). We can show that if the 3-jets of the germs A , B, C are generic, then this system has a solution in an arbitrary small neighborhood of 0 in the parameter space. Q.E.D. 1

§ 18. Classification of germs at points of second degeneration manifolds

In this section we assume that n = 2k + 1 > 5 and prove the following result.

THEOREM 18.1. (1) All germs belonging to PWn -2' 1,h are equivalent to the germ

=0}. (2) All germs belonging to

0+

PWn-2, I.e

(18.1)

are equivalent to the germ

l dz=0}.

(18.2)

IV. CLASSIFICATION OF GERMS OF ODD-DIMENSIONAL PFAFFIAN EQUATIONS

72

COROLLARY 18.1. Every germ belonging to PW,rt2 I 0 is stable and 3determined.

The proof of Theorem 18.1 is divided into several parts. We use the following notation: r = (y, , x2 , y2 , ... , xk , yk) ;

w° = d y, + x2 d y, + ... + xk d yk ;

PWn-2. 1'0 ; e: {w = 01; jm = F'", At' = qRs e is a germ belonging to where S is a manifold {x1 = z = 0}.

18.1. Normalization of j0e . LEMMA 18.1. Let j w = w0 and I > 0. Then the germ {Co = 01 is equivalent to the germ {jrw + f d z = 01where f E 9711+I 1

PROOF. Consider the operator P which sends the germ E A, dx1 +

B,dyi+Cdz into the germ EA,dx,+>B1dyi+(J'C)dz. Let w,=

-

jrw) , t E [0, 1 ] . By Corollary 5.2 in order to prove the lemma it suffices to prove the solvability of the equation

j1 w + t(w

P[X,,dw,+d(X,,w,)+h,w,+w- jrw] = 0

(18.3)

with respect to the family (h,, X,), where X, E 9n,

a

x, = w,rayl +

k 1:

,_2

a

a1

iraxi + w1rayJ

Let u, = X, , w,. Equation (18.3) is equivalent to the system

aut axi au, ay I

j .

au,

wit+...=TI;,

(i=2,...

a yi +h,+...=T3,

au, +...=T4, ax l

,x), (18.4)

au, +...1 =0 az T3

where rli , T21, , T4 E 91-1+1 and ... denotes linear combinations of the germs pit, yri, , h, with coefficients of the form R,(x, y, z) satisfying R, (0) = 0. System (18.4) can be reduced to the following system for u,: a

+v, u,+A,u, =T,,

[(-_+)u,+BU,]1=0,

(18.5)

(18.6)

where v, and µt are germs of vector fields, v,(0) = µ,(0) = 0, A, and B, are some function-germs, T, E W+I .

§ I8. GERMS AT POINTS OF SECOND DEGENERATION MANIFOLDS

73

Arguments similar to those used in the proof of the solvability of equation (10.11) show that (18.5) has a solution Ut E 9,11+2 . Since every function u E 9J 1+2 satisfies equation (18.6), the system (18.5)-(18.6) has a solution ut E 97tf }2 . Thus, (18.4) has a solution (pit , 'Vrt , h,) and pit , Wit , ht E a1+I c 971. Q.E.D. REMARK. In the same manner we can show that if o), T are germs of 1-forms and r E 972°° , j' w = w° , then the germ {w + T = 0} is equivalent

to agerm {w+fdz=0},where fE931°°. 18.2. Normalization of equations for S2(e) and X` .

LEMMA 18.2. We can choose coordinates x, y, z for which a germ e c PWn 2' 1'0 satisfies the conditions

(1) e:{w0+fdz=0}, (2) S2(e): {x1

fE972;

(18.7)

= z = 8x

0} P:-2' 1'h (18.8) (3) )'Xe(x1) = A(r)x1 , j'Xe(z) _ -A(r)z ife E or Xe(x1) = A(r)z, JIX (z) = -A(r)xl ife E PW"-2' I.e. (18.9) PROOF. We can choose coordinates r so that jI e: {co, = 0} and moreover 1.0e: {w0 = 0}. Then Kerw A = (8/8x1 , 611,9z). Therefore, we can choose the coordinates x1 , z for which S2(e) is the intersection of the manifold S1 (e) with the manifold {xI = z = 0} . In these coordinates (dw)k-1I0

.1'Xe(x1) = A11(r)x1 +A12(r)z,

while det(A,j(r))

J'Xe(z) = A2,(r)xi +A22(r)z,

0 and A11(r) +A22(r) =- 0 . By using the transformation

x1 -+ T1I (r)x1 + T12(r)z,

z -p T21(r)xl + T22(r)z

with suitable Ta(r) we can prove the validity of (18.8) or (18.9). Now by Lemma 18.1 we can change the coordinates r and get the normalization (18.7). Conditions (1)-(3) are satisfied. Q.E.D.

18.3. Normalization of ?e

.

LEMMA 18.3. We can choose coordinates x, y, z such that for a germ -2' 1'0 conditions (2) and (3) of Lemma 18.2 are valid and the

ecP

following normalization holds

e: {w0+(x1y2+f)dz=0},

(18.10)

where f E 9123, of/aye E 912°°.

PROOF. Suppose that the coordinates x, y, z satisfy conditions (1)-(3) of Lemma 18.2. The manifold S1 (e) is given by the equation of/ax1 = 0.

IV. CLASSIFICATION OF GERMS OF ODD-DIMENSIONAL PFAFFIAN EQUATIONS

74

implies that

It was shown in § 17 that the condition e E P 02f

axe

S2(e)

= a 02!

S2(e)

I

= 0'

d

(ax) A dy, A 0. 1

(18.11)

°

Let jle: {w0 + (D(r)x, + E(r)z)dz = 0}. The transformation y, y1-E(r)z2/2 reduces this jet to j1e: {wO+D(r)x, dz = 0} . It follows from (18.11) that dD(r)I0Ady, 0. By Theorem 7.4 there exists a transformation of the coordinates r, which reduces jIe to {w0+H(r)x,y2 d z = 0} , H(0) 0. The transformation x, -' x,/H(r) finally normalizes jI e : {w0+x, y2dz=

0}. After an appropriate change of coordinates (by Lemma 18.1) we obtain the following normalization of j2e :

{w0+x,y2+(D1(r)xi +D2(r)x,z+D3(r)z2)dz = 0}.

(18.12)

The transformation y, -' y, - D3(r)z3/3 reduces (18.12) to {co, + x,y2 + (D, (r)x1 + D2(r)x, z) d z = 0}.

We can assume that DI(r) and D2(r) do not depend on y2 (if not, then we can use a transformation x, --4 X, + q1, (r)x2 + rp2(r)x, z with suitable germs

q, (r) and q2(r)). Then by (18.11) we have D, (r) = D2(r) _- 0. Thus, the final normalization of ,j2e is of the form { w0 + x, y2 d z = 0} . Lemma 18.1 implies that the normalization (18.10) with f E 9713 holds true; the condition 19P42 E 932 can be achieved by a change x, -, x, +V, E 9713. Q.E.D.

18.4. Normalization of j e. LEMMA 18.4. In suitable coordinates x, y, z a germ e E PW"-2 ,

1, °

has

the 3 -jet

j

ifeEPWR-2'1h,

(18.13)

dz=0

j3e: w=w0+ IX ,y2+A(r)(3 + x1 z2 J

(18.14)

PRooF. Using the coordinates of 918.3 and the normal form (18.10) we can easily calculate J'Xe , i.e.,

j-IX

(aa2f 2f) a

92f ax,az ax, up to a coefficient depending on r. Then condition (3) of Lemma 18.2 e

1

=.1

az

'

implies that [3L

j3f =A,(r)xiz+.i2(r)z3 or )3f =A, (r)

+x1 z2} +22(r)z3.

918. GERMS AT POINTS OF SECOND DEGENERATION MANIFOLDS

75

The term A2(r) z3 in the corresponding normal form of 33e can be "killed" by the transformation y1 y1 - ).2(r)z4/4 . Q.E.D. LEMMA 18.5. Let ).(0) 0. Then the 3 -jet (18.13) is equivalent to (18.1) and the 3 -jet (18.14) is equivalent to (18.2). PROOF. We prove this lemma for the hyperbolic case (the elliptic case is similar). First we can assume that )(0) > 0 (indeed, if )(0) < 0 we

can multiply the germ by (-1) and change the signs of the coordinates ... , xk ). Moreover, we can take )(0) = 1 (using the change z --

y1 , XI ,

z//). Let A(r) = 1 + f (r) , f(0) = 0. We use the homotopy method (see Corollary 5.2). Let

wt = Wo+[x1y2+(1 +tf)x; z] dz,

t E [0, 1].

It suffices to prove that the equation

J3[X, dw1 + d(X1,w,) + hrwr + f(r)x zdz] = 0 has a solution Xr E 9R, h, of the form Xr

aa

= E rV;r(r) -I

Y;

V,, (r)

+

8

ax

+x1y,(r)

.

;-z

8 aX

hr = hr(r).

(18.15)

(18.16)

1

Let u, = X,,w,, then ur = u,(r) and equation (18.15) can be rewritten as a system for u, , h, , q ,, , iiir and y, which is equivalent to one equation for the germs u1(r) and y,(r) (other unknowns can be eliminated). This equation has the form A(ur , yr)x1 + (B(ur , yr) +

f)x, z = 0,

where 19U ,

A(u,,y,)=Y2'l,(r)+ ax 2

-y2aour

yl

,

B(u,, y,) = 2y,(r) + v,(u,) + Rrur , yr is a germ of a vector field, R, is a function-germ. Now we have to show that the system A(ur, yr) = 0, B(u,, y,) _ -f is solvable. Eliminating y(t) from the second equation we get the equation for u,, viz., vr(u,) + R'1ur = 4, where R, and f are function-germs, yr is a germ of a vector field, (i1 , d x2) Io = 1 for all t E [0, 1 ] . The field yr can be straightened and this equation has a solution ur = u1(r) . Q.E.D. LEMMA 18.6. A germ e E PW11 -2.1 .h is reducible to

f)dz =0}, and a germ e c PWn -2.1 r is reducible to 3

{oo +(x12 +

3

l +x122 +f)dz = 01,

Iv. cLASSIFICATION OF GERMS OF ODD-DIMENSIONAL PFAFFIAN EQUATIONS

76

where f E 9J14 and of/aye E 9Y1°°

.

PROOF. By Lemmas 18.5 and 18.1 the given normalization takes place for f E 9114. We can obtain a f /a y2 E 971°° by a change x1 - xI + rp , V E 9714 . Q.E.D.

18.5. Normalization of the 1 -jet on SI (e) of the form

jfe:

0},

E 9Jt'

LEMMA 18.7. For an even I > 4 and

I = 4, 6, 8...

,

.

E 9711!1 the 1 -jets

and {(+(xly2+x,z)dz=0} on the manifold {x1 = z = 0) are equivalent. PROOF. Let cot = c o o + (xIY2 + x?z + tt) d z , t e [0, 1 ] . According to the homotopy method (see Corollary 5.2) it suffices to prove the solvability of the equation

(18.17)

jf[X1.dwt+d(X1,wt)+htwt]= -adz

with respect to the family (he , Xt

a

where

X1) ,

a

a

wita =Yltax I +72taz+E Yi i>I

YIt=Y1,

V2t=Y2,

71, Y2E9Rf-2

a

(18.18)

+EPftax. i>2 V;t,iVit,h1E951.

(18.19)

Introduce ut = X,-,w,. Equation (18.17) can be rewritten as a system for ut , Pit, wit , h, and y1 , Y2. After the elimination of pi,, wit, ht we obtain a system of two equations for ut , YI , 72, i.e.,

j aut axI - (y2 + 2xI Z)Y2I = 0, aut aut j_f put +(y2+2x1z)y1+x1ax2] -- , r

2

-(x1Y2-x1z)ayl

Notice that (18.19) is equivalent to the condition ut , yl , After the change 1 : y2 - y2 - 2x1 z we have a

a

+2za,

a

8Z axI aye 8xI and system (18.20) is reduced to the form dut aut +2z--Y2y2 =0, aye axI

a

8z

Y2 E 911f -2

.

a

+2x

1 0Y2

j

jr aut +2x Rut I aye [ az

- (xx1 IY2 -

where ut = ut(1), yi = yi(1), I

2

I

2

(18.21)

au + Y2yI +x autl )ayI I ax2J

=-

The condition ut, 7, is equivalent to the condition ut , yI , y2 E 97tI-2 .

I ,

,

Y2 E 9Rf-2

§ 18. GERMS AT POINTS OF SECOND DEGENERATION MANIFOLDS

77

0. Let

Let ur = at + y2flt , where Bar/8y2 = d #1/19Y2 Then (18.21) is equivalent to the system

+ 2zfl, = 0, (18.22)

LL

8ar

Bar 2 d ar ['9Z +2x1flr+xIz

r

7

We seek the solution of (18.22) in the form a =aE931(1+I) E a"'). Then the condition ii, , 71 , 72 E 9Ji!-2 will be satisfied and system (18.22) will become az+2x1f=-.

8x+2zf=0,

(18.23)

1

Under the condition a E

fi E 9Jt(1-1) system (18.23) can be re-

duced to one equation 8a

8a

(18.24)

xiax -zaz where a c ff1} 1)

.

Let z =

(r)xI

z'+I-' .

Equation (18.24) has

solution

a=since l + 1 is odd.

1(r)x z

(l + 1 - 2i),

Q.E.D.

18.6. Normalization of the 1-jet on S1 (e) of the form

1=7,9,11,.... LEMMA 18.8. The assertion of Lemma 18.7 holds for odd 1 > 7.

PROOF. We prove the solvability of equation (18.17) with respect to the family (hr , Xr) , where Xr is of the form (18.18) and the following conditions are satisfied 71t =71,

Yet=72,

Y1,Y2E9J1'

4 ,

V tt , Wit,

ht E 0.

(18.25)

As above, we can show that equation (18.17) is equivalent to system (18.22). We seek a solution of (18.22) in the form ar = a = >I + (x1 z)°p(r) , ft = 3) =q1+72,where p=(1-1)/2, 7Ea(1+1), 71 System (18.22) will become

8z + 2zr11 = 0,

px' z°p(r) + 2z72 = 0,

1

pxv zv l u (r) + 2x172 = 0 ,

87 dp -8z+ 2x171 + xp+I Zp 8z

(18.26)

IV. CLASSIFICATION OF GERMS OF ODD-DIMENSIONAL PFAFFIAN EQUATIONS

l

Taking 72 = -p(x1 z)p-1 p(r)/2 we obtain the following system for 7. 71, ,U(r) :

arl + 2z,1 = 0,

pax

(18.27)

1

t aZ +2x1,1

2 fits!-I)

Under the conditions 7 E 91t(r+I) , 71 E reduced to one equation

ax

system (18.27) can be

xl-za7-(x1z)P+l'9 =z

(18.28)

.

2

I

Let z = Ei
7(r) = E

1)

i#p+I

is a solution of equation (18.28).

Q.E.D.

18.7. Normalization of the 5-jet on S1 (e) of the form

j5e:

E9Jt151.

LEMMA 18.9. The result of Lemma 18.7 holds for l = 5.

PROOF. We can assume that 0h/ay2 = 0 (using the change x1 - x1 + q' , 9 E

a(5) ). We seek the solution of system (18.17) under the conditions 71t f

7211

'Pit f

Wit)

ht E 9J1.

(18.29)

In contrast to the cases considered above ylt and yet depend on t equation (18.17) is not equivalent to system (18.22), but is equivalent more complicated system

r._ JS I

aat az

+

i

.,2r\

1

1 a2 + 2x + t axlaz

+ x as

1

rx z + txl axi

and to a

(18.30)

1 19x2 2

aye + txl ax ax2 f' 2

We seek a solution of the form at = yt + (x1 z) u(r) , 6t = 71 t +,2t , where

7tEa(6),

71tE9)1(41,

172tE9Jtl21.

§18. GERMS AT POINTS OF SECOND DEGENERATION MANIFOLDS

79

Condition (18.29) is automatically satisfied. System (18.30) can be rewritten in the form 197,

8x, + 2zglr + tdX 112, = 0,

xi zp(r) + 172, = 0,

(18.31)

I

z

19

8zt

+2x,17,,+t0a,az'72r+xizzax

=-

System (18.31) is reduced to one equation

ayrLr -za7 -tx;zµ(r)2 +tx z2 U (r )Ox,8Z d2

x,

C3 X,

(3Z

1

19x2I

- (XI z 3 19x2 19,U

)

= zx S

(

18.32 )

with respect to u(r) and y, E 9Ji(6) . Let r(r) be the coefficient of (xI z)3 in the expansion of the right-hand side of (18.32). Simple calculations show that the coefficient at (xI z)3 in the left-hand side of (18.32) equals -8/c/8x2 (it is independent of t). Let u(r) be a solution of the equation Op/Ox2 = Then (18.32) is reduced to the equation

x1- - zt =Rt,

(18.33)

19 z

I

where R, E 9J1(6) and Q.E.D.

0. Equation (18.33) is solved easily.

18.8. 1 -jet normalization on S, (e) (the elliptic case).

LEMMA 18.10. Let 1 > 4, r; E W). The 1-jets

r

3

coo+Ix,y2+ 3 +x,z2+

1co 0+Ix,y2+ 3

l

l

1 dz=0}

+x,zl dz=0} 2

on the manifold {x, = z = 0} are equivalent. PROOF. The proof is similar to that of the hyperbolic case. For even I = 2I, we have the equation

-x,as=T,

TE9Ji(l+I)

instead of (18.24). It is not hard to show that this equation has a solution (`+I) . For odd 1 > 7 we have the system

aE

, 17 3

819,

8x

1

+2x,.8,] =0, J

[aj`+2zf,+x.-_ +S19at 19 z

y,

=-',

115.34)

IV. CLASSIFICATION OF GERMS OF ODD-DIMENSIONAL PFAFFIAN EQUATIONS

80

where g E W', Z E 9Jl(l) (instead of (18.22)). We seek a solution to this system in the form 2

n E'iJt(l+I

at = aqq = 7+,u(r)(x, + zZ)°, 2

Yt = 8 = ;71 - pp(r)(x1 + ZZ)p_I

, p = (l - 1)/2, III(1-I).

'11 E

,

System (18.34) is reduced to one equation for j7, p(r), viz.,

Z9X

xl 017

2

2

x1(x1

+ Z2)ax

(18.35)

x

Let x12

+ z2)p 8x2 +

; (r)xi

x1e =

zr+l-;

i<1+1

(the coefficients j(r) depend on u(r) ). We can show that equation (18.35) is solvable if 1+1

1 - 2i)! = 0.

(18.36)

i=0

The equation (18.36) for /2(r) is of the form 8p/8x2 = Q(r), where Q(r) is some function-germ. Thus, equation (18.35) has a solution µ(r), n E W'+'). The case of 1 = 5 is the hardest one (a and fi depend on 1) and is proved similarly to Lemma 18.9.

Q.E.D.

18.9. Equivalence of the germs {w = 0} and {w+ d z = 0} for c E 971°° . z = O} E 971°0. Then the germ {w0+(x1y2+xi is equivalent to the germ {co, + (x1y2 + x1 z) dz = 0} and the germ {w0 + 3 2 {x1y2 + x1 /3 + x1 z + .) d z = 0} is equivalent to the germ {w0 + (xIy2 + LEMMA 18.11. Let

x,/3+x1z2)dz = 0}. PRooF. Consider at first the elliptic case. Let wt = w0+(x1y2+x3/3+x1z2+tt;)dz. By Proposition 5.1 it suffices to prove the solvability of the equation X t , d w,+

d(Xt,wt)+htwt = -adz with respect to the family (h,, X1), h1 E 9Jt°°, X1 E

Let Xt be of form (18.18). As in the proof of Lemma 18.7 the given equation can be rewritten as a system with respect to the unknowns y11, Yet , 9;t, Wit, h1, and ut = Xt, wt . By eliminating qrt , Wag h1 we ob931°° .

Si

§18. GERMS AT POINTS OF SECOND DEGENERATION MANIFOLDS

tain the system(2) out

2

0,

Ax - [v2 + x1 + z2 + 01t]Ytr Out a

3

+ [y2 + x1 + z2 + 01t]y2t

-

3 + xl zz LxIY2 +

(18.37)

ayi

+xI8z +021ut+z1t(ut)+' =0. Using the transformation 0: y2 - y2 - x1 - z2 + Bet such that °2r + 0 (i.e., the function y2 + x + zz + 01, turns into y2) we arrive at the system

au t -Y271t

(9 U

2 +x1 a Y2

03rur +T2r(ur) + a x t

1

au 041u1 + =31(ur) +

x1y2 -

I

0

(18.38)

+2z-19-'Y2 + Y2Y2,

19Z

2

=,

aut

3ut

3

+ 051 = 0.

3x' ay1 + x1 ax2 We seek ur in the form at+y2fit , where aat/aye = afir/aye = 0 . System (18.38) is equivalent to the system: + r4 t(at) = 0' ax' 8a as 2 3aa +x1ax2 +rS t(at)+08" = azt +2zflt+BZ rat+3xl I + 2x1 fit + 06, 1

1

(18.39) 0

aYI

for at , 6r (in the coordinates x1 , yI , x2 , x3 , y3 , ... , xk , yk ; we have eliminated Y2)' We can now eliminate ftt and reduce (18.39) to one equation for at , i.e., as

2zoxt -2x1 1

as

az

Zaa

4 4aa

t - 3x1 a t -2x1axt +z6 t(at)+09 tat+1910 t = 0. (18.40) z

Y1

We can show that the transformation

Y1 - y1 + 12xl z + 4x123

xz ~ -2x2 + 2x1 z ,

x2(x1 +z 2) (18.41)

reduces the vector field a

0

2

a a

zax l -x1 az - 3x1 a YI

-x 2 a l

axz

to the form

-x1az+(x1+z)aa

vo=z0

2

1

2

'

(2)The letters 6., , 0 ( r,,, r, respectively) denote function-germs (germs of vector fields respectively) belonging to a O° .

82

IV. CLASSIFICATION OF GERMS OF ODD-DIMENSIONAL PFAFFIAN EQUATIONS

Therefore transformation (18.41) reduces the equation (18.40) to the form (18.42)

(v° + rt)(at) + 8tat = 1911. r -

By Theorem 8.4, equation (18.42) has solution at E Lemma 18.11 is proved for the elliptic case. For the hyperbolic case, similar calculations finally lead to the equation (iI° + r,) (a,) + Btar = Bt,

where v° = x1(8/8x1) - z(8/8z) +x1 z(8/8x2) . By Theorem 8.1 this equation has a solution at E' 931°° .

Q.E.D.

18.10. Proof of Theorem 18.1. We choose the coordinates x, y, z so that the germ has the normal form given in Lemma 18.6. Notice that when normalizing 1-jets on the manifold {x1 = z = Of (Lemmas 18.7-18.10) we have solved the differential equations for functions depending on r. These equations have solutions in a neighborhood of the point r = 0 which depends on the germ e: {w = 01, but can be chosen to be independent of 1. Therefore, from Lemmas 18.7-18.10 it follows that a formal expansion (in the degrees of x1 , z) of the coefficients of w can be reduced to (18.1) or { 18.2) by a formal (in the degrees of x 1 , z) transformation

E R(') r xazA

x

r !21

Q

z , ERQ31,(r)xi'z

(18.43)

where RQ 1,(r) are functions defined in a common neighborhood Irk < e. By the Whitney extension theorem we can find a smooth coordinate transformation whose formal expansion coincides with (18.43). This transformation reduces the germ e to (18.1) or (18.2) to within a 1-form belonging to 6R'. Taking into account the remark in § 18.1 we can assume that this 1-form has the form f d z , f c a'. Now we can use Lemma 18.11. Q.E.D. § 19. Point singularities of 3-dimensional Pfaffian equations

In this section we prove the following results. THEOREM 19.1. (1) Any germ e E P W 3 '

1

'h

is equivalent to one of the

germs

dy+(xy+x2z+bx3z2)dz, (2) Any germ e c PW3 '

1.e

b E R'.

(19.1)

is equivalent to one of the germs

dy + I xy+ 3 +xz2+bx3z2 dz,

b c It'.

(19.2)

(3) Two different germs of the form (19.1) or (19.2) are not equivalent, i.e., b is the modulus. (4) Any germ e E PW3 '' is 5-determined.

§ 19. POINT SINGULARITIES OF 3-DIMENSIONAL PFAFFIAN EQUATIONS

COROLLARY 19.1. Any germ e E PW3 '

'

83

is unstable.

THEOREM 19.2. A generic germ e E PW3 is equivalent to the invariant normal form

zdz+(p-2 )ydx+lp+2 Ixdy, modulus

where p is the

Rep>0,

(19.3)

(real or pure imaginary).

We will prove Theorem 19.1 by parts. Let us introduce some special notation for the proof, viz., 3

Wh = dy + (xy + x2z)dz;

cve=dy+ xy+ 3 +xZ) 2

dz.

19.1. 3 -jet normalization. LEMMA 19.1. The 3-jet of any germ e E P W31 ' 1 ' h (e E P ;V3 . I , e respec-

tively) is equivalent to {coh = 0} (to {c)e = 0) respectively).

PROOF. Using the results of § 17.6 we can assume that j3e is reduced to the normal form

j3e: {dy+(xy+f)dz=0},

(19.4)

The functions x and z are coordinates on SI (e) . Consider the field

jIX .

