Real Submanifolds in Complex Space and Their Mappings

Real Submanifolds in Complex Space and Their Mappings M. SALAH BAOUENDI PETER EBENFELT LINDA PREISS ROTHSCHILD

Real

in Complex Space and Their Mappings

Princeton Mathematical Series EDITORS: JOHN N. MATHER

and ELIAS M. STEIN

1. The Classical Groups by Hermann Weyl 4. Dimension Theory by W Hurewicz and H. Wailman S. Theory of Lie Groups: I by C. Chevalley 9. Mathematical Methods of Statistics by Harold Cramer 14. The Topology of Fibre Bundles by Norman Steenrod 17. Introduction to Mathematical Logic, Vol. 1 by Alonzo Church 19. Homological Algebra by H. Cartan and S. Eilenberg 25. Continuous Geometry by John von Neumann 28. Convex Analysis by R. T Rockafellar 30. Singular Integrals and Differentiability Properties of Functions by E. M. Stein 31. Problems in Analysis edited by R. C. Gunning 32. Introduction to Fourier Analysis on Euclidean Spaces by E. M. Stein and G. Weiss 33. Etale Cohomology by .1. S. Mime 34. Pseudodifferential Operators by Michael E. Taylor 35. Three-Dimensional Geometry and Topology: Volume I by William P Thurston. Edited by Silvio Levy 36. Representation Theory of Semisimple Groups: An Overview Based on Examples by Anthony W Knapp 37. Foundations of Algebraic Analysis by Masaki Kashiwara, Takahim Kawai, and Tatsuo Kimura. Translated by Goro Kato 38. Spin Geometry by H. Blame Lawson, Jr.. and Marie-Louise Michelsohn 39. Topology of 4-Manifolds by Michael H. Freedman and Frank Quinn 40. Hypo-Analytic Structures: Local Theory by Francois Treves 41. The Global Nonlinear Stability of the Minkowski Space by Demeirios Christodoulou and Sergiu Klainerman 42. Essays on Fourier Analysis in Honor of Elms M. Stein edited by C'. Fefferman, R. Fefferman, and S. Wainger 43. Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals by Elias M. Stein 44. Topics in Ergodic Theory by Ya, G. Sinai 45. Cohomological Induction and Unitary Representations by Anthony W. Knapp and David A. Vogan. Jr. 46. Abelian Varieties with Complex Multiplication and Modular Functions by Goro Shimura 47. Real Submanifolds in Complex Space and Their Mappings by M. Salah Baouendi. Peter Ebenfelt, and Linda Preiss Rothschild

REAL SUBMANIFOLDS IN COMPLEX SPACE AND THEIR MAPPINGS

M. Salah Boo uendi Peter Ebenfelt

Linda Preiss Rothschild

PRINCETON UNIVERSITY PRESS PRINCETON, NEW JERSEY

Copyright © 1999 by Princeton University Press Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom. Princeton University Press, Chichester, West Sussex All Rights Reserved Baouendi, M. Salah, 1937—

Real submanifolds in complex space and their mappings / M. Salah Baouendi, Peter Ebenfelt, Linda Preiss Rothschild. cm.—(Princeton mathematical series ; 47) p. Includes bibliographical references and index.

ISBN 0-691-00498-6 (cI. : alk. paper) I. Submanifolds. 2. Functions of several complex variables. 3. Holomorphic mappings. I. Ebenfelt, Peter. II. Rothschild, Linda Preiss, 1945—. Ill. Title. IV. Series. QA649.B36 1998 98-44235

51 6.362—dc2 I

ISBN 0-691-00498-6

The publisher would like to acknowledge the authors of this volume for providing the camera-ready copy from which this book was printed The paper used in this publication meets the minimum requirements of ANSI/N ISO Z39.48. 1992 (R1997) (Permanence of Paper) http://pup.princeton.edu Printed in the United States of America I

3

5

7

9

10

8

64

2

to Moungi and Meriem Ann-Sofie, Felicia, and Evelina David and Daniel

CONTENTS

Preface

xi

Chapter I. Hypersurfaces and Generic Submanifolds in

3

Real Hypersurfaces in Holomorphic and Antiholomorphic Vectors CR, Totally Real, and Generic Submanifolds CR Vector Fields and CR Functions Finite 1Srpe and Minimality Conditions Coordinate Representations for CR Vector Fields Holomorphic Extension of CR Functions Local Coordinates for CR Manifolds

3

§ 1.1.

§ 1.2. § 1.3.

§ 1.4. § 1.5. § 1.6.

§ 1.7. § 1.8.

Chapter II. §2.1. §2.2. §2.3. §2.4. §2.5.

Formally Integrable Structures on Manifolds Levi Form and Levi Map of an Abstract CR Manifold CR Mappings Approximation Theorem for Continuous Solutions Further Approximation Results

Chapter III. §3.1. §3.2. §3.3. §3.4. §3.5. §3.6.

Abstract and Embedded CR Structures

Vector Fields: Commutators, Orbits, and Homogeneity

Nagano's Theorem Sussman's Theorem Local Orbits of Real-analytic Vector Fields Canonical Forms for Real Vector Fields of Finite Canonical Forms for Real Vector Fields of Infinite Weighted Homogeneous Real Vector Fields

Chapter W §4.1. §4.2. §4.3. §4.4. §4.5.

Coordinates for Generic Submanlfolds CR Orbits, Minimality, and Finite Type Normal Coordinates for Generic Submanifolds Canonical Coordinates for Generic Submanifolds Weighted Homogeneous Generic Submanifolds Normal Canonical Coordinates vu

6 9 14

17 21

26 30 35 35

40 49 52 57

62 62 68 73 73 87 91

94 94 95 101

108 112

CONTENTS

viii

Chapter §5.1.

§5.2. §5.3. §5.4.

§5.5.

V.

Rings of Power Series and Polynomial Equations

Finite Codimensional Ideals of Power Series Rings Analytic Subvarieties Weierstrass Preparation Theorem and Consequences Algebraic Functions, Manifolds, and Varieties Roots of Polynomial Equations with Holomorphic Coefficients

Chapter VI. §6.1. §6.2. §6.3. §6.4. §6.5.

Geometry of Analytic Discs Hubert and Poisson Transforms on the Unit Circle Analytic Discs Attached to a Generic Submanifold Submanifolds of a Banach Space Mappings of the Banach Space Banach Submanifolds of Analytic Discs

Chapter VIL §7.1. §7.2. §7.3. §7.4. §7.5. §7.6. §8.1. §8.2. §8.3. §8.4. §8.5. §8.6. § 8.7.

§8.8.

128 132 139 145

156

156 162

166 176 178

Holomorphic Extension of CR Functions

184 185

192

196 202 204

205

Criteria for Wedge Extendability of CR Functions 205 Sufficient Conditions for Filling Open Sets with Discs 206 Tangent Space to the Manifold of Discs 212 Defect of an Analytic Disc Attached to a Manifold 218 Ranks of the Evaluation and Derivative Maps 224 Minimality and Extension of CR Functions 230 Necessity of Minirnality for Holomorphic Extension to a Wedge 231 Further Results on Wedge Extendability of CR Functions 238

Chapter IX. §9.1. §9.2. §9.3. §9.4. §9.5. §9.6. §9.7. §9.8. §9.9.

119

Boundary Values of Holomorphic Functions in Wedges 184

Wedges with Generic Edges in CN Holomorphic Functions of Slow Growth in Wedges Continuity of Boundary Values Uniqueness of Boundary Values Additional Smoothness up to the Edge Further Results and an "Edge-of-the-Wedge" Theorem

Chapter VIII.

119

Holomorphic Extension of Mappings of Hypersurfaces

Reflection Principle in the Complex Plane Reflection Principle: Preliminaries Reflection Principle for Levi Nondegenerate Hypersurfaces Essential Finiteness for Real-analytic Hypersurfaces Formal Power Series of CR Mappings Reflection Principle for Essentially Finite Hypersurfaces Polynomial Equations for Components of a Mapping End of Proof of the Reflection Principle Reflection Principle for CR Mappings

241

242 243 246 248 252 255 257 259 265

CONTENTS

§9.10.

§9.11.

Chapter X. § 10.1.

§ 10.2.

§ 10.3. § 10.4.

§ 10.5. § 10.6.

§10.7.

Reflection Principle for Bounded Domains Further Results on the Reflection Principle

270

Segre Sets Complexification of a Generic Real-analytic Submanifold Definition of the Segre Manifolds and Segre Sets Examples of Segre Sets and Segre Manifolds Basic Properties of the Segre Sets Segre Sets, CR Orbits, and Minimality Homogeneous Submanifolds of CR Dimension One Proof of Theorem 10.5.2

281 281 283

Nondegeneracy Conditions for Manifoids Finite Nondegeneracy of Abstract CR Manifolds

315 315 319 322 325 329 335 336

Chapter XI. § 11.1. § 11.2.

§ 11.3.

§11.4. § 11.5. § 11.6.

§ 11.7. § 11.8.

Finite Nondegeneracy of Generic Submanifolds of CN Holomorphic Nondegeneracy Essential Finiteness for Real-analytic Submanifolds Comparison of Nondegeneracy Conditions Compact Real-analytic Generic Submanifolds Nondegeneracy for Smooth Generic Submanifolds Essential Finiteness of Smooth Generic Submanifolds

Chapter XII. § 12.1. § 12.2. § 12.3. § 12.4. § 12.5. § 12.6.

§ 12.7.

§ 13.1. § 13.3.

Holomorphic Mappings of Submanifolds

Jet Spaces and Jets of Holomorphic Mappings Basic Identity for Holomorphic Mappings Determination of Holomorphic Mappings by Finite Jets Infinitesimal CR Automorphisms Finite Dimensionality of Infinitesimal CR Automorphisms Iterations of the Basic Identity Analytic Dependence of Mappings on Jets

Chapter Xffl. § 13.2.

ix

Mappings of Real-algebraic Subvarieties

Mappings between Generic Real-algebraic Submanifolds Some Necessary Conditions for Algebraicity of Mappings Mappings of Real-algebraic Subvarieties

277

289 293 300 305 312

342 349 349 352 358 361 366 370 373 379 379 383 387

References

390

Index

401

PREFACE

The study of real submanifolds in CN has emerged as an important area in several complex variables and has gained independent interest in the past 40 years. In fact, some of the problems studied here go back to H. Poincaré and E. Cartan in the early 1900's. The "modern era" in this subject dates back to the 1950's with the fundamental contributions of H. Lewy, who discovered deep connections between the theory of several complex variables and that of partial differential equations. In this book we focus on a number of important problems for which substantial progress has been made in the last twenty years, such as holomorphic extendibility of Cauchy-Riemann (CR) functions from real submanifolds, holomorphic extendibility of mappings between such manifolds, and the structure of the germs of biholomorphisms which map one real submanifold into another. The case of real algebraic submanifolds is of particular interest and is also addressed in this book. We begin by developing the tools necessary to present complete, self-contained results in the areas mentioned above. These tools come from a variety of fields, including geometry, analysis, commutative algebra, as well as more classical complex analysis. We have tried to include background material which cannot be easily found in other books or monographs.

In the first two chapters we introduce some basic definitions such as that of CR manifolds, CR vector fields, Levi form, finite type and minimality conditions,

and prove some basic properties of these objects. Chapter ifi deals with real vector fields; we prove the theorems of Nagano and of Sussman concerning orbits of such vector fields, and also establish the existence of weighted homogeneous coordinates. These results are used in Chapter IV to give various kinds ofcanonical coordinates for embedded submanifolds in CN. Chapter V, which is independent of the previous chapters, develops mostly algebraic methods for working with rings of formal and convergent power series and polynomials. In Chapter VI we study boundary values of holomorphic functions with slow growth, defined in a wedge whose edge is a generic submanifold in CN, and prove some uniqueness and regularity results. In Chapter VII we develop the theory of analytic discs attached to a generic submanifold of C" and show that the set of such (small) discs forms an infinite dimensional Banach submanifold in the space of all discs. The Bishop equation is xi

xii

PREFACE

treated here in a Banach space setting. This approach is then used in Chapter VIII to prove the theorem of Tumanov showing that minimality implies holomorphic extendibility of all CR functions into an open wedge in CN. The necessity of this condition is also given. Another main topic of this book is addressed in Chapter IX. A reflection principle for CR mappings between real analytic hypersurfaces is proved here, and applications to proper holomorphic mappings between bounded are given. In Chapter X we develop the theory of Segre sets for domains in generic submanifolds. An equivalent condition for minimality in terms of Segre sets, which will be an essential tool for the remainder of the book, is established here. In Chapter XI we introduce a number of nondegeneracy conditions for real submanifolds, including holomorphic nondegeneracy and essential finiteness, and explore the relationships among them. In the last two chapters we study germs of holomorphic mappings which send one real analytic (or algebraic) submanifold into another and show, under some nondegeneracy conditions, that such germs are determined by their jets of finite order. In the algebraic case we give sufficient conditions for all such germs to be algebraic mappings. The material in this book is intended to be accessible to mature graduate stu-

dents; no previous knowledge of several complex variables is assumed of the reader. Although most of the results presented could be found in research articles, we have also included some previously unpublished work. There are many important areas of several complex variables connected with the study of real submanifolds which are not addressed in this book. Most notable is the omission of the Kohn a-Neumann problem and its many consequences. We also do not include any discussion of the Chern-Moser normal forms for Levinondegenerate hypersurfaces. We have been greatly inspired by the work of many mathematicians who havc made fundamental contributions to the topics covered in this book. We will not attempt to list them all here. At the end of each chapter we have included notes giving bibliographical references and historical comments, and we apologize for any errors or sins of omission. Finally, we would like to make special mention here of some of our collaborators in this field over the years, including Steven Bell, Xiaojun Huang, Howard Jacobowitz, Elias M. Stein, Francois Treves, and Jean-Marie Trépreau, from whom we learned a great deal. We would also like to thank Dmitri Zaitsev for numerous useful comments on some chapters of this book. M. SALAH BAOUENDI PETER EBENFELT LINDA PREISS ROTHSCHILD

July 1998

Real Subman!folds in Complex Space and Their Mappings

CHAPTER 1

HYPERSURFACES AND GENERIC SUBMANIFOLDS IN CN

Summary The basic object of study in this book is a smooth real submanifold in CN whose

tangent space has a smoothly varying maximal complex subspace. Here, there is a rich interaction between real and complex function theory as well as geometry. In this chapter, we give basic definitions and properties of such objects. We begin with a brief discussion of real hypersurfaces. In § 1.2, we define holomorphic and antiholomorphic vectors for real submanifolds in CN. These play a crucial role throughout this book. We then define the notions of CR, generic, and totally real submanifolds of and introduce the CR vector fields and CR functions on these submanifolds. The important notions of minimality and finite type are defined in § 1.5; these are discussed in terms of coordinates for hypersurfaces in § 1.6. A brief discussion of holomorphic extension of CR functions on real-analytic generic submanifolds and formal holomorphic power series of CR functions is given in § 1.7. The chapter concludes with a description of local coordinates for general CR submanifolds and the intrinsic complexification for real-analytic CR submanifolds.

§ 1.1. Real Hypersurfaces in begin with some notation which will be used throughout this book. For Z E CN we write Z = (Z,,... , ZN), where Z3 = + iyj, with Xj and yj real numbers; we write 2 = (Zi,... , ZN), where Z1 = Xj — iy1, the complex conjugate of Z1. As is customary, we use i to denote the imaginary unit 12 = The absolute value of is given by + We identify CN with R2N, and denote a function f on a subset of CN as f(x, y), or, by abuse of notation, as We

f(Z,Z). A (smooth) real hypersurface in CN is a subset M of CN such that for every point po E M there is a neighborhood U of p0 in CN and a smooth real-valued function p defined in U such that

(1.1.1)

MflU=(ZEU:p(Z,2)=O},

with differential dp nonvanishing in U, i.e. at every point p of U not all first derivatives of p vanish at p. Such a function p is called a local defining function 3

I. HYPERSURFACES AND GENERIC SUBMANIFOLDS IN CN

4

for M near p0. The hypersurface M is real-analytic if the defining function p in (1.1.1) can be chosen to be real-analytic. EXAMPLE 1.1.2. The hypersurface given by the equation Im ZN = 0 is a "flat" hyperplane in Ci". EXAMPLE 1.1.3. The hypersurface in

given by the equation

ImZN =>1Z112 is

called the Lewy hypersurface.

EXAMPLE 1.1.4. The unit sphere in CN given by 1Z112 = 1 is a compact hypersurface. The reader can check that the holomorphic rational mapping 11(Z) = (Hi(Z),... , HN(Z)) given by

Hj(Z):=

1—ZN

,

the unit sphere minus the point (0, 0,... surface given in Example 1.1.3. takes

HN(Z):=I+l),

j=1....N—1,

I—ZN

,

1)

bijectively to the Lewy hyper-

1.1.5. For any hypersurface M and any P0 M there exist smooth near Po in R2", vanishing at such that M is given , by x = 0 in a neighborhood of p0. Indeed, after an affine change of coordinates we may assume p0 = 0 and ap/ax1(o) 0. Then setting = p. = x1, N, and 1 k N, gives the desired conclusion. Hence 2 = all real hypersurfaces are locally equivalent after a smooth change of coordinates. REMARK

coordinates (x,...

j

However, in general there is no holomorphic change of coordinates which performs this equivalence.

The reader can easily check that if p and p' are two defining functions for M near then there is a nonvanishing real-valued smooth function a defined in a neighborhood of Po such that p = ap' near through P0. Then vanishing a: zE C, and a real-valued smooth function s) defined near 0 in = 0, dçb(0) = 0, such that near M is given by

PROPOSITION 1.1.6. Let M be a real hypersurface in

there are holomorphic coordinates (z, w) near with

(1.1.7)

§1.1. REAL HYPERSURFACES iN CN

5

We may first make a fcr K 0. (0) translation and a linear change of coordinates so that P0 = 0 and Then the Taylor expansion at 0 of p may be written bc u

PROOF, Lct

si+1

p(Z,Z)=>2(ajxj+bjyj)+0(2), j=I where

0(2) vanishes of order at least 2 at 0. Since p is real valued, so are its first

derivatives; hence a1 and b1 are real. We write n+I

n+I

>2 j=I

with c1 = b1

+ bjy3) = Im >2 c1 z1, j=1

We make the linear holomorphic change of coordinates given by I,... , n, and w = c1Z3. A defining equation forM near (0,0)

j= Zj = in the new coordinates (z, w) is given by p(Z(z. w), Z(z, w)) = 0. The smooth function Re w) is obtained by solving the equation p(Z(z, w), Z(z, w)) = 0 for Im w near the base point w) = (0, 0); this can be done in view of the implicit function theorem since w), Z(z, w)) = Im w + 0(2). The properties

(0) =

0 and dçt (0) = 0 are immediate. The equation (1.1.7) has the as p(Z(z, w), w)) = 0 near (0, 0) and, hence, is a defining

same solutions equation for M near po• This completes the proof of Proposition 1.1.6.

0

EXAMPLE 1.1.8. Let M C C2 be the hypersurface through po = (1,0) given by

(1.1.9) p(Z, Z) = 2(Z1 +Z1)+i(Z1

= 0.

We make the initial change of variables

Z=

Z2

to obtain the new

defining function

(1110) Now we may take

= Im

Im

Im (2iZ) +

+ z;i2.

Z and use the implicit function theorem to solve for Im We obtain a defining function as described in Proposition 1.1.6 with w = Z' and z = Z'.

6

1. HYPERSURFACES AND GENERIC SUBMANIFOLDS IN CN § 1.2.

Holomorphic and Antiholomorphic Vectors

of codimension d is a subset M of CN such A (smooth) real submanifold of that for every point p0 E M there is a neighborhood U of p0 and a smooth real vector-valued function p = (p1.... , Pd) defined in U such that

M fl U = {Z E U : p(Z, Z) = 0),

(1.2.1)

with differentials dp1,... , dpi, linearly independent in U. A real submanifold

of codimens ion 1 is a real hypersurface as defined in § 1.1. As in the case of a hypersurface, we shall refer to p as a local defining function for M near p0. One can easily check that if p and p' are two defining functions for M near p0. then there is a d x d smooth invertible matrix a with real-valued coefficients defined in a neighborhood of po such that p = ap' near P0. The reader should note, as in Remark 1.1.5, that by a smooth change of variables near P0. vanishing at p0. such in R2N, we can find new coordimites (xi,... , = ... = = 0 in a neighborhood of However, there is that M is given by in general no holomorphic change of coordinates which does this. For p CN R2N, we put

ip

I

the real tangent space to tangent to M at p if

I

at p. If p

YiP

M and X

ai.bJEIR}. we say that X is

k=l,... ,d,

yj

for a local defining function p = (ps.... Pd) for M near p. The reader can easily verify that this definition is independent of the choice of p. We write

M,

the tangent space of M at p. for the space of all real vectors tangent to M at p. Similarly, we define the complexified tangent spaces and by allowing the coefficients a1 and in the expressions above to be complex numbers. Note that for every p M, dimR = 2N — d. Hence the = and M P define real and complex vector mappings M P i-+

bundles over M, which we shall denote respectively by TM and CTM, the real and the complexified tangent bundles of M. As is customary, we write a

.a\

a

.a

* 1.2. HOLOMORPHIC AND ANTIHOLOMORPHIC VECTORS

7

can be written uniquely in the form

With this notation, any X E

aj,bjEC.