It is uniquely defined up to a numerical coefficient. We can find a transformation of the form x - k11x + k12Z , z , k21x + k22z such that in the new coordinates the relations

j1Xe(z)_-1z jIX (z) _ -,lx (if e E PW3'1'e),

jIXe(x)_Ax,

(ifeEPW31'1"),

.IX(x) _ AZ,

(19.5)

will hold for A E R. In the new coordinates the normalization (19.4) does not occur. However, by Theorem 17.4 and Lemma 17.4 we can obtain this normalization by a transformation preserving relations (19.5). Using the calculation of jIX in the coordinates of the normal form (19.4) from § 17.6., we can find the function f : f = A 1x2z +2223

f = Al

(+xz2)

(if e E

(if e E PW3

PW3I' 1'h),

e)

Notice that the transformation y - y - ).2z4/4 "kills" the term ).2z3 dz in the corresponding normal form. Thus, the normal form {cvh = 0} or {we = 0} holds.

Q.E.D.

-......,o-- ----

yr vcxMJ jr VUD-DIMENSIONAL PFAFFIAN EQUATIONS

19.2. 4 -jet normalization.

LEMMA 19.2. Let e E PW3 ' l .h ( e E PW3 I e respectively). Then j4e is equivalent to {(0h = 0} ({c)y = 0} respectively)-

PROOF. First we consider the hyperbolic case. By Lemmas 19.1 and 17.4 we can assume that j4 e is of the form dy + (xy + x 2 z + g) d z , g E 91t(4) . Let co, = d y + (x2z + ig + xy) d z . By Corollary 5.1 it suffices to prove the solvability of the equation

j4[Xt,dcv,+d(X,,co,)+hrw,]=-gdz

(19.6) with respect to the unknown germ of the vector field X, = qi, 19119x+ V, 8/8y+ 7,8/0z and the function-germ ht . We seek V,, V,, y,, h, satisfying the conditions h1E Ifl 3 . c,,Y,E9R2, (19.7) W,E9A Let uI = Xr , wr . We can write equation (19.6) as a system for ur +{i, ,

yr , hr

,

V, ,

. By eliminating hr and w, we reduce this system to the system

I

j j

a [a_U,

11

a-(y+2xz)7,

=0,

JJ

du,

a

1

az

{ 19.8)

=-g

for ur , qir , y, (we use conditions (19.7)). After the transformation 8

y-y-2xz,

y

a +2z a 8y' 8x

8x

8

az

y

a 8 +2x 8y 8z

the system (19.8) takes they form

j4

axe

- yy1 + 2z aayU,

1

0'

r`

(19.9)

j4 I a , + yip, + 2x au, 1 = -9-g. 19Z Set u1I y_o = a , aIy=o = P, . Then (19.9) is reduced to the system

j

4

da

ax +2zp,) = 0,

(19.10) 8a , 2xf,) = -gly_o. ez + In order to show that system (19.10) is solvable we have to show the solvability of the equation a

5

/ as 1\

aa

Xax - zaZ = zgly_o

(19.11)

with respect to the unknown ar . Since the number 5 is odd this equation has a solution a, = a E D2(5). Then equation (19.6) has a solution (h, , X,) .

§ 19. POINT SINGULARITIES OF 3-DIMENSIONAL PFAFFIAN EQUATIONS

95

An easy check shows that the inclusion a E 911(5) implies that u E 9114 and the conditions (19.7) are valid.

Next consider the elliptic case. It suffices to prove the solvability of equation (19.6), where w, = dy + (x3/3 + xz2 + xy + tg) dz. We seek x, = V, 01,9x + ;v, 810y + y, a/a z and h, satisfying (19.7). Arguing as above we obtain the equation j5(xaa,/az - zaa,/ax) = xg, which also Q.E.D.

has a solution a, E 9R(5) .

19.3. 5-jet normalization.

LEMMA 19.3. Let w = wh+gdz (w = we+gdz respectively), g E 9Jt(5). Then there exists a number b c R such that the 5 -jet {wh + (g + bx3 z2) d z =

0} ({we + (g + bx3z2) d z = 0} respectively) is equivalent to {wh = 0) (to {we = 01 respectively).

PROOF. Consider the hyperbolic case. By Corollary 5.1 it suffices to find a vector field X, = rp, a + W, aY + y, a: , a function h, and a number b E R' such that the equation(3)

j s[X,,dwb ,+ d(X,JWb,1)+hrwb ,] = -(g+bx3z2 )dz

(19.12)

has a solution (X, E 911, h,). We seek V,, w, , y, and h, satisfying E9113,

y,E9R3,

y/,E93t5,

h,E9314.

(19.13)

Arguing as in §19.2 we reduce equation (19.12) under conditions (19.13) to the equation x aa -zaa, zgjy-0 +bx3 z3 (19.14) ax aZ with respect to the unknown a, E fit(6) and b E R. We can choose b such that a6c/ax3aZ3 = 0, then equation (19.14) has a solution

at =a= E x

r

z

6-;t,/(6

- 2i),

ij3

where

'-

1

a

6

T!-(6 I(6

- i)! ax'az6 Next consider the elliptic case. Let wb,t = we + t(g + bx3z2) d z, b e R'. It suffices to prove the solvability of equation (19.12) with respect to (h,, X,) under the condition (19.13). Arguing as in § 19.2 we reduce equation (19.12) to the equation

x 19Z - z -_ = xgly=o + bx4z2 for a E !M"' and b E R' 3)Below wb , = wb + t(g + bz3z2)dz.

(19.15)

so

IV. 1,LASSIFIC:AI IUN OF GERMS OF ODD-DIMENSIONAL PFAFFIAN EQUATIONS

r

Equation (19.15) can be rewritten in polar coordinates as =

(19.16)

where x = rcos q , z = rsin qi , o,(x, z) = r6at(rp) . We have to find a solution of (19.16) satisfying a,(2n) = &,(0). Let 9(c(B)+bcos40sin20)dO.

b = -8

aI(p) = J

Then a,(rp) is a solution of the equation (19.16) and since fo"cos4 1P x sine rp drp = n/8 we have a,(2n) = a,(0) . Q.E.D. LEMMA 19.4. Let e E PW31 ' 1'h (e E PW31

I'` respectively). There exists

a number b E R such that

jseti{wh+bx3z2dz=0}

(jse' {we+bx3z2dz=0}).

PROOF. By Lemmas 19.2 and 17.4 the jet j5e is equivalent to the jet {wh +gdz =01, g E 931151 ({we + gdz =01 respectively). Let b E I be the number we found when proving Lemma 19.3 and let cb be the reducing diffeomorphism. Since 1 is of the form id + (terms of degree > 3) by conditions (19.13), we have

js[4 (wh + (g + bx3zZ) dz)] = bx3z2dz + j5b*(wh + gdz),

js[1 (we+(g+bx3zZ)dz)]=bx3z2dz+j51'(we+gdz). Therefore

j51'(wh+gdz) =wh -bx3z2dz,

j5Ib (60e+gdz)=we-bx3z2 dz.

Q.E.D. 19.4. 1-jet normalization for I = 21, > 6.

LEMMA 19.5. For an even I=211>6 and gE931(1), bER (a) the 1 -jets {wh + (bx3z2 + g) d z =0} and {wh + bx3z2 dz = 0} are equivalent; (b) the 1 jets {we + (bx3zZ + g) d z = O} and {we + bx3zZ d z = 0} are equivalent.

PROOF. (a) As above we are to prove the solvability of the equation

jt[X,,dw, +d(w,,X,)+h,w,] = -gdz

(19.17)

with respect to the unknown (h,, XI), X, E 9)1, where cot

=wh+(bx3zZ+tg)dz,

tE[0, 1].

Let X,=q 0/8x++v,D/8y+y,0/0z. We seek lo,, ytl, y, and h, satisfying the conditions y, = y,

191r=So

V,,,Y,h,E93t-Z

(19.18)

.

§ 19. POINT SINGULARITIES OF 3-DIMENSIONAL PFAFFIAN EQUATIONS

97

By arguments similar to those of § 19.2 we reduce equation (19.17) to the E with respect to at E 9R(I+1) Since (1 + 1) is odd this equation has a solution at = a E 911`+1)

equation x aat/ax - z aat/a z =

(b) Similarly, we obtain the equation x aat/az - z t3at/ax = t E M(/+'), which is solvable as well. 19.5. 1 -jet normalization for odd 1 > 7.

LEMMA 19.6. The conclusion of Lemma 19.5 holds for odd 1 > 7.

PROOF. (1) The hyperbolic case. We seek a solution of equation (19.17) satisfying the conditions q'1, wt, 71, h1 E

(19.19)

n4.

One can rewrite (19.17) as a system for rpt , W1 , y1, ht and u, = Xt j ml _ Wt + (xy +x2z + bx3 z2 + t g)yt . By eliminating wt and ht we get the system

j

dut

[ax

(y+2xz+3bx2z2)yt =0,

az1 - (xy +x2z) y

1

+ (y + 2xz + 3bx2z2)rpt + xut]

(19 . 20)

_ -g

for V, , yt , and ut . After the transformation y -' y - 2xz - 3bx2 z2 (19.20) takes the form

- yy, + (2z + 6bxz2) .9 U, y j = 0, sut aut 2 2 3 2 0u1 az -(xy -x z-3bx z ) ay +yc3t+xut+ (2x + 6bx z) az

j {.irOu'

=-g. 1.2) (91

In order to prove that system (19.21) is solvable we have to prove the solvability of the system

!! [ ax + (2z + 6bxz2)At] = 0,

(19.22)

j [!!'-" +(2x+(6b+1)x2z)ft+xat] for at = ujY_0 and $1 = aut/cOyl ,_o . We seek a1 and fit satisfying at = a E 9It1

=-g1Y=o 9711

3 . We can verify that under these conditions inclusions (19.19) (used for the reduction of equation (19.17) to system (19.20)) are valid. System (19.22) is reduced to the equation

j1+1

1

,

fit = f E

(2x + (6b + 1)x22) ax - (2z + 6bxz2) az - 2xza] = 2zgly=o. (1.23) 9

88

IV. CLASSIFICATION OF GERMS OF ODD-DIMENSIONAL PFAFFIAN EQUATIONS

Let a = ai + C(xz)p where a1 E 911(1+1) , c E IIt' , p = (1- 1)/2. We can

rewrite (19.23) as an equation for a1 and c :

2x . - 2z a + c (1 2

5) (x z)"

(19.24)

= 2 z g I y=0.

Equation (19.24) has solution al E 9R('+'), c E It' since we can choose c so that the coefficient at (xz)p+' in the expansion of the function 2zgIY_o -

`1(xz)p+' equals 0. (2) The elliptic case. Arguing as above we get the equation

1+1 [(2z + 6bx2z + x3 I

/

- (2x + 6bxz2)

- 2x21] = 2xgI_0 (19.25)

for a E Me-' (instead of (19.23)). Let a = aI +c(x2 + z2)', where al E M(1+1),

c E R , p = (1- 1)/2. We can replace (19.25) by the equation -x19 zl

(19.26)

for c and a1 , where A(x, z) = 6bp(x2 + Z2)p-1 (XZ3 - x3Z) - pX4(x2 + Z2)p-' +

X2(X2

+ Z2)p.

Let us rewrite (19.26) in polar coordinates r and 0, viz., 8B

(9)+cA(9),

(19.27)

where x = r cos 0, z = r sin 0, gl y-O = r1+1 e(O) , A = r1+' A(9) . In order to solve (19.27) in the class of periodic functions it suffices to find c satisfying the condition 2s

f((o) + cA(0)) dO = 0.

(19.28)

0

2rz,

We can calculate f A(9) d 9 =

A(22 P)

(it is independent of b ). Hence,

0 2m -

for each 1 > 7 we have f A(9) dO j4 0 and equation (19.28) is solvable. 0

Q.E.D.

19.6. Proof of Theorem 19.1. In the proof below the letter q with indices denotes the function-germs belonging to 912°° and the letter r with indices denotes the germs of vector fields belonging to 9R°° . (1) Reduction of the normal form (19.1) (the hyperbolic case). By Lemmas 19.1-19.6, 17.4 and the Borel theorem we can reduce each germ e c PW3 to the normal form {(oil +(bx3z2+g)dz=0}, gE 9Jl'.

519. POINT SINGULARITIES OF 3-DIMENSIONAL PFAFFIAN EQUATIONS

89

Let w, = wti + (bx3z2 + tg) d z, t E [0, 1 ] . By Proposition 5.1 it suffices to prove the solvability of the equation

X,dw,+d(X,,w,)+h,w,=-g

(19.29)

with respect to the unknowns h, and X, = c, 8/8x + V, 8/8y + Y, 8/Oz. Equation (19.29) can be reduced to the system

1 ax -(y+2xz+3bx2z2+71 ,)Yr=O, Bu11+(y+2xz+3bx 2 Sz

2

z +71 ,)q,+(x+72 1)u,

(19.30)

8u -!{xy+x z+bx z +73 f)=-g ay 2

3

2

for ip,, y, and u, = X1, w, . We can find a germ 14, 1(x, y, z) such that the transformation 0: y y-2xz-3bx2z2+74,,(x, y, z) reduces the germ y+2xz+3bx2z2+71,, to the germ of the function y. After this transformation we can eliminate the coordinate y in system (19.30) (arguing as in § 19.5) and obtain the system

8x + (2z + 6bxz2) flr + T1,,(a,) + ;75,,,8t = 0, (19.31)

azf +(2x+(6b+1)x22+bx3z2)fl,+xa, + 76rar + T2,(at) + 77 A = -781 ,

for a, = u,1y-o and 8, = 8u,/8yIy-o, where u, = u,(1), 878 ,/8y = 0. In order to solve system (19.31) we need to solve the equation V(a,) - (2xz + 6bx2z2)at + T3 ,(a,) + 79,tar

(19.32)

71o,t

for at, where V = (2x + (6b + 1)x2z + bx3 z2 ) a - (2z + 6bxz2)

8x

8

az'

By Theorem 8.1 equation (19.32) has a flat solution, since the vector field V is hyperbolic. (2) Reduction to the normal form (19.2) (the elliptic case). Reasoning along the same lines we have only to prove that the equation

(2z + 6bx2z + 3x3 + 2bx3z2) ax - (2x +6bxz2) «r

- (2x2 + 6bx2z2)at + 111 far + T4, r(at) = 71 2,,

(19.33)

is solvable with respect to the unknown a, . The field - (2x + 6bxz2 )8 /8 z + (2z+6bx2z+(2/3)x3+2bx3z2)8/8x is finitely determined (there are several ways to prove this; we can for example use the results obtained in [Ic2]).

90

IV. CLASSIFICATION OF GERMS OF ODD-DIMENSIONAL PFAFFIAN EQUATIONS

By Theorem 8.3 equation (19.33) is solvable in the class of germs of flat functions. (3) Invariance of the parameter b in the normal form (19.1) Let e1 and e2 be germs having normal form (19.1) with b = bl and b = b2 respectively. .

Let us calculate the form of X,,in the coordinates of the normal form, i.e., Xe = [2x + (6b1 + 1)x2z + 2bx3z2]

aax - (2z +

6bxz2)

a

.

az

These germs are orbitally equivalent to the germs

Xe = C2x+

X2z + 2bix3z2

l +3bixz

a

a

} ax -2zaz.

We assume that the germs eI and e2 are equivalent. Their equivalence implies an orbital equivalence of Xei and Xe2 . Since jSXe

= (2x +x2z - bix3z2)a/ax - 20/az

according to [Bo I] and [Bo2], we have bI = b2. (4) Invariance of the parameter b in the normal form (19.2). Let eI and e2 be germs having normal form (19.2) with b = bI and b = b2 respectively. Then

Xe = 12z + 6bx2z + 3x3 + 2bx3z2 I ax - (2x + 6bxz2) d

.

We can show that the 5 -jets of these germs are reducible to

j5Xe - (-z+x(x2+z2)+f1x(x2+z2))

0

ax

+(x+z(x2+z2)+fl1z(x2+z2)2)

az

,

where .8, = c1 bi + c2, cI 9& 0. By the results of [Bo I] and [Bo2], the orbital equivalence of the germs Xe and X;2 implies that 6I = fl2. Thus, the germs e1 and e2 are equivalent if and only if bI = b2 . The proof of Theorem 19.1 is complete (the fourth statement follows immediately from the statements proved above). 19.7. Proof of Theorem 19.2. (1) Reduction to the normal form (19.3). By Theorem 13.6 each germ e E PW° (4) is reducible to the normal form (19.3) or to the normal form

{ro=-zdz+rU-ydx+ U+2//lxdy=0

(19.34)

Note that the germs (19.34) and (19.3) are equivalent, since we can change

y for (-y) and multiply ro by (-1). (2) Invariance of the parameter a. Let mN = zdz + (,u- 1/2)ydx + (p+ 1/2)xdy. Suppose that {mµ = 0} - it= 0}, Rep > 0, Reµ > 0. (4)i.c., a generic germ e E PW°, see § 13.3.

§20. DEGENERATIONS OF CODIMENSION > 4

91

Then there is a function-germ H, H(0) 34 0, such that Hr o. wµ . In particular, the jets j'(Hw and u are equivalent. Note that j'(HmN) _ H(0)wu . The germs H(0)wµ and wN are equivalent, hence the 1 -jets wu and t N are equivalent as well. By Theorem 13.6 It = µ .

Q.E.D.

§20. Degenerations of codimension > 4

In this section we assume that n - 2k + 1 > 5 and prove the following results.

THEOREM 20.1. No germ e E PW"-2' 1.1 is finitely determined, i.e., the

singularity class PW"-2'

is wild.

THEOREM 20.2. For j < n - 2 the classes PW1 are wild. To prove these theorems we need the following result. THEOREM 20.3. Let k < oo be an arbitrary number. For a generic germ e: {w = 0} E PW"-2, I ' 1 there exists a germ of a 1 form r E Tt4S2 (e) with a vanishing k jet such R that the equation (20.1) 4 ce)[X., dw+d(X,w)+hw]=r Js2 with respect to the unknown function-germ h and vector field X is unsolvable.

Theorem 20.1 follows from Theorem 20.3 and Proposition 6.3. Theorem 20.2 follows from Theorem 20.3, Proposition 6.3 and Theorem 17.7. To prove Theorem 20.3 we need a normalization of js=(e)e . The following theorem shows that the 3 -jet on the second degeneration manifold can be reduced to a stable normal form. I.1 the jet J3 (e)e in THEOREM 20.4. For a generic germ e E suitable coordinates is PW"-2 ,

( ( (w0+lxiy2+zy2+ 2 + 61 Idz=01, 2

3

(20.2)

where S2(e): {x1 = x2 = y2 = 0} .

When using the term "a generic germ" in Theorems 20.3, 20.4 we imply that some conditions mentioned in the proof of Theorem 20.4 are valid for the case of general position. Let us introduce some notation needed for the proof, viz.,

e is a germ belonging to PW"-2'''' , e: {w = 01; r = (Y1, x2, Y21 ... , xk,Yk); r = (Y1 , x3 , y3 , ... , xk , Yk , z) ;

dl f is a set of 1-forms w belonging to 'JR{ - xZ=z=0} such that a /ax1 .i w =

a/ax2'w = a/ay2.w = a/az.m = 0;

S3(e) is the set PWn -2''''(e) , i.e., the third degeneration set; S2 = S2(e); S3 = S3(e).

IV. CLASSIFICATION OF GERMS OF ODD-DIMENSIONAL PFAFFIAN EQUATIONS

92

20.1. Proof of Theorem 20.4. The proof is divided into parts. (1) Reduction of the equations for S2(e) , S3(e) and j'e to a normal form.

2 ' there exist coordinates

LEMMA 20.1. F o r a generic g e r m e e P 4'

x, y, z such that j'e: {w0 = 01 and S2(e) is given by {x1 = x2 = y2 = 0}; S3(e) is given by (xi = x2 = y2 = z = o};

(20.3)

S2Xe(xl) E 9{xj=0} ,

(20.5)

JSZXe(X2) E 9A(x,-01.

(20.4)

PROOF. Let r be coordinates such that j1 o) = wo . Then Kerw A (do)

k_Ito

= (d/dxi , d/dz) .

For a generic germ we have

codim (ToS2 ®(8x_

,

Oz))

= 2.

Then S2(e) can be given by the equations x1 = 0 ,

FI(r, z) = 0,

F2(r, z) = 0,

where

I (o) = 0,

aF2 (o) = 0.

(20.6)

For a generic germ, S3(e) is a manifold and codim (7S3(e)

(_, d ! / = 9XI

2.

Hence S3(e) can be given by the equations

xI = z = 0,

FI(r, z) = 0,

F2(r, z) = 0.

Consider the contact space (R2k+I (r), w0) and the hyperplanes

L: {j'F1(r, 0) = j'F2(r, 0) = 0},

La: (x2 = y2 = 0}.

For a generic germ the restriction w01L is a germ of 1-form in R2k-3 having

a maximum class 2k - 3. This property is valid for the hyperplane Lo as well. By Theorem 7.2 the hyperplanes L and Lo are equivalent in the (r), wo). Therefore by (20.6) there exist coordinates x, y, z such that j'w = wo, T,S2(e) is given by (xI = x2 = y2 = 01, and pa contact sspace

(

R2k-

TOS3(e) is given by {xI = x2 = y2 = z = 01 . Consequently, we can find a transformation preserving j' co and reducing the equations for S2(e) , S3 (e) to the normal forms (20.3), (20.4). Now we can assume that the coordinates satisfy the conditions (20.3), (20.4) and j1 o) = w0 . We can also assume that xI , yI , x2 , y2 , x3 , y3 , xk , yk , z are coordinates on SI (e). Then JS,X (xi) = AI I (r)x, + A,2(r)X2,

§20. DEGENERATIONS OF CODIMENSION > 4

93

Is,Xe(x2) = A21(r)xt + A22(r)x2, A11 +A 22 - 0 (see §17). For a generic

germ A21(0) # 0, and after a suitable transformation x1 -' x1 + R(r)x2 , preserving j 1 w, conditions (20.3), (20.4) and (20.5) are valid. (2) Lemmas on preliminary normal forms.

Q.E.D.

LEMMA 20.2. Let w = wo + r, j 1 r = 0, 1 > 0. Let S be a manifold given by {x1 = x2 = y2 = 0} . Then the germ {w = 0} can be reduced to a normal form

{wo+jst+fdz=0},

fEfi1sl,jlf=0,

by a transformation r - r + 1p(r,.x1 , z) preserving S.

PROOF. Let w1 = w0 + js'r + t(r - jsr), t c [0, 11. Let P be a projector sending a form > At dxt + > Bt d y1 + C d z to a form > At dx1 + BE d y1 + (jsC) dz. By Proposition 5.2 it suffices to prove the solvability of the equation

P[Xt,dwt +d(X1,wt) +ht o,] = -P(r - jsr)

(20.7)

with respect to the unknowns h, and X1 = it>2 (pi! a/axe + Ei>1 Wita/ayi , satisfying the condition t+1

9r, t' Wi,t'htEfits

.

(20.8)

Equation (20.7) can be rewritten as a system for (pt,1 , lyt,t , ht and ut = X1, w1. After the elimination of w1 and h, we get a system of two t equations for u,:

axi 1

+V

1t (u t ) + A l t + u = 8 t 1

(20. 9)

aut r (20 . 10) + v2 t(ut) + A 2 1u1 = 0 , 'S az where via are vector fields, v.(0) = 0, and A,,,, 0, are function-germs belonging to fits t Arguing as in the proof of the solvability of equation (10.11) we can show that equation (20.9) has a solution u1 E fitss 2 (the field a/ax1 + v1,1 can

be reduced to a/ax1 by a transformation depending on t and preserving S). Since u1 E ym1S 2 , ut satisfies (20.10). Therefore, system (20.9), (20.10) has a solution ut E 9Jils 2 . The corresponding solution (h,, X,) of equation (20.7) satisfies (20.8).

Q.E.D.

LEMMA 20.3. Let S: {x2 = y2 = x1 = 0}, 1 > 2 and jsw = wo + >1<;<1 f dz + r, where T E Q('), f E fits) . Then the jet js{w = 0} can be reduced to the jet {wo + T t
(i>3), y;-+y;+w1 (i#2),where rpi,yri EWes.

94

IV. CLASSIFICATION OF GERMS OF ODD-DIMENSIONAL PFAFFIAN EQUATIONS

PROOF. We use Corollary 5.2. Let w, = w0+>;<,_<, f dz+tz, t E [0, 1]. It suffices to prove the solvability of the equation (20.11)

with respect to the unknown h, and

X, = a w,t + Y

ax. +

WY TI

satisfying

gi,,wi,I,ht Efill.

(20.12)

Let T=T,dy,+Ei>3(T,dx,+0,)dy,,where T,,

We can easily

check that the set

SI/,.r=Wi=Ti,

h=-01,

lpi.t=t0i=xio,-0i

w1,i=vI =

(i > 3),

-rxiTi i>3

is a solution of (20.11) satisfying (20.12).

Q.E.D.

(3) Normalization of jse. LEMMA 20.4. For a generic germ e E PW"-2' 1, 1 there exist coordinates

x, y, z such that conditions (20.3)-(20.5) are valid and js2e: {wa + (B(-r)x2 + zy2) dz = 0}.

(20.13)

PROOF. By Lemmas 20.1 and 20.2 we can reduce j . e to {w0 + (B, (i)x, + B2(7)x2 + B3(r)y2) dz = 0}

(20.14)

and assume the validity of conditions (20.3)-(20.5). Then SI (e) is given by

{F(x,y,z)=0},where jj2F=B1(r). Since S2cSI,wehave BI('r)=0. Using condition (20.4) we can show that B2(r)IY_o = B3(r)l2-o - 0. Then for a generic germ we have B3(r) = H(i) z , H(0) # 0. The transformation y2 - y2/H(r) , x2 - H(r)x2 reduces (20.14) to (20.13). Q.E.D. (4) Normalization of jse . LEMMA 20.5. For a generic germ e E

PW"-2,1.1

there exist coordinates

x, y, z such that conditions (20.3)-(20.5) are valid and (wo [(XI C(-)x2 ] d z = 01. .2 Js

+

+ z)y2 +

(20.15)

PROOF. By Lemmas 20.2 and 20.4 we can reduce js2e to {w0 + [B(r)x2 + zy2 + C1(r)xi + C2(r)x1x2 + C3(r)x, y2

+C4(r)x2 + CS(r)x2y2 + C6(r)yz]dz = 0}

(20.16)

§20. DEGENERATIONS OF CODIMENSION > 4

95

and assume the validity of (20.3)-(20.5). The transformations used below preserve these conditions. Condition (20.5) implies that C3(0) # 0. The transformation xI -'x,

reduces C3(r) to 1. Using (20.4) we can show that C,(-r) - 0, B(r) _ y, + C2(r)xz/2 rezC2(r) . The transformation y2 -, y2 - C2(r)x2, y, duces (20.16) to

(0jo+[(x, +z)y2+C4(i)x2+C5(r)xzyz+C6(r)y2]dz+r, =01, (20.17) where r, E Q(2) . For a generic germ C4(0) 0 0. The transformation x2 x2 - y2C5(r)/2C4(i) , y, y, + yZC5(r)/4C4(r) reduces (20.17) to

{n0+[(x,+z)y2+C4(r)x2+C6(r)y2]dz+r2=0},

(20.18)

where r2 E Q(2) . The transformation x, -o x, - C6(i)y2 reduces (20.28) to the form

{coo+[(x,+z)y2+C4(r)xZ]dz+r3=0},

(20.19)

where as above r3 E Q(2) . By Lemma 20.3 the jet (20.19) is equivalent to (20.15) (C(i) = C4(r)). Q.E.D. (5) Normalization of js e. By Lemmas 20.2 and 20.5 we can assume that

jsze: {coo+[(x,+z)y2+C(r)X2+f]dz=0},

f E971(3)

2

2

and conditions (20.3)-(20.5) are valid. By using the transformation x, x, + cp , where cp is a suitable germ from 9Yt , we can obtain d C(r)/d y2 =

of/ayz=0. The field J's. X is of the form 2

2f

a

f

2C(r)x2 a+ax,za8x2-+ dax, dz a dx1 2

2

It follows from condition (20.5) that 82 f /8x2 8x2 = 0. Besides, for a generic germ we have a 3f (0) # o , ax3

C(0) -10

(in the opposite case we have an additional degeneration of X ). The transformation xi - taxi , y; - t;y;, z µz with suitable t,? , µ E R' reduces (0) to I and C(0) to 1/2. Therefore, a generic germ is reducible to the form CD 0 + I(Xl + Z)y2 + L

+ 1 + a(r) x3

I + e(r) 2

2

X2

6

1

+A, ('r)x,xZ + A2 (r) X'2 + g] d z = 0

IV. CLASSIFICATION OF GERMS OF ODD-DIMENSIONAL PFAFFIAN EQUATIONS

96

where g E 931st,

ay = aY = ayl =

c(0) = a(0) = 0,

z

d y2 = 0.

z

z

2

Now we are to show that c(rr), a(r), A1(r) and A2("r) can be reduced to 0. We can use Corollary 5.2 and prove the solvability of the corresponding equation by equating the functional coefficients of xIx2' (i + j <_ 3) in the right- and left-hand parts. The proof of Theorem 20.4 is completed. Q.E.D.

20.2. Normalization of jse. LEMMA 20.6. Let S2: {x1 = x2 = y2 = 01, f E 9315, Then the 4-jet .

2

3

{o+[(xt+zY2+++fJdz=o} on S2 is reducible to the form 2

o

l

dz = 01

+

whe re A(r) is a function-germ.

PROOF. Let Pi be a projector sending a function C to a function jsz C Is2 xi /i! and let P be a projector sending a form E Al d x1 + E B+ d y1 + C d z to a form r Al dx1 + E B1 d y1 + P4(C) d z. Consider a family w, = a

coo+[(x1 +z)y2+x2/2+x,/6+tf] dz. By Proposition 5.2 it suffices to prove the solvability of the equation

P[X1,dw, + d(Xt, w,) + hla,,] = -P4(f) dz (20.20) V, 8/8yi +Y 8/8z

with respect to the unknown h1, and XI = > c0, d /8x1 + satisfying the condition

(20.21) 9;i) +V, , y , h E 97122. Arguing as in the proofs of solvability of similar equations (§ 18) we can show that the proof of the solvability of equation (20.20) under condition (20.21)

can be reduced to the proof that the equation P5

-x2ax1 +X1z2 I =-(P5r)

(20.22)

is solvable with respect to the unknown a E//TO) . The germ z in the right-

hand side belongs to 97ts2) and does not depend on

Y2.

Thus, T has the

form r = ES_0T,(r)xx2 `.Then the germ )+4zT05(,))x5+T32(r)+2Z=14`'Jx4x G- T4 1(')+3z(r23( 5 2 q 1

1

+

r23(?)+3 z705(') x13x22

4

+T0S(#)x1X2 +

1

2

2

3

T14(' )xlx2

320. DEGENERATIONS OF CODIMENSION > 4

Q.E.D.

is a solution of equation (20.22). LEMMA 20.7. Let g E 9A(')

97

The 5 -jet

.

2

j w0 + [(Xi + z)y2 +

3

l

1

2 + 6 + Ar)x4 + gJ dz = 0 }

on S2 is reducible to the form 1

(

(0 +

[(x1

l

+ z)y2 + z + 6 + A(r)xi + B(r)x dz = 0 }

.

1

The proof is similar to that of Lemma 20.6.

JJJ

REMARK. We can prove the following statement:

for a generic germ e E PWn-2, 1 '' the jet js3e is equivalent to the jet 2

I

m

3

ll

w0+ (xI+z)y2+ 2 + G +EA,(r)xi

dz=0(m=4,5,i=4 JJ

20.3. Proof of Theorem 20.3. From Lemmas 20.2 and 20.7 we can assume that for some f E'Mts, 2

3

e: { CO = w0 + (xi + z)y2 + 2 + 6 + A(r)x" + B(`r)xl + jJ d z = 0} . fl

We can also assume that 8A/8y2 = 0, 8B/8y2 = 0. We prove that for an arbitrary s there exists a 1-form t = y(r)x4dz E Vt° such that equation (20.1) is unsolvable. To prove this fact we rewrite (20.1) as a system of equations for h and

wi=X dyi,

ci=X,dxi,

y=X,dz, u=X,w.

We can eliminate 92 1 ... , tpk and y/, , ...

y/k , h in this system and obtain

,

an equivalent system for u, rp, and 7, viz., 2

/

\

1

jsz [dx - `Y2 + 2 + 4A(r)xi + 5B(r)x4 I yJ = 0, I

\ ++ 2 4Ar)x + 5Br)xi //l t01 2

.ls1

z+

(Y2

ou

+x4

8xi' 8u

8yi =

r)X4

- x2

aY + (xi + z) dye z

oA(r) _ x 0A(r) _ 0u 0A(r)} +x40A(?)u 8y! ' 8yi ) 8yi 8xi ay,

_(x, + z)y2+ _ zi 2

2

6

_

+x4 E xJBA( 1>3 8xi (20.23)

98

IV. CLASSIFICATION OF GERMS OF ODD-DIMENSIONAL PFAFFIAN EQUATIONS

After the transformation (P: y2 Y2 - x2 /2 - 4A('r)x' - 5B(r)x4 and the elimination of the coordinate y2 we reduce system (20.23) to the system

I js

I

as + (xl +

aXI

12A(r)x2 + 20B(r)x3 )fl] = 0,

,fl.. I .

=Y(r)x

js2 O$+v(a)+x4a YI

LL

with respect to the unknown function-germs c k, .8, where

du

fi =

a = uIy2=o,

u = u((D),

aY2 y2=o

0 = 5x 4aB(r) +4x 30A(?) Oz aZ

-x 2'

1

1

3

2

v = az + 13 + 3A(i)x' +

2

r

+ 4AQ)xi z + 5B(r)x4z

+ xZ2 +X4 EX, Xr i>3

aA a 4 aA ay! -X,ayl a-,.

4aA(r) 0 i>3X1 dxi dyl

a +(x1 +z)aX2

axi

a d YI

i>3x1

System (20.24) is reduced to the equation for a -xal1

js2- (x+ 12A(x+20B(x)v(a)

J

= (20.25)

Let a = r-,+,<,

We use G(a) to denote the left-hand side of (20.25) and consider the system J d[G(a)] axi axi

= 0,

i+j<4

(20.26)

X -X 2-o

for a,,(r). This system follows from (20.25). We can show that equations (20.26) imply the relations

all =0 (i+j<3),

a13=a04=0,

a22 = -2za. , a40 = Z2a04 , 0a00 0a00

az By (20.25) we have

as°r)l x,=x2=o

ay1

=

(20.27)

=0.

-5!y(i). At the same time (20.27)

implies that 1

51

a[G(a)] axls X,-X2-:.o

dA OA aao.o = 1\d -X1a 1) axI \Y Y/

aA aao.0

1

r>3

11-3

ax! Ya +10Aanoo. YI

§21. POINT SINGULARITIES OF PFAFFIAN EQUATIONS IN R2

l

99

Thus, we arrive at the equation

r

/dA

xi

aY,

0A 8a°,0 aYl / 8x1

8A da°,0

8A

axi 8yt + ayl a0.0 = -y(r)IZ_0. (20.28)

Since a0,0 depends on n - 5 variables xi, yi (i > 3) and y(i)I Z-o is an arbitrary function depending on n-4 variables y1 , xi , yi (i > 3), equation (20.28) is unsolvable if y(r) is a suitable germ with a vanishing s -jet. The proof is complete. REMARK. It is clear from the proof that even the problem of 4 -jet classification on the second degeneration manifold is "wild"; an invariant of the 4-jet

on the manifold S2 is a germ of a function depending on n - 4 variables. §21. Point singularities of Pfaffian equations in R2k+I

In this section n = 2k + 1 >_ 5. We prove that a generic germ e e W"-2.1.2

P

is 5-determined and a generic germ e E PW° is 1-determined.

A generic germ belonging to one of these singularity classes is reducible to an

invariant normal form with k parameters. In order to present the invariant normal forms we use the following notation.

Let u = (u2 , ... , uk) be a (k - I)-tuple satisfying the conditions

(20 Reui>0, Imui>0, pi=ui+p (1+m+1 0

Imp,=O,

Here k=I+m+2p-1. We use Gµ = GO(x2 , Y21 1

xk , Yk) to denote the function

,

I uixiyi+f[(Imui) x +Y] M

2

2

2

2

1+1

l+m+p

+ E [Reui(xiyi + xp+iyp+i + Im ui(xiyp+i - xp+iY()] 1+M+1

(note that it is a normal form of a quadratic Hamiltonian in the symplectic dx2 A dYz + + dxp A dyp , see [A2]) and use Q(u; Fr) to denote the function space

(R2k-2

1

k

Q(u;r)= 2r, xiyi - Gp (r=(x2, y2,... ,xk,Yk)) 2

100

IV. CLASSIFICATION OF GERMS OF ODD-DIMENSIONAL PFAFFIAN EQUATIONS

THEOREM 21.1. A generic germ e E PW and reducible to the invariant normal form

-2, 1,2 is unstable, 5-determined.

{wo+[x1(y1+Q(µ; r`))+x,z+bx3z2]dz=0} if e c r

PW"-z.

(21.1)

or to the invariant normal form

wo + [x1Y1 + Q(/t; 'r))+

x

2

3 + x1 z +b

(Xl 5

3 2 + 23xlz +xlz 4)]J dz =011

(21.2)

ifeEP

Wn-2. I

(b E R1 is the modulus).

THEOREM 21.2. A generic germ e E PW° is unstable, 1-determined, and reducible to the invariant normal form

{zdz+(13.1)}=0.

(21.3)

In order to prove Theorem 21.2 it suffices to notice that the normal form (21.3) coincides with the normal form (13.10) (with the term +zdz ). After that we can argue as in § 19.7.

We use the term "a generic germ" e: {w = 0} E PWn-2' I ' 2 in Theorem 21.1 to stress the validity of the conditions (1) S2(e) = PWn-2' 1'0 U {0} is a smooth manifold (near 0). Recall that S2(e) = PWn-2' 1(e) is the set of the second degeneration; (2) the vector field Xe is nondegenerate at the singular point 0 E Rn (3) for the form w = WIs2(e) the inclusion w E W 3 holds true (see 13). 2 1.2

PW"-2,1'2 We denote by PW,, the set of germs e E 'Vsatisfying n_2,1,2 conditions (1), (2), and (3). We can easily check that the set Wn depends only on the germ e (it does not depend on the choice of the 1-form

w representing the Pfaffian equation). It follows from the results of § 17 that if e E n-2.1.2

PWn-2.1.2

is a germ of a

generic Pfaffian equation then e c PWn REMARK. Let a3 = W I s . Consider the vector field XX (see § 13). Let A,_31 be the spectrum of its linearization at the singular point 0 E R'. It follows from condition (3) that there are no resonances, viz., mR_3An_3 # 0

(mENu{0} ,

m? 0 O).

(21.4)

The proof of Theorem 21.1 is divided into several parts. We begin with normalizing a 3 -jet on the manifold S2(e) (§§21.1 and 21.2). Let us fix a

§21. POINT SINGULARITIES OF PFAFFIAN EQUATIONS IN R2k

l

101

special notation for the proof of Theorem 21.1:

S is a manifold {x1 = z = 01; l = i , m= 7)'ls , gy (m) _ 9N } (m>0); S1 = SI (e) ; S2 = S2(e) (SI = PW, -2(e) , S2 =

ms

(e)) ;

PW"-2, 1, 2 ;

e: (co = 0} is a germ belonging to

r=(Y1,x2,y2, ,

PW"-z''

r=(X2,y2,... ,xk,Yk),

,Xk,Yk);

Q = Q(u ;

,1;

wti =wo+[x1(YI +Q(u; r

3

we=('o+ IxI(YI+Q(U;F))+ 3 +x1zzI dz. 21.1. Preliminary choice of coordinates.

LEMMA 21.1. For a generic germ e E

PW"-2 ,1.2

there exist coordinates

x, y, z such that j'e: {wo = 0} and the following conditions are valid: (a) S2 is the intersection of Si and the manifold {x1 = z = 01 (b) {x1 , x2 , y2 , ... , xk , yk } is a coordinate system on S1 ; PW"-2, 1 2,h (c)

j'Xe(x,) = A(r)xl, j'X(z) _ -A(r)z (if e c

'X (x,) = A(r)z, .'X(z) = -A(r)x1 (if e E PWn -z, 1 ,2,e i

PROOF. We can choose coordinates r such that j co - wo . Since T0S1 = Ker wo10 , condition (b) is valid. It was shown in § 17 that condition (2) of gen-

eral position (nondegeneracy of X, at 0) implies that T0S2e (8/8x, , d/az) = T0S, . Hence we can choose the coordinates x, and z satisfying condition (a). Condition (c) can be reached by a suitable transformation x, -+ A12(r)z , z - A21(r)xi + A22(r)z (see the proof of Lemma 18.2). Q.E.D.

LEMMA 21.2. For a generic germ e E

PW"-2.1.2

there exist coordinates

x, Y, z such that conditions (a), (b), and (c) of Lemma 21.1 are valid and e is of the form

{wo+(XI(Q(u; r)+y1)+ f) dz =0},

f E9A

(21.5)

PROOF. By Lemma 18.1 we can change the coordinates r and obtain the following normalization of j' e :

Fe:{coo +(A,(r)x,+A2(r)z)dz=0}.

(21.6)

The transformation y, -. y1 - A2(r)z2/2 "kills" the term A2(r)zdz (in j'e ). Since T0S, = Kerdy1 we have dA1Io # 0, [coo A dA,]lo = 0. The restriction iii = oils, is of the form woI{A,(r)=o} The condition &i E W° 3 implies that the function A, (r) can be reduced to the form y1 + Q(u ; r) by

102

IV. CLASSIFICATION OF GERMS OF ODD-DIMENSIONAL PFAFFIAN EQUATIONS

a transformation r , di(r) preserving the Pfaffian equation {wo - 0} (see Appendix A). This implies, in its turn, that j'e is reducible to j' e : {w0 + xI (y1 + Q(,u ; r)) d z = 0}

and by Lemma 18.1 we obtain the normalization (21.5).

Q.E.D.

21.2. 3-jet normalization on S. PW"-21I ,2

LEMMA 21.3. For a generic germ e E

there exist coordinates

x, y, z such that (a) e: {wo + [XI (y1 + Q(/t; r) + g)] dz = 01, (b) condition (c) of Lemma 21.1 holds.

gE

9R3

;

(21.7)

PROOF. Consider the normal form (21.5). Let j 2f = BI (r)x, +B2(r)x1 z+ B3(r)z2. Using condition (a) of Lemma 21.1 and arguing as in the proof of Lemma 18.5, we can show that BI(r) = B2(r) 0. The term B3(r)z2dz can be "killed" by the transformation yl - yI - B3(r)z3/3 and by Lemma 18.1 we get the normal form (21.7). Q.E.D. PW"-2.I.2,h (e E PWn-2'''2,e LEMMA 21.4. For a generic germ e E respectively) there exist coordinates x,y,z in which j3e : {wh = 0} ( j3e {we=0} respectively).

PROOF. Consider the normal form (21.7). Let T g = CI (r)x3 +C2(r)xI z+ C3(r)x1 z2 + C4(r)z3 . Using condition (c) of Lemma 21.1 and arguing as in the proof of Lemma 18.4, we obtain the relations

if

e c:

C3(r) - 3C,(r) if

eE

CI(r) = C3(r) = 0

C2(r) = 0,

PW"-z,I,z,h

PW"-2, I.2.e.

Since the term C4(r)z3 d z can be "killed" by the transformation y1 -,

yI - C4(r)z4/4 , we have the normalization of j3e : (wo + xI [yI + Q(µ; r) + C(r)x1 z] dz = 0} w0 + x1 yI + Q(,a ; r) + C(r)

(:,

PW"-2'''2'h ,

if e E

l

+ z2 ] dz = 0 }

if e E

PW"-2''

' 2' e

Condition (2) of general position §21 /implies that C(0) /- 0. Using the homotopy method (Corollary 5.2) one can reduce C(r) to 1. Q.E.D.

21.3. 4-jet normalization on S.

LEMMA 21.5. Let g E X71(4) be a function-germ, I and 2 be the 4 -jets on S o f the f o r m I : {wh = 0} , 2: {wh + g dz = 01. Then I and 2 are equivalent.

§21. POINT SINGULARITIES OF PFAFFIAN EQUATIONS IN RZk't

103

PROOF. Let w, = wh + tg d z , t E [0, 1 ] . By Corollary 5.2 it suffices to prove the solvability of the equation

]4[X1 dw,+d(X,,w,)+h,w,]=-gdz

(21.8)

with respect to the unknown h1 and X, E a, Xt =

We seek h,, yit ,

a YItax

(pit ,

k

k

+Y2tdz +Vita +EVitax r=1

.

(21.9)

.

j=2

orir satisfying the conditions

Y2,t=72Ea2,

Y,.t=YIEa2,

pi.twr.tE9R.

(21.10)

Equation (21.8) can be rewritten as a system for h, , yi, , 'Pit , wit and u = After the elimination of h, , (pi, , and lyl, we obtain the following system of two equations for u,, Y,t , yet

X, J Wt .

J4[

0XI

- (YI +Q(it; i)+2x,z)Y21 = 0,

l4l a , +(y, +Q(,u;

Y XI(YI +Q(,u;

aut +x,u-x,Exa+x, i>2

aQ(a;

+xlz)

ax(_t+X°'1 ay ,ay,

i>2

-F) Out +XIE aQ(u; eyi ax'] i>2

=(21.11)

We can simplify this system by the transformation (D: y, -, y, - Q(.u; i) 2x1z . Denote u,(O) by ii,. Let at = ut y _o , 6t =

System (21.11) is reduced to the system

a

, g = g(,D)jy,=.

I

Y,

14[axt +2zfi,] =0, J

41Oa,

az

ax +,8,(2x,+x2,z2)+x,,,-XIExi aat I>2

-x

1

i>2

for a1,,

aXi

aye

ay;

(21.12)

i

aXi

We seek a, satisfying a, = a E i(s) and 6, satisfying 8, =

fi E a(') Under these conditions inclusions (21.10) hold true and at the same time system (21.12) is reduced to the system i9a

ax

1

+2zf-o,

a+2x,Q

1V. CLASSIFICATION OF GERMS OF ODD-DIMENSIONAL PFAFFIAN EQUATIONS

104

which is equivalent to the equation x18a/ax1 - z as/az = zgg . This equation has a solution a E a(S) . Q.E.D. LEMMA 21.6. Let g E a(4) be a function-germ, , and 2 be 4 -jets of the form 4f,: {we = 0} , r;2: {we + g dz = 0}. Then , and 2 are equivalent.

PROOF. Let co, = we + tg d z , t c [0, 1 ] . Arguing as in the proof of Lemma 21.5 we can reduce the equation J4[X,,dw, +d(X,,cot) +h,w,] = -g dz to the equation x1 as/a z - z as/ax, = r E a") which has a solution Q.E.D.

aE

COROLLARY 21.1. For a generic germ e E

PW"-2.1.2

there exist coordi-

nates x, y, z such that j4e: {wh = 0) or j4e: {we = 01 . PROOF. This statement follows from Lemmas 21.4, 21.5, 21.6, and 18.1.

21.4. 5-jet normalization on S. LEMMA 21.7. For an arbitrary function-germ g E a(5) there exists a number b c R such that the 5 -jets on S {wh = 0}

and {wh + (g + bxi z2) dz = 01

are equivalent.

PROOF. We can assume that ag/ay, = 0 (we can do this using a transformation x, -. x, + rp with a suitable rp E (5) ). Let w, b = Wh + t(g + bxi Z2)dz, t E [0, 1]. We will prove the solvability of the equation

j5[X,,dw, b+d(X,,w, b)+h,w,

b]=-(g+bx;z2)dz

(21.13)

with respect to the unknown b E R1 , function-germ h, , and vector field Xt E 931 of form (21.9).

Equation (21.13) can be rewritten as a system for b , (pit , w , Vii , y2, , h,, and u, = X, , w, . After the elimination of rptt, wir , and h, we obtain the system 151 ax` - (y, + Q(u ; r) + 2x, z + 1 1r ax + 3bx1 z2) l // //

+

(,

l Yet, \ + Q(µ ; r} + 2x1 z + t (_!- + 3bxi z2))

= 0L111

atxl(Y1+Q(12;r)+x,z)+xlu,-xiEX;ax1 au

y,

1>2

I

(aQ(k; r) (_au, au` J + aQ(u; r) au, 1l 1l +X, F 1 ax, ` ay; +x gay,/ ay; ax; i>2 _

-g-bx3z2

(21.14)

S21. POINT SINGULARITIES OF PFAFFIAN EQUATIONS IN Ru"1

105

of two equations for b, Yii , v2,, and u,. We seek ut with vanishing 1 -jet on S. The coordinate y, can be eliminated by the transformation 2x, z - t

0: y, - y, - Q(. u;

(L09gxi

+ 3bx1 z2)

and we reduce system (21.14) to the system

'S aa,

2z + t a2g + 6btx, z2 fit] = o, [ax. + ax2

l

-5 [aat

j az+

a2g (2x1 +tax1

2

r>2

aQ(12; i>2

_

aa!

1

az+(6bt+1)xzll $,+x,at-

ax;

ax i

aQ(,U; ir dat ax;) ay;

aa, ay;

g-bx;z2

for at = ut((I))Iy,-o and fi, =

(21.15) In order to show the system (21.15)

aym} I

Y, =o

is solvable we have to prove the solvability of the following equation for at z

r

z

j5l 2x, x

+(6bt+ 1)xiz ax,

2z+tax +6btxz2

(azt +x,at-x,Ex;axt i>2

i

i>2

yi

I

+x, 2

aQ(U; ay;

= 2z(g + bx z2) ,

aat

ax1z> ) (21.16)

which is obtained by eliminating 8t .

Let us seek at in the form a, = a, + a2 , a, E

(6)

,

a2 = y(r)xi z2 .