(1.2.2)

A tangent vector X, given by (1.2.2), is holomorphic (antiholomorphic) if = 0, for j = 1,... , N (a1 = 0, for j = 1,... , N). By the chain rule, this definition is independent of the choice of holomorphic coordinates. For p E CN it is customary to denote by .OCN the space of holomorphic tangent vectors at ICN the space of antiholomorphic tangent vectors. Note that p and by dimc l.LOCN = dimc 7.O.ICN = N.

p M we denote by V,, the space of antiholomorphic vectors tangent to M at p, i.e. := fl CT,,M. For

The space is sometimes written to check that

=N

(1.2.3)

M. By elementary linear algebra, it is easy

—

rank

az1

P)) I

where p is a local defining function for M as above. Note that

is independent of the choice of holomorphic coordinates in CN and the defining function p. We write V,, for the space of holomorphic vectors tangent to M at p, the complex conjugate of V,,. It should be noted that may vary with p E M.

1.2.4. Let M C C2 be given by x2 =

Y2

=

0.

Then d =

2

and

Y2

=

0,

then d =

2

and

= forall p EM. 1

If M C C2 is given by =Oforall p EM.

EXAMPI..E 1.2.5.

=x2andp2 = =p2=0,thend=2,and EXAMPLE 1.2.6. Let

P1

=

ax1

,,'

IfM CC2 isgivenby

Y2

11, ifZ1=0;

(0,

Another viewpoint for the construction of basis of the vectors a

=

is the following. We take as a

a

a

a

aXN

ay1

ayw

I. HYPERSURFACES AND GENERIC SUBMANIFOLDS IN CN

8

into itself determined by

and introduce a real linear mapping J from

ía \i=— , it— ía

\1=—— , j=1,...,N. ax1 a

a

,,

=

,,,,

,,

where I is the identity. The reader can check, by using the Cauchy-Riemann equations for holomorphic functions, that the definition of the operator J is independent of the choice of holomorphic coordinates. The operator which corresponds to the identification of R2N 1 is the complex structure on with CN. By linearity, I can also be extended as a C-linear operator from CTPCN onto itself, denoted again by J, with I2 = —I. If M is a submanifold of CN and p E M, we let Note that

— I,

:= (X E

(1.2.7)

1(X) E

TM into itself and hence defines a complex structure on TM. If CTM is the complexification of then J extends also as a C-linear operator from CTM onto itself. We have the following proposition. PROPOSITION 1.2.8. For every p M the following hold.

V,, = (X

(1.2.9)

(1.2.10)

: 1(X) = —iX},

E

V,,

=

= {X +iJ(X): XE

(1.2.11)

(1.2.12) Re

=(X+X : X€

PROOF. A direct calculation shows that

J(fr)=_ifr. Hence, we obtain

', The proposition easily follows from these relations. We leave the details to the

reader.

0

The space is often called the complex tangent space of M at p. It follows from (1.2.10) or (1.2.12) that (1.2.13)

§1.3. CR. TOTALLY REAL, AND GENERIC SUBMANIFOLDS

9

§13. CR, Totally Real, and Generic Submanifolds We shall restrict our attention to real submanifolds of CN for which the mapping p i-+ V,, (defined in § 1.2) determines a subbundle of CTM, i.e. those submanifolds for which is constant as p varies in M.

DEfINITION 1.3.1. A real submanifold M C C" is CR (for Cauchy-Riemann) is constant for p E M. For aCR submanifold, will be called the CR dimension of M. The complex subbundle V C CTM whose fiber at p E M is called the CR bundle of M. Similarly, the real subbundle TCM C TM is whose fibre at p E M is TM is called the complex bundle of M. Examples 1.2.4 and 1.2.5 are CR submanifolds, while Example 1.2.6 is not. CR submanifolds of CR dimension 0 are called totally real. The CR submanifold given in Example 1.2.5 is totally real, while that of 1.2.4 is not. If f(Z, 2) is a smooth function defined in an open Set of C"", recall that the differential df(p) is given by (1.3.2)

df(p) =

+

= y1

+ where

(1.3.3)

=

+i

and

=

—

df(p) =

I

dy1. We also write

df(p) = >

so that df(p) = af(p) + af(p). It follows from (1.2.3) that if M is CR and of codimension d and p

=

(pa,

. . .

Pa)

is a local defining function for M near p. then the CR dimension of M is as small as possible if a p1.... , are linearly independent near p. (The reader should note that the fact that dp11... , dp, are linearly independent does not imply that ap1,... , are linearly independent, as can be seen by Examples 1.2.4 and 1.2.6. The reader should also note that the linear independence of ap1,... , aPd is equivalent to that of ap1 dpa by the reality of the functions Pj.) DEfiNITION 1.3.4. A real submanifold M c is generic if near every p E M there is a local defining function p = (pie . . Pd) such that the complex differentials dp1,... , are C linearly independent near p. .

if the linear independence of Definition 1.3.4 holds for a local defining function p. then it holds for any other local defining function. The reader can check that a

tO

I. HYPERSURFACES AND GENERIC SUBMANEFOLDS IN C"

real hypersurface is necessarily generic. A generic submanifold of codimension d may be viewed as an intersection of d real hypersurfaces in "general position:' It follows from (1.2.3) that a generic submanifold of codimension d is necessarily CR of CR dimension N — d. PRoPosmoN 1.3.5. Let M be a totally real submanifold of C" of codimension Mis generic. thend = N d. Thend NandhencedimR M N. If

= N. PROOF. Since M is totally real, for any p E M the dimension of the span of ap1(p),... , is N by (1.2.3), which implies d?: N. If M is also generic, then ap1,... , Opd are linearly independent, which implies that N d. 0 If M is totally real and generic, we shall say that it is maximally totally real. For example, the submanifold embedded in the standard way in Ce" is maximally totally real. As in the case of a hypersurface (see Proposition 1.1.6), we shall show that there exist local holomorphic coordinates for generic submanifolds and totally real submanifolds for which the defining equations take a particularly simple form. PROPOSITION 1.3.6. Let M be a generic of codimension d in through p0. and let n = N — d = CR dim M. Then there are holomorphic coordinates (z, w) near P0. vanishing at p0. with z E w = s + it E anda real vector-valued smooth function s) defined near 0 in with values in Rd, = 0, = 0, such that near P0. M is given by

(1.3.7)

Im w =

Re w).

PROOF. As in the proof of Proposition 1.1.6, we may assume P0 = 0 and write are complex linear forms. Since the are linearly independent at 0, so are the The proposition follows by taking the L1(Z) as coordinates w1 and applying the implicit function theorem.

P1 = Im (t1(Z)) + 0(2), j = 1,... , d, where the £j

0 We shall refer to coordinates as in Proposition 1.3.6 as regular coordinates for M at Po. The reader should note that regular coordinates are not unique. In §4, we shall prove the existence of regular coordinates which also satisfy additional conditions. For totally real submanifolds we have the following.

PRoPosmoN 1.3.8. Let M be a totally real submamfold of codimension d in C" through P0. and let r = 2N — d. Then there exist holomorphic coordinates (Z', Z") near P0. vanishing at po, with Z' E C'S, Z" E CN_r, and a real vectorvalued smooth function (Re Z') defined near 0 in W with values in Rt, (0) = 0.

§ 1.3.

II

CR, TOTALLY REAL, AND GENERIC SURMANIFOLDS

dçb(0) = 0, and a complex vector-valued smooth function *(Re Z') defined near M is given = 0, d*(0) = 0, such that near 0 in R' with values in by

Im Z' =

(1.3.9)

in addition,

Z'), Z" =

Z').

M is real analytic, then we can choose (Z', Z") such that

PROOF. Since M is totally real it follows from Proposition 1.3.5 that d

0,

N and,

is of dimension N. Hence in view of (1.2.3), the span of (ap1(p),... , we may assume {ap1(p),... , apN(p)} are linearly independent. Since M is of codimension d, we also have (dp1,... ,dpd) linearly independent. As in the proof of Proposition 1.3.6, we may assume p0 = 0 and after a linear holomorphic change N. After subtracting real linear of variables, we have Pj = Im + 0(2), 1 combinations of(p1,... , PN) from the j = 1,... , d — N, we may assume = Lj(x) + 0(2), where the L1 are a set of linearly independent real linear PN+j forms. After reordering, we may assume that Yi Yr, Li(y),... , r are linearly independent as real linear forms. We now put = Pj for 1 d — N. We put and N + 1 d, and = L1(p1,... , pN) for 1 z; = Zj, L1(Z), 1 j j sd—N. Thisgives

j

j

j

j

1

IinZ+0(2), (1.3.10)

p=

ImZ7...,+0(2), ifr+1 Re + 0(2), if N + 1

j

d.

The first part of the proposition, i.e. (1.3.9), then follows from (1.3.10) by using the implicit function theorem. It remains to prove the last claim of the proposition. If M is real analytic, we may choose the local defining functions Pj to be real analytic. The construction performed in the first part of the proof using the real analytic version of the implicit function theorem will give (1.3.9) with and real analytic. We now define new holomorphic sets of coordinates 2', 2" in cr x as follows. Since and are real analytic, we may extend them to be holomorphic in a small neighborhood of 0. By the implicit function theorem, we may define 2' implicitly by

= 2' + that Im 2' = 0 if and only if Im Z' =

Z'). With the coordinates (2', Z"), M is given by Im Z' = 0, Z" = *(Re 2'), where is real analytic. Putting 2" = Z" — iji(2') yields the desired result in the coordinates (2', Z"). D

so

For the case of a maximally totally real submanifold, the defining equations take a simpler form, as indicated in the following result.

L HYPERSURFACES AND GENERIC SUBMANIFOLDS IN

12

C

C" be a smooth maximally totally real sub-

M. There exist holomorphic coordinates Z near P0 vanishing man bid and po at p0 and a real vector-valued function defined in a neighborhood of 0 in R", Z). = 0, such that near p0. M is given by Im Z = with = 0, Furthermore, if h is a germ at P0 of a holomorphic function vanishing on M, then 0. h 0. If M is real analytic, the coordinates Z can be chosen so that

PROOF. The first and last parts are immediate consequences of Proposition 1.3.8. Now let h be a germ at P0 of a holomorphic function which vanishes identically on M. By the first part, we have h(x + i4>(x)) 0 for x in a neighborhood of 0 in RN. Differentiating the latter equation, the reader can easily check, using induction on the length of the derivatives, that all derivatives of h must vanish at 0 0. The proof of Proposition 1.3.1 1 is complete. In §8 we give a local description of a CR submanifold as a CR graph over a generic submanifold. At the other extreme from the totally real submanifolds are the holomorphic submanifolds (or complex submanifolds) defined below. DEfiNITION 1.3.12. A holomorphic submanifold of C"' of holomorphic codimension £ is a subset M of C"' such that for every point P0 E M there is a neigh-

borhood U of P0 and a holomorphic vector-valued function h = (h1,... defined in U such that (1.3.13)

,

Mfl U = (Z E U: h(Z) = 0),

linearly independent in U. A holomorwith complex differentials , phic submanifold of holomorphic codimension 1 is called a (smooth) holomorphic hypersuiface.

Note that if M is a holomorphic submanifold as in Definition 1.3.12, it is a CR submanifold of C"' of real codimension d = 2t and CR dimension N — £, which is the same as the complex dimension of M. Holomorphic submanifolds can be characterized in terms of their CR bundles as follows. PROPOSITION 1.3.14. Let M C CN be a CR subman:fol4. Then the following are equivalent

(i) M is a holomorphic submanifoid, (ii) TCM = TM, (iii) V V = CTM, where V V is the bundle over M whose fibre at any p Mis (iv) dimRM=2CRdImM.

PROOF. The fact that (i) implies (ii)—(iv) follows from the comments preceding

the proposition. The equivalence of (ii), (iii) and (iv) follows essentially from

§1.3. CR. TOTALLY REAL, AND GENERIC SLJBMANIFOLDS

13

Proposition 1.2.8. It suffices therefore to show that (iii) implies (i). Since (iii) implies that dimR M (and hence also codimension M) is even, we write d = for the codimension of M. For any p0 M, let p = (P1,... , be a local defining function of M near p0. Since the rank of (ap1 (0),... , is £ (in view of (1.2.3)) whereas the rank of (dpi(0),... , is 2t, we may assume, after performing a holomorphic affine change of variables in the Z space and a real linear change of the Pj if necessary, that po = 0 and that we have

p1(Z,Z)=ReZ1+0(2),

(1.3.15)

-

Z) = Im Z1 + 0(2),

1

j

write Z = (Z', Z"), with Z' E Ct and Z" E By applying the implicit function theorem we conclude from (1.3.15) that M is given by We

l<j
(1316)

1

with Uj and

v1 smooth, defined near the origin in Ct and vanishing of order at least 2. We claim that the functions 2") + iv1(Z", Z") are holomorphic for all (Z, Z),... , Z E M, j, 1 j £. Since the rank of it follows that

(1.3.17)

1

rkl

I

=&

2")J where '(xt is the £ x £ identity matrix, u = (u1,... , ug) and v = (Vi holds for all Z" near 0. Elementary linear algebra then shows that

(1.3.18)

az"

2") +

az"

Vi),

2") = 0,

which proves the claim. To show that (1) holds, we use (1.3.16) and observe that

M is given by the vanishing of the holomorphic functions

=

— (U1(Z",

2") + iv1(Z", 2")), j = 1,...

This completes the proof of the proposition.

,

0

To conclude this section, we give the following characterization of the different classes of real submanifolds in terms of the operator I introduced in § 1.2.

1. HYPERSURFACES AND GENERIC SUBMANIFOLDS IN CN

14

PROPOSmON 1.3.19 Let M be a real submanifold of CN. The following equivalences hold.

fl JT,,M) is independent of p

(1) M is CR (f and only if

(ii) M istotally real if and only if (iii) M is generic and only + (iv) M is holomorphic (f and only

M.

=0, Vp EM. =

Vp E M. Vp E M.

=

PROOF. Statements (i)—(iii) follow from the definitions and Proposition 1.2.8, while (iv) is a consequence of Proposition 1.3.14. The details of the proof are left 0 to the reader.

§ 1.4. CR Vector Fields and CR Functions Recall that if U is an open set in CN, a smooth complex vector field on U is a mapping X which assigns to every p E U a complex tangent vector at p, varying smoothly with p. Therefore we can write (1.4.1)

X=

bk are smooth, complex valued functions defined in U. Similarly, ak if M is a submanifold of CN, a vector field on M is a mapping X which assigns to every p E M a complex tangent vector to M varying smoothly with p. Hence, we can write

X=

(1.4.2)

+bk(p)—.t—, azk

where ak and bk are smooth, complex valued functions defined on M and satisfying N

+

(1.4.3) k

azk

=0,

(Pi,... , Pd). Note that if X is a vector field defined in an open neighborhood U of M in Ce" which is tangent to M at every p E M, i.e. (1.4.3) holds for p E M, then the restriction of X to M defines a vector field on M. A vector field X on M is therefore a smooth section over M of the bundle for any (and hence every) defining function p =

CTM. We write X E r(M, CTM). Let Ck(M) denote the space of all k times continuously differentiable, complex valued functions on M. If X is a smooth vector field on M, then X maps Ck(M),k 1, into by f Xf.

AND CR flJNO'IONä

CR

Similarly, X maps D'(M), the space of distributions on M, into itself. (For basic definitions and properties concerning distributions on manifolds, the reader can consult standard texts e.g. Schwartz [1] or Hörmander [3). However, we shall not make extensive use of distributions in this book.) DEfiNITION 1.4.4. Let M be a CR submanifold and let V be its CR bundle. A vector field L on M is a CR vector field if for every p E M, is in Vi,.

If M

is of codimension d and L is a CR vector field on M we may write

L=

(1.4.5) with

c1

smooth functions on M and

>2Cj(p)(p) =0,

1

k

d,

for all local defining functions p on M. We may regard a CR vector field L as a smooth section over M of the bundle V ; we write L E f'(M, V). If h is a holomorphic function defined in an open neighborhood of a CR submanifold M and f = hIM, the restriction of h to M, then Lf 0 for any CR vector field L on M. Indeed, since j = I,... , N, any linear combina-

tion of the tangent to M annihilates f. The following definition incorporates this notion and is basic to the subject of this book. DEfiNITION 1.4.6. Let M be a CR submanifold. Then f E Ck(M), k > I, is a CR function if Lf 0 for every CR vector field L on M. Similarly, a CR distribution on M is one which is annihilated by every CR vector field on M. We observe here that if M is a totally real CR submanifold, then any function

defined on M is a CR function, since there are no nonzero CR vector fields. If M is a CR submanifold and h (Z) is a holomorphic function on an open subset U c C's' for which M c U, the closure of U, and h extends smoothly to a neighborhood of M in CN, then h determines a unique smooth function f on M, written f = bvMh, the boundary value of h on M. It is clear by continuity that f is a CR function. We shall discuss boundary values of holomorphic functions more thoroughly in Chapter VII. We give here some examples of smooth CR functions which are not restrictions of holomorphic functions (and some which are not even boundary values of such).

Let M be the real line, considered as a totally real submanifold of C. If j (x) is a nonzero smooth function on R for which all derivatives vanish at EXAMPLE 1.4.7.

I. HYPERSURFACES AND GENERIC SUBMANIFOLDS IN C"

16

j(x) =

then j cannot be the restriction of a holomorphic function on a neighborhood of 0 inC. If, in addition, j(x) is real-valued, then the classical Schwarz reflection principle shows that j(x) cannot be the continuous boundary value, in any neighborhood of the origin, of a holomorphic function in the upper or lower half plane. This gives an example of a CR function which is not the boundary value of a holomorphic function. o (e.g.

EXAMPLE 1.4.8. Let M C C2 be the "flat" hypersurface given by Im Z2 = 0, as in Example 1.1.2. Any CR vector field on M is a multiple of L = The smooth CR functions on M are those of the form f(Z1, Re Z2), where f and all its derivatives are holomorphic in Z1 for all fixed Re Z2. If j(x) is as in Example 1.4.7, then .

f(Z1,Re Z2) = j(Re Z2) a CR function on M which is not the restriction of a holomorphic function in C2 (or in any neighborhood of 0 in C2). In addition, if j(x) is real, then! is not the boundary value of a holomorphic function near the origin in either half space, is

(IinZ2 >0}or{ImZ2 <0). EXAMPLE 1.4.9. Let M be the Lewy hypersurface InC2 given by Im Z2 = I Zi 12,

in Example 1.1.3. Let be a function holomorphic in the upper half plane 0, smooth up to the real line, which does not extend holomorphically in any neighborhood of 0 in C. Since the coordinate function Z2 restricted to M is given by Z2(M = Re Z2 + ilZi 12 and hence has a nonnegative imaginary part, the function J+(Z2IM) is a smooth CR function which is the boundary value of a function holomorphic in the open set Im Z2 > IZ1 12, smooth up to M, which does not extend holomorphically to any neighborhood of 0 in C2. To give an example of a function 4(z) satisfying the above conditions, we let z determined by as

Im z >

(1.4.10)

—r/2 <arg(z) <3jr/2.

Then J+(z) =

satisfies the desired conditions. It will be shown in Chapter VIII that any CR function on the Lewy hypersurface extends holomorphically to one side of the hypersurface; this was shown to be false for the flat hyperplane (Example 1.4.7). DEfiNITION 1.4.11. Let M be a CR submanifold with CR dimension n, and P0 E M. A local basis of the CR vector fields on M near P0 s a set of n CR vector fields L1,... , defined in an open neighborhood U of P0 in M such that the are linearly independent at every p E U.

* 1.5. FINITE TYPE AND MIN1MALITY CONDITIONS

17

Elementary linear algebra shows that if L,,... ,

is a local basis of the CR vector fields near p0, then any other CR vector field L may be written uniquely in a neighborhood of p0 as (1.4.12)

L

= Ecj(p)Lj,

where the c1 are smooth functions of p defined near p0 on M. We note that if f is

then f is CR if and only if L1 f a function on an open neighborhood U of j = 1,... , n, where the are any local basis of the CR vector fields on U.