Equation (21.16) takes the form

2x, axi - 2z da + L(y(i)) = 2z(g + bxi z2)

(21.17)

,

i

where 2 a2g1 a2g L(Y) = Y( [21X1Z2ax,az - 2tx, zaxz

J

33 +2xz

[Ex;+ 8u; y

i>2

axi

i>2

axi

ayi

-

i>2

; r)

ayi

axi

106

IV. CLASSIFICATION OF GERMS OF ODD-DIMENSIONAL PFAFFIAN E94UAI IVIN5

Let x(f) be a functional coefficient at (x1 z)3 in the expansion of f E JJP" in the degrees of xlz. Equation (21.17) is reduced to

/c[2z(g + bx, z2) - L(y)] = 0

(21.18)

with respect to the unknowns b E Rl and y = 7(-r). Since

z

z

x

0,

21

1

we can rewrite (21.18) as (21.19)

V(y) _ fi(r) + 2 where

fi(r) = x(zg), v = E X, 1>2

a

i oxi

+E i>2

Q('U -,

axi

ayi

OQ(u; r) a ayi aXi

where Z) = wls2 . Since there Choose b = -2re(0). Note that V = are no resonances (21.4) and the right-hand side of (21.19) vanishes at 0, equation (21.19) has a formal solution y = y(_r). By the Borel theorem we

can reduce (21.19) to V (y) = T , where T is the germ of a flat function in the variables r . Since there are no reasonances (21.4), the field V is hyperbolic. By Theorem 8.1 the equation Vy = T is solvable. Thus, equation (21.13) has a solution (b, ht' X,). We can verify that the inclusion at E 9R4 implies f, Then by (21.14) we have y1 ,y2 , E 9Ji, whence X, c 9Ji. Q.E.D.

LEMMA 21.8. Let g E W), 1

,

y2

be 5 -jets on S, i.e., I : {wh = 01,

2: {wh + (g + bxi z2) dz = 01, where b E III' is chosen according to Lemma 21.7. Then there is a diffeomorphism D reducing y2 to t1 and such that j5((D'(bx; z2 dz)) = bx3 z2 dz.

PROOF. Let y11, yet , 'Pit , Wit, h,, and u, be a solution of equation (21.13). It follows from the proof of Lemma 21.7 that we can choose u,

of the form u, = 17(r)xl z + f,, f, E 9R3 . Then we can show that y1, = U(r)x, +fi, , y21 = -rt(r)z+ f2t , where fl, , f2t E a2. Therefore the reducing e_g(')z and z into + diffeomorphism ' sends xl into e'(')xl + where . . . stands for terms from 9Ji2 . Hence the relation j5Z"(xi z2 dz) _ xi z2 dz is valid.

Q.E.D.(5)

COROLLARY 21.2. For a generic germ e e P Wn -2.1, 2. h there exist coor-

dinates x, y, z such that j5e : {wh + bxi z2 d z = 01, b E Rl . (5)We have also used the fact that h, c Iii

.

§21- POINT SINGULARITIES OF PFAFFIAN EQUATIONS IN R""'

107

PROOF. By Lemmas 21.5 and 18.1 75e can be reduced to {wh+gdz = 0} ,

g E R(5) . By Lemma 21.8 there are b E R' and c E Diff (n) such that j (D '(bx1 z2 d z) = bx, z2 dz, {wh + (g + bx, z2) d z = 0} = {wh =01.

These relations imply that

jsc' {wh +gdz = 0} = {wh - bx13 z2 d z = 0}. Q.E.D.

Now we prove a similar result for the elliptic case. LEMMA 21.9. For an arbitrary function-germ g E 9R(5) there exist a number b E R' , a diffeomorphism (D E Diff (n) and a function-germ H = 1 + h, h E W such that

is (HcI,S e+ I g+b (5 l

+

L

ja4V.[(xI + z2)2

xz2+x1z4)] dz}) =

dx1 A dz] = (x + z2)2 dx1 A dz.

COROLLARY 21.3. For a generic germ e c PW,.

(21.20) (21.21)

-2' 1'2'e there exist coor-

dinates x, y, z such that j5e: {we

l

+b 5 + 3x1z2+x,z4) dz=0}.

(21.22)

PROOF OF COROLLARY 21.3. By Lemmas 21.6 and 18.1 the 5 -jet on S of , 2. a is reducible to {we + g d z = 0), g E a(5} a generic germ e E Let 0 = (xi /5 + (2/3)x1 z2 + x1 z4) dz. By (21.21) d8 , therefore PW"-2.1

?0 *6 = 0 + (OA/ax1) dx1 + (8A/d z) d z where A E a(6). Then relation (21.20) implies

.5HV(we+gdz)=we-bj5Hc*9-we, 1

jsc'0=we-b(0+aAdx,+Ozdz).

(21.23)

The transformation y, -, y, + bA reduces the right-hand side of (21.23) to 'e9 up to the terms from 9R6. Thus, the 5 -jets of the germs {we+g dz - 01 and {we - b0 = 0} are equivalent. Q.E.D. PROOF OF LEMMA 21.9. Let W h,r

= Coa + t

2 3 2 +x,z 41 l` 55 + 3x,z /! J dz, g+b (x

tE [0, Q.

108

IV. CLASSIFICATION OF GERMS OF ODD-DIMENSIONAL PFAFFIAN EQUATIONS

We prove that the equation s

js(XX,dwb,,+d(XX,ojb ,)+htwb t] _ -g-b

(( 5 +

2

3

z2+x,z4)

(21.24)

is solvable with respect to the unknowns b E R` , h, , and X, off the form (21.9) satisfying the conditions ht E 9J1, X, E 9J1 and the conditions

j 72, = -C(r)xl.

j711 = C(r)z,

`

`

(21.25)

If the solvability of equation (21.24) under these conditions is proved, then by Corollary 5.2 there exists a function germ H = 1 + h, h E 9J1, and a diffeomorphism c such that (21.20) is valid. Using condition (21.25) one can show that relation (21.21) is valid as well (the diffeomorphism 0 sends x, into x1 cos C(r) + z sin C(r) + and z into -x, sin C(r) + z cos C(r) + , where . stands for terms from 9Ji2 ). Thus, to prove the lemma it suffices to prove the solvability of equation (21.24) under the conditions h,, X, E 91 and condition (21.25). Arguing as in the proof of Lemma 21.7 we can reduce equation (21.24) to the equation . .

J6

2

112z + 4tbz(x2 +Z 2 )+ 3x1 +t 3

axt z

- (2x, + 4tbx, (x2 + z2) + t-X d Z

\

1

d

+ x1a, - X;-(ad)

=x,g+b Cx6/5+ 3x4 z2+x2z4)

(21.26)

for at E 9R6

As in the hyperbolic case we can assume that ag/ay1 = 0. We seek a,

in the form a, = a = a1 + y(r)(x; + z2)2, a, E a(6) . We can show that this form of a, implies h,, Xt E 9R and condition (21.25). We can rewrite (21.26) as the equation 2z

-1 axi

- 2x,

i9a

r

6

5 + 3 xi z2 + x1 z4/I l

1 = -L(Y) + x1 g + b l

(21.27)

for a, and 7(-r), where =

3X s

a2g a2g a(x+z ez )-2x,(x,+z')' +4t(x+z } x,ax,az-zax2 z

z

z

z

y

2

+ 2x2 (x2 +z )XZ(F).

Let x, = pcos9, z = psinO, x1 a2g/ax1az-za2g/axe = p4B1 (r, 0), 6). In the coordinates p, 0, r equation (21.27) takes the

x, g =

521. POINT SINGULARITIES OF PFAFFIAN EQUATIONS IN R-"

109

form 1

801

-

Icos40-2cos20+4tB1(r, 0)1 r

+ B2(B) + b Lco5

o

+

2

cos4 0 sine 0 + cost 0 sin4 B

(21.28)

3 and we have to prove its solvability under the condition a1 B=o = 1 e=2x Since fox Bl (r, 0) _- 0 (for each function g E 9Y1(51 ), and 2x

2x

=2t,

f cos4odo= 4tr, 0

0

2x

1 0

s

(coo

+

2

cos4 0 sins 0+ cos20 sin4 0}

/

d o= 3

then in order to prove that equation (21.28) is solvable under the condition a1 1e=0 = al Ia=2x we have to prove the solvability of the equation 6 + --2 ,

(21.29)

2x

where 4(r) = f B2(r, o) do . Set b =

Then equation (21.29) is

0

solvable with respect to y(r) (we argue as in the proof of the solvability of equation (21.19). Q.E.D. 21.5. 1 -jet normalization on S for 1 > 6. LEMMA 21.10. For an even I = 21, > 6, g E 9R1" the 1 -jets on S

{Wh+(bxiz2+g)dz=0} and {wh+bxiz2dz=0} are equivalent.

PROOF. Let m, = wh + (bxiz2 + tg)dz, t E [0, 1]. We have to prove the solvability of the equation !1 [X1, dw1 + d (X,, w,) + hlw,] = -g d z with respect to the unknown hl and vector field XX E JR . We seek X1 belonging to SIR!-2 . Arguing as in §21.4 and using this condition we reduce the given

equation to an equation for a1 E a('+'), i.e., x1 oat/8x1 - z 8c!/8z = a('+'). Since (I + 1) is odd, this equation is solvable. Q.E.D. the 1- jets on S

LEMMA 21.11. For an even 1 = 21, > 6, g c 1

g)

+ s

we+b( 5 are equivalent.

dz=0

dz-0}

,

E

110

IV. CLASSIFICATION OF GERMS OF ODD-DIMENSIONAL PFAFFIAN EQUATIONS

PxooF. Consider a similar equation for h, and X,. As above we seek . Transformations similar to those in the proof of X, belonging to JJt(/-2)

Lemma 21.9 reduce our equation to x1 as/Oz - zaa/ax, _ Since (I+ 1) is odd this equation is solvable.

E

611(1+1)

Q.E.D.

LEMMA 21.12. The assertion of Lemma 21.10 holds if I > 7 is odd.

PROOF. We can assume that ag/ay, a 0 and seek a solution of the equation

/[X,,dco,+d(X,,to,)+h,a,] = -gdz

(21.30)

tE[011) satisfying

the condition h, X, E 911'. Arguingas in the proof of Lemma 21.7 and using this condition we can prove that equation (21.30) is solvable by proving that the equation z)ai9a jr+I f (2x, + (6b + 1)x2 xl

- (2z + 6bx1 z2) {

+ x1 a - X1 Xw(a)1

} = 2zg

(21.31)

for a E fii1-1 is solvable.

p = (1 - 1)/2. We can

Let a = al + Y(r)(xlz)p, where a, E rewrite (21.3 1) as 2x,

ax

l - 2z Zl + [(p - 2)y(r) + 2XX(y)1(x, z)p+1 = 2zg.

(21.32)

1

We use ('r) to denote the coefficient of (x1 z)p+' in the expansion of 2zg in the degrees of x1 and z. Equation (21.32) is reduced to the following equation for y(r) : 2y (21.33) X;(Y) + p 2 Equation (21.33) has a formal solution because for any integers m1 > 0, ... , mn_3 > 0 we have

p-

j60.

(21.34)

Indeed, if (21.34) is violated, then

(m1+p22)A]+lm2+p22\

i12+m3A3+...+mrt_3Zn_3=0

(assuming that Al + Z2 = 1 ). Since (p - 2)/2 > I , this relation contradicts condition (21.4).

Thus, equation (21.33) can be solved as a formal series. By the Borel theorem and Theorem 8.1 it has a smooth solution. Q.E.D.

III

§21. POINT SINGULARITIES OF PFAFFIAN EQUATIONS IN R2 "

LEMMA 21.13. The assertion of Lemma 21.11 holds if I > 7 is odd.

PROOF. We seek a solution of equation (21.30) satisfying the condition hl , Xl E M Using this condition we reduce (21.30) to the equation for a E ai -I

r+I

2

K 2z + 4bz(x1 + z2)

- 3x1) /

- (2x1 + 4bx1(x2 + z2))

`

Ox

1

dZ +x1a - x1X;(a))] = 2x1 g.

Let a = al + y(?)(xI + z2)P, where aI

E

1(r+I)

(21.35)

, p = (I - 1)/2. We can

rewrite (21.35) as I1

[x1 L

1 - 2z+[Px(x y+ z2)P- 2X1 (4

+ Z2)P J

(21.36)

= 2x1 g.

Introduce coordinates p, 9 such that x1 = p cos 9 , z = p sin 8 . Let

x1g = pr+1Q(9, r),

a1

= pr+I&1(9

Equation (21.36) takes the form

+ y(r) [43 cos4 9 - 2 cos2 0] + 2 cos2 9Xw(y) = 2Q(O, r)

(21.37)

97-

and we have to prove its solvability under the condition at a=o = a119=z

In order to solve (21.37) under this condition it suffices to solve the equation 2s

2n

y('r) f 3pcos49-2cos20)dO+2X;(y) f0 cos29dO=2 (r) or (what is the same) equation (21.33). The solvability of equation (21.33) has already been shown. Q.E.D. 21.6. Proof of Theorem 21.1. Note that when normalizing 1-jets on S we have solved differential equations with respect to functions depending on r . These equations have solutions defined in a common neighborhood of the

point r = 0. Therefore, we can reduce a generic germ e E PW"-2' I.2 to the normal form (21.2) or (21.1) up to a germ of a 1-form z E'JJI°° (see the proof of Theorem 18.1 in J18.10). Moreover, we can reduce a germ to the

normal form up to a germ g dz, g e a'. In order to show that the term g d z can be "killed" we use Proposition 5.1 and prove the solvability of the corresponding equation with the parameter t e [0, 1 ] . When reducing this equation to an equation for a function germ al (see the transformations of a similar equation in § 18.9) we obtain the equations (V + V,)(a,) + (9 + 91)at = 'r,

(21.38)

in the coordinates xi , z , F, where 0, 01, TI are function-germs, 0,' T1 C 9ROD , V and V are vector fields, and V E 9R°O . The field V and the germ 0 are of the form z2] a V = [2x1 + (6b + 1)x2 z + 2bx3 axi - [2z + 6bx1 z2]z + (2x1 z + 6bx2z2)XZ ,

6= -(2x1z+6bx2z2), n-2,1,2,h

if e E P W

; or of the form

V = [2z + 3x, +4bz3 +4bx2z+ Sbx' + 3bxiZ2]

ax,

- [2x, + 4bxi + 4bxiz2] az + (2x2 + 4bxi + 4bx2z2)X;,

9= -(2x2+4bx; +4bx2z2), if

PW,"-2'1'2'r

eE

In the hyperbolic case the solvability of equation (21.38) follows from Theorem 8.1 (the field V is hyperbolic on the manifold S). It remains to prove the solvability of equation (21.38) in the elliptic case. Then this equation can be reduced to the form [z + xl (x2 + z2 )lax'

+ [-x, + z(x2 + z2)] 8zr

2

2

+A[x1 +Z ]

(A+1 -.)yi au

k

k

ax;

=2

i=2

all t ay,

+ (VI + V) (ur) + (61 + 01)ul = T,,

(21.39)

where

AER,

61 E 9

2

,

4 V1E9AS, T,E91is, VE9Jts, 6,E9Yts.

It is not hard to show that we can assume that u, , Bi , Br , r, (V, , V respectively) are functions (vector fields respectively) defined in a neighborhood U of the point 0 E R' ' and equation (21.39) belongs to the class H(U ; S ; 2) (see the notation before Theorem 8.7). We cannot use Theorem 8.7 directly because the term A(x2+ z2)X; violates the validity of estimate (8.4)(6). To "kill"this term we use the transformation 4), viz., X; = x;(x2 + z2)

a. =

2

(6)with r - 2.

y; = y;(x2 + z2)fl' A(21 - 1)

,

A.1,'

Q

J-

2

§22. BASIC RESULTS AND COROLLARIES

113

which is a diffeomorphism of U\101. After this transformation we obtain an equation of the same form (21.39), but now A 0 and it belongs to H(U; S; 2), where U = '(U) (generally speaking, U is not bounded). For the new equation estimate (8.4) is valid (r = 2) and by Theorem 8.7 it has a solution u, satisfying (8.5) in F J. Then the function u1(xi, z, x2,Y2,... ,xk,Yk) 2 0 2 a =ut(xi,z,x2(x1 +z)',Y2(x1 +z )',--.,xk(x1 +z) Yk(xl+z) 2

2

2

2

2

2

satisfies (21.39) in the neighborhood U and estimates (8.5) ensure the smooth-

ness of u, in U. The proof is complete. §22. Basic results and corollaries; table of singularities; list of normal forms; examples

We use (1), (2), ... for the numbers of the normal forms in the list of normal forms below in Table 20.2 (see page 116).

THEOREM 22.1. Let n = 2k + 1 _> 5. (i) Any stable germ of a Pfaffian equation on an n-manifold is equivalent to one and only one of the normal forms (1)-(4). (ii) Any unstable but finitely determined germ of a generic Pfaffian equation on an n-manifold is equivalent to one and only one of the normal forms (5)(7).

THEOREM 22.2. Let n = 2k + I > 5, M be an n-manifold. (i) For a generic Pfaffian equation E on M there exists a subset M C M of codimension > 4 such that the germ of E at each point M\M is stable. (ii) There exists an open set A C PA1(M) such that for any Pfaffian equation E E A there exists a codimension 4 subset M c M with the following property: the germ of E at any point a E M is not finitely determined. (iii) Let T be a set of points at which germs of a generic Pfaffian equation on M are unstable but finitely determined. Then T consists of isolated points. THEOREM 22.3. (i) Any stable germ of a Pfaffian equation on a 3-manifold is equivalent either to the normal form {d z +x dy = 0} or to the normal form

{dy+x2dz=0}. (ii) Any unstable but finitely determined germ of a Pfaffian equation on a 3-manifold is equivalent to one and only one of the normal forms

(5) (fork= 1), (10), (11). THEOREM 22.4. (i) At each point of a 3-manifold the germ of a generic Pfaffian equation is finitely determined. (ii) For a generic Pfaffian equation E on a 3-manifold the set of points at which germs of E are unstable either consists of isolated points or is empty.

IV. CLASSIFICATION OF GERMS OF ODD-DIMENSIONAL PFAFFIAN EQUATIONS

1 14

Notation in Table 20.1 of singularity classes. codim codimension of a singularity class I minimum jet order which enables us to determine the given singularity class

index of finite determinacy of the germs from the given class (when there is no sufficient jet, p = oo ) m number of moduli that distinguish close nonequivalent germs from the given class

p

st *

**

sign of stability: + (the germs from the given class are stable), (the germs from the given class are unstable) takes place only for generic germs from the given singularity class preliminary (not invariant) normal form

Comments on the list of normal forms. x1 , y1 , ... , xk , yk, z in (1)-(9) ;

Coordinates:

x, y, z in (10), (11). Numerical moduli:

b c llt in (6), (7), (10), (11) ; µf c C in (5), (6), (7) (generally speaking),

Rep, > 0. Functional parameters: f, f , g1 in (8), (9) are germs of functions in the variables x1

fE

M?{as

, Y1 , 1

=xz=O} , J

... , xk Yk , z ; 1

.fi = J gi = 0.

EXAMPLE 22.1. For a generic Pfaffian equation E on a 3-manifold M there exist closed submanifolds MI (of dimension 2), M. (of isolated points) and L (of isolated points) such that

M2cMI cM, LcMI, LnM2=0 and for the germ of E at a point Q E M the following properties hold: 1. If a E M\M1 , the germ is stable and reducible to the normal form

{dz+xdy=0}.

2. If a c MI\(L U M2) , the germ is stable and reducible to the normal

form {dy+x2dz=0}. 3. If a E M2, the germ is unstable but 5-determined; it is reducible to the normal form (10) or (11). 4. If a E L, the germ is unstable but 1-determined; it is reducible to the normal form

{zdz+ (µ+2}ydx+ (p-1)xdy=0J, where Rep > 0, p is a real or pure imaginary modulus.

§22. BASIC RESULTS AND COROLLARIES

115

TABLE 20.1. Typical singularity classes Singularity class

codim

PW2k2k++1 1

0

1

1

2

3

P

m

st

Nor-

Remarks

Z. form

PW2k-1.°

2k+1

2k-1'1'h PW2k+I 2k-1, I ,e PW2k+.

0

+

(l)

2

0

+

(2)

3

3

0

+

(3)

k>2

+

(4)

k> 2

(8)

k > 2, r

(9)

s
1

3

3

3

0

21-1,1,1 PW2+1

4

3

00

00

PWk+11

(2k-2s-1)

1

00

00

-

x (k - s)

f.

codim < n

= 2k + I PW2k,-, 1.1.2.h

2k + 1

3

5

k

-

(6)

k > 2, .

PW' X1.1 2 `

2k + 1

3

5

k

-

(7)

k > 2, .

PW+1

2k + 1

0

1

k

-

(5)

PW31'1'h

3

3

5

1

PWj 1'`

3

3

5

I

-

(10) (11)

EXAMPLE 22.2. For a generic Pfaffian equation E on a 5-manifold M there exist closed submanifolds M1 (of dimension 4), M. (of dimension 2), M3 (of dimension 1), L (of isolated points) and L1 (of isolated points) such that

Lc M1, MDM1 DM2DM3, L1nM3=o L1 cM2, LnM2=B, and for the germ of E at a point a E M the following properties hold true: 1. If a E M\M1 , the germ is stable and reducible to the normal form

{dz+x1dy1+x2dy2=0}.

2. If a E Ml\(M2 U L), the germ is stable and reducible to the normal form {dy1 + x2 dy2 + x; d z = 0}. 3. If a E M2\(M3 U L1), the germ is stable and reducible to the normal form (3) or (4) (for k - 2 ).

4. If a E M3, the germ is not finitely determined; it is reducible to the normal form (8)(7) (for k = 2). 5. If a E L1 , the germ is unstable but 5-determined; it is reducible to the normal form (6) or (7) (for k = 2). 6. If a E L, the germ is unstable but 1-determined; it is reducible to the normal form (5) (for k = 2 ). ()only for a generic point a E M,

.

116

IV. CLASSIFICATION OF GERMS OF ODD-DIMENSIONAL PFAFFIAN EQUATIONS

TABLE 20.2. List of normal forms No

Normal form

(1)

dz+xldy,+ .+xkdyk=0

(2)

dy1+x2dy2+

+xkdyk+xidz=0

(3)

dy1+x2dy2+

+xkdyk+(x1y2+xiz)dz=0

xty2+ 3 +x122 dz=0

(4)

(1s,+')xEdy,=0

(5) k

dye + Er2 x, dy,

(6)

+ {x1 Y1 + r-k=2

1

p1) x.y,] + xj z + bx3z2

dz = 0

-A,)xiY3ll

dy1+E`_2x;dy,+ x1 [y1+r-j=2(I'

(7)

J+

s

dz=0

+x1z2+b S

+j dz0

(8)

dy1+F-k2Xidy,+ IX 1y2+zy2+ 2 +

(9)

dz + Ei=1 x, dyj + 8, dy,+1 + Ek=,+2(8 dx- + ff dy1) = 0

(10)

dy+( xy+x2z+bx3z2)dz =\\0

(11)

dy+(xy++xz2+bx3z2} dz=0

/

REMARK. In the examples we assume that for the Pfaffian equation E all typical singularities >2 are realized, viz., >2(E) j4 0. §23. Commentary §§17.1,17.2.

Similar results were obtained by Martinet [Mar] (the normal

form (17.1) is equivalent to the normal form z dz+x1 dy1 + +xkdyk = 0, which was obtained in [Mar]). §17.5 and §20. Supposedly, the class of orbital equivalence of the field Xe is the only invariant of the germs e c PW(n) (for arbitrary singularities), but we do not know how to prove this fact (even for the case n = 3 ). However,

this is true for singularities of codimension < 3, which follows from our classification results. For these singularities the field X. is divergence-free. PWzk+It' 1.0 , then in suitable coordinates x1 , ... , X2k For example, if e E

Xe=x1 l

-x2 ax

orXe=x'aa

+XIX219

2

3

2

-x2 ax +(x'+x2)'9X3. 1

Note, that the fact that div Xe equals zero does not follow from the definition

of the field X,. §18.

The results of §18 were announced in [Z8].

§19. The existence of modulus in the classification of germs e E

PW31'

I

was proved by Jakubczyk and Przytycki [JP1], [JP2] (they gave a negative

§23. COMMENTARY

117

answer to the question posed by Martinet [Mar]). In §19 we have shown that

this modulus is unique (we mean the modulus b in the invariant normal form (19.1) or (19.2) ). The results of §19 were announced in [Z8]. Normal forms of generic vector fields vanishing on a submanifold were obtained in [Cha]. The work [Cher] is devoted to a local classification of Pfaffian equations

having a degeneration at every point in a neighborhood of the point under consideration. Such germs are reducible to stable normal forms. [MouJ is devoted to a local classification of singularities of integrable Pfaffian equations.

CHAPTER V

Classification of Germs of Even-Dimensional Pfaffian Equations In this Chapter we classify singularities of Pfathan equations on a manifold

M of dimension n = 2k > 4. The first occurring singularity class has codimension 3; we show that if n > 6, the corresponding germs are stable relative to C'-equivalence (for arbitrary r < oc ) in spite of the existence of one modulus in the invariant normal form. The singularities of codimension _> 4 and the first occurring singularities of 4-dimensional Pfaffian equations are "wild" (except the point singularities associated with the case of vanishing 0-jets). We give preliminary normal forms with functional parameters in the

main "wild" cases; moreover, for n = 4 we give an invariant normal form with a functional modulus. For the most part we retain the notation of Chapter II and introduce the following notation for this chapter only:

n=2k; y=(y1,...,yk_1),u,v} is the coordinate system in R" ; COO =dy1

+x2dy2+...+Xk-ldyk-1 ;

r = (y1,x2,y2,...,Xk_1IYk-1); D(e) = PWn-3(e) (recall that it is the set of points near 0, at which the class of the germ e is n - 3 , i.e., for the case of even n = 2k it is the first degeneration set). §24. Singularities associated with the decrease of germ class; preliminary normal form

The class of an even-dimensional Pfaffian equation {w = 0} at a point

a E M and the class of a germ e E PW(2k) are defined in exactly the same way as for the odd-dimensional case. We retain the notation (§16.1) connected with the germ class. In contrast to the odd-dimensional case the maximum class value equals n - 1 , so, for a generic germ e E PW (n) we have cl (e) = n - 1 . The class calculation in local coordinates can be done using Lemma 16.1, 119

120

V. CLASSIFICATION OF GERMS OF EVEN-DIMENSIONAL PFAFFIAN EQUATIONS

which is valid for the even-dimensional case. The conclusions of Theorem 16.1 are valid for the even-dimensional case as well. At the same time, preliminary normal forms of the germs belonging to PW?k and PWik+I are different. THEOREM 24.1. Any germ e E PW2P+1 is reducible to the normal form k

{dYi+x2dY2+...+xdY+x+idY+i+

l (fidxi+idyi)=0}j=p+2

J

where f and gi are functional parameters with vanishing 1 -jet.