§1.5. Finite

0,

and Minimality Conditions

In this section we shall introduce the notions of finite type and minimality for CR submanifolds of Ci", which will play an important role in subsequent chapters. We begin with some preliminaries. If X and Y are complex vector fields defined on a smooth manifold M, i.e. X, Y E r(M, CTM), we denote by [X, Y] the commutator of X and Y defined by [X, YJf(p) = X (Yf)(p) — Y(Xf)(p) for any smooth function f defined on M. It is easy to check that [X, Y] is again a complex vector field on M. We may therefore define iterated commutators [X,,, [X12,... , i, k, for any X•1] . ]J, I set of vector fields X a,... , We shall say in this book that such a commutator is of length j, so that [X, Y] is of length 2. (We warn the reader that some authors refer to [X, Y] as a first commutator and go on to further confusion for higher . .

commutators.) If X and Y are real vector fields (i.e. X, Y E F(M, TM)) then their commutator [X, Y] is also a real vector field. Recall that if M c is a CR submanifold, then TCM, the complex bundle of M given in Definition 1.3.1 is a real subbundle of TM and for p E M we have (1.5.1)

dimR TM = 2CR dim M

dimR M.

DEfiNITION 1.5.2. Let M C be a CR submanifold and p0 E M. Then M is of finite type' at the point Po if there are smooth vector fields X,,... , E

r(M, TCM) such that the X,,...

together with their commutators of all lengths span the tangent space of M at po. ,

If M c

is a CR submanifold, then M is of finite type at p0 if and only if there exist CR vector fields on M, L,,... , L, such that the vector fields L1,... , L1, their complex conjugates L,,... , L1, together with their commutators of all lengths span the complexified tangent space of M at p0. Indeed, this 11n

the literature, this notion is also called finite type in the sense of Kohn and Bloom-Graham.

1. HYPERSURFACES AND GENERIC SUBMANIFOLDS IN CN

18

follows immediately from the fact that a section of TCM is the real part of a CR vector field on M. (See Proposition 1.2.8.)

1.5.3. If M c CN isa CR submanifold and P0 E M, let L1,... , a local basis of the CR vector fields in a neighborhood of p0, as in Definition 1.4.11. Then it is easily checked that M is of finite type at po if and only if the vector fields L1,... , L1,... , and their commutators of all orders span CTM at p0. In particular, if M is a hypersurface in the and Lk together span a space of dimension 2n, namely (see Proposition 1.2.8). Since M is of dimension 2n + 1, it is of finite type at p0 if and only if the commutators of the L1 and Lk span a new direction (i.e. one which is not in often referred to as the missing direction.

be

DEfiNmON 1.5.4. If M C is a hypersurface of finite type at and , a local basis of the CR vector fields near po, the minimum length of any commutator of the is the type of M and Lk which, at Po is not in at P0.

L1,...

It can be easily checked that the type of a hypersurface is independent of the choice of local basis of the CR vector fields and is therefore invariant under a holomorphic change of coordinates. The type can also be characterized in terms of local coordinates and local defining function for M; this result will be given in § 1.6.

EXAMPLE 1.5.5. If M is the flat hyperplane in CIV of Example 1.1.2, given by = 0, then TCM is spanned at eveiy point by a

a

a

a

8Yw—i

,

It follows that r'(M, TCM) is closed under the operation of taking commutators. Hence there is no point of finite type in M, since the direction a/axN is missing at eveiy point. 1.5.6. The Lewy hypersurface given in Example 1.1.3 is of type 2 at all points, as can be easily checked by the reader.

EXAMPLE 1.5.7. Let M be the generic submanifold of codimension 2 in C3 given by M = (Z E C3: Im Z2 = 1Z112, Im Z3 = 1Z114}. Then the dimension of M is 4 and its CR dimension is 1. The vector field

L=

—t.. — 2iZ1 -2az1 az2

—

-2-. az3

§ 1.5. FINITE TYPE AND MINIMALITY CONDITIONS

19

spans the CR vector fields, where the coefficients are regarded as functions on M. Note that L and L are linearly independent at every point p in M. By computing are commutators of L and L, one checks that the two missing directions of

obtained by commutators of lengths 2 and 4 respectively. Hence M is of finite type at every point. REMARK 1.5.8. If M is a real submanifold of CN, and if X and Y are smooth tangent to M then the vector field [X, Y] defined in complex vector fields on is again tangent toM. If L and L' are antiholomorphic vector fields on Ci", then so is [L, L']. Hence if L and V are CR vector fields on M, then their commutator

is again a CR vector field on M. (The reader can verify these observations by calculations in local coordinates.) We may write this briefly as [V. V] C V.

(1.5.9)

where V is the CR bundle of M. For real-analytic CR submanifolds, we have the following. THEOREM 1.5.10. Let M c C's' be a connected real-analytic CR Suppose M is offinite type at some point p0 E M. Then,for every pi E M, there 0, such that M exists an open neighborhood U of in M and f E is offinite type PROOF.

at p E U if and only if f(p)

f

0.

Pickpi EM.

are defined in an open neighborhood U of p' in M. We may assume that the real-analytic in an open neighborhood of U, the closure of U. First, assume that be the collection of all commutators of the vector fields po E U. Let (C, X3, including the X1 themselves. Also, denote by the collection of all determinants, not identically zero, of the dim M x dim M matrices obtained by choosing dim M real-analytic vector fields in the collection (C, } and forming the matrix of their coefficients. Note that the collection } is not empty, since there exists such a determinant which does not vanish at P0 by the assumption that M is of finite type at P0. Then the real-analytic function in U defined by

f(

— )—

(f1(p))2 2i supq€O 1f1(q)12

precisely on the set of infinite type points in U. If P0

U, then P0 can be connected to P1 by a curve in M, and a standard connectedness argument combined with the above completes the proof. 0 vanishes

1. HYPERSURFACES AND GENERIC SUBMANIFOLDS IN C"

20

1.5.11. It follows from the Theorem 1.5.10 that the set of points where M is not of finite type is a proper real-analytic subvariety of M for definitions).

(see §5.2

We return now to the general situation of a smooth CR submanifold to introduce a notion more general than that of finite type. DEfiNITION 1.5.12. Let M C CN be aCR submanifold and p0 E M. We say

that M is minimal at p0 if there is no (germ of a) real submanifold S C M through

P0 such that TM is tangent to S at every p E S and dimp S
M. If M is

S C M, where S is a submanifold for which TM C for all p E S. Then all the commutators of vector fields in F(M, TCM) are also tangent to Sat all p S. Since M is of finite type at po. the space spanned by such commutators at po is of dimension m = dimRM. Hence dimRS = dimRM, which proves that M is minimal at 0

PROOF. Suppose that po

1.5.14. Let M c

C2

be given by Im w =

Re

wIzI2. Then M is

neither of finite type nor minimal at 0. In fact, M contains the complex hypersurface w = 0, which is a CR submanifold through 0 of real dimension 2 (< 3 = dimR M) and CR dimension I (= CR dim M).

Minimality has the following important equivalent description for hypersurfaces.

be a smooth hypersurface and p0 E M. THEOREM 1.5.15. Let M C Then M is not minimal at p0 if and only if there exists a (germ of a) holomorphic hypersurface S C M through p0. PROOF.

Since M is not minimal at po there exists a CR submanifold S C

M through P0 with CR dim S = CR dim M = n. Hence dimk S ? 2n, but since dimR S < dimR M = 2n + I, we must necessarily have dimR S = 2n. By Proposition 1.3.14, S is a holomorphic submanifold of codimension 1, i.e. a holomorphic hypersurface. 0

EXAMPL.E 1.5.16.

Let

be the smooth real valued function given by

§1.6. COORDINATh REPRESENTATIONS FOR CR VECTOR FIELDS

*0')

(1.5.17)

I

if y

>0,

=10

if y

0.

Let M C C2 be given by {(z. w) : Im w

=

where z, w E C,

z

21

= x + iy.

Then we claim that M is minimal, but not of finite type at 0. Indeed, suppose

that M is not minimal at 0. Then by Theorem 1.5.15, there exists a holomorphic hypersurface S c M through 0. Since a/az is tangent to S at 0, it follows that S must be given by an equation of the form w = h(z) with h holomorphic in a neighborhood of 0 in C and h(0) =0. Since S C M we must have

h(z)

for z = x + iy in a neighborhood of 0. This is impossible, since

is not a realanalytic function. The fact that M is not of finite type at 0 follows easily from the fact that is flat, i.e. all its derivatives vanish, at 0. We observe also that M is not minimal (or finite type) at any point (z, w) for which Im z <0, whereas M is of finite type, and hence minimal at any point (z, w) for which Im z > 0. This is in contrast to the real-analytic case, as stated in Theorem 1.5.10 and Remark 1.5.11.

It will be shown in Chapter IV (Theorem 4.1.3) that for a real-analytic CR submanifold, finite type is in fact equivalent to minimality. § 1.6.

Coordinate Representations for CR Vector Fields

In this section we assume that M is a generic submanifold of codimension d in CN and give a local basis for the CR vector fields on M. The CR dimension of M is n = N — d. Let a of p M near po. Since M is generic, we have ap1,... , linearly independent (Definition 1.3.4). By relabeling the coordinates in Cs"' we may take (z, w) E w E z E C" and assume that •

PwI

aU)4

:

aw1

is invertible near p0, assumed to be the origin. Similarly we let

I

PzI

az.

I. HYPERSURFACES AND GENERIC SUBMANIFOLDS IN CN

22

Using matrix notation, we write

considered as an n x 1 matrix, with similar notation for

PRoPosmoN 1.6.1. A local basis of the CR vectorfields near the origin is given by(L) = T(L1,... , La), with

(L)

(1.6.2)

=

(h),

—

where TA stands for the transpose of a matrix A.

PROOF. It is clear that the L1 are linearly independent, since for each j,the partial appears only in L1. In order to show that the vector fields are differentiation

tangenttoM,wemustshowthatLjp& =OonMforj=1,... ,n,k=1,...d. = We regard p as a 1 x d matrix, so that in matrix notation Multiplying the right hand side of (1.6.2) to the right by p yields that (L)p and hence the tangency of the L1.

0

0

It will be convenient to express the L1 in terms of local coordinates of M. For this, we assume (z, w) are regular coordinates (see Proposition 1.3.6) and that M tv)= Themapping isgivenby(1.3.7).

'Il:

(1.6.3)

is adiffeomorphism from an open neighborhood c of the origin in

For j = 1,...

,

n

s) the vector fields A1, are given in

vector field L1. In the local coordinates (z, matrix notation by

(A) =

(1.6.2') where

given by

is

onto M.

we put A1 = '4ç' (L1), the pushforward by '1'' of the CR

—

(—),

=

the n x d matrix given by

= a(/)J/aZk.

= a411/ask and

asa

is

the d x d matrix

§ 1.6. COORDINATh REPRESENTATIONS FOR CR VECTOR FIELDS

23

We leave the details of the calculation to the reader. In later chapters, we shall to denote the CR vector fields of M in any abuse the notation and use L1,... , set of local coordinates. It is worth noting separately the case where M is a hypersurface. Recall the form of the defining equation (1.1.7) for M in regular coordinates (z, w) given by Proposition 1.1.6. PROPOSITION 1.6.4.

If M is a hypersurface in

given by (1.1.7), then the

basis (1.6.2) of the CR vector fields on M is given by

j=l,...,n.

(1.6.5)

In the parametrization

:

(z.

s)

i—+

(z.

s+

s)), their pushforward by

is given by

(1.6.6)

The form of the CR vector fields given in (1.6.6) is particularly useful for studying commutators of vector fields. For a hypersurface M, this basis is used to determine if M is of finite type (Definition 1.5.1). A hypersuiface M is of (finite) type m at çf and only there exist regular coordinates (z, w) vanishing at with M given by (1.1.7) near where THEOREM 1.6.7.

=

(1.6.8) Pm (z,

is a nontrivial real homogeneous polynomial of total degree ns in Pm(°'Z)

zandzsatisfying Pm(Z,O)

To prove Theorem 1.6.7, we shall use the following proposition concerning commutators, which will be useful in other contexts as well. PRoPosmoN 1.6.9. If

Ak are given by (1.6.6), then

(1.6.10) and there are smooth functions (1.6.11)

[A1, Ak] = 0, (z,

s) such that

j,k= I,... ,n.

24

I. HYPERSURFACES AND GENERIC SUBMANIFOLDS IN CN

IfA=(Ai,...,Aq)withAjE(Ai,...,An,Ai,...,An),then [Aq, [Aq_i,... , [A1, [Ai, Ak]]... ]1 =

(1.6.12)

with some smooth function

(z,

s). If in addition,

(1.6.13)

with Pr a real homogeneous polynomial of degree r, r fijk(z,Z,O) =

(1.6.14)

(1.6.15)

aZJ 8Zk

0) = AqIo. . . Ai

fijtcA(Z,

2, then

+

aZk

(fl PROOF. To prove (1.6.10), we note first that, by Remark 1.5.8, [A1, Ak]isaCR vector field and hence a linear combination of the A,, with smooth coefficients.

On the other hand, from the form of (1.6.6), [Aj, Ak] is a multiple of

so

that the multiple must be 0. The form of (1.6.6) also gives (1.6.11) and (1.6.12) immediately. To prove (1.6.14), we use a direct calculation to show (1.6.16)

+

2i8jk = -

(

1

\

.

'h,

(1

(.

.

(.

On the other hand, by (1.6.13) we have, (1.6.17)

0)

=

dZk az1

+

= O(IzI'2),

and similarly,

(1 6 18)

0) = O(IzIt_I),

0) = = O(IzI).

Putting s = 0 in (1.6.16) and using (1.6.17) and (1.6.18) yield (1.6.14). In order

to prove (1.6.15) we use an induction on q, beginning with q = 0, for which the

25

§ 1.6. COORDINATE REPRESENTATIONS FOR CR VECTOR FIELDS

result is given by (1.6.14). The details are left to the reader. This completes the 0 proof of Proposition 1.6.9. The following is also needed for the proof of Theorem 1.6.7. PROPOSITION 1.6.19. Given a hypersurface M C

p0

M. and a positive

1, there exist regular coordinates (z, w) near P0. vanishing at p0. such that M is given locally by (1.1.7) with

integer J

(1.6.20)

4'(z,

0) = q(z,

+

is a real polynomial (not necessarily homogeneous) of total degree J satisfying q(z, 0) q(0, 0.

where q(z,

PROOF. We prove the proposition by induction on J. By the definition of regular coordinates, we may choose (z, w) so that (1.6.20) is satisfied for J = (in which case q(z, 0). Assume (1.6.20) for J replaced by J — 1; we shall prove it for J. Writing the Taylor expansion of (z, 0) at the origin we have where = is a polynomial of total + degree J. Using the induction hypothesis, we have £(z, = Im hj(z) + q(z, where h1 is a homogeneous, holomorphic polynomial of degree J and q(z, a real polynomial (not necessarily homogeneous) of total degree J satisfying q(0, 0. We consider the holomorphic change of coordinates q(z, 0) defined by z' = z, w' = w — hj(z). Now M is defined by Im w' = 4"(z', s'), where 1

4"(z',

s')

=

+ Re hj(z')) — Im

It follows from the assumption on

4"(z',

0) =

Re

hj(z'), s' =

Re

w'.

that — Im

= q(z',

+

which proves (1.6.20) and completes the proof of Proposition 1.6.19.

0

In Chapter IV (Theorems 4.2.6 and 4.2.16) we give a stronger version of Proposition 1.6.19 which is also valid for generic submanifolds. We return now to the proof of Theorem 1.6.7. PROOF OF THEOREM 1.6.7. Suppose first that (1.6.8) holds. By Proposition 1.6.9, all commutators of the A1, of length less than m vanish at the origin, since by (1.6.15) they involve derivatives of Pm of order less than m. Also, applying (1.6.15) again, by an appropriate choice of A1,... , Aq, we can find a commutator of length m which gives the missing direction at the origin.

I. HYPERSURFACES AND GENERIC SUBMANEFOLDS IN CN

26

Conversely, assume that M is of type m at the origin. We apply first Proposition 0) = q(z. 1.6.9 with J = m, i.e. we can assume that + 0(IzIm+I), with a (not necessarily homogeneous) polynomial of total degree m satisfying q (z, q(z, 0) = q(0, = 0. We claim that in fact q(z, is homogeneous of degree 0 a homogeneous m. Indeed, if q(z, with PT(Z, = pr(Z, + polynomial of degree r with r < m, then by the assumption on q, we have = 0. The first part of the proof would then show that M is of Pr(Z, 0) = Pr(O, type r. This proves the claim and completes the proof of the theorem. 0 EXAMPLE 1.6.2 1. Let Mk+I be the hypersurface inC2 given by

+

Im w =

fork =

1,2

ThenthetypeofMk+I at0isk+ 1. Hencethehypersurfaces

Mk and Mk' are not biholomorphically equivalent, at 0, unless k

= k'.

For generic submanifolds of higher codimension, unlike the case of hypersurfaces, it is much more difficult to determine finite type in terms of local coordinates. This problem will be addressed in Chapter IV. Also, for real-analytic generic sub-

manifolds, a condition equivalent to finite type will be given in Chapter LX (see Corollary 9.4.5). § 1.7.

Holomorphic Extension of CR Functions

In this section we explore the relationship between CR functions on a generic submanifold and the restriction of holomorphic functions to such manifolds. We shall first prove that a real-analytic CR function on a real-analytic generic submanifold M c CN extends as a holomorphic function in a neighborhood of M in CN (Corollary 1.7.13). Note that if M is a real-analytic generic submanifold, then in the parametrization (1.6.3) the function s) is real-analytic and hence, in particular, the variable s can be complexified. Thus, we can extend the mapping 4', given by (1.6.3), to a local diffeomorphism '1': —+ (defined in a neighborhood of 0) as follows:

(1.7.1)

We shall abuse the notation and denote by 'I' both the function given by (1.6.3) and its extension '1' (given by (1.7.1)). If f is a CR function on M, then the function defined in a neighborhood of 0 in (1.7.2)

u(z,

s)

given by

= f(z, s + iØ(z,

s))

= f(4'(z,

s))

§17. HOLOMORPHIC EXTENSION OF CR FUNCTIONS

27

will also be referred to as a CR function on M. We shall say u(z, s) extends holomorphically to a neighborhood of 0 M 1ff does, i.e. if there is a holomorphic function h (z, w) in a neighborhood of 0 E CN such that u(z,

(1.7.3)

s)

= h(z, s +

Observe that the extension of u(z,

s)).

s), given by

u(z,z,s+it)

(1.7.4)

for t E Rd small, is in general not a holomorphic function of (z, s + it), but only

of s + it. PRoPosmoN 1.7.5. Let M be a real-analytic generic submanifold of C's" of codimension d. If u(z, s) is a CR function on M, then u(z, s) extends holomorphically to afull neighborhood of 0 çf and only there exists e > 0 such that for every z E C's, Izi < €, the function s i—* u(z, s) extends holomorphically to the open set (s + it C", IsI < E, hi < E) in such a way that the extension u(z, s + it) is a bounded, measurable function of all its variables. PROOF. As mentioned above, to say that u (z, s) extends holomorphically to a neighborhood of 0 means that there is a holomorphic function h(z, w) in a neighborhood of 0 in s)) u(z, s). Hence, if satisfying h(z, s + u extends holomorphically to a neighborhood of 0, then u (z, s) extends holomorphically in s for fixed z by putting (1.7.6)

u(z, z, s + it) = h(z, s + it +

s + it)) = h(41(x, y, s, t)),

and satisfies the statement in the proposition provided that E is sufficiently small. This extension of u(z, s) is made possible by the real-analyticity of s). Conversely, suppose that s i—* u(z, s) extends holomorphically for fixed z, Izi <E, and satisfies the assumptions of the proposition. In light of the first part of the proof, we define a function h(z, w, ri), by h(z, w, th) = u('I''(z, w)). By the hypotheses, h is a measurable, bounded function. We shall show that h is holomorphic. For this, we introduce vector fields j = 1,... , n + d, defined in a neighborhood of 0 in as follows. Put

Afl+k=-L.+i-!-, k=1,... ,d,

(1.7.7)

ask

and (1.7.8)

(A)

=

—

s + it)T(I +

s + it)Y' (i-)

1. HYPERSIJRFACES AND GENERIC SIJBMANIFOLDS IN CN

28

j

where we have used matrix notation and the components of (A) are A1, = = A, j = 1,... , n, where the A are given by 1,... , n. Note that are linearly independent and annihilate the functions (1.6.2'). Since A1,... ,

z,j =

1,... ,n,andwk =sk+itk+i(I)k(z,z,s+if),k= 1,... ,d,itfollowsthat the pushforwards j = 1,... , n +d, form a basis for the antiholomorphic Thus, to prove that h(z, w) is holomorphic, it suffices

vector fields in CN near to show that

j=l,...,n+d

(1.7.9)

in the sense of distributions, since h is bounded and measurable. By definition of the pushforward, we have

w)) = A1u(z,

w, iii) =

(1.7.10)

s

+ it).

We must therefore show that the right hand side of (1.7.10) vanishes identically. We consider first the case j = n + k, for some k =I,... , d. We claim that

(1.7.11)

\aSk

atk,

in the sense of distributions in the open set YE {IzI < €, IsI < ti < C E To prove the claim (1.7.11), it suffices to show that for any xi(z, both with sufficiently small support, we have and x2(s, t) E (1.7.12)

s

+ it), (-p-- + \ask a:k,

t))\

0.

i

By the boundedness and measurability of u and Fubini's theorem, the left hand side of (1.7.12) is

—f [f R2u(z,

s

+ it)

ask

+

alk

X2(S,

t)dsdt] Xi(z,

Integrating by parts and using the holomorphy in s + it, we see that the inner integral vanishes identically, which proves (1.7.12) and hence the claim. Since the coefficients of A1 are holomorphic in s + it it follows from (1.7.11) that we have

atkl

j= 1,... ,n, k= I,... ,d,

in V( in the sense of distributions. The latter implies that A1u(z,

s

+ it) can be

viewed as a holomorphic function in s + it valued in the space of distributions in

*1.7. IIOL.OMORPHIC EXTFNSION OF CR

s) in view of (1.7.11). The same holds for the open ball of radius E in It follows, in particular, that u(z, s + it) (as well as Aju(Z, s + it)) can be viewed as a real-analytic function of t valued in the space of distributions in the ball

= {(z,s,t): t = coincides with the applied to the restriction of u. In particular, since u(z, s) to restriction of (given by (1.6.2')), we is CR and the restriction of A1 to flo coincides with to), it follows that the restriction of the distribution

to

deduce that the holomorphic function A1u(., s + it), valued in the space of distributions in the ball of radius in R2', vanishes at t = 0. A simple uniqueness 0 in VE, s + it) argument for holomorphic functions implies that which in view of (1.7.10) yields (1.7.9). The proof of Proposition 1.7.5 is now

0

complete.

The following important corollary is an easy consequence of Proposition 1.7.5.

COROLLARY 1.7.13. Let M C C" be a real-analytic generic submanifold and f a CR function in a neighborhood of p E M. Then, f extends as a holomorphic

function in a neighborhood of p in CN if and only if f is real-analytic in a neighborhood of p in M. Proposition 1.7.5 and Corollary 1.7.13 will be used to study the holomorphic extension of CR mappings between real analytic hypersurfaces in Chapter VIII. We conclude this section by showing that any smooth CR function on a smooth generic submanifold M has a unique formal holomorphic Taylor series at every

pointpE M. PROPOSITION 1.7.14. Let M be a smooth generic of CN of codimension d and CR dimension n. Assume that 0 E M and f is a smooth CR function

defined in an open neighborhood of 0 in M. Then, there exists a unique formal

holomorphic series (1.7.15) U

Z(u)

a local parametrization of M into CN (with Z(0) = 0), then the Taylor series of the function u i-+ f(Z(u)) at 0 is given by

such

that

(1.7.16)

u

is

f(Z(u))

Note that the composition of formal power series with no constant tenns is well defined and, hence, the right hand side of (1.7. 16) is also well defined. If u is the restriction to M of a holomorphic function U defined in a neighborhood of 0 in CN then the series(1.7.15) is the Taylor series ofU atO; that is, ca =

I. HYPERSURFACES AND GENERIC SUBMANIFOLDS IN C"

30

PROOF. It suffices to prove Proposition 1.7.14 in regular coordinates Z = Cd near 0 M and with the embedding (1.6.3). Thus, w (z, w), z E for the remainder of this proof we assume that M is given by (1.3.7). We claim there exists a that, given a smooth function h(z, s) defined near 0 E unique formal series in 2n + d variables (1.7.17)

with

x =

(xis...

,

such that the Taylor series of h(z,

(1.7.18)

s) at 0 is given by

—'

This is straightforward to verify since 4(O, 0, 0) = 0 and d4(0, 0,0) = 0, and we leave the verification to the reader. To complete the proof, we must show that if h is CR, then dapy = 0 for all multi-indices fi with IfiI > 0. For this, we let be the basis for the CR vector fields on M given by (1.6.2'). Observe

thatwehave,foralla,8,y,andj=1,... ,n, A1Za

(1.7.19)

E0,

A1(s

1,...

shall use the notation > To show that dafl, = 0, for for any multi-index s) and set to each term in the Taylor series (1.7.18) of h(z, apply = 0 for every This completes the proof in view of (1.7.19), since

=

where

—

,

We

= 0, we

= 0. with

0 § 1.8. Local

Coordinates for CR Manifolds

In this section we give a convenient system of local coordinates for arbitrary CR submanifolds and we prove that any smooth CR submanifold is locally a graph of a CR mapping over a generic submanifold in a lower dimensional space. We also show that any real analytic CR submanifold can be embedded as a generic submanifold of some complex submanifold. We have the following. THEOREM 1.8.1. Let M be a CR subman (fold of codimension d in CN thmugh

Then there exist an integer d1, holomorphic coordinates near po. vanishing at po, a generic man jfold M1 in and d — d1 CR functions on = 0, such that, near P0. M is the graph , M1, *(0) *= of over M1, i.e. p0.

i/i

(1.8.2)

M

= ((m,

*(m)) E CN

m

M1}.

§ 1.8. LOCAL COORDINAThS FOR CR MANIFOLDS

31

if M is real analytic, then the above holomorphic coordinates can be chosen so that%t' PROOF.

Let Pt,...

of {dp1,...

and d1 the rank , Pd be defining functions for M in We now proceed as in the proofs of Propositions 1.3.6 and

,

1.3.8. By linear algebra and an application of the implicit function theorem we may choose coordinates (Zi, vanishing at

W'i,...

. . . ,

W1, .

,

. .

,

so that after a linear transformation the P1 have the form —d,

1

(1.8.3)

I

I

P1+d1

are smooth, real valued functions, and the where the are smooth complex valued, all vanishing at the origin together with their differentials. Let M1 be the generic manifold in defined near p0 by

1<j<2d1—d.

(1.8.4)

Denote by V and V1 the CR bundles of M and M1 respectively. Note that dimcV = dimcVi =N—d1.

Itremainstoshowthatthe*1areCRfunctionsonMi. Le14, 1

j

be antiholomorphic vector fields in CN -(d—d1) such that their restrictions to

a local basis of the sections of V1 near po = 0. Since (1.8.5)

£jpk(P)=O, pEM1,

any section L of V , i.e.

I

1

L E r'(M, V), is necessarily of the form dd1

(1.8.6)

L

=

a

Nd1

+

From the equations

(1.8.7)

1

and using (1.8.3) and (1.8.6),

it

follows that

N—d1

(1.8.8)

k=1,...,d—d1,

—d,

form

1. HYPERSURFACES AND GENERIC SUBMANIFOLDS IN CN

32

j

d — d1, and using the fact that the holds on M. Applying L to Pj+741—d, 1 vanish at 0, it follows that the functions Aj in (1.8.6) satisfy differentials of the = 0, j = 1,.. .d — d1. Hence we may choose a basis for the sections of V of the form d—d1

(1.8.9)

a W*

are smooth functions near 0. It then follows from (1.8.8) that the axe annihilated by the Since the 4, j = 1,... , N — d1, span functions the CR vector fields on M1 near it follows that the *k are CR functions on M1. This proves (1.8.2) with a CR vector-valued function on M1. It remains to show that if M is real-analytic then one can choose the in (1.8.2) to be zero. For this, we note first that if M is real-analytic the 'fri in (1.8.3) are are also CR functions on M1, they real-analytic. By Corollary 1.7.13, since the Let (z, w') extend to be holomorphic in some neighborhood of M1 in are replaced by the w'). Then if the w7 be the holomorphic extension of (z, new holomorphic functions w7 — (z, w'), j = 1, . d — d1, we achieve the 0 desired form, i.e. = 0 in (1.8.2). where the

.

.

As an immediate corollary of Theorem 1.8.1, we obtain the important result that a real-analytic CR submanifold is locally a generic submani fold in a (possibly) lower dimensional complex space.

through COROLLARY 1.8.10. Let M be a real-analytic CR submanifold in cmv, with m' + po. Then there exist holomorphic coordinates (Z', Z") E cm' rn" = N, near p0 and vanishing at p0, such that M c (Z" = 0} cm' near PG with M a generic subman (fold of {Z" = 0} PROOF.

cm'.

By Theorem 1.8.1, there are local coordinates (z, w', w") near po,

vanishing at po, so that M

is

given by

j=1,...2d1—d, w"=O. If we take Z' = (z. w') and Z" = w", the result immediately follows.

0

The following result shows that a generic submanifold is not contained in any proper complex submanifold.

PROPOSITION 1.8.11. Let M c be a smooth CR real-analytic, then the following two conditions are equivalent.

If M is

(i) M is a generic submansfold of ci'. (ii) Any germ of a holomorphic function at any point in M which vanishes on M must vanish identically.

NOTES FOR CHAFFER 1

33

If M is assumed to be only smooth, then (i) implies (ii). PROOF. The fact that if M is smooth then (1) implies (ii) is a consequence of Proposition 1.3.11. Indeed, through any point po M, there exists a maximally totally real submanifold of C" contained in M, as can be seen by taking regular coordinates (z, w) vanishing at p0 (Proposition 1.3.6) and restricting to Im z = 0 in (1.3.7). Now suppose M is real-analytic. We shall show that if M is not generic, then (ii) fails to hold. If M is not generic, there is a point P0 M around which M is not generic. By Corollary 1.8.10, since M is CR, we can find holomorphic M is coordinates Z = (Z', Z") in cm' x cmi' with m" > 0, such that near contained in {Z" = 0). Hence the components of Z" are holomorphic functions which vanish identically on M contradicting (ii). The proof of Proposition 1.8.11 0 is complete. Using Proposition 1.8.11, one can check that the submanifold given by Z" = 0 is the unique holomorphic submani fold of C" of smallest possible dimension which contains M as a generic submanifold. Indeed, if h is a holomorphic function which vanishes on M, then by Proposition 1.8.11, the restriction of h to Z" = 0 vanishes

identically. Hence any local holomorphic submanifold containing M must also contain the submanifold given by Z" = 0, i.e. the latter is the smallest germ of holomorphic submanifold containing M. This motivates the following definition. DEfINITION 1.8.12. Let M be a real-analytic CR submanifold in CN through of smallest possible po. Then the germ at P0 of the holomorphic submanifold of dimension containing the germ of M at P0 is called the intrinsic of M at po. 1.8.13. By making use of Corollary 1.8.10, many questions concerning local properties of real-analytic CR manifolds can be reduced to the case of generic submanifolds.

Notes for Chapter I The study of real hypersurfaces in complex space goes back to Poincaré [I] and Cartan [11, [2] early in the twentieth century. It was Lewy [11, [2] who discovered the importance of the CR vector fields of a real hypersurface in connection with his study of boundary values of holomorphic functions. This led to his example of a first order partial differential equation without solution. The notion of finite type was introduced in PDE theory by Hormander [1]. Finite type in several complex variables, as defined in this chapter, was studied by Kohn [1] for hypersurfaces and later by Bloom and Graham [1] for higher codimensional generic submanifolds. Minimality was defined by Tumanov [1] in connection with the extension of CR

34

I. HYPERSURFACES AND GENERIC SUBMANIFOLDS IN CN

functions into wedges (see also Chapter VII and its notes) and earlier by Trépreau [1] for the case of a hypersurface. Most of the other definitions and results in this chapter are by now standard. We mention some early works where these concepts appeared, including Rossi [1], Tomassini [1], Greenfield [1], Andreotti-Hill [1]. It would be difficult to pinpoint to whom each definition and/or result is due. In this book we have restricted attention to CR manifolds, i.e. those for which the CR dimension is constant. However, important results regarding submanifolds for which the CR dimension varies from point to point have been obtained by a number of mathematicians, including Bishop [1], Moser and Webster [1], Kenig and Webster [1]. For basic material dealing with real submanifolds in complex space, we mention the recent books by Boggess [1], D'Angelo [3], Jacobowitz

[11, and Treves [2]. The reader can consult e.g. the books by Helgason [1] or Kobayashi and Nomizu [1] for standard material on differential geometry.

CHAPTER II

ABSTRACT AND EMBEDDED CR STRUCTURES

Summary In this chapter, we define formally integrable and integrable structures on differentiable manifolds. Abstract CR manifolds are a special case of these structures. In §2.2 we define the Levi form and Levi map of an abstract CR manifold and give some basic properties. We also define pseudoconvexity for CR manifolds of hypersurface type and prove some basic results for embedded hypersurfaces. A discussion of CR mappings between CR manifolds is given in §2.3. In §2.4 and §2.5 we prove an approximation theorem for solutions of integrable structures. This result, which implies that CR functions on generic submanifolds of CN can be uniformly approximated by holomorphic polynomials, will be crucial in Chapter VIII, where holomorphic extension of CR functions is studied. §2.1. Formally Integrable Structures on Manifolds

In this section we consider a subbundle V of CTM, the complexified tangent bundle of M, where M is a smooth manifold. (Here we do not assume that M is embedded as a submanifold of CN, as in Chapter I.) For p E M, let n be the complex dimension of the fiber V,, of V at p and write dimR M = n + m. We shall say that V is formally integrable (or involutive) if the space of smooth sections of V. C°°(M, W) , is closed under commutators, i.e. if L1, L2 E W) , then EL1, L21 E W). We shall abbreviate the property of formal integrability by writing (2.1.1)

[V,V]cV,

and refer to (M, V) as aforinally integrable structure. An important subclass of the formally integrable subbundles are the integrable ones.

A subbundle V of CTM of dimension n is integrable if for any p E M there exist m = dimR M — n smooth complex-valued functions Zm defined in an open neighborhood C M of p with C-linearly independent differentials dZ1,... , such that L E r(M, V) and DEfiNITION 2.1.2.

35

11. ABSTRACT AND EMBEDDED CR STRUCTURES

36

j = I,... ,m. For

E Mfixed,any such setof functions will be called a family of basic solutions in

vanishingat po,

We note that if V is integrable, then (M, V) is necessarily a formally integrable structure. Indeed, if L1, L2 E W) , then ELi, L21 also annihilates all the functions Z1 given in Definition 2.1.2. By dimension, at any point p E the space of complex tangent vectors annihilated by the complex covecHence, the vector field EL1, L2] is tors dZj(p),... , dZ,,(p) coincides with a section of V. If the subbundle V is integrable, we shall refer to (M, V) as an integrable structure on M. For example, a nowhere vanishing complex vector field L on a manifold M defines a formally integrable structure with fiber dimension 1. l'his structure is

integrable if in a neighborhood of every point in M there exist dimk M —1 solutions of the equation Lh = 0 with linearly independent differentials.

DEfiNITION 2.1.3. A formally mtegrable structure (M, V) is called aformal

CR structure if for allp EM (2.1.4)

flY,, =

(0).

We shall also refer to a formal CR structure as an abstract CR man (fold, and to V as its CR bundle. A smooth section of V will be called a CR vector field on M. A function (resp. distribution) on M is a CR function (resp. CR distribution) if it is annihilated by all the CR vector fields on M. The number n = dimc for any p E M, is called the CR dimension of M. Ifdimk M = m + n as above, then the numberd = m — n will be called the CR codimension of M. Ifd = 1, the CR structure is said to be of hypersurface type. The significance of the CR codimension is the following. If (M, V) is integrable, then the basic solutions Zi, .. . Z,, provide a mapping from a neighborhood of p0 into Ctm. This mapping is in fact an immersion (as is shown following Proposition

2.1.5), and hence the image manifold is of real dimension m + n and its real codimension in CM coincides with the CR codimension d = m — n. If a formal CR structure (M, V) is integrable, we shall refer to it as an integrable CR structure or a locally embedded CR manifold. The following result connects the definitions of this section with those given in Chapter 1. PROPOSITION 2.1.5. If M is a CR submamfold in CN and V is the CR bundle of M defined in Chapter 1, then (M, V) defines an integrable CR structure. PROOF. By Definition 1.3.1, V is a subbundle of CTM and it is easy to see that (M, V) is a formal CR structure. We must show that it is integrable. Let p0 E M and consider coordinates Z = (z, w', U)") E CN as in the proof of Theorem 1.8.1

37

§2.1. FORMALLY INThGRABLE STRUCTURES ON MANIFOLDS

(see (1.8.3)). With the notation of (1.8.3), dimc V,, = N — d1, for all p E M nearpo,anddimRM = 2N—d. Hence,herewehavem = N—d+d1. It follows from (1.8.3) that the restrictions of the coordinate functions Zi,... , ZN—I,I, have independent differentials near P0 = 0 and, by definition, they —d are annihilated by any CR vector field on M. This shows that V is integrable and 0 completes the proof. The reader should note that, for an integrable structure (M, V), the mapping are a Z(p) E Ctm, with Z(p) = (Zi(p),... , Zm(p)) where the 3p family of basic solutions in C M as in Definition 2.1.2, is in general not of need not be a real submanifold of Cm, constant rank, and hence the image even if is small around Po• (Here, cm is viewed as R2".) However, if we have an integrable CR structure, i.e. (2.1.4) holds, then the mapping p '-* Z(p) is an is of constant rank m + n and hence is an immersion. In this case embedded real submanifold of Ctm of dimension m + n, provided is sufficiently small around P0. To see that p i-+ Z(p) is of constant rank m + n, we observe that (2.1.4) implies that dimc = 2n for every p E M. Moreover, we have

= are respectively spanned by {dZi(p),... , fdZ1 (p),... , dZm(p)). It follows that the dimension of the span of

and also that

and

and

(dZ,(p),... fl = 2m — (m + n — 2n) = m + n. This observation justifies the interchangeable use of "locally embedded" CR manifold for "integrable" CR manifold. The following definition gives an important class of integrable structures which are not CR. in

is 2rn —

DEfiNITION 2.1.6. Aformally integrable structure (M, V) is called real çffor

all p

M

(2.1.7) We have the following.

THEOREM 2.1.8. Any formally integrable real structure (M, V) is integrable. PROOF. We first claim that there is a subbundle e of TM, the real tangent bundle of M, such that e = with e also formally integrable, i.e. [E, El c £. (=

38

II. ABSTRACT AND EMBEDDED CR STRUCTURES

Indeed, if L1,...

isabasisof r(U, V), where U isanopensubsetofM,then = 1,... , n also span r(U, V) over C by (2.1.7). Hence one ,

Re L1, Im j can choose a local basis of real vector fields. The formal integrability of e is an immediate consequence of that of V. The proof of the theorem now follows from 0 the local Frobenius integrability theorem stated below. THEOREM 2.1.9 (LocAl FROBEN1US INTEGRABILrrY). Let c C smooth real vector fields in If L1,... ,

set and L11... ,

independent at every p E

and satisfy

[L1, Lk] =

>cikrLr, j,k = 1,... In,

where cj*r are smooth functions in c, then for any P0

coordinates x =

R' be an open are linearly

there exist smooth local

(xi,... , xg) near P0 such that

j=1,...n, where (air) is an invertible n x n matrix of smooth functions near P0.

The proof of the above Frobenius Theorem can be found in many references and is based on elementary ordinary differential equation techniques. See, e.g. Chevalley [I] orTreves [2]. The following is an example of a structure which is neither real nor CR. EXAMPLE 2.1.10. Let M = R2 and L =

Note that L is nowhere vanishing and, if x = (x1, x2) E R2, then if and only if x2 = 0. The = structure defined by L is integrable since we have LZ = 0 with Z(x) = x1 +

The corresponding structure with n =

1

ax2

—

—

ax1

is often referred to as the Mizohata

structure. One may ask whether a formally integrable structure is necessarily integrable. This is true in the real-analytic category as is proved in Theorem 2.1. 11 below. If M is a real-analytic manifold and the subbundle V of CTM is real-analytic, i.e. near every pOint P0 E M there exist real-analytic vector fields which form a local basis for F'(M, V), and also formally integrable, then the structure (M, V) is said to be a real-analytic formally integrable structure.