The proof of this statement is almost the same as the proof of Theorem 16.3. For the maximum class (e c PWn -1 , p = k - 1) the following result is obtained: THEOREM (Darboux). All the germs e E PWn

are equivalent to the germ

{dyl +x2dy2+ +xkdyk =01 (the "quasicontact" structure in R"(xl , ... , xk, y1 , ... , yk) Theorem 17.1 holds for even n too. So, the first singularities we meet (i.e.,

the singularities belonging to PWn-3) are realized on (n - 3)-dimensional manifolds. In §§25, 26, 28 we study such singularities in detail. §25. Other singularities (of the class n - 3)

In this section we study the singularity class PW"-3 and divide it into singularity subclasses.

25.1. The invariant field of directions. To each germ e: {w = 0} E PWrt-3 the following objects are invariantly related:

(a) the set PWn-3(e) , i.e., the first degeneration set (we denote it by D(e) );

(b) the field of directions in R" \ D(e) defined as the direction at a point (dw)k-11,,. E R" \ D(e) (near 0) that is the kernel of the form w n This 0 and field is correctly defined since if a f D(e) , then co n (dw)k-1IQ

dim Ker co n (dw)k-11. = 1

.

The field of directions is generated by the vector field Xe defined in R" by the equation

X,u= wn(d(D)k-1,

(25.1)

where u is a germ at 0 of some nondegenerate volume form in R" . THEOREM 25.1. (1) If two germs el, e2 E PWn -3 are equivalent, then the

fields Xe and X: are orbitally equivalent (in other words the field X. defined u p to the orbital equivalence is the invariant o f each germ e E PW, -3 ).

(2) The field X, vanishes on the set D(e).

§25. OTHER SINGULARITIES (OF THE CLASS n - 3)

121

(3) Let a E D(e). Then the spectrum of the Xe-linearization at the point a contains n - 3 zero eigenvalues; three other eigenvalues ZI , A2, '13 satisfy the condition kI +22 +'13 = 0.

PROOF. (1) The freedom in the definition of X. is associated with the choice of the 1-form representing the Pfaffian equation and with the choice of the volume form u. But if we replace w in equation (25.1) by H1 w and µ by H2µ (HI (0) $ 0, H2 (0) 0) then the field X. is replaced by the field HZ 1HiXe. (2) Since w n (dw)k-I ID(e) = 0 it follows that X I D(e) - 0 (3) If a E D(e), then (dw)kIQ = 0 and hence d(Xe,µ) = 0. This relation implies that the sum of all the eigenvalues is 0. Since codim D(e) = 3 and XeID(e) = 0, we have n - 3 zero eigenvalues. Q.E.D. 25.2. Singularity subclasses distinguished by a condition on 2 -jets. Let us introduce the singularity subclasses of the class PW"-3 , with which the following properties of the eigenvalues of the linearization of the vector field Xe are associated: (1) There are three different nonzero real eigenvalues. We denote the set of all e E PW"-3 of corresponding germs by PW" Re (2) There are three nonzero eigenvalues a, b+ci , a, b, c # 0. We denote the corresponding set of germs by PW." Im (3) There are only two nonzero real eigenvalues. We denote the corresponding set of germs by P W 03" Re (4) There are only two nonzero pure imaginary eigenvalues. We denote the corresponding set of germs by PWno1m 3 (5) There are two equal nonzero eigenvalues. We denote the corresponding set of germs by PWn I3 . (6) All the eigenvalues are zero. We denote the corresponding set of germs by PW" 030 . It follows from Theorem 25.1 that PWn-3 is the union of the singularity classes. THEOREM 25.2. All the singularity c l a s s e s above are distinguished by a con-

dition on 2 -jets. The singularity classes PWn, Re3 and PW" IM have codiPWn 031m , and PWn 3 have 3, the singularity classes PW" 03Re

codimension 4, and the singularity class PWn o 0 has codimension 5 (in PW(n) ). PROOF. By Theorem 24.1 an arbitrary germ e E PWn -3 is reducible to the normal form

{w0+fdu+gdv =0},

jI f = j1g=0.

(25.2)

V. CLASSIFICATION OF GERMS OF EVEN-DIMENSIONAL PFAFFIAN EQUATIONS

122

We can calculate a 1 jet of the vector field Xe in appropriate coordinates a _ , ag a (aa.r I a Xe _

ag_af

- J (au

3

av

Consider the matrix a28

(axI) au

ax1

a2f

ax,au (°) - ax,av (°)

xI

2

ax,a-(0) o

a ? (°)

(25.3)

2

ax,au(O)

ax12(°)

av

aZf a28 auav (°) - av2 (°)

a2f (°) (°) - auav a2g 8u2

28

A(e)

+1

(25.4)

f

axlav (°)

aaxlat[ (°)

The spectrum of the Xe-linearization at 0 consists of zero eigenvalues and 11, A2 , A3 that are the eigenvalues of A(e) . Notice that for an arbitrary matrix A with zero trace we can find a germ e E PW: -3 such that A = A(e) .

For each singularity subclass S C P Wn-3 we denote by Q(S) the set of matrices {AAA = A(e), e c S1. We denote by Q the set of all matrices with zero trace. One can easily see that

codimQ(S)=0(inQ)ifS=PW"-3

or S=PW"Im;

codim Q(S) = 1 (in Q) if S = PW'_3Re or S = PW" 03Im or S = PW, ,3 codim Q(S) = 2 (in Q) if S = PW" 030

;

Since codimPW"-3 = 3 (in PW(n)) we have codimS (in PW(n)) = codim Q(S) (in Q) + 3, and the theorem follows. Q.E.D. Next we formulate some properties of germs belonging to the singularity class PW"Re U PWn ,m . THEOREM 25.3. Let e E PW,"-3 U PW" Im . Re

.

Then

(1) D(e) is a smooth manifold in a neighborhood of 0; k-2 10 = 3 and this kernel is transverse to D(e) at 0. (2) dim Ker (a A (dco) PROOF. In the coordinates of the normal form (25.2) the set D(e) is given by the equations F1 = F2 = F3 = 0, where F,

-1

ag

au

of av)

,

,

,

ag

J FZ= (axI

J F3=1 , (Of ax, I

,

It follows from the proof of Theorem 25.2 that for a germ e E PW" Re U PW im we have detA(e) 96 0, where A(e) is of form (25.4). This implies dF, n dF2 A dF3I0 /- 0, hence D(e) is a smooth manifold near 0. In order to prove that dim Kerto A (dw)k-2 = 3 it suffices to consider the normal form (25.2) and notice that in appropriate coordinates Kerw n (dw)k-2 = (a/ax, , a/au, 8/Ov). The transversality condition for KerwA and D(e) is equivalent to the condition detA(e) /- 0 and is there(dw)k-2

fore valid.

Q.E.D.

§25. OTHER SINGULARITIES (OF THE CLASS n - 3)

123

25.3. The real invariant. In this section we define the real invariant of the germs of PW" 3 and PW," I,;, (1) Let e E PW, RC . Denote by 'l l , '12 , 13 the nonzero eigenvalues of the Xe-linearization at 0. We can assume (changing the order if necessary) that Al , IZ2 > 0. We denote min(A1 , '12)/ max(,k1 , '12) by #(e). (2) Let e E PW ",1m . Consider the nonzero eigenvalues a, at +bi, a1-bi

,

where a + 2a1 = 0, b > 0. We denote the number b/Ial by p(e). THEOREM 25.4. µ(e) is an invariant of a germ e E PW"

Re

U PW"

Im

PROOF. Let e1 and e2 be equivalent germs from PW" R3UP Wn rm and let 21(e1) ,

't2(e,) , 'Ye,) be nonzero eigenvalues of Xe -linearization, i = 1, 2.

Then the set {21(e1),.k2(el),A3(e1)} coincides with the set {kA1(e2),U2(e2),

k23(e2)} for some k E R \ {01, since by Theorem 25.1 the fields Xe and Xe are orbitally equivalent. We can assume that ( 1 ) for e1 , e2 E PW" Re :

23(e1) = -1 - k(e,) , i = 1, 2 ;

Z1(el) = A1(e2) = 1 , A2(e;) = µ(e1) ,

(2) for e1e2 E PWn Im'

'lI(e;)=1,'t2(e1)_-2+V_-_1µ(e;),

k3(e,)=-2-

,u(el),

Simple analysis shows that in both cases µ(e1) = µ(e2). 25.4. Two lemmas about preliminary normal form.

i=1,2.

Q.E.D.

Let D be a manifold

given by {x1 = u = v = 01. LEMMA 25.1. Let e : { w = 01, j I w = COO' m > 0. There exists a trans-

formation r - cp(r, x1 , u, v), which reduces e to the form {jDm + f du +

f,gEM

gdv=01,where

PRooF. Introduce the projector P sending a 1-form ) a, dx1 + E b; dyl

+c1dxl+c2du+c3dv toa 1-form >al dx;+>b,dy1+cldxl+(jDC2)du + (jDc3) dv Let 0a, - jDw + t(co - jDa ), t E [0, 1]. According to Propo.

sition 5.2 it suffices to prove the solvability of the equation P[X,,dco, + d (XI , w,) + h,w,] = -P(w - jD co) with respect to the unknown functiongerm h, and vector field X, E 9RD having the form E1>2 w1 aX; +E1> t Vi iay, This equation is reduced to a system of three equations for the function-germ

t=X"w,: __L +v, , ON

'

jD I u

( ) +bl ,

t ,

(25.5)

= Tt ,

t] =0,

(25.6)

0,

(25.7)

124

V. CLASSIFICATION OF GERMS OF EVEN-DIMENSIONAL PFAFFIAN EQUATIONS

where the v are vector fields, vt1(0) = 0, bit, T. are function-germs, and T, E EA'im+l . Equation (25.5) has a solution E 97tD+2 Since relations (25.6) and (25.7) are satisfied for any function from 9fD+2, the germ , is a solution of system (25.5)-(25.7). Q.E.D. LEMMA 25.2. Let j 1 w = coo . The germ {w = 0} is reducible to the form

{wo+fdu+gdv=0},where f,gE9JlD. PROOF. By Theorem 24.1 the germ {w = 0} is reducible to {wo + f du + g dv = 01. Let jDf = fo(r) , jDg = go(r) . After the transformation y1

y1 - Au - gov the germ has the form {wo + T = 01 with 'o r = 0, and we can use Lemma 25.1. Q.E.D. 25.5. Normal form of a 2 -jet on the first degeneration manifold.

THEOREM 25.5. (1) For each germ e E PW"R there exist coordinates

r, x1, u, v such that D(e) : {x1 = u = v = 0), z

{wo + B(r)x1v du + C(r)xludv = 0}.

(25.8)

(2) For each germ e E PWn tm there exist coordinates r, x1, u, v such that D(e): {x1 = u = v = 01 and

JD(e)e: {wo+[x1uB(r)-x1vC(r)]du+[x1uC(r)+x1vB(r)]dv =0}. (25.9) PROOF. (1) Choose the coordinates r such that j 1 w = w0 . Then Ker to n (dw)k-2I0

= (a/ax1 , a/au, a/av), and we can choose the coordinates x1 ,

u and v such that D(e): {x1 = u = v = 01 (by Theorem 25.3 D(e) is transverse to (a/ax1 , c1/au, a/av) ). In the coordinates r, x1 , u, v we have the relations fD(e)Xe(x1) = A11(r)x1 +A12(r)u+A13(r)v, JD(e)Xe(u) = A21(r)x1 + A22(r)u + A23(r)v, J ID(e)X (V) = A31(r)x1 + A32(r)u + A33(r)v.

The matrix A(r) _ (A1,(r)) is nondegenerate; its eigenvalues Al(r) , A2(r) ,

and A3(r) are real, pairwise different, and EA,(r) _- 0. Therefore, there T(r)(x1 , u, v)' reducing A(r) to diexists a transformation (x1 , u, v)' agonal form: A(r) diag(A1(r) + A2(r), -A1(r), -A2(r)). By Lemma 25.2 there exists a transformation r , jp(x1 , u, v, r) reducing f (e)e to the form {wo + f du + gdv = 0}, where f, g E 9R(D(e I) . Since D(e) is given by

{x1=u=v=0} we have

of ax1

= ag

ax1

- of avv

ag

- au

_0

(25 . 10)

§25. OTHER SINGULARITIES (OF THE CLASS n - 3)

125

The transformation y1 -+ y1 - y reduces 3D(e)e to {w0 = 01. By Lemma 25.1 after a suitable change of the coordinate r we normalize the 2-jet on D(e) : so jD(e)e = wo +.JD(e)d u for some function-germ u E

f,

JD(e)e : { co0 + f du + g d v = 01,

(25.11)

JJio(e).

Using the normalization of ID(e)Xe (i.e., the form of the matrix A(r) ), we obtain the relations of = of eg ag ax1

-A2V

,

- A1u,

8x1

av

au

= -(AI +Az)z

I

These relations imply that JD(e)e: {coo - A2x1v du + Al xI u dv + JD(e) du = 01, where u is a function-germ belonging to 9JiD(e) . The transformation y1 y1 -,u "kills" the term j) (e) d u and we arrive at (25.8) with B(r) = -A2(r), C(r) = AI (r) .

(2) the arguments are similar but we get another normalization of the matrix A(r), viz., JDXe(xi) = -2A1(r)x1

,

JDXe(u) = AI (r)u + A2(r)v,

jDXe(v) = -A2(r)u - AI(r)v. As above we can obtain normalization (25.11) with the germs f and g

satisfying the relations

.I Of

.1 ag

JD ax = JA(v) , JD

ag

1

JD ax

-JDXe(U)

of=JDX

(au-'f

(25.13)

1

(x1)'

Using (25.12) and (25.13), we obtain the normal form (25.9) up to the term fD(e)d u , U E 9JtD(e) which can be "killed" by the transformation y1 -'

y1 - y. Q.E.D. THEOREM 25.6.

e E PW, R. {w0 + B(r)x1v du + x1 u dv = 0} ,

(25.14)

e E PWn Im: {0i0 + [x1uB(r) - xlv] du + [x1u + xivB(r)] dv = 0) (25.15) are normal forms of a 2 -jet on D(e)

.

PROOF. Let e E PW" x and {ui = 0} be the normal form (25.8). Then

the germ e is generated by the form 61C(r) (since C(O) 96 0). By the Darboux theorem for 1-forms there exists a transformation r -' 4)(r) such

that b*(wo/C(r)) = m0. This transformation reduces the form ai/C(r) to coo + B(r)xly du + x1 u d v . For e E PWn 3 the arguments are the same.

Q.E.D.

126

V. CLASSIFICATION OF GERMS OF EVEN-DIMENSIONAL PFAFFIAN EQUATIONS

25.6. Singularities distinguished by a condition on 3 -jet.

Let E be an

arbitrary Pfaffian equation with a given germ e E PW, Re at 0. Consider a map from the germ D(e) to (0, 1), viz., a - ,u(r E), where µ(e) is the invariant mentioned in §25.4. Analogously, for a germ e E PW"Im we

can consider the map a -+ u(n.E) from the germ D(e) to (0, oo). It is clear that these maps do not depend on the choice of the Pfaffian equation E. Thus, for each germ e E PWn,R or e E PW"Im we have an invariant function on the manifold D(e) (defined locally). We denote this function by µe . Note that µe(0) = µ(e) . Let n = 2k > 6. For a generic germ e c PWn.Re U PW"Im we have (dµe A ))I0 j4 0, (25.16) where {w = 01 is a restriction of the germ e on D(e) (a contact structure on D(e) ). In the coordinates of the normal forms (25.8) and (25.9) we have ai = wo . In the coordinates of the normal forms (25.14) and (25.15) condition (25.16) means that dB(r)l0 A dy1 y-1 0. We denote by PWn Re'° the singularity class consisting of germs from PW",Re for which condition (25.16) is valid. We denote the singularity class PWR-3 \PW' by PWn .Similar subclasses of PW, Im are denoted by PWn ,3.'0 and I

PKn.-3Im ,1

We denote the set of germs E PW4I Re (E PW4I 1m respectively) such that µe(0) # 0 by PW41'QRe (PW4I im respectively), and use PW4I,'Re (PW41,Im respectively) for the singularity classes PW4, Re \ PWa,'xe (PW41,1m \ PWa,'1m respectively).

It is clear that for n = 2k _> 6 we have 30

codim PW" Re (Im) = 3,

-3 codim PWn Re (Im) = n - 1 , codim PW4I RIe 1

Im)

=4

and these classes are distinguished by a condition on 3 -jets.

THEOREM 25.7. Let n = 2k > 6. F o r each germ e E PW" Re' ° (e c PW" im'0 respectively) there are coordinates r, x1 , u, v such that D(e) is given by {x1 = u = v =01 and JD(e)e: {w0 + (1 + µ(e) + x2)xl v du + xl u dv = 01,

(25.17)

(jD(e)e: {w0 + [x1 u(2µ(e) + x2) - x1 v] du + [x1u + x1 v(2µ(e) + x2)]dv = 01 (25.18) respectively).

PROOF. (1) We use normal form (25.14). Since d BI0 n dy1 # 0, by Theorem 7.7 there exists a transformation `I': r -+ 1p (r) such that 41'w0 = Hw0 ,

B(T) = [B(0) + x2] , where H = H(r) is a function-germ, H(0) # 0. The transformation `P reduces (25.14) to

{Hw0+[B(0)+x2]xlvdu +xludv = 0)

§25. OTHER SINGULARITIES (OF THE CLASS n - 3) or

127

(c° + B(0) x2x,v du + xH dv =0 } . x1H we obtain the normal form

After the transformation x1

{w°+(A+x2)xlvdu+x,udv =0},

(25.19)

where A = B(0) .

LEMMA 25.3. The normal form (25.19) is equivalent to the normal form

[coo +(.CI +x2)xlvdu+xludv =0}

(25.20)

and to the normal form

{w°+(1 -2+x2)xlvdu+xludv = 0}.

(25.21)

PROOF. The normal form (25.19) is equivalent to {

+x (Wo

X2

+ x, vd u +

l

xlu +x

d v= 0 }. 2

( 25 . 22)

111

Let us interchange u an d v in (25.22):

{

too

+ x2

+

Z

x1v + x2 d u+xu d v

l

=0 }.

( 25 . 23)

Since the function 1/(.Z + x2) has the value Z-1 at 0, the germ (25.23) is equivalent to (25.20). Let us show that the germs (25.19) and (25.21 ) are equivalent. Indeed, we can verify that the transformation y, -. y, +x, uv reduces (25.19) to a germ equivalent to (25.21). Q.E.D. It follows from Lemma 25.3 that each germ e c PW"'0 is reducible to normal form (25.19) where A E (1, 2). The spectrum of the X -linearization

at 0 consists of n - 3 zero eigenvalues and -A, -1 + A, 1 ; therefore, l = µ(e) + 1. (2) We can obtain the normal form

{w°+[xlu(A+x2)-xlv]du+[x,u+x,v(A+x2)]dv=0}

(25.24)

by similar arguments and easily show that by replacing A with (-A) in (25.24), we obtain a germ equivalent to (25.24). We can thus assume that in (25.24) A > 0. The spectrum of the Xe-linearization at 0 consists of n - 3 zero eigenvalues and 2, -1 ± iA ; therefore, A = 2µ(e) . The proof of Theorem 25.7 is completed. 25.7. The scheme of adjacencies.

THEOREM 25.8. Germs at all points of a generic Pfaffian equation on an n-manifold belong to one of the singularity classes given in the scheme below, where all the adjacencies of typical singularity classes are shown.

..

a.waajrikAl'UN UY GERMS OF EVEN-DIMENSIONAL PFAFFIAN EQUATIONS

n=4:

PWI'1 W4. Re 1

PW(4)

3

1 / PW4,0,Re P W4

,0

R l

PW4

PW

1

\_

PW44,1

4,Im D

PW4

PW4, 0,Im

PWI'1 4,Im

n=2k> 6: 3,1

h-PW,Re -n

PW"-3,0 R, Re

PW(n)

4-

`-

Wn-3 n,0,Re

PW"n T

PW°

PW300

W" rt,I

n-3.0 P WnIm

PWn-3

n,0,I. PW'I

PWn 1m,1

-5

I

I PWn-2s-1

n

s=L- + 48n+1 1. We omit the proof of the adjacency PWno30 (it is rather long; the idea is the same as in the proof of the adjacency W"-'- 1 . W,, -2 (see Theorem 11.4)). Note that a generic germ e E PWR , e: {w = Of satisfies +- PW,° does not the condition (dw)kI0 # 0, hence the adjacency hold. All other adjacencies (or their absence) follow from the results of this PW,"-5

PWn-3

section.

§26. Classification of first occurring singularities

of Pfafan equations in R", n = 2k > 6

In this section n = 2k > 6. First occurring singularities represent the singularity classes PW,"

' ° and P W" Im' ° . We prove the following results.

§26. CLASSIFICATION OF FIRST OCCURRING SINGULARITIES

THEOREM 26.1. Any germ e E P Wn'."

13., 0

129

is equivalent to the normal form

(too +[xlu(2,t+x2)-x1v]du+[x,u+x,v(2A+x2)]dv =0}, where 2=µ(e).

(26.1)

COROLLARY 26.1. Any germ e E PW," im'0 is stable and 3-determined.

THEOREM 26.2. Let s < ac be an arbitrary number, let e E K, Re , and let 2 =#(e) be an irrational number. Then the germ e is C3-equivalent to the normal form

{too+x1v(1 +A+x2)du+xIudv =01.

(26.2)

COROLLARY 26.2. Any germ e E PW" R3'0 with an irrational invariant u(e) is stable and 3-determined relative to C3-equivalence (for arbitrary s < 00 ).

COROLLARY 26.3. For a generic Pfaffian equation E on _a manifold M,

dim M = n = 2k > 6, there exists a codimension 4 subset M c M such that the germ of E at any point a E M\M is stable relative to C3-equivalence and is C3-equivalent to the normal form (26.1) or (26.2) (for arbitrary s < oo ). THEOREM 26.3. Any germ e E PW" R3'0 is not finitely determined (relative to C°°-equivalence).

THEOREM 26.4. Let e E PW"-Re'0, u(e) be a rational number. Then there exists s < oo such that the germ e is not finitely determined relative to C3-equivalence.

The difference between the cases e E PW°Re'0 and e E PW im'0 is associated with the structure of resonances. Recall that the field Xe has three nonzero eigenvalues µ(e), 1 , and eigenvalues -(1 +µ(e)) for the case PW,,-3'0 , and -1 , 1/2 ± iµ(e) for the case e E PW" m'0 . For any eE µ(e) the relation 11(-1) + 12 G + iµ(e)

+ 13 [

- iµ(e)

0

(26.3)

is valid if and only if 11 = 12 = 13 . At the same time for rational µ(e) the relation (26.4) 11µ(e) + 12 + 13(-1 -#(e)) = 0 is valid for an infinite number of 3-tuples (11 ) 12, 13) such that 11 12 or 11 # 13

.

Moreover, if e E PWn Re'0 and µ(e) is an irrational number, then the relation (26.5) 11 [µ(e) + x2] + 12 + 13[-1 -,u(e) - x2] = 0 holds for some x2 , 11 , 12 , 13 such that 11 96 12 or 1, j4 13 and Ix21 < e , where

e is an arbitrary small number.

v. c;i.Aa,s2FICATION OF GERMS OF EVEN-DIMENSIONAL PFAFFIAN EQUATIONS

The relations (26.5) are obstacles to the equivalence of a generic germ ' , ' 0 to the normal form (26.2). Relations (26.4) do not allow a germ e E PW° 3,0 with a rational invariant M(e) to be reduced to the normal form (26.2) by a C3-diffeomorphism for sufficiently large s. In the proofs below we will use the notation

eEP

wo+[x,v(l +A+x2)]du+x,udv; w1m,a = 0jo+[x,u(2A+x2) -x,v]du+[x,u+x,v(2A+x2)]dv. wRc A =

26.1. Two auxiliary lemmas. We prove two auxiliary lemmas on the solvability of degenerate linear systems with respect to function-germs. LEMMA 26.1. Let f1

f(o) = 0,

' A'

f2

.

T1 , T2 ,

T3 be function-germs at 0 and let

(df, n df2 A df3)I0

(26.6)

0.

Then the system 0

fi

f2

-fL

0

f3

-f2 -f3

l X

-

0

TL

T2

(26.7)

T3

with respect to the unknown function-germs 1 '2 , 3 is solvable if and only if TI, 72113 satisfy the relation

f12-f3T1- f T3=0.