THEOREM 2.1.11. If (M, V) is a real-analytic formally integrable structure, then it is integrable. near every Po M one can find real-analytic basic solutions Z1,... , (as in Definition 2.1.2). We first state the Frobenius local integrability theorem for holomorphic structures, which is the analog of Theorem 2.1.9 in the smooth case and whose proof

§2.1. FORMALLY INTEGRABLE STRUCTURES ON MANIFOLDS

39

is based on similar arguments using ordinary differential equations techniques in the holomorphic setting. THEOREM 2.1.12 (LOCAL HOLOMORPHIC FROBENIUS INTEGRABILrrY).

c CK be an open set and L1,... , holomorphic vector fields in holomorphic coefficients. If L i,... and satisfy

,

L,,, are linearly independent a: every p

Let with cz

j,k=1,... ,n, where cjkr are holomorphic functions in then for any P0 E there exist holomorphic local coordinates Z = (Z1,... , Zk) near P0 such that

j=1,...n, where (aj,) is an invertible n x n matrix of holomorphic functions near P0.

PROOF OF THEOREM 2.1.11. Let po E M and L1,... , real-analytic sections of V, linearly independent at every p in an open neighborhood of P0. By going to

withk = n+m = dimM. We write

j=l,...,n,

(2.1.13)

with a1C(x) (complex-valued) real-analytic functions in Let be an open subset of C' with n Rk = such that all the coefficients (x) extend holomorphically to 2. We can then consider the holomorphic vector fields, with holomorphic coefficients in

(2.1.14)

=

j

= 1,...

,

n.

Since (2.1.1) is satisfied, it follows that the assumptions of Theorem 2.1.12 hold for the vector fields L1 in Hence, by that theorem, after a holomorphic change of coordinates in C* around 2 = 0(Z), we may write the vector fields in the form (2.1.15)

=

j = 1,...

,

n.

Clearly we have L1 2r = 0, for j 1,... , n, r = n + 1,... , k. It is now easy to see that if 0 = (0k,... , 0j), then the functions Oj (x), j = n + 1,... , k, are

II. ABSTRACT AND EMBEDDED CR STRUCTURES

40

real-analytic basic solutions near po in

for the original structure. This completes

0

theproofofTheorem2.l.ll.

Another important class of formally integrable structures which are integrable are those CR structures which are of CR codimension 0. We shall say that a formal CR structure (M, V) is an almost complex structure if m = n, i.e. if dim M is even and the CR dimension n is 1/2 dim M. For completeness, we state the following

celebrated result, which will not be used in the rest of this book. We refer the reader to Nirenberg [2, Theorem 41 for a proof. Any (smooth) almost

THEoREM 2.1.16 (NEw

complex structure is integrable. The following example, due to Nirenberg, is perhaps the simplest to show that there exist structures which are formally integrable, but not integrable. In fact, there are smooth complex vector fields L for which the equation Lu = 0 has no nontrivial solutions.

2.1.17. There is a smooth function a(x, y), defined in R2 and vanishing to infinite order at the origin, such that for any neighborhood U of 0 in R2 the partial differential equation (2.1.18)

+ix(l

=0, V(x,y)

U

no C' solutions but the trivial ones, u constant. The reader is referred to Nirenberg [2, Theorem 21 for details of the construction. The reader should also note that this example uses a perturbation of the Mizohata operator of Example 2.1.8. has

Although Example 2.1.17 does not define a CR structure on R2, there are nonintegrable smooth CR structures as well. (See the end of §2.2.)

§2.2. Levi Form and Levi Map of an Abstract CR Manifold In this section, we shall define the Levi form of a hypersurface and, more generally, the Levi map of an abstract CR manifold. We also give some of their properties, especially for the case of a hypersurface. Let (M, V) be an abstract CR manifold, where V is the CR bundle of M. For the natural quotient map p M, we denote by (2.2.1)

-÷

41

§2.2. LEVI FORM AND LEVI MAP

and by

(2.2.2)

the corresponding map

ir: CTM -÷

We observe that if X and Y are CR vector fields on M, defined near p, with X(p) = X,, VP and Y(p) = Y,, E V,,1 then ([X, Y](p)) depends only on the values X,, and Y,, and not on the choice of the vector fields X and Y extending X,, and Y,,. (The proof of this is easily obtained by taking a local basis of the CR vector fields near p. The details are left to the reader.) DEfINITION 2.2.3. The Levi map at p M is the Hermitian vector valued form

4,:

V,,

x V,, -÷

(2.2.4)

:=

i

K,,

-

([X, Y](p)),

where X and Y are CR vector fields on M extending the CR vectors X,, and Y,,. Note that the definition above does not use the fact that V is formally integrable (involutive), see (2.1.1). However, we consider here only CR structures. Definition 2.2.3 yields a smooth Hermitian bundle map

(2.2.5)

£: V x V —+

CTM/(VeV).

We call C the Levi map of (M, V). DEfINITION 2.2.6. The Levi map C,, is nondegenerate all Y,,

if

= 0 for

V,, implies X,, =0.

If (M, V) is of hypersurface type, i.e. dimc V,, = (dilnR M — 1)12, then the vector space on the right hand side of (2.2.1) is one dimensional. The Levi map at p is then a Hermitian form on Vi,, called the Levi form. In this case, we say that (M, V) is Levi nondegenerwe at p if the Levi form is nondegenerate. Furthermore, a CR manifold of hypersurface type, (M, V), is called pseudoconvex at po if the Levi form is either positive semidefinite at all p in an open neighborhood of P0 M or negative semidefinite at all p in such a neighborhood. Similarly, (M, V) is said to be stri ctly pseudoconvex at p0 E M if the Levi form is (positive or negative) definite at p0 E M (and, hence, for all p E M in a neighborhood of p0). We shall discuss embedded pseudoconvex hypersurfaces further below. For a smooth generic submanifold M C CN of CR dimension n and codimension d (see Definition 1.3.4), and a point p0 E M, we shall describe the Levi map at in regular coordinates Z = (z, w) for M vanishing at P0 (as given by

42

II. ABSTRACT AND EMBEDDED CR STRUCTURES

Proposition 1.3.6). Thus, we assume that M is given near p0 = (0,0) by (1.3.7). We parametrize M near the origin by

(2.2.7)

€M.

R2"

A basis for V,,0 is then

,... ,

(2.2.8)

-2-. az10

and as a basis for

® V,,0) we can take

(2.2.9)

as10

,...

,

asd0

with the obvious identification. Given such bases, the Levi map can be represented

by d Hermitian matrices of size n x n. PROPOSITION 2.2.10. Let M C CN be a smooth generic submanjfold of CR dimension n, codimension d, and po M. Let (z, w) C" x Cd be regular holomorphic coordinates for M vanishing at P0 50 that M is defined near P0 = (0, 0) by (1.3.7). Then, the Levi map can be represented by the Hermitian matrices (2.2.11)

a

(\azfazk

,

£= 1,... ,d.

PROOF. By using the local basis Li,... , L,, for the CR vector fields given by (1.6.2') and computing [L1, Lk](0), the proposition follows. 0

2.2.12. One can also represent the Levi map of a generic submanifold M in an arbitrary coordinate system Z near po E M by using any defining function p for M near Po• However, we will only use such a representation in the case of embedded pseudoconvex hypersurfaces and, hence, we shall only discuss it in this context (see Proposition 2.2.19 below); we refer the interested reader e.g. to the book by Boggess [1] for a description of the Levi map in local coordinates. We now give an alternative characterization of pseudoconvexity for embedded hypersurfaces which will be useful in the study of the reflection principle between bounded pseudoconvex domains in §9.10. For this, we need some notation. Let M C C's' be a smooth hypersurface and p0 E M. Let C Ci" be a (sufficiently small) open neighborhood of Po• We may assume that \ M has two connected

43

§2.2. LEVI FORM AND LEVI MAP

components, function

which we shall refer to as the two sides of M. Define the and be the distance to M, i.e. 2) to

8M(Z, 2) := inf IZ

(2.2.13)

pEM

— p1,

I denotes the Euclidian distance in CN. Also, define the function a1 (Z, Z) as follows

where

in

-

a1(Z, Z) :=

(2.2.14)

and

I (—cSM(Z,Z),

the function o2(Z, 2) by o2(Z, 2) := —a1 (Z, 2).

beasnwothhypersurface with poE M,and LetM C a1(Z, 2), and a2(Z, 2) be as above. Then, there is a open neighborhood U

PROPOsITION 2.2.15.

let

of po such that a1(Z, 2)and a2(Z, 2)are smooth in U. Moreover, if U is chosen sufficiently small, then da1

0 and da2

0 in U.

PROOF. Obviously, it suffices to prove the statement for a1 (Z, 2). We

identify

CN with the real vector space R2N and ignore the complex structure on R2N for the remainder of this proof. Since the Euclidian distance in R2N is invariant with respect to translations and rotations, we may assume that Po is the origin and that M is given as the graph y = f(x), where the coordinates in R2N are y) and df(O) = 0. We may also assume that is the (x, y) = (Xi,... , component where y — f(x) > 0. If (x, f(x*)) is a point on M minimizing the distance to a point (x, y), then (2.2.16)

x7 — x1

=

—

f(x*)),

j = I,...

,

2N

—

I.

By the implicit function theorem, there is an open neighborhood U of the origin and a smooth function f = x*(x, y) such that (x, y) = (x*(x, y), f(x*(x, y))) is the unique point on M closest to (x, y), for every (x, y) U. The proposition follows from the fact (easily verified using (2.2.16)) that, in the coordinates (x, y), (x, y) =

where x =

(y —

+

x(x, y).

It follows from Proposition 2.2.15 that

defining

+... + fx2NI (x*)2,

0 a1(Z, 2)

(and

similarly a2(Z,

function forM near po(as in § 1.1) such that, by shrinking

= (Z: a1(Z, 2) >0).

2))

is a

if necessary,

II. ABSTRACT AND EMBEDDED CR STRUCTURES

44

A function u, defined in some open set D c taking values in [—oo, co) is called plurisubharmonic (in D) if it is upper semicontinuous and if the restriction of u to every complex line (intersecting D) is subhannonic. If the function u is smooth, then (as is easy to see) u is plurisubharmonic if and only if the Hermitian form N

- (Z,Z)ajâk,

j.k=I 8ZJ8Zk

a=(a1,... ,aN),

is positive semidefinite on for every Z D. If this form is positive definite on CN for all Z E D, then u is called strictly plurisubharmonic. THEOREM 2.2.17. Let

Mc

CN be a smooth hypersurface and p0 E M. Let

ÔM(Z, Z) be the distance to M as above. Then, M is pseudoconvex a: po if and

only if there is an open neighborhood U of P0 such — log 8M is plurisubharmonic fl U or fl U, where either in and are the two sides of M as defined above. REMARK 2.2.18.

(1) If M is pseudoconvex at —

then we shall refer to a side

in which

log 8M is plurisubharmonic as a pseudoconvex side. Observe that both

sides may be pseudoconvex but, as is easily deduced from Proposition 2.2.19 below, this only happens if M is Levi flat, i.e. if the Levi form is identically 0 in a neighborhood of P0. (ii) It follows from Proposition 2.2.15 and Theorem 2.2.17 that if M is pseudoconvex at M, then there is a neighborhood U of p0 and a smooth defining function p(Z, Z) in U such that — log p(Z, Z) is plurisubharmonic in (Z E U: p(Z, Z) > 0). This will be used in §9.10.

The proof of Theorem 2.2.17 will be given in two propositions. We begin with the following proposition, which also hints at the description, mentioned in Remark 2.2.12, of the Levi form of a hypersurface in terms of a local defining function. PROPOSITION 2.2.19. Let M c C" be a smooth hypersurface with po E M, and let p(Z, 2) be a local defining fwzction for M near as in §1.1. Then, M is pseudoconvex at p0 if and only for all p in an open neighborhood of po in M,

either N

(2.2.20)

a2

(p,

0

§2.2. LEVI FORM AND LEVI MAP

45

ft(p, j)aj = 0, or

for a/la E C" with

N

82

>0

(2.2.2 1)

foralla

=0.

with

PROOF. Leti: M -+ SinceLdp = ,*(i8p) d = a+ã, itfollowsthatO isreal. (Wereferthereaderto §8.4and §11.2 for further discussion of the relationship between local defining functions for M and the intrinsic differential geometry on M.) Also, since 8(p) annihilates V,, and V,, (see § 1.2), we may identify the one-dimensional vector space V,,) with C as follows. For X,, E its image in V,,) is identified with C where

:= (8(p), Xe).

(2.2.22)

Thus, a representation of the Levi map (2.2.4) as a complex-valued Hermitian form on V,, is obtained by

(2.2.23)

x V,,

(8(p), [X, Y](p)) E C,

(Xe,

where, as in Definition 2.2.3, X and Y are CR vector fields extending X,, and Y,,. Using the identity (see e.g. Helgason [1, Chapter 1, §21) 2 (dcv, X A Y)

(2.2.24)

=

X ((cv, Y))

—

Y

((cv, X)) — (cv, [X, Y]),

which holds for any smooth 1-form cv and vector fields X and Y, and the fact that

dO = it(id8p) = z*(iaap),

(2.2.25)

we deduce that (2.2.26)

(0(p), [X,

= x((e, Y)) = —2

—

Y((8, X)) 1,,

—

2(dO(p), X,

A

X,, A

The last equality in (2.2.26) is a consequence of (2.2.25) and the identities

46

II. ABSTRACT AND EMBEDDED CR STRUCTURES

which, in turn, follow since X, Y are CR vector fields and 0 is a real 1-form annihilating all CR vector fields and their complex conjugates. Writing a CR vector in the local coordinates Z, a,'a Zk, we may identify CN subspace V c of a E CN for which

with the complex

=

(2.2.27)

The conclusion of Proposition 2.2.19 now follows from the definition of pseu-

doconvexity by writing the expression on the second line of (2.2.26) in the local coordinates Z. U By Proposition 2.2.19, the conclusion of Theorem 2.2.17 is a direct consequence of the following proposition. PROPOSITION 2.2.28. Let M C CN be a smooth hypersurface with P0 M. Let p(Z, Z) be a local defining function for M near P0 and 8M(Z, Z) the distance to M as above. Then, there is an open neighborhood U C CN of Po such that (2.2.20) holds for all p M fl U and only if there is an open neighborhood U' C Ce" of p0 such that — log 8M is plurisubharmonic in {Z U': p(Z, 2) > 0}.

PROOF. It is straightforward to verify that if we replace the given defining function p(Z, 2) by p'(Z, 2) = c(Z, 2)p(Z, 2), where c(Z, Z) > 0 in an open neighborhood of then (2.2.20) holds for all p E M near P0 if and only if the corresponding inequality with p replaced by p' holds for all p M near po. We leave this calculation to the reader. Thus, we may replace p with one of the special defining functions or 02 introduced in (2.2.14) (and the line following that equation). We choose or 02 according to whether the given defining function p is positive or negative in We shall assume, without loss of generality, that p is positive on and, accordingly, we shall prove Proposition 2.2.28 with p replaced by a1. For brevity, we shall write a = Suppose that — is plurisubharmonic in c1 n U', for some open neighborhood U' of po. We observe that — log 8M = — logo in Moreover, a direct calculation shows that, for any a CN,

(2.2.29)

logo)

a

k

N

2

=

aj

—

a2a a z1 azk

aJak

47

§2.2. LEVI FORM AND LEVI MAP

fl U'

Thus, we have in

N

N

2

2

°

(2.2.30)

0,

We deduce that

for every a E

N

82a

j.k=I

aza2 * J

(2.2.31)

holds for Z E

jk

fl U' and every a E

a3ak

with

(2.2.32)

a in a neighborhood of po with do 0, we conclude, by passing to the limit in (2.2.3 1) and (2.2.32), that (2.2.20) holds, with p = a, for every p E M sufficiently close to po. This proves one implication in Proposition 2.2.28. To prove the opposite implication, we assume that the function — log tSM is not plurisubharmonic in flU' for any neighborhood U' of po. Let U be an arbitrary open neighborhood of p0. We shall prove that (2.2.20), with p = a, does not hold for all p E M fl U. In order to show this, we first construct an analytic disc (i.e. a holomorphic mapping of the unit disc in C into C's'; see Chapter VII) which is contained in fi U except at one point pi E M fl U (interior to the disc) where the disc has quadratic contact with M. Using this disc, we then show that (2.2.20)

fails to hold at p'. By the assumption on — there are points Z E U fl arbitrarily close to Po. and A E CN such that the function of a single complex variable

t

(2.2.33)

is not subharmonic near r = 82

—

0.

log

+ At, Z + Ai)

We choose Z0 E

- (log SM(ZO + tA0,

8r8r

+

such that

fi U, Ao E

=E >

0.

By Taylor expanding the function in (2.2.33), we then obtain (2.2.34)

log SM(ZO + tA0,

+ fA0) =

log tSM(ZO, Zo) + Re (at) + Re (br2) + E1r12 + O(1rJ3),

II. ABSTRACT AND EMBEDDED CR STRUCTURES

48

for some a, b E C. This can be rewritten (2.2.35)

SM(Z0 + TAo, Zo + fAo) = SM(ZO, Zo)

I

Since such Z0 can be chosen arbitrarily close to po. we may assume that there is E U fl M which minimizes the distance from Z0 to M, i.e. a point

Zo)

(2.2.36)

=

Ipi — Zoo. .•-* CN,

Consider the family of analytic discs A,:

A,(r) := Z0 + rA0 + (Pi —

(2.2.37)

= (t: < r), and r > 0 is a number to chosen. Observe that the distance to M has the following property where I E [0, 1],

8M(Z +

(2.2.38)

Z

Z) +

+

Thus, we obtain

+ rA0, Zo + IA0) — Ipi

(2.2.39) SM(A,(T), A,(r))

—

ZoI

By using (2.2.35), (2.2.36), and choosing r > 0 so small that

E1t12 + O(1r13) ? eIrI2/2,

Vt

we obtain

(2.2.40)

8M

(At(r), A,(r))

(5M(Z0,

—

By shrinking r further if necessary, we deduce from (2.2.40), since A0 (A,.) is

n U for all t [0, 1). Thus, (2.2.40) fl U, that A, (Ar) c fl U, for all r (0). By definition, we have implies that A1(r) E A1(0) = p' E MflU. Weclaim that

contained in

(2.2.41) jjc—I

where a =

PI)ajãk >0,

azfazk

,31)a1 = 0,

To prove the claim, let h(r, f) = a(Ai(r), A1(r)). By the chain rule, (2.2.41) is equivalent to (2.2.42)

(0). This would complete the proof.

a2h

_(0, 0) >

arar

0,

ah

—(0,0) = 0.

at

§2.3. CR MAPPINGS

Observe, since SM and a coincide in constant C > 0 such that

h(r,

(2.2.43)

49

that (2.2.40) implies the existence of a

f)> C

—

t).

By Thylor expanding at r = 0, we deduce that

(2.2.44)

Re

0)r) +

0)r2) +

(0, 0)1r12>

+ Clearly, (2.2.44) implies that the linear term on the left side must vanish identically, i.e. ah/aT(o, 0) = 0. Also, since Re (82h/ar2(O, O)r2) is harmonic and vanishes

at r = 0, (2.2.44) and the maximum principle implies that a2h/ar2(0, 0) 0. Hence, (2.2.42) has been proved. This completes the proof of the claim and, hence, of the proposition. 0 Although the focus of this text is, in some sense, on manifolds with degenerate Levi maps, we mention here without proof an important local embeddability theorem. 2.2.45 dim M 9, dim M = 7). Let (M, V) be an abstract smooth CR of hypersurface type which is strictly pseudoconvex. Then (M, V) is integrable dim M 7. The proof of this deep theorem, which is outside the scope of this book, is due to Kuranshi [1] forthe case dimM 9andtoAkahori[l]forthecasedimM = 7. We should mention here that the case dim M = 5, with the notation of Theorem 2.2.45, is still open. When dim N = 3, the conclusion of Theorem 2.2.45 does not hold, as shown by a counter-example of Nirenberg [3], whose construction follows the lines of that of Example 2.1.15.

§2.3. CR Mappings In this section, we shall discuss mappings between CR manifolds. We consider only those mappings that respect the additional structure induced on the manifolds

by their CR bundles. As we will see, for embedded CR manifolds the typical example of such a mapping is the restriction of a holomorphic mapping of the ambient complex spaces. Let (M, V) and (M', V') be abstract CR manifolds with CR bundles V and V'. respectively.