(26.8)

PROOF. It is clear that (26.8) follows from the solvability of system (26.7). Suppose now that (26.8) is valid. Thanks to condition (26.6) we can assume that f = x, (i = 1, 2, 3). Then r, E 9t{X,=X2=°} , T2 E 9A{x,=X3=0} and we can represent T, and T2 in the form

Ti = Ax, + Bx2,

T2 = Cx1 + Dx3 ,

where A, B, C, and D are some germs, 8B/ax, = 8D/8x1 = 0. It follows from (26.8) that (B - D) E DX{x, =o} ' whence B - D. Now we can verify that system (26.7) has the solution

l=-C, 2=A, 3=B. Q.E.D. The following statement can be proved similarly.

LEMMA 26.2. Let D be a manifold given by if, = f2 = f3 = 0), (df1 A df2 n df3)10 # 0, Ti E !Jlol JD

with respect to

y1' 2 ,

The system

.

0

f1

-fl

0

-f2 -.l3 S3

f2 f3 0

2-

T!

t3

T3

1

is solvable if and only if

JD [f2T2-f3T1-fT3]=0.

T2

§26. CLASSIFICATION OF FIRST OCCURRING SINGULARITIES

26.2. 1-jet normalization. prove the following result.

131

First we consider the case e E PW"

and

THEOREM 26.5. Let 2 E R \ Q , I > 3, and f, g c 9J1D) be arbitrary function-germs. Then the !jet on D { wRe

to {w,=0}.

2

+ f du + g d v = 0) is equivalent

{ 3m - 1 , m = 2, 3, 4, ... } . Let PROOF. Consider three cases. (1) 1 wt = WRe A + t (f d u + g d v) , t c [0, 1 ] . By Corollary 5.2 it suffices to prove

the solvability of the equation

jD[XI,dw, +d(X,,w,) +htw,] = -f du - gdv

(26.9)

with respect to the unknown vector field X, E 911D and function-germ h,. Here T C Mo+,) . We seek a family (h,, X,) satisfying (h,, XI) E 91to'.

(26.10)

Let a

a

k-,

a

xl =y1,Ox +y2rau +y3tav

k-,

a +Witay,.

(26.11)

Equation (26.9) can be rewritten as a system for function-germs ht , Vi,, Wit, y11 , and R, = Xt j o, . After the elimination of h,, (p,, and vi, we obtain the following system of three equations for R, , y,t, yet and Y3,:

v(1 +2+x2)

0

JD I -v(1 +2+x2)

-u

u

0

-(.1+x2)x,

(A+x2)X,

0

OR,

711

)(

72r V3,

2i-x- -I

OR

(26.12)

ou r+f

aR

av + gl By Lemma 26.2 system (26.12) is reduced to the equation

-v(l+t1+x2)R1] =T

JD'

for the function-germ R,' where r E T -

`miD+,)

(26.13)

is some function-germ. Let a1(r)xlIua=Val.

T-I "3

Consider the germs y, o, a, (r) _ (A +x 2)al + a2

+ 1 + x2)a3 . Since

A is an irrational number and (l + 1)/3 is not an integer, there exists a

GERMS OF EVEN-DIMENSIONAL PFAFFIAN EQUATIONS

a a (r)l > 0 for any positive

neighborhood U: Irl < e such that inf f ya rEU integer a, < 1 + 1 . The function

'ra,'a2'ar(r)

L

RI=R=

Xai ua2Va3 1

7a,'a2'a7(r)

defined in the neighborhood U is a solution of equation (26.13). The corresponding solution (h1, X1) of equation (26.9) satisfies (26.10). (2) 1 = 3m-1 , m E {3, 4, 5, ... } . We seek a solution of equation (26.9) satisfying (26.14)

(hr , XI) E 931Q

Let Xr be of the form (26.10) and R1 = X1,co,. Equation (26.9) under condition (26.14) can be reduced to the following system for the functiongerms R1 ,

j,

y1r ,

y21 and y3r

0

-v(1 +Z+x2)

-u

71t

v(1 +a.+x2)

0

x1(k+x2)

721

u

-x1(2. + x2)

0

y31

-aR1 ax1

r + xlv aY2 - x2

= JD

l

aR1`) -f

.

(26.15)

- Ri - g

au

By Lemma 26.2 system (26.15) can be reduced to the equation

-v(1 +A+x2)1

JD [u our +x1(t = r + x1 uv

I

(OR,

(26.16)

- x2 aR

for the function-germ R,' Here r E smD+I). We seek R1 in the form

Rr=R=(xluv) m- y(r)+R, 1

RE931

+

Denote the function (l/m!3)(ar/ax''aumav'") by K(r). We can find y(r) satisfying the equation K(r) + ay(r) aye

- x ay(r)

= 0.

2 0y1

Let y(r) = 7(r). Then equation (26.16) can be written down in the form aR - au +x1(t+x2) ax1 X1 -v(1+x+X2)-5v au

a,

(26.17)

§26. CLASSIFICATION OF FIRST OCCURRING SINGULARITIES

where i E

9Jto+I),

133

8i/8x"'8umavm = 0. Due to the last condition on z,

equation (26.17) has a solution R E M('+" in a sufficiently small neighborhood I rl < c. (3) 1 = 5. We seek a solution of equation (26.9) satisfying (hl , X,) E TD Arguing as above we can reduce equation (26.9) to the equation 6

7D

{u Bur +xI(/

+x2)aR _V(1 +.1+x2) d_ } 1

=1 + JD {XI uv

(OR

1 - x2 Rl

(26.18)

OY2

(Of 8g

Og OR

+ ` [8x1

au + av - au

OR ax1

OR Of

av axI

for the function-germ Rl . Here r/ is some function-germ belonging to 9J16 (t E [0, 1]). We seek R, in the form R, = xI uv y(r) +,k,, R1 E 9J1D . Denote by K1(r) the function $(arl/8x28u28v2) . We can find y,(r) satisfying the equation

y2 - x2 y1 + K, (r) _- 0. O

Let y(r) = y(r). Then equation (26.18) can be written in the form u

'` +x

Ou

I (Z

+ x2)8 R' - v0 +t + x2) OR' = a1,

(26.19)

where it E 9J1D . We can easily check that (Or/8xI 8u28v2) = 0 for each

t E [0, 1], f, g. Therefore, equation (26.19) has a solution Rl defined in a sufficiently small neighborhood Irk < e. It is clear that the corresponding solution (h,, X,) of equation (26.9) belongs to 9JiD The proof of Theorem 26.5 is complete.

THEOREM 26.6. Let 1 > 3 and let f, g E 9RD be arbitrary function germs- Then the 1 -jet on D {wlm + f du + gdv = 0} is equivalent to A

{ WIm

1 = 01.

PROOF. Consider the operator T : 9J:D - 9J1D , viz.,

T(R) = [-u - v(2) + x2)]

Ru

+ [u(2) + x2) - v] OR + 2x1

OR .

Let 1 # 5. Arguing as in the proof of Theorem 26.5 we reduce the proof to demonstrating the solvability of the equation T(R) = _ (I

aR T(R) = r - xI (u2 + v2) aye

+x x 1

3m - 1), 2(U2 +

(26.20) OR

v2) ay,

(l = 3m - 1) (26.21)

V. CLASSIFICATION OF GERMS OF EVEN-DIMENSIONAL PFAFFIAN EQUATIONS

134

for an unknown function-germ R E 1J1D 1

.

Here r is a germ belonging to

(t+1) D

Equation (26.20) is solvable since there are no resonances (26.3) for 11 +

12 +13 = 1 + 1 ; the solution exists in a sufficiently small neighborhood I r1 < e

.

Equation (26.21) is solvable as well; we can find its solution in the form

R=R+

}'(r)Xm-1(u2

+

V2M-1

,

where R E 9Ttg+I) (we use the fact that any function of the form y(r)x;"-1 x (u2 + v2)'"-1 belongs to the kernel of the operator T).

For the case I = 5 Theorem 26.6 is also proved in the same manner as Theorem 26.5.

Q.E.D.

26.3. Proof of Theorems 26.1 and 26.2. (1) Let e E P W" Im' ° . By Theorem 25.7 we can assume that jDe: {w1m.2 = 0). By Theorem 26.6 we can reduce jDe to the normal form {w1m a = 0) for an arbitrary I < oo.

We have solved differential equations (26.20), (26.21) for functions depend-

ing on r by normalizing 1 -jets on D. These equations can be solved in a neighborhood Irl < E(1) , where the number /e = E(1) is such that 11(-1)+12 f 2 +i(1c(e)+x2)

\ J + 13

(2 - i(1c(e)+X2) J 00

for every 11 + 12 + 13 = I + 1 3m and x21 < e(1) . Therefore, (26.20), (26.21) can be solved in a neighborhood Iri < e which does not depend on 1.

Arguing now as in § 18.10 we can show that the germ e is reducible to the form {WIm

d

+ r = 0), where r is a germ of 1-form belonging to MD . To

prove the equivalence of the germs {WIm A + r = 0) and {WIm 2 = 0) we use Proposition 5.1. It suffices to prove the solvability of the equation do 1 , X1 + d (X, , w,) + h,w, = -r,

(26.22)

where co, = WIm' z + tr , Xt is an unknown germ of a vector field, and h,

is an unknown function-germ. As in the proof of Theorem 26.5, equation (26.22) can be reduced to the equation V,(R,) + Q,R, = f

(26.23)

for R, = X, , co, , where V, = V + v, , v, is a vector field vanishing on D, v, E MOO, the vector field V has three nonzero eigenvalues at 0, viz., - 1 , 1/2 ± i u (e) , Q, and f, are function-germs, f E 9JID . By Theorem 8.1 equation (26.23) has a solution R, E R"". The proof of Theorem 26.1 is complete. (2) Next we prove Theorem 26.2. By Theorems 25.7, 26.5 for an arbitrary

to the form {wRf.l + z = 0) , 1 < oo we can reduce a germ e E PW,' where r is a germ of a 1-form, r E D . By using Proposition 5.1 we reduce

626. CLASSIFICATION OF FIRST OCCURRING SINGULARITIES

135

the proof of C°-equivalence of the germs {WRe I + r = 0} and {CoRe x = 0} Cv+1-solution to equation (26.23), where V, = to that of the existence of a V + V , yr E --'D , the vector field V has three nonzero eigenvalues at 0, viz., +1, µ(e) , -1 - µ(e) , Qr and f are function-germs, f E'J1tID . Notice that

the vector field V is independent of 1. By Theorem 8.2 we can choose a number I = 1(s) such that (26.23) has a solution of class C'`+I . Q.E.D. 26.4. Proof of Theorem 26.4. Let e E PW" Re'0 , e: {w = 01, µ(e) _ .l = p/q. First, we consider the case 2p + q > 6. To prove Theorem 26.4 it

suffices to prove the following statement.

THEOREM 26.7. For an arbitrary s < oc there exists a function-germ 0(r) such that js0 = 0 and the equation J2 +g-1 [X, dot + d (X . co) + hw] = -0(r)x°+9vp-I A

(26.24)

with respect to the unknown vector field X and function-germ h is unsolvable.

If Theorem 26.7 is proved, we can use Proposition 6.2 which implies Theorem 26.4. Since the manifold D is defined invariantly, we can replace (26.24) by the equation jD+q

1(X,dw+d(X,io)+he]=-0(r)xiqvp-Idv,

(26.25)

where w = jD +q-1 w

Note that if relation (26.4) is valid, then either 11 + 12 + 13 > 2p + q or 11 = 12 = 13 . Therefore, we can argue as in the proof of Theorem 26.5 and prove that jD +g 2e - {wR z = 01. Moreover, using Proposition 5.2 we can prove that 2p+q-le ,v {co Rell + dv = 0), JD where y(r) is a function-germ. We can thus assume that in (26.25) Co = V" A. (D Re,! + 7(r)xj+q Arguing as in the proof of Theorem 26.5, we can reduce (26.25) to the equation y(r)X°+9vp-1

o(r)Xp+yvp

jD +9Q(R) =

(26.26)

for a function-germ R = X, 6j. Here Q(R)

u au

+ x2)x1 d

R 1

+V(1 +/1+x2)dR +x1uv +

[X2OR

(p+q)y(r)xp,+g-1

vp-1 8u

- OR]

+'xi+'vDG(R),

where G(R) = a0(r)R + v(R) and v is some vector field.

136

V. CLASSIFICATION OF GERMS OF EVEN-DIMENSIONAL PFAFFIAN EQUATIONS

We can easily show that the relation jDQ(R) = 0 which follows from (26.26) implies that JDR = R0(r) + R1(r)x1 uv

(26.27)

and, moreover, the germs Ro(r) and R1 (r) satisfy the relations x2 0Ro

- 0yo = 0,

x2 SRI

2

1

1

- aRI

= 0.

(26.28)

2

It follows from (26.26) that

a(Q(R))

= (p + q)!p !0(r)

I

axp+Qav°

and, in particular,

a(Q(R)) axp+Qav°

_ (p + q)!p!e(r), X, =u=v=x2 =0

where 0(r) = 8Ix2_0 . At the same time 1 .9 (Q(R)) (p + 4) ! p ! axp +Q av °

_ (p + 4)Y(r)R1(r) + G(R0(r)) , X, =v=v °x2=0

where Y(r) = llx2_o , R1(r) = R1Ix2=0 , Ro(r) = Rolr2=0

Thus, we arrive at the equation (26.29)

(p + q)y(r)R1(r) + G(Ro(r)) = 0(r).

Notice that R1 and Ro depend only on the variables y1 , x3 , Y31 ...

,

xk-1 , yk_1 (this follows from (26.28)), while 0(r) depends on the variables yl , Y21 x3 , Y31 ... , xk _ 1 , yk-1 So, the left-hand side of (26.29) depends on two functions in 2k - 5 variables and the right-hand side of (26.29) can

be an arbitrary germ of a function in 2k - 4 variables. Therefore, for a suitable 0(r), j$B = 0, equation (26.25) is unsolvable. Thus, for 2p + q >_ 7 we have functional moduli in the classification of (2p + q - 1) jets on D. Using similar arguments we can show that for 2p + q < 6 functional moduli appear in 6 -jets on D (we omit the calculations). Q.E.D.

26.5. Proof of Theorem 26.3.

germ e e PW,,Re 3

.

Suppose that there exists an s-determined

Let I = µ(e) . Then A E R \ Q and by Theorem 26.5

the germ {WRe 2 = 0) is s-determined as well. Let w be a 1-form in lEt" such that now = WRe A . Denote the orbit of {jl naw = 01 in the space J1 by Ql . It follows from Theorem 26.4 that we can choose sequences k -+ 0 and Ik -+ oc such that the codimension of Q1 Q in J" would exceed n. Let Q3 = {w E A1(lR")I jsrow = jswRe.; I . Arguing as in the k

k

§27. DEGENERATIONS OF CODIMENSION > 4

137

proof of the transversality theorem (see [AVG]), we can show that for each k > ko there exists a thick set Ak C Q. such that if w E Ak then j Ik {,row = Qlk Q4 for any point a E RP? and sufficiently large ko . The intersection 0) A= flk>ko Ak is not empty. Consider an arbitrary form w E A. It is clear that the germ {Row = 0} is not equivalent to the germ {wRe,. = 0} . At the same time, j rr0w = J' Re. 4 . Contradiction. Q.E.D. §27. Degenerations of codimension > 4 In this section n = 2k > 6. We prove the following results. THEOREM 27.1. Functional moduli appear in the problem of the classification of 1 -jets on the first degeneration manifold of germs belonging to the singularity classes n-3, 1

n-3, 1

PWn,Re

'

PWn,Itn'

PWn-3 ,Re, 1 ,

n-3

PW ,Re,0

PWn-3

1

= 2 for singularity classes PW" Rc' I and PWn 1m' 1 3 t = 3 for singularity classes PW,, Re3 .0 and PWn Im, o t = 5 for singularity class PW" Re, I . I

THEOREM 27.2. Functional moduli appear in the problem of the classification of 3-jets on the first degeneration manifold of germs belonging to the

singularity classes PW"-o 0 and PWn (j < n - 3) 27.1. Singularity classes PW" Re I and PW"Im'

.

I

.

It was shown in §25

that the 2 -jet on D(e) of each germ e c PW" -3' I can be reduced to the normal form (25.14) and the 2 jet on D(e) of each germ e E PW" IM, I can be reduced to the normal form (25.15). In both cases [dB(r) A wo]lo = 0 . Recall, that the Pfaffian equation two = 0) and the germ B(r) are defined invariantly, i.e., {wo = 01 is the restriction of e on D(e) (and is a contact

structure on D(e) ); the germ B(r) in the coordinates of the normal form (25.14) or (25.15) coincides with the germ µe transforming D(e) to (0, 1) or to (0, oc) (see §25.3). Thus, the problem of the classification of germs belonging to PW" Re' 1 or PW" Im' I includes the problem of the classification of function-germs f: D R under the action of the group of contactomorphisms (contact space (D(e), w0)) . A pair (f, wo) in this problem is not generic, i.e., there is a singularity (df A wo) 10 = 0. We shall indicate functional moduli in the classification of such singularities. I

)To be exact, we mean that for an arbitrary germ e from the pointed singularity classes and for an arbitrary a < 00 there exists a germ e` such that j'e = j 'F but jo(eIe * jD'(e)e . (

V. CLASSIFICATION OF GERMS OF EVEN-DIMENSIONAL PFAFFIAN EQUATIONS

138

Denote by P the set of function-germs f : D --, R' such that (df ncoo) 10 =

0. Assume that two generic germs f, g E P are equivalent in the contact space (D, wo). Then the hypersurfaces Fe : {rl f (r) = f(O) + e} and Ge: {rig(r) = g(O) + e} are equivalent for small a as well. For each small c there exists a unique point a(e) E Fe such that T (t)Fe = KerwoJ.(e) . Analogously, for each small c there exists a unique point fl(e) E Ge such that Therefore, for each small c the germ of Fe at the Tfl(e)Ge = Ker w0 point a(e) is equivalent to the germ of Ge at the point f(e) . But for each B(e).

e the equivalence of such germs of hypersurfaces in a contact space implies the equality of [(dim D)/2] moduli: A, (FF)

= ) (Ge) , ...

,

A[dimD/2](Fe) - A[dimD/21(Ge)

(see [L1] and Appendix A). These moduli depend smoothly on e. Thus, we have pointed out [(n - 3)/2] functional moduli (functions in one variable) in the problem of the classification of germs from PWn Re (1m)

27.2. Singularity lass PW" 3 . First we prove that the 4 jet on the first degeneration manifold of a generic germ e E PW" 3 is reducible to a stable normal form. THEOREM 27.3. For a generic germ e E PW" i3 one can choose coordinates such that 2

iDe= {wo+ 2x2+x,v du+(2x,u+uv)dv=0}. (

(27.1)

PROOF. As in the proof of Theorem 25.5 we can choose coordinates such that jDe: {wo = 0}. D(e): {x, = u = v = 01, Consider the field jDXe . We can assume that its eigenvalues at 0 are -2, 1, 1.

It is not hard to show that by a linear transformation (x, , u, v)'

T(x, , u, v)' with a suitable matrix T = T(r) the jet jDXe can be reduced to the normal form jDXe = K(r)

{2u-+ (x, + v)+ (B(r)x, + v)]

,

(27.2)

where B(r) and K(r) are some function-germs. At the same time, arguing as in the proof of Theorem 25.5 we can obtain the normalization

{wo+fdu+gdv = 01,

f, g E 9RD),

(27.3)

of the 2 -jet on D(e) by a transformation preserving (27.2). In the coordinates of the normal form (27.3) of a ag 0 ag of a (27.4) i X, _ (OU av) ax, - ax, au + ax, av'

§27. DEGENERATIONS OF CODIMENSION > 4

139

Relations (27.2) and (27.4) imply that

f (_)-2+xv)K(r)+W(uv) g = (2xlu + uv)K(r) + W2(u, v) , where W1 , v2 E M`D' and OW1 /8v - 8v2/8u - 0. Therefore, we obtain the normal form of the 2 -jet on D(e) : ll

je:

{wo+K(r) B(2x1 +x1v du+K(r)[2xlu+uv]dv+dµ=01 ,

. The transformation y1 where z E qn(3)) D(e

y1 - z "kills" the term d U and

we obtain the normal form

je:

{wo+K(r)

[Bxi

11

+xiv] du+K(r)[2xlu+uv]dv=0

.

(27.5)

Note that K(O) yl 0. We can find a transformation r - c(r) such that 1"(w0/K(r)) = we . Therefore, the normal form (27.5) can be reduced to

jDe. {ao+

[62(r)

xz+xlv1 du+[2x1u+uv]dv =0}.

(27.6)

For a generic germ e E PWln 13 we have B(0) =0, and dB n w010 By Theorem 7.7 we can reduce the normal form (27.6) to (

0.

z

jDe: {Hw0+Lx122+xiv

du+[2xlu+uv]dv=0

,

where H(O) 0 0. Now, to get the final normalization (27.1) it remains to fulfil the transformation x1 _ X I

u

v

u ''H _(r) ,

v ''H _(r).

Q.E.D.

Now we are to prove that the 4 -jet on D(e) has the same normalization (27.1). To prove this statement, we can use the homotopy method (Corollary 5.2). The arguments are similar to those used in the proof of Theorem 26.5 We use the absence of resonances (27.7)

for 4<m1+mz+m3<5. Theorem 27.1 now follows from Proposition 6.2 and the following state-

ment. Let {w = 0} be the normal form (27.1)

.

140

V. CLASSIFICATION OF GERMS OF EVEN-DIMENSIONAL PFAFFIAN EQUATIONS

THEOREM 27.4. For an arbitrary germ w + r r c MD and an arbitrary s < oe there exists a function-germ a e 9RO such that jsa = 0 and the ,

equation

jD[X,dw+d(X,w) +hw] = adv, with respect to the unknown vector field X and function-germ h, is unsolvable.

The proof of this theorem is similar to that of Theorem 26.7 and we omit it. Notice only that the proof is based on the existence of resonances (27.7)

oforder6(m1 =2, m2+m3=4). The proof of Theorem 27.1 for the singularity class PW"13 is complete.

Theorem 27.1 for the case of the singularity classes PW", f o and PWR.Im,0 follows from Proposition 6.2 and the following statement. 27.3. Singularity classes PW". Re

o

and P W n, Im, o

THEOREM 27.5. For a generic germ e: {w = 0} E PW" Re3 o or e: (w = 0} E P WR,Im. and an arbitrary s < oo there exists a function-germ f E 92D such that js f = 0 and the equation

jD[X,dw+d(X,w)+hw]= fdv, with respect to the unknown vector field X andfunction-germ h, is unsolvable.

We omit the proof of Theorem 27.5 (it is similar to that of Theorem 26.7 and rather cumbersome). As in the proof of Theorem 26.7 we need some germ normalization. The following statement will help us conduct calculations necessary for the proof of Theorem 27.5. THEOREM 27.6. For a generic germ e E PW"x2,o or e E PW" Im,o there exist coordinates such that

e: {wo+fdu+gdv=0},

jDf= jDg=0,

D: {x1 = u = x2 = 0}.

This theorem can be proved by arguments similar to those used in the proof of Theorem 20.4. Theorem 27.2 is a corollary of Theorem 27.1, because singularity classes P W° 030 and PWJ(j < n - 3) are adjacent to the singularity class PW," It,, o (Theorem 25.8) and we can use Proposition 6.3. §28. Normal forms of Pfaffian equations in R4 In this section we give a complete classification of first occurring singularities of Pfaffian equations in i . We use x , u, v , y as coordinates in R4

and denote by e a germ {w = 0} a PW41 . We assume that the manifold D = D(e) is given by the equations {x = u = V = 0). We also need the

§28. NORMAL FORMS OF PFAFFIAN EQUATIONS IN R4

141

notation

wlm,z= dy+[x(2)+y)-xv]du+[xu+xv(2A+y)]dv, A>0, WRe,z= dy+[(1+)+y)xv]du+xudv, AE(0, 1), (Im ,1,?. = WIm.d +

n(y)X2uvz

2

WRe,A,q = WRC,A

uv

2

du, du,

where q = q(y) is a germ of a function in one variable. THEOREM 28.1. Each germ e E PW41 im is equivalent to the normal form {wIm 1 q = 0}. THEOREM 28.2. Let r < oc be an arbitrary number, e E P W41,Re , al(e) E R \ Q . Then the germ e is C'-equivalent to the normal form (o)Re z

R

= 0}.

THEOREM 28.3. The function-germ q(y) is the invariant of a germ e c In other words, two normal forms { ° (Im) z q = 0} and {wRe(Im), i,- = 0} are equivalent if and only if A = 2 and n - jt is a germ of PW4,'Re U P W4''Im .

the zero function.

So, the normal forms {wRe(Im),l,q = 0} are invariant. COROLLARY 28.1. There are no stable and finitely determined germs of Pfafan equations in R4 except those equivalent to the quasicontact structure

{dy+xdv =0}. Proofs of Theorems 28.1 and 28.2 are divided into several parts. Below we assume that if e E P W41'R°e , then µ(e) E R \ Q .

28.1. Normal form of a 2 -jet on D(e). 1,0 THEOREM 28.4. A 2 -jet on D(e) of a germ e E PW4 00 E PW4.im respectively) is equivalent to the jet {wRe z = 0} ((W1m ,1 = 0} respectively), where A = µ(e) .