II. ABSTRACT AND EMBEDDED CR STRUCTURES

50

DEaNrrIoN2.3.1. ACR mapping (of class Ck,k 1)11: (M,V) —÷ (M',V') is a C mapping H: M -÷ M' such that, for all p E M, (2.3.2)

C

denotes the usual tangent map (çushforward) induced by H. where

—÷

It is immediate from the definition that if

H: (M, V) —÷ (M', V'),

G: (M', V')

—÷

(M", V")

CR mappings, then the composition G o H is a CR mapping (M, V) —÷ (M", V"). The reader should observe that even if a CR mapping H: (M, V) —÷ (M', V') M —÷ M', the inverse H': M' —÷ M need is adiffeomorphism as a not be a CR mapping (M', V') —÷ (M, V). Indeed, consider the following two CR manifolds: M = C x R2 with V,, at each point p being the complex tangent space of the first factor, and M' = C2 with V,, at every p' being The identity mapping of the underlying space R4 is a CR mapping (M, V) -÷ (M', V'), but

are

is not a CR mapping (M', V') —+ (M, V). In this example the CR dimensions of the two manifolds are different. It is easy to see, however, that if a CR mapping H: (M, V) -÷ (M', V') is a diffeomorphism M -+ M' and if the CR dimensions of(M, V) and (M', V') are equal, then the inverse mapping H' is aCR mapping. When the target manifold (M', V') is embedded in CN, i.e. M' c is a CR submanifold and V' is the induced CR bundle as in Chapter 1, then we have the following. PROPOSITION 2.3.3. Let (M, V) be an abstract CR manifold and M' C C's' a CR submantfold with induced CR bundle V'. If H: M -+ M' is a k 1, with components H = (H,,... , HN), i.e. H1 = Z3 oH (for some choice of coordinates Z in then H is a CR mapping (M, V) -÷ (M', V') and only if each component H1, j = 1,... , N, is a CR function.

We shall reduce the proof of Proposition 2.3.3 to that of a simpler case. Recall that the induced CR bundle V' of M' C CN is defined as follows (2.3.4)

for each p' E M'.

V,. Hence,

=

fl

if a mapping H: M -÷ M' C CN is a CR mapping

(M, V) —÷ (M', V'), then H considered as a mapping M —* is a CR mapping (M, V) —÷ (CN, T0.ICN). Conversely, if a mapping H: M -÷ M' C CN is a CR mapping (M, V) —÷ (CN, To ICN) then it follows from (2.3.4) that it is also a CR mapping (M, V) —÷ (M', V'). Thus, Proposition 2.3.3 is a consequence of the following.

§2.3. CR MAPPINGS

51

PROPOSITION 2.3.5. Let (M, V) be an abstract CR manifold and

1. ThenHisaCRmapping(M,V) -* (CN,TOICN)Lfand only (f each component H1, j = 1,... , N, is a CR function.

PROOF. Let L be a CR vector field on M. We have the following, for any p E M,

N/

((LHj)(p)

=

(2.3.6)

j=I Hence, E followsthat H: M —*

\

a

+ (LHj)(p) J

H(p)

a J

if and only if

11(p)

= 0 for j = 1,... . N.

It (CN, T°' M) if andonly if C's' is aCRmapping (M, V) —*

Oforj = 1,...

N and foreachCR vectorfield L on M. Thiscompletes the proof of Proposition 2.3.5. LH1

,

When it is clear from the context what the CR bundles V and V' of M and M' are, weshallsimplysaythatamappingH: M —÷ M'isCRifH: (M,V) —÷ (M',V') is a CR mapping.

It follows from Proposition 2.3.3 and the basic material on CR functions in §1.4 that if M C CN and M' c CN' are CR submanifolds, H is a holomorphic mapping from an open subset U c C's' into CN' such that M c U, and H extends mapping to M with H(M) C M', then the restriction of H to M is a CR as a mapping from M into M'. We conclude this section with a result concerning embedded CR submanifolds. (A similar result can be formulated for abstract CR manifolds, but since we shall not need it, we treat here only embedded submanifolds.) Recall that if M c C" is a CR submanifold, then the complex structure J of the ambient space restricts to a complex structure on for every p E M. The following proposition states essentially that the tangent map induced by a CR mapping is complex linear on the complex tangent space. PROPOSITION 2.3.7. Let M c C", M' C CN'

be

CR submanifolds, and let

H: M -÷ M'beaCRmapping. Then,foreachp€ (2.3.8)

(J' o

=

E TM,

o

where J and i' denote the complex structure maps on fmm C"' and CN', respectively.

and

induced

II. ABSTRACT AND EMBEDDED CR STRUCTURES

52

E TM. By Proposition 1.2.8, there exists

M and PROOF. Pick p E V,, such that X,, = +

=

(2.3.9)

By linearity, we have

=

+

+

E E V,,, it follows, again by Proposition 1.2.8, that Using the fact that and V,, are the —i and +1 eigenspaces for J (and similarly for J', see Proposition 1.2.8), we compute

Since

+ La)) =

(2.3.10)

+

=

+

= = =

+ + + Lu)).

0

This proves Proposition 2.3.7.

Let us conclude this section with the following result, which is an easy consequence of Proposition 2.3.7 (and Proposition 1.2.8) and whose proof is left to the reader. COROLLARY 2.3.11. Let H: M —÷ M'

Mc

(2.3.12)

M' c CN' be

CR submanjfolds and let

Then,

C

Vp

M.

§2.4. Approximation Theorem for Continuous Solutions Let (M, V) be an integrable structure as given in Definition 2.1.2. In this section we shall study the solutions of this structure, i.e. the functions or distributions on

M which are annihilated by all the vector fields L e r(M, V). Note that by the chain rule, any holomorphic function of the basic solutions Z1,... , (given in Definition 2.1.2) is in particular locally such a solution. Conversely, the main result of this section shows that in fact all solutions can be locally approximated by holomorphic functions of the basic solutions Z1.

2.4.1. Let (M, V) be an iiuegrable structure, p0 M, and Z = (Z1,... , Zn,) a family of basic solutions near po. Then there exists a compact neighborhood K of P0 in M such that for any continuous solution h in M, there is a sequence of holomorphic polynomiaLs

in m complex variables with the

property that (2.4.2)

h(u) =

urn I'—'OO

u E K,

§2.4. APPROXIMATION THEOREM FOR CONTINUOUS SOLUTIONS

53

where the convergence is uniform in K.

h is a solution of class Ck, k?: 1, then the convergence in (2.4.2) is in C*(K) as will be shown in §2.5. If

PROOF OF THEOREM 2.4.1. We begin by making an appropriate choice of local

coordinates near the central point Po. LEMMA2.4.3. Let(M,V), po,andZ = (Z1,... ,Zm) as in Theorem 2.4.1. after an invertible complex linear transformation in the Z1 one can find local coordinates (x, y) near po. vanishing at po, with x = (x1,... Xm) and Then,

,

y=(yi,... Zj(x,

(2.4.4)

where the

k

y)

= x1

+

y),

j

= 1,...

,

m,

are smooth, real-valued functions defined near the origin in Rk,

= n + m, with

4'

(0) =

PROOF OF LEMMA

= 0.

2.4.3.

Since dZ1,...

,

dZm are linearly independent near

po, we can find a system of coordinates (u1,... , vanishing at p0 such that the determinant of the matrix A(u) = (8Zi/8ue(u))i
x := Re

Y, := uj+m,

j=

1,

. .

. fl

It is then immediate to see that Z(u(x, y)) is of the form given by the right hand side of (2.4.4). This completes the proof of the lemma. D

We continue the proof of Theorem 2.4.1. We choose the coordinates (x, y) given by Lemma 2.4.3 and assume that Z(x, y) is of the form (2.4.4). Let r and d be two positive numbers to be determined later. We choose x with r. Weput X(x)=— 1 forlxi

:= f(x,y): IxI S Let h be a solution of class C' on M. We shall indicate later how to pass to the case where h is assumed only to be continuous, as in the statement of the theorem. We define an rn-form a(x, y) = y; z) for z C'" and v Z' by p

y; z)

(—)

m/2

exp[—v(z

—

Z(x, y))2]

where

dZ(x, y) = dZ,(x, y) A ... A dZm(x, y),

y)dZ(x, y),

II. ABSTRACT AND EMBEDDED CR STRUCTURES

54

(vi,... ,

and, for a vector v = We

fix y E R" with 0 <

Cm, we use the notation v2 = v• v = be the (m + 1)-dimensional < d and let D = E

manifold

(2.4.6)

D

{(x', y') E

:

Ix'I

< r, y' = ty, t

E (0, 1)).

By Stokes' Theorem we can write

[

(2.4.7) We

y'; z) =

JD

I

y'; z).

JaD

0 for

shall first compute the right hand side of (2.4.7). Since x (x')

r

we have

y')

(2.4.8)

P

= (—) JT

I exp[—v(z

—

y)

Z(x', y))21

P

— (—)

exp[—v(z — Z(x',0))2]

means taking the differential with respect to the x' variables. In order to compute the left hand side of (2.4.7) we need the following lemma.

where

V) an LEMMA 2.4.9. Let c be an open subset of Rk, k = m + n, and A distribution defined in integrable structure, with basic solutions Z,,... , = 0 in with dZ(u) = dZ1(u) A f in is a solution and only

PROOF. Assume first that f is of class C' in Then f is a solution if and only The latter is equivalent to the statement that if, for every p E df(p) df(p) is in the span of fdZ, (p),... , dZm(p)}. Since d(fdZ) = df A dZ, the lemma follows in the case where f is of class C'. By shrinking if necessary

and using Lemma 2.4.3 we may assume that the Z1 's are given by (2.4.4). Then at every point (x, y) E form a basis of dy,,... , dZ,,... , m and 5q' Hence if follows that there exist vector fields R1, 1 < q such that for any f of class C' we may write

j

(2.4.10)

df =

+

1

Sqfdyq.

I

From the above it follows that f is a solution if and only if 5q = 0, q = and hence the vector fields 5q form a basis of the sections of V in

1,...

, n, For a

§2.4. APPROXIMATION THEOREM FOR CONTINUOUS SOLUTIONS

55

(2.4.10) still holds, and hence f is a solution if and only if with distribution coefficients. The rest of is a linear combination of the

distribution f in

dl

the argument for a general distribution is the same as in the C' case. This proves 0 Lemma 2.4.9.

We return to the calculation of the left hand side of (2.4.7). We observe that (x', y') and that since exp[—v(z — Z(x', y'))2] is a holomorphic function of the the product of two solutions is again a solution, we conclude by Lemma 2.4.9 that

da(x', y')

(;) = v

m/2

A dZ(x', y').

exp[—v(z — Z(x', y'))2] h(x',

We introduce the sequence of entire functions in cm defined by

(2.4.11)

14(z) :=

f

(—) IT

I

Jw'

We shall show that

exp[—v(z — Z(x',0))2]

y)) converges uniformly to h(x, y) on K as v —÷ cc.

For this, we shall prove, after replacing z by Z(x, y), that the left hand side of (2.4.7) converges uniformly to 0 on K as v -+ cc, while the first integral on the right hand side of (2.4.8) converges uniformly to h(x, y). This will be proved in the following two lemmas.

If r and d are chosen to be sufficiently small, then for any continuous function f in LEMMA 2.4.12.

(2.4.13) f exp[—v(Z(x, y) — Z(x', y'))21 f(x', uniformly for (x, y) E K as v —÷

y') —÷ 0

00.

PROOF. Note that since = 0 for and lxI > r, the integral in (2.4.13) is evaluated over the set r and y' [0, yl. The main point is Ix'I to estimate I exp[—v(Z(x, y) — Z(x', y'))2]I. Using (2.4.4) we have

Re (Z(x, y) — Z(x', y'))2 = (x

(2.4.14)

—

—

(4(x, y) —

y)I +

y) —

x')2

y'))2.

By the mean value theorem we have 1q5(x, y) —

a

y')I

sup

y) —

A

:=

sup

y')I

II. ABSTRACT AND EMBEDDED CR STRUCTURES

56

Since choose d

= 0 we choose r and d sufficiently small so that lal

We then

to obtain for (x, y) E K

exp[—v(Z(x, y) — Z(x', y'))2]I

e_7v(*)2,

0

from which the lemma easily follows.

LEMMA 2.4.15. If r and d are chosen to be sufficiently small, then for any continuous function f in

(2.4.16)

f

lim

exp[—v(Z(x, y) — Z(x', y))21x

Jx'€R

unçformlyfor(x,y)

f(x',

y)

y) =

y)]dx'. We make the the integral then becomes

K.

PROOF. Note first that we have

change of variables of integration (2.4.17)

6f

y)

exp [—

—

+

x)

=

(x +

y)

—

)))] x

whereg(x', y') = x(x')f(x', y'). Here again we chooser andd sufficiently small so that

a=

1/2.

sup

It is then easy to check that, as v —+ oo, the integral (2.4.17) converges to (2.4.18)

f

+

f(x, y)

y) E K. The lemma follows from the following simple result whose proof is left to the reader. A

=I+

2.4.19. Let B be an m x m real matrix with norm less than 1. iB, where I is them x m identity matrix, then

det[A] ,r m/2

f

,

2

]

= 1.

This completes the proof that the entire functions

have the property that

If

defined by (2.4.11) y)) —÷ h(x, y) uniformly on K. Since any entire

§2.5. FURTHER APPROXIMATION RESULTS

57

function is a uniform Limit of hoLomorphic polynomials on compact sets of Cm, the theorem is proved for the case that h is of class C'. If h is merely continuous,

we observe that the same proof works. Indeed the form of Stoke's Theorem used, (2.4.7), is still true since in this case the coefficients of the form aM and are continuous functions (although the coefficients of the form are not necessarily C'). This completes the proof of Theorem 2.4.1. 0

§2.5. Further Approximation Results In this section we shall sharpen Theorem 2.4.1 in several ways. First, we shall show that if the solution h in Theorem 2.4.1 is of class with r > 1, then the convergence is in the c' norm. We shall also give a more general sequence of approximating functions. Finally, we state a uniqueness result which is a corollary of the approximation result. The following definition generalizes, for a formally integrable structure, the notion of totally real submanifolds in CN introduced in Chapter I. DEfiNITION 2.5.1. Let (M, V) be a formally integrable structure as defined in §2.1 and E a submanifold of M. We shall say that E is totally real if, for eveiy p E E, we have V,, fl = (0), and maximally totally real if E is totally real and of maximum dimension, i.e. dim E = m. Let (M, V) be an integrable structure, and Z

= (Z,,... , Z,,,) a family of

basic solutions defined near p0. For a submanifold E c M, let i: E -+ M be the embedding of E into M and LdZ1 the pullback of dZ1 to E. It is easy to check that E is the orthogonal to the span of (idZ, (p),... , (p)). Therefore a submanifold E of dimension m is maximally totally real if and only if

the ftdZ,(p),...

are linearly independent over C. This observation is independent of the choice of family of basic solutions. ,

2.5.2. Let (M, V) be an integrable structure, p0 M, and Z = Zm) afamily of basic solutions near P0. Let E be a maximally totally realsubmanifoldofM with po E, andy sufficiently small support (p) 1 in a neighborhood of with x Then there is a compact neighborhood K of p0 in M such for any solution h of class c' in M, r 0, the sequence of entire functions defined by

(Z,,...

,

:=

(2.5.3)

1'

(—)

I exp[—v(z

JE

—

Z(p))2] x(p)h(p)dZi(p) A... AdZm(P),

for z E Cm, has the property that

C'(K).

o

Z converges to h in the Banach space

II. ABSTRACT AND EMBEDDED CR STRUCTURES

58

For the proof of Theorem 2.5.2 it will be convenient to construct a local basis of commuting complex vector fields near P0 which are matched to the family of basic solutions Z = (Z1,... ,

M. and 2.5.4. Let (M, V) be an integrable structure, P0 Zr,) a family of basic solutions near po• Let E be a maximally

PROPOSITION

Z = (Z1,...

,

totally real subman (fold of M with P0 E E. Then there exist local coordinates (x, y) as in Lemma 2.4.3 such that (2.4.4) holds and so that, near Po. E is given by y = 0. Furthermore, after an invertible complex linear transformation in the Z3. and R1,... , if necessary, there exist unique complex vectorfields L1 defined near P0 such that

(2.5.5)

LJZk—O,

Ljyq_=öjq,

(2.5.6)

In

I

R1,...

addition, the vector fields L1,... ,

,

commute pairwise.

PROOF. As in the proof of Lemma 2.4.3 we start with a system of coordinates uk),vanishingatpo,suchthatnearpo, E = ... = u,, = (UI is invertible for 0. Since E is maximally totally real, A(u) = u near 0 in IRk. After making the change of variables (2.4.5), and after taking a linear combination, the Z3 are given by (2.4.4) and E is given by y = 0. This proves the first statement of the lemma. Since we must have rje(x, of the form R3 = We take in

azs

=

(2.5.7)

(=1

=

we may take (r4x. y))i<j (2.5.8)

If we take L1 of the form —i

=

>

y)R,,

then (2.5.5) and (2.5.6) hold. In order to prove the last statement, note first that

(dyi,...

,

dy,,,

dZ1 (x, y),... , dZm(X, y)) is a local basis of the cotangent space

to M. To check that all the commutators vanish, it suffices to show that they annihilate the Z1 and the yk; this is an immediate consequence of (2.5.5) and (2.5.6). The proof of Proposition 2.5.4 is now complete.

0

PROOF OF THEOREM 2.5.2. By Proposition 2.5.4, we may find coordinates

(x, y) near p0. satisfying the conditions of Lemma 2.4.3, such that E is given

59

§2.5. FURTHER APPROXIMATION RESULTS

is defined by by y = 0. Then the proof of Theorem 2.4.1 shows that if o Z converges to h in C°(K), where K is a compact (2.5.3), the sequence

neighborhood of po independent of h. This proves the theorem for r = 0. In order to complete the proof of the theorem when r > 0, we shall make use of the vector fields L1 and R1 of Proposition 2.5.4. For simplicity we assume r = 1. it Since h(p) and Hr(Z(p)) are solutions, and hence are annihilated by the o Z)(p) converges suffices to show that, for j = 1,... ,m, the sequence uniformly to (R1h)(p) for p E K. Note that by (2.5.6) we have

o Z)(p) =

(2.5.9)

az1

On the other hand, by the commutation property of Proposition 2.5.4, R1h is also

a solution, which is continuous since h E C'. We may then apply the result for r = 0 to obtain that if (2.5.10) where

f exp[—v(z

(—)

JE

—

Z(p))2] x(p)(R1h)(p)dZ(p),

dZ(p) = dZ,(p) A ... A dZm(p), then the sequence

o

Z converges

to RJh in C°(K). The desired convergence of Rj(Hr o Z)(p) to (R1h)(p) is then a consequence of (2.5.9) and the following lemma. LEMMA2.5.ll

.

If

is defined by (2.5.10)

lim

for p

—

az1

then,forj = I,... ,m,

/

= 0,

K.

PROOF. We note first that if f is a function, defined in a neighborhood of 0 where (2.5.5) and (2.5.6) hold, then the differential of f is given by (2.5.12)

df

=

+

Since E is given by y = 0, for any function f of compact support and any j, 1

j

(2.5.13)

m, we have, by making use of Stokes' theorem,

fd(f(P)dZI(P)A ...

A... AdZ,n(p)) = 0,

where dZ1(p) indicates that this term is omitted. Taking

1(p) = exp[—v(z — Z(p))2] x(p)h(p)

11. ABSTRACT AND EMBEDDED CR STRUCTURES

60

in (2.5.13), we may make use of (2.5.12) to obtain for j (2.5.14)

f

Z(p))2]

—

= 1,... A

,

... A dZm(p) = 0.

Hence we obtain from (2.5.10), (2.5.14) and (2.5.6),

=

—

f

A... AdZm(p)

exp[—v(z — Z(p))2]

(p)) exp[—v(z—Z(p))2] x (p)h(p)dZ1 (p)A. .AdZm (p), .

—

which yields (2.5.15)

=

—

f — (—)

J

8Zj

exp[—v(z

—

A... AdZm(p).

Z(p))2]

vanishes there. By the argument used in for lxi
x(x)

1

The proof of Theorem 2.5.2 in the case r = 1 is now complete. Since the case 1 is very similar, the details are left to the reader. 0

r>

We now give a useful uniqueness result which is an immediate corollary of Theorem 2.5.2. Let (M, V) be an integrable structure and E a maximally totally real subman(fold of M with p0 E. Then any solution of class C° near po which vanishes on E must vanish identically in a neighborhood of p0 in M. COROLLARY 2.5.16.

The existence of the local vector fields R1,...

given by Proposition 2.5.4 for an integrable structure can also be used to give a criterion for integrability of a formally integrable structure. We state here the following result, which will not be used in the rest of this book and refer the reader to Baouendi-Rothschild [2] for the proof of a slightly more general result. ,

THEOREM 2.5.17. Let (M, V) be aformally integrable structure. Then M is integrable (fand only iffor every P0 M there exist smooth, pairwise commuting, complex vector fields L1,... , R1,... , R,,, defined in a neighborhood of p0 such that they span the complex tangent space to M at every p e M near and L1,... , form a basis for the sections of V in a neighborhood of p0.