PROOF. By Theorem 25.6 each germ e E PW41 Re (e E PW4 I. respectively) has the following 2 -jet on D(e) in suitable coordinates:

{dy + B(y)xv du +xudv = 0} ({dy + (xuB(y) - xlv) du + (xu + xvB(y)) dv = 0} resp.). Condition e E PW41 Rllm, implies that B'(0) 0 0, therefore, we can obtain the normal forms

jdy+(1+)+y)xvdu+xudv l

H(y)

-0

{dy+(xu(2).+y)-xv)du+(xu+xv(22+y)) H(y)

( 28.1 )

=o}

,

(28.2)

V. CLASSIFICATION OF GERMS OF EVEN-DIMENSIONAL PFAFFIAN EQUATIONS

142

where B(O) = 1 +A for the normal form (28.1) and B(0) = 2,k for the normal

form (28.2), H(0) # 0. Arguing as in the proof of Theorem 25.7 one can show that it is possible to meet the condition A E (0, 1) for the normal form (28.1) and A > 0 for (28.2). Then 2. = µ(e) . The transformation H(y)x reduces (28.1) and (28.2) to {WRe 1 = 0} and {wim = 0} x respectively.

Q.E.D.

28.2. Normal form of a 4-jet on D(e) .

THEOREM 28.5. A 4 jet on D(e) of a germ e E PW4 It. (e E PW41, Im respectively) is equivalent to the jet {wRe 4 = O} ({WIm.A = 01 respectively), where A = µ(e).

PROOF. Let I = 3 or 1 = 4, and f, g be arbitrary germs belonging to 911D)

. By Theorem 28.4 it suffices to show that an 1 jet on D of the form

(wR. 4+fdu+gdv=0)

({wlm z+fdu+gdv=0} resp.)

is equivalent to the jet {wRe,z = 01 ({WIm x = 0} respectively). To prove this fact we use Corollary 5.2 and argue as in the proof of Theorem 26.5. The corresponding equations are solvable because there are no resonances

-µ(e))=0

11

for 11+12+13=1+1. Q.E.D. 28.3. Normal form of a 5-jet on D(e) . THEOREM 28.6. A 5 -jet on D(e) of a germ e E P W41 xe (e E PW41 'Im respectively) is equivalent to the jet {wRe 2 = 01 ({WIm,A. = 0} respectively)

for some function-germ q = q(y). At first we shall prove a preliminary statement.

THEOREM 28.7. For arbitrary germs f, g c 9fD) there exists a function-

germ q(y) such that a 5 -jet on D of the form {WRe,4 + fdu + gdv + i(y)x2uv2 du = 01 is equivalent to the 5 -jet {wRe 2 = 0} on D. PROOF OF THEOREM 28.7. Let co, = wRe.4+t[f du+g dv+rl(y)x2uv2 du]. By Corollary 5.2 it suffices to prove the solvability of the equation

jD[X,dwt+d(X,,w,)+h,wt]= (f+q(y)x2uv2)du-gdv,

tE[0, 1], (28.3)

with respect to the vector field X, E TID and function-germs q(y) and h1, t E [0, 1] . We seek a solution of equation (28.3) in the form (ht , X,) E 9%'D Let 8

X =Y]t ax

+Y2t au j

+7st ay

8 +Ytaay

(28.4)

§28. NORMAL FORMS OF PFAFFIAN EQUATIONS IN R4

143

Equation (28.3) can be rewritten as a system for function-germs 7(y) , h1, and R1=Xr ,cor:

D[a,+h,]=0, .5 [BR, rBR ,

ax

5

Y,r ,

5

1 DR, = JD741

- (1 +A+ Y)V72, -

= 0,

UY31]

jD aU1 + (2 + y)xy31 + (1 + 11 + y)vy11] _ -f - 7(Y)x2 UV 2, L

7D

[

a

+ y)xy21 + UY11]

-g

After the elimination of y4, and h1 we obtain the following system for y11, Y211 y31, R1, and 7(Y)

is

0

-[1 +)+ y]v

[1 +A+y]v

0

u

- (. + y) x

-u

yI' Y21

0

(731

OR, Ox

- OR - f - 7(Y)x2Uv2 ORI

av

(28.5)

g

By Lemma 26.2 system (28.5) can be reduced to the equation

[U 7D

our +

('t +Y)x aR1 - (1 + 1l +Y)v a = t + 7(Y)x2U2v2 , ]

(28.6)

for R, and 7(y) , where r is a germ form 9RD) . We can choose 7(y) _ -6, (aT/ax2au2av2)I

, then equation (28.6) has a solution R1 = R E

moo)

(see §26.2). Q.E.D. PROOF of THEOREM 28.6 (for the case e E PW41x ). By Lemma 25.1 we can assume that a 5 jet on D is of the form { cWRC A + f d u + g d v = 01 , f, g E 9RD) . By Theorem 28.7 there exists a germ 7(y) such that

(f+ 7(y)x2UV2) du + g dv = 0} = .i {o Rf d = 0}

,

where b is a diffeomorphism, and, moreover, it is clear from the proof of Theorem 28.7 that = id+ (terms belonging to T14).

,rya

v. CLASSIFICATION OF GERMS OF EVEN-DIMENSIONAL PFAFFIAN EQUATIONS

Therefore,

j5J'{wRc A+fdu+gdv= 0}= j5{wRc I-17(y)x2uv2du=0}. Q.E.D.

Theorem 28.7 holds also for germs belonging to PWa''Im (we can replace

in its statement). As above, this theorem implies Theorem 28.6 for the case e E Im WRc.A by W1m.Z

28.4. Normal form of 1 -jet on D(e) (l > 6).

THEOREM 28.8. Let l > 6, 196 3m - 1. f, g E 9XD' . Then the l jet on D(e) {wRe,z,,r + fdu + gdv = 0} is equivalent to the !jet {wRc A a = 0) on D(e) and the 1 -jet {wlm.x, a + f du + g dv = 01 on D(e) is equivalent to [Co, i q = 0} .

PROoF. Let wr = io+t(fdu+gdv), t E [0, 1], where WE = wReIA,R or U1 = wIm,I equation

7.

By Corollary 5.2 it suffices to prove the solvability of the

JD[XX.dwr+d(X,,wr)+hew,)=-fdu-gdv

(28.7)

with respect to the unknown vector field Xr of form (28.4) and the functiongerm hr . We seek a solution satisfying the condition (h,, X,) E 9JrD' . Arguing as in the proof of Theorem 28.7 we reduce (28.7) to an equation for

uaRrl

Rr = Xr co, . For the case iw = wRe,r it looks like 3D

where r is a germ form

9J4D+1) .

= r,

(28.8)

Since 1 + 1 0 3m, equation (28.8) has a

solution R, = R E 9J1D+') (see §26.2). For the case w = wIm ,r we obtain a similar equation for R, which is also solvable. Q.E.D.

THEOREM 28.9. Theorem 28.8 holds for 1 = 3m - 1 > 6. PROOF. We shall consider only the germs from Pw," Re . The case e E PW4' I,°n can be treated in a similar fashion. For the case 1 = 3m - 1 > 6 we seek a solution (28.7) satisfying the condition (h,, Xr) E 9JZ a . Then equation (28.7) can be rewritten as a

S28. NORMAL FORMS OF PFAFFIAN EQUATIONS IN R'

h, and y :

system for R, = Xr JD

I

49R`

OR

145

xvY21 + hr]

= 0'

`-[(1+A+y)v+27(Y)xuv21'2I-uy3r] =0,

R )D I aut + [(1 + A +y)v + 27(y)xuv2]y1,

+[(A+y)x+27(y)x2uv]y2,+xvy41+h,(1 +A+y)xv+f] =o,

RI + h,xu + g + U71, - (A + Y)xY21 - 21(y)x2uv721]

jD

0,

L

R, = Y41 + (1 +k + y)xvy2, + xuy31.

After the elimination of y4t and h, we obtain a system for y,,, y2, and R, of form (26.7), where =

z 1

z3

_

-a-`

/3,

= -aR` z -xvR ' +xvaR`(1 +A+y), ax 2 au ay OR aR' f1 = -(1 +A+y)v - 2g(y)xuv, 2 avr +xu -Y , (2g(y) - 1)x2uv. f3=(A f 2 = -u, ,

By Lemma 26.2 this system can be reduced to the equation {uaur+(A+Y)xax'-(1+A+y)va'`+xuvR,

'D1

OR + 2q(y)x 2U2v20R r +27(Y)xuv2aa' + (27(Y) - 1)x 2uv ax' aY

= z, (28.9)

9R(1+1). We for R, , where r E seek a solution in the form R, = y(y)(xuv)"i-1 1) + R, R E 9 4 . Then (28.9) can be rewritten as

u OR + (A +Y)xax-R

Let z = tion

1,+r2+1,=r+1 1711,121 r, (y)x1, u12v'3 -rm,m,m

Y(Y)

(Y) =

m-2

.

2)(xuv)m.

(28.10)

Since m > 3, (28.10) has solu-

R=

zl 12 !3 (Y) 11(A +Y) +'12-13(1

R'.m,m(Y) = 0. Q.E.D.

- (1 +A + y)v OR = r + Y(y)(m -

av

Ou

R r r 2r (Y)xlu12v 1,+12+1,=1+1

+A+Y)

for (11, 12,13) j4(MI M' M)'

146

V. CLASSIFICATION OF GERMS OF EVEN-DIMENSIONAL PFAFFIAN EQUATIONS

28.5. Proof of Theorems 28.1 and 28.2. We will use the obtained normal-

ization of an r -jet on D for an arbitrary r < oc . Arguments showing the possibility of passing to C°°- or C'-normal forms are quite like those for n > 6 (see §26.3). Arguments for e E PW41 ' o and e E PW41 im are similar. We will prove only the first case. Assume that the germs e 1 : (aiRe 2 8, = 0} and e2: (WRe A 92 = 0} are

28.6. Proof of Theorem 28.3.

equivalent. Then the fields X;, and Xe are orbitally equivalent by Theorem 25.1. Let us calculate these fields in the coordinates of our normal forms, i.e., a49

Xe. = [('1+y)x + (2g1(y) - 1)x2uv - g;(Y)X3u2v21 x +g;(y)x2u2V2a

+[-(1+.1+y)v-2g1(y)Xuv2] a +uaua av

ay

The substitution y -, y - xuvg1(y) transforms these fields to 2

a

Xe, = [(2+y)x+(g1(Y)-1)x uvlax+[-(l+)+y)v-g(y)xuv

2 a

a ]av+uau+r,

(28.11)

where jDT=O. We can prove that if D,X = HX , H(O)

0, then there exists a diffeo-

morphism CD of the form 00

00

x -ix>p,(y)(xuv)',

u-' u1: n1(y)(xuv)', ;=o

=o

00

00

v - v E B1(y)(xuv)' ,

y -'

;(y)(xuv)' ;=o

=o

and a function-germ H of the form 00

H = >h1(y)(xuv)', ;=0

such that JD6'Xe, = J.6 (i.e., we state that it is possible to choose a resonance pair (H, (D) transforming Xe to Xe ; a close statement is proved in [Br]).

Relation jDDj.Y = jD (HXe=) implies that b0(y) = y, ho(y) = 1. Consider now the relation jDdy,X = jD(HX,) . We can show that it implies g1(y) - 1

h1(Y) = 0,

R(y)

where R(y) _ p0(y)n0(y)00(y) g2(y). Q.E.D.

gI(Y)

= g2(y) - 1,

= 92(Y),

(28.12)

R(y) It follows easily from (28.12) that g1(y)

§30. BASIC RESULTS AND COROLLARIES

147

§29. Point singularities THEOREM 29.1. A generic germ e E PWZk is unstable, 1-determined, and reducible to the invariant normal form {w = 0) where to is of form (13.1) .

Thus, the classification of germs of Pfaffian equations {w = 0) E PW0 is the same as for the 1-forms w E W k (§13). Conditions of general position in Theorem 29.1 coincide with those for a germ of a 1-form: a germ e: {w = 0} E PW k is generic if CO E W? (see § 13). PROOF OF THEOREM 29.1. It suffices to prove that the equivalence of the

Pfaffian equations {w, = 0} and {w2 = 0} , where wI and w2 have form (13.1), implies the equivalence of 1-jets of the 1-forms co, and w2.

Assume that c'w1 = Hw2, H(0) 0 0. Then j'(Wwi = H(0)w2. Since the 1-forms H(0)w2 and w2 are equivalent, there exists a diffeomorphism IF such that jl `Y'wI = w2 . Q.E.D. §30. Basic results and corollaries; table of singularities; list of normal forms (1), (2), ... refer to the normal forms in Table 30.2 (see page 150). THEOREM 30.1. Let k > 3. 1. Any stable germ of a Pfaffian equation on a 2k-manifold is equivalent

either to the normal form (1) or to the normal form (3). 2. Let r < oo. Any stable (relative to C'-equivalence) germ of a Pfaffian equation on a 2k-manifold is equivalent to one and only one of the normal

forms (1), (2), (3). 3. Any stable germ of a Pfaffian equation on a 4-manifold is equivalent to the standard quasicontact structure {dx + u dv = 01. THEOREM 30.2. Any unstable but finitely determined germ ofa generic Pfaf-

fran equation on a 2k-manifold (k > 2) is equivalent to the normal form (10).

_

THEOREM 30.3. Let k > 3, M be a 2k-manifold. 1. For a generic Pfaffian equation E on M there exists a subset M C M of codimension > 4 such that a germ of E at each point a E M\M is stable (relative to C'-equivalence for any r < oc ). 2. There exists an open set _A C PA' (M) such that for any Pfafflan equation

E E A there exists a set M C M of codimension 4 with the following property: the germ of E at any point a E M is not finitely determined (relative to C3-equivalence). 3. Let S be a set of points at which the germs ofa generic Pfaffian equation on M are unstable but finitely determined. Then S consists of isolated points.

THEOREM 30.4. For a generic Pfaffian equation E on a 4-manifold the set of points at which a germ of E is unstable has codimension > 3. There exists an open set A C PA' (M) (dim M = 4) such that for any Pfaffian equation

V. CLASSIFICATION OF GERMS OF EVEN-DIMENSIONAL PFAFFIAN EQUATIONS

148

E E A there exists a curve with the following property: the germ of E at each point of this curve is not finitely determined. Notation in the table of singulslrity classes. (See Table 30.1.) codim I

codimension of a singularity class; minimum jet order which enables to determine the given singularity class;

p

index of finite determinacy for the germs from the given class (p = 00 when there is no sufficient jet);

m number of moduli that distinguish close nonequivalent germs from the given class; st

sign of stability: + (the germs of the given class are stable), - (the

*

germs of the given class are unstable); takes place only relative to C'-equivalence for arbitrary r < 00 and

for an irrational value of the invariant ) - u(e) ; **

***

takes place only for generic germs of the given singularity class; preliminary (not invariant) normal form.

Comments on the list of normal forms. Coordinates :

x] YI , ... , xk_I , y,r_I , u, v in (2)-(6) ;

XI,Yl,... IXk,Ykin(1), (7), (10); x, y, u, v in (8), (9). A E (0, 1) in (2), (8) ; A, E (0, 00) in (3), (9) ; p J C C (generally speaking), Re yj > 0 in (10). Functional modulus : ,7(y) in (8), (9). Functional parameters :B = B(y1 , x2 , y,, ... , xk-I , Yk-1) in (5) ; Numerical moduli :

fj=fj(x1,Y1, - ,Xk,Yk), gJ=gj(x1,y],... ,xk,yk) in(7); T is a germ of a 1 - form from 0 (from MD respectively), where D is the first degeneration manifold {x] = u = v = 0} in (5), (6) (in (4) respectively).

§31. Commentary §24. The proof of the Darboux theorem (the equivalence of a generic germ of a 2k-dimensional Pfaffian equation to the quasicontact structure {d y] + x2 d y2 + + Xk dyk = 0}) can be realized in various ways (see, for

example, [Stl], [MorRol).

§31. COMMENTARY

149

TABLE 30.1. TYPICAL SINGULARITY CLASSES

Singularity

codim

l

p

m

St

class

Normal

Remarks

form

PW2k-1 PWZk-3,o

e 2k , R.' PW2k-3,0

2,lm PWZk-3

2k,1

PWZk-3, 1

2k,Re

0

1

1

0

+

(1)

3

3

3

1

+

(2)

k> 3, *

3

3

3

1

+

(3)

k> 3

4

2

o0

oo

-

(4)

k > 3: 2k - 1

3

o0

00

-

(5)

k=2:4 PW2k-3,1

2k,lm

fork> 3

-

k > 3: 2k - 1

3

00

00

-

k=2:4 PW2k-3

(6)

***

fork 2: 3

2k, 0, Re

4

2

00

00

PWk,031m

4

2

oc

oo

PWZk+1

(2k-2s-1)

1

00

00

s
x(k-s-1)

-

see §27.3 see §27.3

(7)

k>3, codim> 10, codim < 2k, ***

PW`,Re

3

3

00

00

PW41,Im

3

3

00

00

-

PW k

2k

0

1

k

-

(8)

*

(9) (10)

§26. The results of §26 were announced in [Z9], where there is an inaccuracy: a germ from PW k ,Im'0 is stable relative to C°°-equivalence (not only

relative to C-equivalence for an arbitrary r < oc ). For a germ e E PWZk Rep ° , k > 3, the field Xe in suitable coordinates ,

looks like

a Xlaxl

a a +(2+y,)uau -(1+A+yl)vaa +x,uvay2.

PW2k-3'0 A similar normal form can be distinguished for the case e E In 2k,lrn both cases the field Xe is divergence-free (the fact that Xe is divergence-free, .

V. CLASSIFICATION OF GERMS OF EVEN-DIMENSIONAL. PFAFFIAN EQUATIONS

150

Table 30.2. List of normal forms No

Normal form

(1)

dy,+x2dy2+...+xkdyk =0

+xlv(1+A+x2)du+x,udv = 0

(2)

dy,

(3)

dyL + x2 dy2 + ... + xk-, dyk-1

+[xlu(2A+x2) -x,v]du+[x,u +xlv(2A+x2)]dv = 0 l!

l du

(4)

+(2xlu+uv)dv+r=0 (5)

111//

\

Bx,v du+x1udv+r=0,

dy1+x2 dBI0 A dyL = 0

(6)

+(x1u+Bx,v)dv+r=0, dBJ0Ady1=0 (7)

dy,+x2dy2+. +x,+Idys+L+ i S+2(fJdx) +g1dy!)=0

(8)

dy+(1 +A+y)xvdu+xudv+jf(y)x2uv2du=0

(9)

dy + [xu(2A + y) - xv] du + [xu + xv(2A + y)] dv +17(y)x2uv2du = 0

(10)

does not follow from the definition of this field). §27. Conjecture: two germs e1 , e2 are equivalent if and only if the fields Xe and Xe are orbitally equivalent (for arbitrary singularities). This holds true for the first occurring singularities. Supposedly, the following statement holds: the 2 -jet on the first degeneration manifold uniquely defines the C'-orbit of a germ e E PK'Zk,RHIm) (for an arbitrary r < oo ). §28.

Singularities of Pfaffian equations in R4 were also considered by

Mormul. It is shown in [Mor3] that the field X{w-o} has three nonzero

eigenvalues if and only if Kerw is transverse to the curve D (the first degeneration manifold). In this work the invariant of the 1 -jet of the field X1,01 is defined by the relation 3/2

A = (AIA2 + AIA3 +.Y3)

1AI 112A31

§31. COMMENTARY

151

where Al , A2, and 2.3 are nonzero eigenvalues of X{w-0) Normal forms for the first occurring singularities were announced in [Z9]. §29.

Theorem 29.1 was proved by Lychagin [Li].

APPENDIX A

Local Classification of First-Order Partial Differential Equations 1. In this appendix we give some results of [L1] where first-order partial differential equations are considered as hypersurfaces in the contact space of 1 -jets. These results can be obtained as corollaries of the classification of 1-forms and Pfaffian equations.

2. A germ of a first-order partial differential equation 8u F XI,--- ,xn,uax ,... ii8u n 1

is treated as a germ of the hypersurface E: {F(x1, ... , xn , u , y1 , ... , yn) _

0} in the contact space (12I, w), where w = du - E y; dxi. Two germs E1 and E2 are said to be equivalent if there exists a germ of a contactomorphism sending E1 to E2. By Theorem 7.1 they are equivalent if and only if the germs {wI,,, = 0} E PW(2n) and {wJEZ = 0} E PW(2n) are equivalent. R2n+1

3. Assume that the plane of w at 0 E R2n+1, w), i.e., dF A w 0 . Then c

is transverse to an equation 01 = 2n - 1 and the germ {wI E = 0} is equivalent to the germ du - y2 dx2 - y3 d x3 - - - . yn dxn (in the coordinates u, xi , x2 , Y21 x3 , Y31 ... , xn , yn ). Therefore, by Theorem 7.1 the germ E is reducible to the normal form

E

y1 = 0

Cl w

or az = 0.

(A.1)

1

4. Singularities in the problem occur at points where dF A Co = 0, or, equivalently, where the restriction iw = wlE vanishes. The germ {ui = 0} at other points satisfies the Darboux condition w A (dw)n-1 96 0. Let 610 = 0. Then the germ of iw at 0 can be an arbitrary germ of a 1form belonging to W0 and satisfying the condition rank d& 0 = 2n. Thus, the classification of singularities of first-order partial differential equations

is reduced to the classification of the germs {w = 0} E PW , such that rankdwlo = 2n . The following theorem is a corollary of the results in §13, and §29. 153

APPENDIX A. LOCAL CLASSIFICATION

154

THEOREM A.1. A generic germ at 0 E R2n+1 of a first-order partial differ-

ential equation F (XI , ... , xn , u, au/ax1 , ... , au/ax,) = 0 satisfying the condition dFIo n du = 0 is reducible to the form

u-

).;xiyi = 0

u

au

1 E ax; = Aix u

(A.2)

i

(generally speaking, in complex coordinates, see § 13). Here Ai are moduli('), i.e., two different normal forms (A.2) are not equivalent.

Notice that A. = 1/2 + pi (see Corollary 13.1). Since codim PW, 0 = 2n, we can formulate the following result. THEOREM A.2. A germ of a generic first-order partial differential equation

on a manifold M (considered as a hypersurface in the set of 1 jets on M) is either stable (at generic points) and equivalent to the normal form (A.1) or unstable but 2-determined (at isolated points) and equivalent to the normal form (A.2).

The case of nongeneric germs from W° and nontypical singularities of first-order partial differential equations is considered in [Z1, Z5, Z6].

(I )appearing in the classification of the Pfaffian equations {io = 0).

APPENDIX B

Classification of Submanifolds of a Contact Manifold Appendix A is concerned with the classification of hypersurfaces in a contact space. In this appendix we consider submanifolds of codimension > 2. The notion of equivalence of submanifolds of a contact manifold is given in §7.

The following theorem shows that singularities from PW p i' and are realized as restrictions of a contact structure on submanifolds in a contact PWZp-3

space.

THEOREM B.1. For an arbitrary germ e E PWZD ,' (e E PW pp -3) and an arbitrary contact 1 -form w E W(2n + 1), n > p, there exists a manifold E C 1[i2n+l such that the germ of the restriction {w = 0}IE is equivalent to the germ e. PROOF. We can assume that w = d z + x1 d y1 + + xn d yn . By Theorem 16.3 the germ e c PWZp+i' is equivalent to the germ

j'f=0

e1:

(in the coordinates z, x1 , Y,, ... v , yp ). By Theorem 24.1 the germ e E is equivalent to the germ PWZp-3

j'g1 =0

e2:

(coordinates z, x1 , Y1 , ... , xp_ 1 , yp_ 1 , yp ).

Let E1 be a manifold given by the equations {xp+2 = yp+2 =

{

= xn = Yn = 0, xp+1 = J - xp , Yp+1 = Yp}

and let E2 be a manifold given by

{xp+2=yp+2=...=xn =yn =0,xp+1 =g1 -xp_1,Xp =g2,Yp+1 Then {WIE. = 0} = e1 , {WIE, = 0} = e2.

=Yp-1}.

Q.E.D.

The following statements are corollaries of Theorem B. 1, Theorem 7.1, and classification results in Chapters IV and V. 155

APPENDIX B. CLASSIFICATION OF SUBMANIWWLDS

156

THEOREM B.2. Let l > 2, n > 1. The germ of a generic (21 + 1) -dimensional submanifold E of the contact manifold (R2n+1 , dz + x1 dyl + - + x,, dyn)

at a generic point a E E is stable and equivalent to the germ of one of the following manifolds: (1) {x1+1 = YI+1 = ... = xn = Y =

0); 2

(2) {x1+2 = Yt+2 = ... = X. = Yn =0' x1+1 = xt - x1, Y1+1 =Y1 }

x,+l=xtyl+xIY,-x1,Yt+1=Yt}; (3) {x1+2=YI+2= (4) {x1+2 =,+2 = ... = xn = yn = 0. x1+1 = x1y1+X, /3+xry1 -x1, Yt+1 =

yt} .

Nongeneric points form a set of codimension > 4 in E. THEOREM B.3. Let 1 > 3, n > 1. The germ of a generic 2p-dimensional submanifold of the contact manifold (R2n+1 , d z + x1 dy1 + - - . + x,, dy,,) at a generic point of this submanifold is stable (relative to C'-equivalence for an arbitrary r < oo) and equivalent to the germ of one of the following manifolds : `1) {Xp = Xp+l = Yp+l = = Xn = Yn = 01; {x,+2 = YP+2 = .. = X. = Yn = 0. (2) xP+1 = xP-lYp - XP-1, Yp+l =Yp_1 . XP = XP-1YP-I(1 +A+X1)}

(A E (0, 1));

(3) {X,,+2 = Yp+2 = ... = Xn = Y. = 01 _1(21.+x1)-xp_1. xP+l =xp-1Yp+xp-1y x.=xP-IYp(2.k +x1)-xp_1Yp_I,Yp+I-Yp-1} Nongeneric points form a set of codimension > 4.