NOTES FOR CHAPTER II

Notes for Chapter II The concept of involutive structure for real subbundles is classical, appearing already in the statement of the Frobenius Theorem (which can be found in many standard books, see e.g. Chevalley [1]). Complex involutive structures and locally integrable structures appeared in Andreotti and Hill [1]. We refer here to the book Treves [2] for a more detailed account on this topic. Let us just mention here that Theorem 2.1.16 is due to Newlander and Nirenberg [1] The notion of pseudoconvexity has a long history. The reader is referred e.g. to the book by Range [1] for an outline. Plurisubharmonic functions were first introduced by Oka [1] (VI and IX), and Lelong [I]. The content of Theorem 2.2.17 in C2 is due to Oka [1] (VI); proofs that are valid in C" were given by Oka [1], Lelong [2], and Bremermann [1]. The proof given here follows that given in the book by Hormander [2]. As mentioned in the text, Theorem 2.2.45 is due to Kuranishi [1] for dim M 9 and to Akahori [1) for dim M = 7. Webster [4], [5] has given an alternative proof of this result using integral formulas. Further, more recent results on integrability of CR manifolds can be found in Catlin [1J. A class of counterexamples to local integrability, containing the one due to Nirenberg [3] mentioned in §2.2, has been given by Jacobowitz and Treves [1]. The approximation results (Theorems 2.4.1 and 2.5.2) are due to Baouendi and Treves [1]. Generalizations of some of these results can be found in the book Tteves [2].

CHAPTER III

VECTOR FIELDS: COMMUTATORS, ORBITS, AND HOMOGENEITY

Summary This chapter, which is independent of the first two, is devoted to the study of real vector fields and their commutators. We begin by giving a proof of Nagano's theorem which states that for every Lie algebra of real vector fields in there is a unique submanifold, through a given point, whose tangent space coincides with the Lie algebra at that point. In §3.2, we prove Sussman's theorem, which describes the minimal manifolds (orbits) tangent to a collection of smooth real vector fields. The remainder of the chapter deals with special coordinates for collections of real vector fields. In §3.4, we define the HOrmander numbers for a set of real vector fields; these numbers represent the different lengths of commutators needed to obtain different directions in the tangent space. We discuss the notion of weighted homogeneity for vector fields and prove the existence of "canonical coordinates" (associated with these numbers) for a linearly independent set of vector fields. The proof makes use of exponential coordinates and the Baker—Campbell—Hausdorff formula. We consider separately the case where the vector fields are of finite type, i.e. the vector fields and their commutators of all lengths span the tangent space, and the case of infinite type. We also give sharper results in the situation where the coefficients of the vector fields are real-analytic. The results of this chapter (but perhaps not the methods) will be of importance in later chapters.

§3.1. Nagano's Theorem In this section we shall prove Nagano's Theorem, which establishes a one-toone correspondence between Lie subalgebras of real analytic real vector fields on and real analytic submanifolds of As noted in Remark 3.1.34 below, Nagano's Theorem may be regarded as a generalization of the Frobenius Theorem (Theorem 2.1.9). Let be an open set in and denote by the Lie algebra of all realanalytic real vector fields in c, i.e. real-analytic sections of the real tangent bundle We denote by the real-valued real-analytic functions in For 62

63

§3.1. NAGANO'S THEOREM

X

E

and P0 E

a we define the flow t

tX of X from po as the

i-÷

integral curve y (1) satisfying the initial value problem

= X(y(t))

(3.1.1) y(O)

=

p0.

and p E c,

If)) is a subset of

then

we write ))(p) for the subset of

obtained by evaluating the vector fields in )) at p. We say that 3) is a Lie if 31 is a real vector subspace and is closed under brackets, subalgebra of

i.e.

(3.1.2)

[Y1, Y2] E 3),

V Y1,

}'2 E 3).

Furthermore, we say that)) is a C'°(f)-module if

(3.1.3)

fY

3),

V

f

THEOREM 3.1.4 (LOCAL NAGANO

of Po in

and let g be a Lie subalgebra of Then there exists a real-analytic (3.1.5)

T,,(W) = 9(p),

Y

3).

Let c be an open neighborhood which is also a C°'(c2)-module. with P0 E W, such that W in

V p E W.

the germ of W at p0 is unique with this property.

REMARK 3.1.6. We shall refer to the germ of the submanifold W at

given

at Po• The reader should by Theorem 3.1.4, as the local Nagano leaf of is an immersed, connected, real-analytic submanifold note that if t: W' —+ = g(t(w)) holds for all w in an open containing P0 = L(wO) such that subset in W', then it holds for all w in W'. This follows easily from the realanalyticity of the vector fields in g and the real-analyticity of W'. This observation allows us to construct the global Nagano leaf of at po. p0), as

the union of all connected real-analytic (embedded) submanifolds of c whose germ at po coincides with the local Nagano leaf at p0. The reader can check that W(X(c2), p0) is an immersed real-analytic connected submanifold (but may not be an embedded submanifold of PROOF OF THEOREM 3.1.4. We choose coordinates x = at po and vector fields X1,... , X,. in g,

(3.1.7) k=I

axk

(x,,...

j=1,... ,r,

,

vanishing

UI. VECTOR FIELDS: COMMUTATORS, ORBITS. AND HOMOGENEITY

64

such that Xi (0),... , X,.(O) isa basis of g(O) and

j = 1,... ,r.

=

(3.1.8)

It follows that (3.1.9)

0

det(aj&(x))1<j.*
for x in some neighborhood of 0. Since the conclusion of the theorem is local, we shall assume that this neighborhood is Thus, after a C°'(c)-linear transforinawe may assume that tion on the X,,... , a

(3.1.10)

j1,... ,r,

a

i

k

k=r+I

with bJk(O) =0. Consider the vector space )) over IR spanned by the vector fields X1,... , given by (3.1.10). Also, denote by 90 the set of vector fields in g which are of the form

(3.1.11) k=r+I

axk

Note that Øo is a Lie subalgebra of g and also a

Moreover, all

vector fields in go vanish at 0. We define the set of vector fields

Z:=Y+go.

(3.1.12)

It is easy to check that (3.1.13)

[Z1,Z2]€g0cZ, VZ1IZ2EZ.

Thus, Z is a Lie subalgebra of g, proper if r > 0, and Z(x) = g(x) for all x E Now, we claim that, for any I E Y and X E go, X vanishes on the integral curve t i.-+ t I. This claim is a consequence of the following lemma. LEMMA 3.1.14. Let A be a Lie subalgebra of X(1) with the property that all commutators of vector fields in A vanish at 0. If X E A vanishes a: 0 then it vanishes on any integral curve t exp0 t Y for Y A.

PRooF. Let X E A and I E A be as above and assume that 1(0) (otherwise, there is nothing to prove). We write (3.1.15)

X=

I=

0

NAQANO'S THEOREM

CLAIM 3.1.16. The following hokLN

(3.1.17)

(ad Y)k(X) =

+

j=I

I

j=I p=I

I

where

(3.1.18)

(ad Y)k(X)

= [Y,

[Y,

.

.. [Y, X]]...],

is a string V1V2... with Y appearing k times in (3.1.18), and where k with V1 A such that at least one of the V1 vanish at 0. length PROOF OF

of

CIAII1 3.1.16. The proof is by induction on k and is left to the reader.

0 Cuu,i 3.1.19. Let S = V1 V2...

V1 be a string of length j with V1

E A such

that at least one of the V1 vanish at 0. Then all the coefficients of the differential operator S vanish at 0.

The proof is by induction on the position of the V1 that vanishes at 0. 0 This proof is also left to the reader. PRooF.

We now conclude the proof of Lemma 3.1.14 by observing that Claims 3.1.16 The 0 lemma follows from the real-analyticity of the aj(x).

and3.1.l9implythat,forj = 1,... ,n, Ykaj(0)=oforeveryk= 1

We now return to the proof of Theorem 3.1. We define the manifold W C by the following parametrization (3.1.20)

(t,,...

,

t,) s—*

exp0

for (tj,...

in a sufficiently small neighborhood U of 0 in (Recall that the , are defined by (3.1.10)). It is immediate to see that the rank of the map (3.1.20) is r in U if U is chosen sufficiently small and, hence, (3.1.20) defines a manifoLd in aneighborhood ofO in Lemma 3.1.14 implies that go(x) = Oforallx E W and, hence by (3.1.12),

(3.1.21)

g(x)

= Z(x) =

for x E W. To prove (3.1.5) it suffices then to show, by dimensionality, that

Y(x) c

T1(W) for all x E W. For this, we choose p e W and X E Y. We shall show that X(p) E Since p E W, there exists , E U such that

(3.1.22)

p = exp0

66

III. VECTOR FIELDS: COMMUTATORS, ORBITS. AND HOMOGENEITY

In other words, p is a point on the integral curve of Y from 0 with time one for Consider the mapping Y= (3.1.23)

f(s, t) :=

exp0(t(sX + Y)).

It is defined for t E (—8, 1 + 8) and Isi < rectangle R C R2 into W.

for suitable 8, e > 0, and maps this

CLAtM 3.1.24. The following holds fort e (—8, 1 +8) (3.1.25)

as

t) = tX(f(O, t)).

PROOF. We think of f(s, t), as well as X(f(s, t)), Y(f(s, t)) (and all other We first differentiate f(s,:)

vector fields), as vector-valued functions R —+ with respect to t

(3.1.26)

t) = sX(f(s, t)) + Y(f(s, ()),

and hence (3.1.27)

t) = X(f(0, t)) +

f).

Also, note that (3.1.28)

1(0,0) =0. as

It follows from (3.1.27—3.1.28) that the function t ordinary differential system (3.1.29)

= X(f(0, t))

i-+

af/aS(o, t) satisfies the

t))u1,

u(0) = 0.

+ The proof will be completed, in view of the uniqueness of solutions of differential equations, by showing that the function t i-+ tX(f(O, t)) also solves (3.1.29). Clearly, this function satisfies the initial condition. We differentiate it and obtain (3.1.30)

t))) = X(f(0, t)) + t

t).

67

§3.1. NAGANOS THEOREM

Using (3.1.26) and the definition of the commutator, we obtain (3.1.31)

t))Yj(f(O, t))

t))) = X(f(O, t)) + t

:)).

= X(f(O, :)) + t[Y, X](f(O, r)) + j=1

In view of Lemma 3.1.14, [Y, X](f(O, t)) = 0 for all t, and hence it follows from

(3.1.31) that tX(f(0, :)) satisfies (3.1.29). Claim 3.1.24 follows from uniqueness 0 as explained above.

Since f(s, :) maps the rectangle R into W, [af/as](s, t) is in the tangent space Since In particular, it follows from (3.1.25) that X(p) is in p E W and X E Y were chosen arbitrarily, the proof of (3.1.5) is complete in view of (3.1.2 1). It remains to prove the uniqueness part of Theorem 3.1.4. through P0 satisfying Suppose that W' is another real-analytic manifold in (3.1.5). Necessarily, dim W' = dim W = dim g(0). Hence, it suffices to show that there is an open neighborhood V of 0 in such that

(3.1.32)

WflVCW'flV.

Let U be a convex open neighborhood of 0 in U C W (where the map (3.1.20) is defined) and define C W to be the manifold defined as the image under (3.1.20) of U. We choose an open neighborhood V of W in such that W fl V = W. We can choose U and V so small that W' fl V is closed in V. In order to prove g such E W. By the construction of %V, there exists Y (3.1.32), we pick that the integral curve y(r) = exp0(tY) goes through P1 at time one. Note that y(t) W C V fort [0, 1] since U was chosen to be convex. Since (3.1.33)

Y(p) E

by assumption, it follows that y(:)

prove that y(l) E W'fl V. Let E =

Vp

W',

W' fl V for: sufficiently small. We need to

E [0,1]: y(t) E W'fl V ,V: E

The set E is open by (3.1.33). It is easy to check that E is also closed since W'fl V is closed in V. The inclusion (3.1.32) is proved, and the proof of Theorem 3.1.4 0 is complete.

REMARK 3.1.34. If, in addition to the assumptions of Theorem 3.1.4, one assumes that dim g(p) = dim for all p then the conclusion of the theorem is a consequence of the Frobenius Theorem (Theorem 2.1.9) in the real-analytic setting.

III. VECTOR FIELDS: COMMUTATORS, ORBITS. AND HOMOGENEITY

68

§3.2. Sussman's Theorem In this section we prove a theorem of Sussman, which may be viewed as an analog of Nagano's Theorem for sets of smooth real vector fields. It may also be regarded as a generalization of the Frobenius Theorem (see Remark 3.2.24 below). Let V be a collection of smooth real vector fields in an open set c By a polygonal path off integral curves of vector fields in V joining p to q we mean a piecewise smooth curve y: [0, 1] with y (0) = p. y (I) = q, and 0 = <SI < ... <Sj = 1 such that, for J = 1,...

= expY(S,I)t(s)XJ. 5j—l

s

Si,

where X, E V and t (s) is a smooth diffeomorphism of onto some closed interval of R with t (sj_ I) = 0. If the collection V consists of real-analytic vector fields then we require the curve y to be piecewise real-analytic. THEOREM 3.2.1

SUSSMAN

Let cz be an open set in R' and

Let V be a collection of vector fields in Then there exists a W of C°° submanifold with P0 E W such that every vector field in D is tangent

let P0 E

to W at all points of W and such that the following hold:

submamfold of containing P0 and to which all vector (i) If W' is a fields in V are tangent at every point of W', then there exists an open

neighborhood V C of po such that W fl V C W' fl V. (ii) For every open set U C containing po, there exists a positive integer I and open neighborhoods V1 C V2 C U of such that every point p W fl V1 can be reached from P0 by a polygonal path off integral curves of vector fields in V contained in W fl V2.

Moreover, if the vector fields in V are real-analytic, then W is also real-analytic.

REMARK. Since, by (i) of the theorem, the germ of the manifold W at is unique, we shall refer to it as the local Sussman orbit of (relative to 2)). Note that the local Sussman orbit does not change if we replace by any other open subset £2' C £2 containing P0. Before we enter the proof of Theorem 3.2.1, we need to introduce some notation.

We shall use the notation = (X1,... ,Xk) e and T = (ti,... ,tk) where k is a positive integer that will vary throughout this section. We denote by (T, p) '-÷ the mapping Rk x £2 -+ £2, defined forT near 0 in Rk, by (3.2.2)

:= exptkXk exptk_IXk_I ...

where we use the notation (3.2.3)

exptX. p :=

. exp f1X1

p.

69

§3.2. SUSSMAN'S THEOREM

For fixed p

Now, let U be an open subset of

U and as above, the mapping

T i-+

(3.2.4)

is defined and maps into U for T in an open neighborhood of 0 in Rk. We denote by 4)) E ,

C Rk the largest open neighborhood of 0 such that (1) if T° = (1?,... then the segments

(3.2.5)

j=l,...,k

((4),...

and such that (2) the mapping (3.2.4) is defined for T E are contained in and q = and maps into U. Conditions (1) and (2) imply that if T° then q can be reached from p by a polygonal path, contained in U, of integral curves of vector fields in D, as described in Theorem 3.2.1 (ii). Taking a disjoint union C Rk x U in which the map (3.2.2) is over p U, we obtain an open subset defined and maps into U. the pseudogroup of local diffeoFor U C as above, we denote by consists morphisms generated by the vector fields in V. More precisely, of all the mappings

(3.2.6)

p

for all possible choices of a positive integer k, E Dk and T UPEU C Rk. Note that the mapping (3.2.6) is not necessarily defined for all p U. We take the domain of the mapping (3.2.6) to be precisely those p E U for which T is in This means that we require not only that the image of p under (3.2.6) is in U but that the entire polygonal path of integral curves joining p to stays in U. The mapping (3.2.6) is a diffeomorphism from its domain onto its image. The inverse of the mapping is given by (3.2.7)

... . exp —tkXk

= exp —ti X1 . exp

.

where '11.T is given by (3.2.2). We shall use the notation

(3.2.8)

:= (Xk, Xk_1,

. . .

,

X1),

?

(tk, tk_I,

.

where and T are as in (3.2.2). It follows from (3.2.7) that For p we denote by the linear span C

. .

=

of D(p), i.e. the tangent vectors at p obtained by evaluating the vector fields of V at p. These subspaces define a C°° distribution in although we shall not make use of this fact here. Now, let U be an open subset of as above. We shall define another

III. VECTOR FIELDS: COMMUTATORS, ORBITS. AND HOMOGENEITY

70

which will depend on 1.) and U. For p collection of linear subspaces of be the smallest Gv(U)-invariant subspace of T,,(U) containing let is the linear span of the union of all i.e.

U,

(3.2.9)

Dk, T E UqEU q in the domain of as described above, = p. Observe that if p and q are as above, i.e. connected by a polygonal path contained in U of integral curves of vector fields in V. then is also in Gp(U). since the inverse of dim = dim we denote For p0 in the domain of ro, andqo = with and

C Tqo(U) the subspace

by

:=

(3.2.10)

for T E near T°. We shall need the following two lemmas for the proof of Theorem 3.2.1.

with 0(T) =

LEMMA 3.2.11. For

T°, p0. and qo as above, the following holds

(3.2.12)

C '-D(qo).

PROOF. Since

is spanned by

(3.2.13)

where 0 is as in (3.2.10), it suffices to show that each tangent vector in (3.2.13) is

Fori = l,...k, we write = (X1, . ( .2.1 )

. .

,

X,,

.

= (t1,... ,Ij,...

. .

,

,

tk)

Xk)

= (f', X,, f"),

= (T',

T"),

and similar notation for T'° and T"°. Hence, (3.2.15) T°

)=

This completes the proof since Gp(U), and is in the domain of O.(d/8t1) E

E

4)r.r.o which together imply that

0

it

§3.2. SUSSMAN'S THEOREM

Vt, and T°

LEMMA 3.2.16. There exists a positive integer k, a vector

with p0 in the domain

=

E GD(U) such that

E PROOF. We start by choosing a positive integer k1, and such that such that po i5 in the domain of (Pt 'T° E

and T,° E

=

P0

Now, in view of and T1° = 0 (we can take e.g. any k1, any E Lemma 3.2.11, either T° equals Pg(p0) or there is a tangent vector \ t'T0• In the first case, the proof is complete. Thus, we assume there is \ V,,0tii.o. By the construction of that there is E E

T2° E Rk2, a point q in the domain of

E

and Yq E

such that (3.2.17)

Yp0 =

Let Y be a vector field in V such that V (q) = Yq. Consider =

I,

2y

2) E

where is defined by (3.2.8), and T = (T1, 5,:, T2) E is in where S Rk2 and 1 E R. Observe that T° = 0, = P0. We claim that VTh.t.TO contains the linear span of and and To see this, we differentiate the map T denoted

by 0(t) as in (3.2.10), with respect to T1 and t at T°. As in the proof of Lemma 3.2.11, we obtain, in view of (3.2.17), (3.2.18)

di To!

=

=

and it is easily verified from the definition of

(3.2.19)

and T° that

E

This proves the claim. Now, either VPOtTO = in which case we are done IRk with k = k1 + 2k2 + 1 and Vk and T° as above, or we repeat the above

procedure to add another missing vector to terminate after a finite number of steps since dim the proof of Lemma 3.2.16.

Clearly, the process will n. This completes

0

PROOF OF THEOREM 3.2.1. We choose an arbitrary open subset U C Q con-

taining p0. We shall define a submanifold W(U) c U, containing such that for all p W(U), and such that all points p E W(U) = can be reached from p0 by a polygonal path contained in U of integral curves of vector fields in V. Let k be a positive integer, Dk, and T° IRk be given by

fl

III. VECTOR FIELDS: COMMUTATORS. ORBITS. AND HOMOGENEITY

Lemma 3.2.16. We claim that there is an open neighborhood ofT0 in Rk such that is a submanifold of U. We call this its image under the mapping T image W(U). To prove the claim, i.e. that W(U) is a submanifold of U, we denote = mforallp E W(U) since, by bym = construction, p and p0 are connected by a polygonal path in U of integral curves. It follows from Lemmas 3.2.11 and 3.2.16 that the mapping T 4t.T(Po) has maximal rank m at T° and, hence, constant rank for T in an open neighborhood of T°. The claim follows from the rank theorem. By construction, all points in W(U) can be reached from po by a polygonal path in U consisting of k integral curves for all p W(U) of vector fields in V. Moreover, we have = and, in particular, all vector fields in V are tangent to W(U). To construct the desired submanifold W satisfying the conclusions of the theorem, we shall choose an appropriate sequence of open neighborhoods Uj C of po decreasing to (P0) and submanifolds Wj = W(U1) such that (3.2.20)

...

C

C 14', C

C ...

C W1.