(AE(0,00)).

We can formulate other corollaries, for example, the normal forms of 3and 4-dimensional submanifolds of a contact manifold can be distinguished.

APPENDIX C

Feedback Equivalence of Control Systems Consider a system of ordinary differential equations k

x = F(x) +

(C.1)

u1 (x)G1(x) ,

where x , F(x) , G; (x) E R" , uI , ... , uk are scalar functions (controls) the choice of which must ensure some properties of system (C. 1). When studying qualitative local properties of a control system it is expedient to replace it by an equivalent simpler system. The introduction of new controls vl , ... , Vk , i.e., k

u; = fle(x) +

h,1(x)v1

i = 1, ... , k,

,

det(hij) A 0,

(C.2)

i=1 and new coordinates

e(0) # 0,

(C.3)

3' = F(Y)+>v;(Y)G,(Y)

(C.4)

x = ma(y),

leads to another system k

that is also linear with respect to controls. Control systems (C. 1) and (C.4) are said to be feedback equivalent if there exist transformations of the form (C.2), (C.3) sending (C. 1) to (C.4). This definition can be simplified by passing to invariant terms. A control system of form (C. 1) defines an affine module of vector fields k

V=

if+uigi, uj E C00(n)

,

(C.5)

where the field f corresponds to the system x = F(x) and the fields g, correspond to the systems x = G;(x), i = 1 , ... , k . THEOREM C. I [J]. Two control systems of the form (C. 1) are locally feedback equivalent if and only if the corresponding q,Jine modules of vector fields 157

APPENDIX C. FEEDBACK EQUIVALENCE OF CONTROL SYSTEMS

158

are equivalent, i.e., there exists a diffeomorphism sending each vector field of the first module into a vector field of the second one.

Using this theorem we can apply classification results for differential 1forms to control systems of the form (C. 1) for the case k = n - 1 . Assume that k = n - 1 and a system of form (C. 1) is nondegenerate, i.e.,

dim(F(x), G1(x), ... , Gi_1(x)) = n,

x E R' .

An affine module (C.5) corresponding to such system can be defined as

V = {v C Vect(n)Iw(v) - 1},

where w is a differential 1-form such that co(f) _-

, ... , n -

1

,

w(gi) - 0, i =

There exists a unique 1-form cv satisfying these properties. Now, using Theorem (C. 1) and results of Chapter III we obtain the following 1

1.

result.

THEOREM C.2. Let k = n - 1 . A germ of a nondegenerate control system (C.1) at a generic point x E lll" is stable and feedback equivalent to one of the germs

{Xi=ui,yi=ui(i=1, ...,k),2

1-XI U1-

Xkuk}

(n = 2k + 1);

{xj =u,,yi=(I+x1)ui(j=1,... ,k,i=2,... ,k), 2=(1+x1)u, y1 =

1/(l+x1)-X2u2-...-Xkuk±ZU}

(n=2k+ 1);

=(1+x1)ui (j=1,... ,k,i=2,... ,k), y1 = I/(1 + x1) - x2U2 - .

- xkuk} (n = 2k);

{zi=ui (i= 1,...

(j=2,...k), y1 =

1/(I±x1)-x2u2-...-xkuk)

(n = 2k);

where u, ui ui are functional control parameters (functions in n variables). ,

Nongeneric points form a set of codimension 2. Each stable (with respect to the feedback equivalence) germ of a nondegenerate control system is feedback equivalent to one of the germs given above.

APPENDIX D

Analytic Classification of Differential Forms and Pfaffian Equations 1. In this appendix we consider germs of real-analytic and holomorphic 1-forms and Pfaffian equations, i.e., germs of the form

w=Eai(x)dxt,

e:

{a1'x)dxi =0},

where the a, (x) are germs of functions in n variables x = (x1) ...

, xR)

that are real-analytic at 0 (of functions in n complex variables, holomorphic at 0). Two analytic (holomorphic) germs of 1-forms are said to be equivalent if there exists a germ at 0 of an analytic (holomorphic) diffeomorphism, sending one germ into another. Two analytic (holomorphic) germs of Pfaffian equations {w1 = 0} , {w2 = 0} are said to be equivalent if there exists a germ H at 0 of an analytic (holomorphic) function such that the forms Hm1 and w2 are equivalent. For the analytic case all the singularity classes considered in Chapters PW2k-''1'h III-V are defined. For the holomorphic case the classes and 2k+I coincide. 2k+ 2. The Darboux and Martinet analytic classifications coincide with the smooth classification. We mean that Theorems 12.1, 12.2, 17.2, and Darboux theorems (§§10, 16, 24) hold also in the analytic category. The holomorphic classification is the same, but the normal form w+ is equivalent to the normal

form w- (see Theorems 12.1 and 12.2). The results on the preliminary normal forms (Theorems 10.2, 16.3 and 24.1) hold for analytic and holomorphic cases as well. Proofs of all these statements are exactly the same as for the smooth case. 3. Analytic and holomorphic classifications of singularities from the classes PW ''' and PW L -3 differ essentially from the smooth classification. First let us consider the singularity class PW3''' 159

APPENDIX D. ANALYTIC CLASSIFICATION

160

THEOREM D.1. For an arbitrary holomorphic germ X of a vector field on a plane with 3 -jet of the form a a 2 - iy ay (D.1) (ix + x y) 8x

there exists a germ of a holomorphic Pfaffian equation e c PW3 XQ = X (to within orbital equivalence)-

.1

such that

Theorem D.1 shows that the holomorphic classification of singularities from PW31' I includes the holomorphic orbital classification of germs of resonance vector fields on a plane with a 3 -jet of form (D.1). Such classification

was obtained in [MarRa], [EI]. It follows from these works that the orbits of the holomorphic classification are parametrized not only by a real number (the only invariant of the formal and smooth classifications), but also by functional moduli. We do not know if these functional moduli, together with the mentioned invariant of the smooth classification, give a complete system of holomorphic invariants of germs of Pfal ian equations belonging to PW3' '' ; supposedly, this is true. We have obtained the following result. THEOREM D.2. None o f the germs f r o m the singularity c l a s s P finitely determined (in the holomorphic category).

1.1

is

This statement holds in the real-analytic category as well. 4. Let us consider the singularity class PW:+I ''' , k >- 2. For the smooth classification, there exist stable normal forms (18.1), (18.2). We announce the following result.

THEOREM D.3. None of the germs erms from rom the singularity class PW2k zk

+I

is

finitely determined (in the analytic or holomorphic category).

The scheme of the proof is as follows. is finitelY determined in the analytic holoec 8 O1 If a germ zk+I morphic) category, then by Theorem 18.1 the germ (18.1) or the germ (18.2) is 3-determined in the analytic (holomorphic) category. (2) Let (w = 01 be the normal form (18.1) or (18.2). If the germ {w = 0} is 3-determined in the analytic (holomorphic) category, then the infinitesimal PW2k-1.1

equation (6.2) with w = t i has an analytic (holomorphic) solution (h, v) for each right-hand side with vanishing 3-jet. Arguments proving this statement are close to those used in [Ill.

(3) The infinitesimal equation (6.2) with w = Fw is unsolvable in the analytic (holomorphic) category. To prove statement (3) we reduce the infinitesimal equation to an equation

of the form Xu + bu = r, where X is a vector field, b and r are functiongerms, and u is the unknown function-germ. Such a reduction was realized in §18.

APPENDIX D. ANALYTIC CLASSIFICATION

161

It is noteworthy that for a generic germ e E PW k -1 1 the corresponding vector field X in suitable coordinates looks like a x1ax1

a

a

Or

-X2ax2+XIX2aX+...

a

a

2

2

49

x18x2-x2ax1+(xl+x2)Ox3+...

(D.2)

where ... denotes terms of the form x' z'6 f (xI , ... , X20 , a+ f >- 3. These terms can be "killed" in the smooth category; but for the analytic or holomorphic classification of germs (D.2), one can point out functional moduli close to those obtained in [MarRa], [Ell, [She] (S. M. Voronin, private communication). This can be done by reduction to the problem that was solved in [Vol.

5. Now consider the case of 1 -forms w, w1o = 0. Using the notation of g 13 we state the following result.

THEOREM D.4. Suppose that for some constants C, v > 0 the invariants -u1, i=k+1,... ,2k,ofthe 1-jet , i = 1 , . . . , k , and k,. of an analytic (holomorphic) 1 -form w C W k satisfy the estimate 1 = g + u;

(al-11 + ... +

1) > Clad-V

(D.3)

for all integers ai > 0 such that IaI = a1+ +a2k > 3. Then w is equivalent to the normal form (13.1) (for the holomorphic case to is equivalent to the normal form (13.3) ).

This result was announced in [Z4], where the case w r Wk +1 was considered as well. A similar result in terms of 1-order partial differential equations was proved by Webster [W]: if the invariants {A1 , ... , 12k } _ { 1 /2 f z1, i = ] , ... , k} of a 2 -jet of a holomorphic partial differential equation U

(see Appendix A) satisfy the estimates (D.3) for some C, v > 0, then the equation is reducible to the form u = A1x10ulax1 + ... + Akxk aulaxk .

APPENDIX E

Distributions and Differential Systems Let w c A' (M) be a 1-form which does not vanish at any point of M. Then a germ of a Pfaffian equation {co = 0} can be treated as a germ of an (n - 1)-distribution in R" (n = dimM) [VG1]. An (n - 1)-distribution in R" can be given by (n - 1) vector fields X1 , ... , X , in R" such that w(X,) = 0 and rank (X i 1,,, ... , X"_,1 o) = n - I for each point c r. For example, one can treat the contact structure (g2k+1 , d z+x1 dy1 + +xk dyk) as a modulus of vector fields over a ring of smooth functions; this modulus is generated by the 2k vector fields a a a s a a I... ... ayk - xkaz ' 8xk' ay1 ax, -x18z,

Similarly, the quasicontact structure (a2k , d y, +x2 d Y2 + distribution generated by a

,...

,

a aXk

,

a ay2

a -x2-,...

a

xk

+ xk dyk) is the a

ay, and the normal form (18.1) can be treated as a distribution generated by a a a 0 a a a 2 ... a ... , xk (x'y2 + x1 z) ay, ax1 ° ayk ay1. axk ' ay2 ayI az aX1

ay1

ayk

For n = 3 and n = 4 the Darboux condition (cl {co = 0} = maximum =

3) can be formulated in terms of vector fields. Let X1, ... 'X,-, be an arbitrary system of vector fields such that w(X,) = 0, 1 = 1, ... , n - 1, and rank (X, I., ... , X"_, 1Q) = n - 1 . Let us denote by V{.=o) = V the modulus consisting of germs of vector fields of the form > f, Xj where f; are arbitrary functions. Then V = {VI,} where V C T 1(P" is a hyperplane. We use [V, V] to denote the modulus generated by vector fields [VI, V2]1 V1 , V2 E V. For n = 3 and n = 4 the Darboux condition can be formulated as dim[V, V]10 = n. In terms of vector fields we can formulate other conditions for a germ {w = 0} to belong to various singularity classes: {w = 0} E PW3 '0

if S(UB) is transversal to V1._01 at 0, 162

APPENDIX E. DISTRIBUTIONS AND DIFFERENTIAL SYSTEMS

{w = 0} E PW4,Ite u PW4I,Im

163

if D(w) is transversal to V{01 at 0.

Here S(w) and D(w) are the first degeneration manifolds.(') At the same time, for n > 5 even the Darboux condition cannot be formulated in terms of the growth vector of a distribution [VG1]. It is also interesting to consider differential systems V = (XI , ... , X _1) which differ from distributions only by the possibility of linear dependence of X1 Io , ... , Xn_ 11. at some points a E M. Various questions connected with singularities of differential systems were studied in [JP2, Morl-Mor3, MorRo]. A 2-tuple of vector fields in R3 and 3-tuple of vector fields in R4 were studied in detail. Here we formulate two classification results from the mentioned works. Let S, be the set of germs at 0 E R" of differential systems V = (X1, ... , X _1) such that rank(X1Io, ... , Xii_11o) = n -2. THEOREM E.1 [JP2]. A generic germ V E S3 is reducible to the normal

form ((9/ax,x8/8z+z8/ay). THEOREM E.2 [MorRo]. A generic germ V E S4 is reducible to the normal

form (8/8x, 8/ay, x8/0z+ya/(9u). The problem of the classification of germs of k-distributions in R" is not "wild" only for the following three cases: k = 1 (a direction field), k = n - 1 (Darboux's case), and k = 2, n = 4. The normal form of generic germs of 2-distributions in R4 was obtained by Engel in 1889 [En] (see also [VG2]). The work [Z10] is devoted to the classification of singularities of 2-distributions in R4 . If 2 < k < n - 2 and (k, n) # (2, 4), then the orbit of a generic germ of a k-distribution in R" has an infinite codimension.(2) Preliminary normal forms with functional moduli can be found in [Z I I ]. The classification of germs of regular distributions with a fixed (but not generic)

growth vector [VG I] can be found in [KR], [Z12]. For the growth vector (k, k + 1 , k + 2, ... , n), n - k < 3, a germ is reducible to a stable normal form. If the growth vector has another form, then the classification problem remains "wild". Singularities of differential systems (X1 in [JP2].

, ... Xk) in R" were considered ,

(I )i.e. the manifolds {a E M 'dim (VI.-o} ' V(.-O)] I. = n - 1). (2)This fact was proved in (JP2), then also in [VG2]; moduli appear not only in formal series but also in the C'-classification, see (Val.

APPENDIX F

Topological Classification of Distributions Consider a k-distribution E in R" (it can be given either by (n -k) differential forms wl, ... , w"_k or by a k-tuple of vector fields V = (X11... , Xk )) . A differentiable curve y = y(t) such that y(t) E E (i.e., '(t) E Kerwi(r(1) a

'(t) E spanVJ,(,) ) is said to be a trajectory of E. Two germs E1 , E2 of k-distributions are said to be topologically equivalent if a germ of a homeomorphism maps the germs of the trajectories of El into the germs of the trajectories of E2 [JP2]. In [JP1,JP2] the classes of the topological equivalence were discussed. We announce the result for 2-distributions in R . THEOREM F. 1. The germ at each point of a generic 2-distribution on a 3manifold is topologically stable and topologically equivalent to one and only one of the germs a a a (1){dz+xdy=o}a r ax'ay-xaz

(2) {dy+x 2dz = 011*

(a

a

20

ax' 9z// -x ay

(3) {dy+(xy+x2z)dz=0} a2z)a) ; (4)

1dy +

x2

l (XY+ 3 +xz ) dz=0} s JJJ

axa , azs -

x'2`I a)

xy+ 3 +xz

J

Y

In order to prove Theorem F. I it suffices to prove the topological equivalence of germ (19.1) to germ (3) and of germ (19.2) to germ (4) (for an arbitrary b E 1a ). As for the global classification of 2-distributions on a 3-manifold, it was

shown in [Ben] that there exists a contact structure (l 3 , w) which is not equivalent to the standard contact structure (R3 , d z + x d y) .

164

APPENDIX G

Degenerations of Closed 2-Forms in 1

2k

In this appendix we give the list of normal forms of closed 2-forms in Ru` , k > 2. The normal form of first occurring singularities

x1dx,ndy,+dx2Ady2+- +dxkAdyk

(G.1)

was obtained by Martinet [Mar] who also announced the normal forms for the next degeneration (of codimension 3):

r

2

d I xl - z n d y1 + d

`

3

(12±12_ 3) n dye + /

k

dx1 n dye. (G.2) ;=3

The theorem on reduction to these normal forms was proved by Roussarie [Ro]. The germs, corresponding to the next degeneration (of codimension 4) are unstable [GT2] and, supposedly, they are not finitely determined. At generic points of a 2k-dimensional manifold a germ of a generic closed 2-form is stable and equivalent either to the standard Darboux model dxl n + xk dyk , or to the germ (G.1), or to the germ (G.2); nongeneric dy1 + points form a set of codimension > 4. The description of degenerations can be found in [AG, Mar, Ro, GT2]. The Darboux and Martinet normal forms hold in the analytic (holomorphic) category as well. At the same time, analytic germs that are CO°-equivalent to the germ (G.2) are not (generally speaking) analytically equivalent to (G.2) (arguments are similar to those used in Appendix D).

165

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Z 10.

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24 (1990). , Normal forms of germs of smooth distributions, Mat. Zametki 49 (1991), no. 2, 36-44; English trans]. in Math. Notes 49 (1991). , Normal forms of germs of distributions with a fixed growth vector, Algebra i Z12. Analyze 2 (1990), no. 5, 125-149; English transi. in Leningrad Math. J. 2 (1990). , Classification of germs of regular distributions with a minimum growth vector, Z13. FunktsionaL AnaL i Prilozhen. 25 (1991), no. 1, 80-81; English transl. in Functional Anal. Appi. 25 (1991). Z14. , Typical singularities of differential i forms and Pfefequations, Seminar on supermanifolds (D. Leites, ed.), Reports of Math. Dept. Stockholm Univ., 1990, pp. 1285. Z11.

Author Index Arnold, V. I., xi

Belitskii, G. R., x, xi, 21, 26, 29 Bogdanov, R. I., 29 Chen, K., 23, 29

Darboux, J: G., ix, 1, 31, 32 Engel, F., 163

Givental, A. B., x, 24, 25 ll'yashenko, Yu. S., xi Jakubczyk, B., 116 Leites, D., xi Lie, S., 25 Lychagin, V. V., xi, 1, 4, 25, 29, 31, 47, 57, 151

Martinet, J., ix, x, 1, 2, 6, 31, 57, 116, 117, 165

Mormul, P., 150 Pfaff, J. F., ix Przytycki, F., 116

Roussarie, it, x, 26, 165 Sternberg, S., 29

Takens, F., 30 Voronin, S. M., 161 Webster, S. M., 161

Zhitomirskii, M. Ya., I

171

Subject Index Jet, 11 normalization of, 20

Adjacent class, 15

C'-equivalent germs, 13

k-determined, 16

Class

k -jet, 11

of a 1-form, 31

sufficient, 16

of a Pfaffian equation, 59, 119 Contact manifold structure, 10, 25, 61, 155,

Modality of a germ, 17 Modulus (internal), 17

164

Darboux condition, 153 Darboux theorem, 20, 61, 120 on 1-forms, 34 relative, x, 25 Degeneration

Normal form, 17 invariant, 17 preliminary, 17 Normalization of a k -jet, 20

first, 5, 62, 119, 120

Pffalan equation, 10 represented by a 1-form, 10

second, 5, 64

third, 5, 91 Distribution, 10, 162, 164

Quasicontact structure, 10, 120

Rank of dw, 31 Resonant, 22

Equivalent germs, 13, 25, 153, 164 Feedback equivalence, 157 Finite determinacy, 16 index of, 16 Flat function, 12

Singular point, ix Singularity, 14 first occuring of, 14 isolated, 15 point, 15 Singularity class, 14 adjacent, 15 codimension of, 15 point, 15 typical, 15 wild, 22 Source of a germ, 10 Stable germ, 17 Straightened, 35 Sufficient k -jet, 16

General position, 7 Generic, 13, 14 Germ C'-equivalent, 13 elliptic, 68 equivalent, 13, 25, 153 finitely determined, 16 generic, of a hypersurface in a contact

space, 25 hyperbolic, 68 infinitesimal equation for, 21 k-determined, 16 modality of, 17 parabolic, 68 source of, 10 stable, 17

Thick set, 14 Transversality theorem, 16 Transformation with unit linear part, 21

Unit linear part, 21 Whitney topology, 13 Wild, 22

Infinitesimal, equation, 27 Internal modulus, 17

173

List of Symbols A (M)

the space of external differential 1-forms on a

manifold M PA1(M) W (n)

PW(n)

the set of Pfaffian equations on a manifold M the set of germs at 0 E 1R" of 1-forms in IR" the set of germs at 0 E ]R of Pfaffian equations in Ql:°

C°°(n) Vect (n) Diff(n)

the space of function-germs at 0 E IR° the space of germs of vector fields at 0 E Qf"

the group of germs at 0 E IIf" of diffeomorphisms (Rn, 0) (1R°, 0)

nQ

translation of a germ's source point a into

Vs

0E1R° the set of germs of functions vanishing at each

point of a manifold S or the set of germs of vector fields (differential l-forms) of the form

v=>jv,,

v,EVect(n), JEMS

fw;,

cv;EW(n), f,.E9ns)

(w= 9Jls

the kth power of 9JIS or the set of germs of vector fields (differential 1 -forms) of the form

v= (w Jk s

Jsk k Js

Mk,

Jk,

jk

f v,

,

= > jcoj,

v, E Vect (n) , f E

DRcs

Wi E W(n), j. E Mk

the set of k -jets on a manifold S C°°(n)/97ts 1, JS = Vect(n)/9Jts 1, JS = W(n)/9Jlk-1 S

a map from C°`(n), Vect (n), W(n) into J$ k

k

M10) , J{o} , j

175

k 01(resp.)

LIST OF SYMBOLS

176

S(w) (S(E))

the set of points at which the germ of a differ-

ential form w (Pfaffian equation E) belongs to the singularity class S S(w) (S(e))

the germ of S(w) (S(E)) at 0 E R", where w is a l-form (E is a Pfaffian equation) with

cl

W (P W')

W. I ' ° , P Wzk+11

the germ co (e) at 0 E Q(" the class of a l -form or Pfaffian equation the singularity class consisting of a germs of I-forms (Pfaffian equations) of the class j

etc.

various singularity classes of l-forms and

X. , X,,

Pfaffian equations the field of directions invariantly connected with singularities w E W(n), e E PW(n)

1

,

if W is a k-form and X is a vector field, then X, W is a (k - l)-form W such that W(Y1 , ... , Yk-1) = W(X, Y1 , ... , "k_1) for arbitrary Y1 , ... , Yk_ 1 ; if W E A' (M) , then X, W = W (X) is a function

Recent Titles in This Series 78

(Continued from the front of this publication)

A. V. Skorokkod, Asymptotic methods of the theory of stochastic differential equations, 1989

V. M. Filippov, Variational principles for nonpotential operators, 1989 Philip A. Griffiths, Introduction to algebraic curves, 1989 75 B. S. Kuhin and A. A. Saakyan, Orthogonal series, 1989 74 V. 1. Yadovieh, The linearization method in hydrodynamical stability theory, 1989 73 Yu. G. Resbetnyak, Space mappings with bounded distortion, 1989 72 A. V. Pogorelev, Bendings of surfaces and stability of shells, 1988 71 A. S. Markus, Introduction to the spectral theory of polynomial operator pencils, 1988 70 N. I. Akkiezer, Lectures on integral transforms, 1988 69 V. N. Sa1Ir, Lattices with unique complements, 1988 68 A. G. Postnikov, Introduction to analytic number theory, 1988 67 A. G. Dragalin, Mathematical intuitionism: Introduction to proof theory, 1988 66 Ye Yan-Qian, Theory of limit cycles, 1986 65 V. M. Zolotarev, One-dimensional stable distributions, 1986 Ill-posed problems of 64 M. M. Lavrent'ev, V. G. Romanov, and S. P. mathematical physics and analysis, 1986 63 Va. M. Berezanskil, Selfadjoint operators in spaces of functions of infinitely many variables, 1986 62 S. L. Krushkal', B. N. Apanasov, and N. A. Gusevskii, Klcinian groups and uniformization in examples and problems, 1986 61 B. V. Shabat, Distribution of values of holomorphic mappings, 1985 60 B. A. Kushaer, Lectures on constructive mathematical analysis, 1984 59 G. P. Egoryckev, Integral representation and the computation of combinatorial sums, 77 76

1984

L. A. Aizenberg and A. P. Yuzhakov, Integral representations and residues in multidimensional complex analysis, 1983 57 V. N. Monakhov, Boundary-value problems with free boundaries for elliptic systems of equations, 1983 56 L. A. Aizeaberg and Sb. A. Dautov. Differential forms orthogonal to holomorphic functions or forms, and their properties, 1983 55 B. L. Ro3.destvenskil and N. N. Janenko, Systems of quasilinear equations and their applications to gas dynamics, 1983 54 S. G. Kreln, Ju. I. Petunin, and E. M. Semenov, Interpolation of linear operators, 1982 53 N. N. (eaeov, Statistical decision rules and optimal inference, 1981 52 G. I. Eskia, Boundary value problems for elliptic pseudodifferential equations, 1981 51 M. M. Smirnov, Equations of mixed type, 1978 50 M. G. Krern and A. A. Nudel'man. The Markov moment problem and cxtrcmal problems, 1977 49 I. M. Milin, Univalent functions and orthonormal systems, 1977 48 Ju. V. Linaik and 1. V. Ostrovskii, Decomposition of random variables and vectors, 58

1977

47 M. B. Nevel'son and R. Z. Has'minskil, Stochastic approximation and recursive estimation, 1976 46 N. S. Kurpel', Projection-iterative methods for solution of operator equations, 1976 45 D. A. Suprunenko, Matrix groups, 1976 (See the AMS catalog for earlier titles)

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