Choose U1 C to be any open neighborhood of po. Assume that we have chosen U1 c ... C U1. Now, we choose U1.,.1 sufficiently small such that W1 (1 Uj.,.i is Indeed, by construction, consists closed in We claim that C of points reached by polygonal paths, contained in U1.,.1, of integral curves of the vector fields in V. Since the vector fields in V are tangent to W1 and W1 fl Uj.,.i it follows that all such paths stay in W1. This proves the claim. is closed in In view of (3.2.20), the dimension of W, stabilizes, i.e. there exists Jo such that Jo. Take W = W,0. By construction, property (ii) dim W1 = dim Wj.,.1 for all j is satisfied. Property (i) follows easily from (ii). In case the collection V consists of real-analytic vector fields, the manifold W is real-analytic since the exponential 0 map is real-analytic. This completes the proof of Theorem 3.2.1. we can also REMARK 3.2.21. Given and V as in Theorem 3.2.1 and po of po in as the set all points construct the global Sussman orbit W(D, that can be reached from Po by some polygonal path of integral curves of p vector fields in V. The global Sussman orbit W(V, P0) is an immersed (but not necessarily embedded) submanifold of The proof of this goes along the same lines as the proof of Theorem 3.2.1. The reader can consult Sussman [1] for the proof. Observe that, unlike the real-analytic case (see Remark 3.1.6), the germ at P0 of the global Sussman orbit of po contains, but is in general not equal to, the local Sussman orbit at P0 as can be seen by the following example.

ExAMPII 3.2.22. Let

be the open square (_1,1)2 C R2 and V the two

§3.4. CANONICAL FORMS FOR REAL VECTOR FIELDS OF FINITE TYPE

73

vector fields,

—, X(xi)—, ax2 ax1

(3.2.23) where x

isa smooth function in (—1,1) such that x(xi) =

x(x,) >

0 if x1 > 0. It is easily checked that if x0 = (x?,

4)

0 e

if x1 < 0 and with x1 <0,

then the local orbit of x0 is the line (x2 = 4}, whereas the global orbit of x0 in is all of ci. REMARK 3.2.24. If g is the Lie algebra and C°°(c)-module generated by the vector fields in V. and if dim g(p) is independent of p in ci, then as in Remark 3.1.34, the conclusion of Theorem 3.2.1 is also a consequence of the Frobenius Theorem (Theorem 2.1.9).

§3.3. Local Orbits of Real-analytic Vector Fields In this section, we will combine the statements of the local Nagano and Sussman theorems in the real-analytic case. Let V be a collection of vector fields in X(ci), is an open set containing p0 and where X(ci) is as defined in §3.1. where ci C be the smallest C°'(cl)-module containing the Lie subalgebra of X(ci) Let g = generated by D; it is easy to see that g is also a Lie subalgebra. We can now apply the local Nagano theorem to g and the local Sussman theorem to V. be as above, then the local Sussman orbit THEoREM 3.3.1. Let V and g = of P0. relative toD, given by Theorem 3.2.1 coincides with the local Nagano leaf of g at po, given by Theorem 3.1.4.

PRooF. Denote by WN a representative of the local Nagano leaf of g at p0. and by W5 a representative of the local Sussman orbit of po, relative to V. Since the vector fields in Dare tangent to WN, it follows, by Theorem 3.2.1 (i), that there is an open neighborhood V of po such that W5 fl V C WN fl V. We conclude that for every p E Ws fl V, since g(p) c Tp(Ws) at every p e W5, g(p) =

andg(p) =

E WN.

TheequalityofthelocalNaganoleafand

the local Sussman orbit follows from the uniqueness part of Theorem 3.1.4.

0

We shall refer to the local Sussman orbit of po, relative to V. or the local Nagano at po as simply the local orbit of relative to V. leaf of

§3.4. CanonIcal Forms for Real Vector Fields of Finite Let cibean open subset ofR" and V = (X1,... , X,} a collection of real vector are smooth and linearly independent fields in ci. We shall assume that X1,... , at a distinguished point p., E ci. The aim of this section is to show the existence of

III. VECTOR FIELDS: COMMUTATORS, ORBITS, AND HOMOGENEITY

74

in which the vector fields vanishing at weighted smooth coordinates near in V take a special form. We shall first consider the case where the collection V is that is, the values at P0 of the vector fields in V together with of finite type at their commutators of all lengths span We will introduce smooth coordinates (x, s) = (x, s1,... , Sh) E R", with and define the vanishing at = n near r+ x E W, E dilations

&(X,St,... ,S,,)=(IX,lmlSI,...

(3.4.1)

being fixed integers. We assign the m1 < for t > 0, with 2 < ... < weight one to x, the weight m1 to s1, the weight —l to a/ax, and the weight —nz1 is said to be weighted homogeneous of to a/as1. A polynomial p(x, s1,... , degree k if

p(8,(x, s)) = rkp(x, s),

(3.4.2)

t > 0,

(x, s) E

with polynomial coefficients is Similarly, a differential operator Q(x, s, D1, said to be weighted homogeneous of degree k if, for t > 0,

=

(3.4.3) where

q(x, s,

a) denotes the symbol of the operator Q(x, s, D1, D5). The

product of two differential operators, weighted homogeneous of degree k and k' respectively, is weighted homogeneous of degree k + k'. Similarly, a weighted homogeneous differential operator of degree k acting on a weighted homogeneous function of degree k' yields a weighted function of degree k + k'. Any differential operator Q(x, s, with smooth coefficients can be expanded as a Taylor series (3.4.4)

where

each Q•(x, s,

Q(x, s, k. If Q(x, s,

is

weighted homogeneous of degree j. We say that 0(k) if the series (3.4.4) contains only terms of weight at least is weighted homogeneous we shall denote by its D3) is

degree.

Recall that V is assumed to be of finite type at Po. We define the integers mh, also called the Hörmander numbers of V at as follows. The

m1,...

,

the smallest integer for which there exists a commutator C of vector fields in V of length (see § 1.5) such that C(po) is not in the span of V(po) C number m1

is

§3.4. CANONICAL FORMS FOR REAL VECTOR FIELDS OF FINITE TYPE

75

to be the linear span of D(po) and the We define the subspace E1 c values at p0 of all commutators of vector fields in V of length m We define £ to be = dim E1 — r.

(3.4.5)

< ... < and subspaces E1 C We define inductively the numbers m1 < as follows. The number mk÷I is obtained as the smallest E2 c ... c Eh = integer for which there exists a commutator C of vector fields in 7.) of length The subspace Ek÷l is then defined as the span of E* and such that C(po) the values at p0 of all commutators of vector fields in V of length mk+I. We define

(3.4.6)

—dimEk =dimEk÷l

6+1

It is clear that this process terminates after a finite number of steps and that Eh = by the finite type condition. We shall call the number 4 the multiplicity of the Hörmander number For convenience, we shall denote by E0 the span of the values at p0 of vector fields in V. THEOREM 3.4.7. Let c be an open subset LetD = (X1,... , X,.) Po E be a collection of real vector fields with smooth coefficients in that are linearly independent and of finite type at po. Then there are smooth coordinates

(x, s) = (x, s1,... , Sh) near po, vanishing at (3.4.8)

where

axk

(x, si,...

j.q ,

—

such that in these coordinates

ô5jq

k=

1

r,

i) are weighted homogeneous polynomials of degree m1 —

1; here, x = (Xj)i<j
where the

are the Hörmander numbers of V at P0 as defined above and the £j are their multiplicities. if the vector fields in V have real-analytic coefficients, then the coordinates (x, s) may be chosen to be real-analytic coordinates.

Expanding Xk in weighted homogeneous vector fields as in (3.4.4) and using Theorem 3.4.7, we find that Xk = Xk._I + XkO + Xk.I + . . ., where REMARK.

= The content of the theorem is essentially that all weighted homogeneous terms of

degree less than —1 are identically zero.

HI. VECTOR FIELDS: COMMUTATORS. ORBITS. AND HOMOGENEITY

76

We shall refer to coordinates in which the collection of vector fields V takes the We also refer to vector fields form (3.4.8) as canonical coordinates for V at in the form (3.4.8) as being in canonical form. The proof of Theorem 3.4.7 is quite technical, and while the theorem itself will be used in future chapters, in particular Chapter X, the techniques of proof will not be essential for the readability of the rest of the book. Readers who would like to see the idea of the proof in a simpler case are referred to Remark 3.4.67 below. In the proof of Theorem 3.4.7 (and also in the proof given in Remark 3.4.67 ) we shall need the following well-known result (see e.g. Varadarajan[1] or HOrmander[1J), which may be applied to noncommuting differential operators. THEOREM 3.4.9 (BAKER-CAMPBELL-HAUSDORFF). Let A and B be noncom-

muting symbols, and denote by eD the formal exponential series of any symbol D. Then,

eAeB

(3.4.10)

=

where C is aformal series, with real coefficients, in commutators of A and B of all lengths.

OF TIIE0REM 3.4.7. For j = 1,... ,h, we choose commutators S;k. , £j, of length m1 such that their values at P0 span E1 modulo E1...1. Let u W be any coordinates near We may now introduce the smooth change of coordinates PROOF

k = 1,...

(3.4.11) where

X = (Xk)1
(3.4.12)

x X=

Cj

5j . S1 =

that 8(0) = po. so that in the u-coordinates corresponds toO in the (x, s)-variables. In order to avoid confusion, we shall refer to X acting in the (x, s)-variables as X and X acting in the u-variables as simply X. Thus, the expression for X, in the local coordinates (x, s) is determined by Observe

(3.4.13)

8.(k,) = X,,

K

= 1,...

,

r.

We now fix such a ic. We shall calculate the expression for

kK =a(x,s)•

ax

asj

p3.4. CANONICAL FORMS FOR REAL VECTOR FIELDS OF FINiTE TYPE

77

in the following way:

(3.4.14)

(a(x, s), b(x, s)) = (XK8')(8(x, s))

=

where b(x,s) = (bi(x,s),... In what

Z =

,bh(x,s)).

follows, we need to introduce the exponential series of a vector field. Let be smooth real vector fields in Q and = (ii,... , E Rk. ,

We define

Z) :=

(3.4.15)

1=0

with coefficients that are differential operators in If f(u) is a smooth function defined in a neighborhood of then the formal power series in regarded as a formal power series in

(3.4.16)

.

Z)f)(po) =

coincides with the Taylor series at = 0 of the smooth function given by Z) po). defined for sufficiently small E Rk, i.e. (3.4.17)

.

Z) . p0)

(Exp(C . Z)f)(po).

f(u) is real-analytic and the vector fields Z1 have real-analytic coefficients, then the right hand side of (3.4.16) is convergent for sufficiently small and (3.4.17) becomes an equality for real-analytic functions. Hence, from the above comments and (3.4.14) it follows that, as formal power series in (x, s) we have If

(3.4.18)

(a(x,s),b(x,s)) X +

We shall analyze the expression

78

III. VECTOR FIELDS: COMMUTATORS. ORBITS. AND HOMOGENEITY

using the Baker—Campbell—Hausdorff formula in Theorem 3.4.9. Applying Theorem 3.4.9 once, we obtain

(3.4.19)

Exp(x . X +

Exp(tXK)

and Sjp. where C(x, s, C) is a formal series of commutators of t with the XC Observe that C(x, s, t) is a formal power series in (x, s, t) with coefficients that The equality in (3.4.19) is understood in the sense are differential operators in of formal power series in (x, s, t). Since S1,, is a commutator of vector fields in V of length m1, any commutator that appears in the formal series C(x, s, t) is a sum of commutators of the form

(3.4.20)

Y := p(x, s,

]...]],

. . .

for some i, where p1(x, s, C) is a weighted homogeneous polynomial of degree i, provided that (x, s) are assigned weights as in (3.4.1) and t is assigned the weight one. Since the vector fields £ = 1,... , r, and Si,,, j = 1,... , h span for all p in a neighborhood of po in and since and p = 1,... , the statement of the theorem is local, we may shrink and we shall assume this collection of vector fields spans Hence, we may write at every point p the commutator Y in (3.4.20) as

(3.4.21)

Y

+

= p,(X, s,

YJp(U)SJp),

where the and yjp(u) are smooth functions in We introduce the following notion. If q is a nonnegative integer or q = 00,

we say that a smooth function y(u) vanishes to order q at p0 relative to V if the following is true: (I) For any 0 of j < q and any string P = vector fields in V of length (j considered as a differential operator), .

. .

(Py)(po) = 0, and (2) if q < oo, there exists a string P° =

.. .

of elements in V of

length q such that

(P°y)(po)

0.

Observe that a function y(u) vanishes to order 0, relative to V, at P0 if and only 0. Since V is assumed to be of finite type at a function vanishes if Y(po) to order oo relative to V at P0 if and only if all derivatives of y(u) at P0 vanish. We shall denote the order of vanishing of a smooth function y(u) at po relative to V by o(y) so as not to confuse it with the weighted order 0 of vanishing at 0 for differential operators Q(x, s, defined following (3.4.4). Observe that if P = is a string of length j and y(u) is a function . . .

with o(y) = k, then o(Py)

k

—

j.

§3.4. CANONICAL FORMS FOR REAL VECTOR FIELDS OF FINITE TYPE

LEMMA 3.4.22. Let Z := vectorfields in D of length i and write

Z = a(u)

(3.4.23)

wherea(u) =

X

bea comniutazorof

..

+ >2

.

Si,

are smooth in Q. Then

and y1(u) =

(3.4.24)

forallj=

.

79

O(Yjp) 2 m1 — 1, 1,

.

.. ,h

and p= 1, ...

,

We begin by proving that Yq(U) vanishes to order at least one at Po relative to V. i.e. Yq(Po) = 0, for all q such that PROOF.

(3.4.25)

mq

? 1.

S

— I, it follows from the definition Since Z is a commutator of length i of the Hormander numbers that Z(po) E Eq_i. In view of the fact that

I

=

1

q— landp= 1,... ,ej,spanEq_1,we

have q—I

Z(po)=cr(po) X(po)+>2yj(po).Si, i=I and hence, Yq (P0) = 0. We complete the proof by induction on

for the following

property. (*)e For any commutator Z of any length i as in Lem,na 3.4.22, y1(u), defined — i > €. by (3.4.23), vanishes to order at least £for all j such that

holds for £ = 1. Clearly, Lemma 3.4.22 We have already proved that follows for a given commutator Z of length i and a given j from (*)c with £ = the holds fore = 1,... ,€0for€0 — I (fore0 mj —i. is vacuous for all commutators). We shall prove it for £ = £o + 1. condition For this we fix a commutator Z of length i and assume that (3.4.23) holds. We must show that, for any string P = X, . of length Lo and any Jo such that .

.

(3.4.26) we have

(3.4.27)

= 0.

80

Iii. VECTOR FIELDS: COMMUTATORS. ORBITS. AND HOMOGENEITY

We fix a string P of length Lo and Jo as above. Consider the commutator of length

i + to R :=

(3.4.28)

..

.

z]...]],

and decompose it as

R = &(u)• X

(3.4.29)

Observe that, since mh — I implies

S1.

Lo + I implies m10 — (to

+ i)

1, condition (*))

(3.4.30) On the other hand, using the decomposition (3.4.23) of Z and expanding the right hand side of (3.4.28) we obtain

R = (Pcr)(u)

(3.4.3 1)

x

+

h

where the dots... denote a finite sum of terms of one of the following two forms

(Pqa)(u)

(3.4.32)

[x',

[x",

5k,,],

with Pq, strings of length q
X+

(Pqa)(u)[X', (3.4.33)

—

q

s3) h

.X+

Skp] =

.

si).

To prove (3.4.27), in view of (3.4.30) and (3.4.3 1), we must show that (3.4.34)

First, since [x',

= 0,

= 0.

isacommutatoroflengthto—q+1

1 by (3.4.26), it follows from condition (*) that (Pa) = 0. To prove the i+q second identity in (3.4.34), which will complete the induction, we recall that )'kp(U) is the coefficient of 5k,, in the decomposition of the commutator Z, which is of

§3.4. CANONICAL FORMS FOR REAL VECTOR FIELDS OF FINITE TYPE

81

length). s k hold by nduef iofl it follows that (P4ykp)(po) = 0, provided that mk — I q + 1. On since q theotherhand, if —i
LEMMA 3.4.35. Let D(x, s, t) be a formal power series in (x, s, t), with no constant term, with coefficients thai are vector fields in

(3.4.36) D(x, s, t) = P(x, s, t) + O(mh + 1) =

Write

Pk(x, s,t) + O(mh + I),

where Pk(x, s, t) is a weighted homogeneous polynomial of degree k with coefficients that are vector fields in For each k, 1 k
Pk(x,s,t) =

(3.4.37) Denote by

the number of terms in (3.4.37) for which the corresponding

coefficient

a Assume that, for every k =

I,... (3.4.38)

>_mj,

Also, assume that for some

Vj = 1,... ,h, Vp= 1,...

1,

Vi
(3.4.39)

0 let r(x, s, t)fl(u)Z be a vector field chosen among s, t)a,CO(u)Xj)., (P€':(X, s,

such that (3.4.40)

is nonconstant in

Then

Exp(D(x, s, t))

= Exp(r(x, s, t)(fl(u) — fl(po))Z) Exp(D'(x, s,

82

111 VECTOR FIELDS: COMMUTATORS. ORBITS. AND HOMOGENEITY

where D'(x, s, t) has the same properties as D(x, s, t), i.e. satisfies the analagous

assumptions (3.4.36—3.4.39). In addition, we have

=#,0(D)— 1

(3.4.41)

and D(x, s, t) — D'(x, s, t) = 0(to). PROOF. We shall prove the lemma only in the case where r(x, s, t)f3(u)Z = s, for some j, p, and leave the other case to the reader. Applying the Baker—Campbell—Hausdorff formula in Theorem 3.4.9, we obtain (3.4.40) in which (3.4.42)

D'(x, s, t)

= P(x, s, t) —

s,

t) + O(mh + 1),

where C(x, s, t) is a finite sum of commutators of s,

—

with the remaining terms in P(x, s, t). It is easy to see that C(x, s, t) is O(€o+ 1), which proves that D'(x, s, t) — D(x, s, t) = 0 (so). Hence, since we have removed one nonconstant term in P€0 (x, s, t), it is clear that the proof of the lemma reduces to checking that each commutator in C(x, s, t) is decomposed as (3.4.43) for some k > + 1, where each s, t) and geneous polynomials of degree k, and such that

s, t) are weighted homo-

(3.4.44)

forallj=1,... ,handp=1,... We complete the proof as follows. Let p*(x, s, t)a(u)Y and q1(x, s, satisfy where Y, Z E

(3.4.45) where

k + 0(a)

m(Y),

I

+ o(fl) > m(Z),

p(x, s, t) and q (x, s, t) are weighted homogeneous polynomials of degree

k and £, respectively, and 1

(3.4.46)

nz(U)

nz3

if U = X1 for some i = 1,... , r, if U = Si,, for some j = 1,... ,

andp=1,...,€1.

83

§3.4. CANONICAL FORMS FOR REAL VECTOR FIELDS OF FINITE TYPE

Expansion of their commutator gives (3.4.47)

[pk(x. s, t)a(u)Y,

s,

t) (a(u)(Y,5)(u)Z — fi(u)(Za)(u)Y + a(u)P(u)[Y, z]). Decomposing the commutator [Y. Z] according to (3.4.23), we obtain

(3.4.48)

[p*aY, qefiZ]

= where

+afl(8 X +

= €3(u). By Lemma 3.4.22, we have

8 = 8(u) and

0(E1) > m1 — (m(Y) + m(Z)).

(3.4.49) We

—

claim that each term on the right side of (3.4.48) can be written in the form where s, t)a'(u)Y', for some Y' {(X1),

k' + o(cr') ? m(Y').

(3.4.50)

a Z, pk a'. Y' = pkqea

We leave it to the reader to verify this for the terms

pkqtaf3ö X, and we shalicheck itforthe term

fi

(Za)

Y,

S3.

In

.

this case, we have (3.4.51)

k' + o(a') ? k + I + o(a) + o(fi) + o(EJ)

? m(Y)+m(Z)+m1 —(m(Y)+m(Z))=m1 =m(Y'), where in the second line we have used (3.4.45) and (3.4.49). This completes the proof of the claim. Lemma 3.4.35 follows from the claim by induction. U

In order to use Lemma 3.4.35 in the proof of Theorem 3.4.7, we need the following observation. LEMMA 3.4.52. Let D(x, s, t) be a formal power series in (x, s, t) with coefficients that are differential operators in c, and let Z be a vector field in such that Z(po) = 0. Then the following identity holds, as formal power series in (x, s, t,

(3.4.53)

Exp(D(x, s, t))f)(po)

(Exp(D(x. s, t))f)(po).

III. VECTOR FIELDS: COMMUTATORS, ORBITS, AND HOMOGENEITY

84

for every smooth function f(u) in PROOF.

By expanding the formal power series, we obtain

D(x;s, I)))

Exp(D(x, s, t))

(3.4.54)

= D(x,s,t)J\ j! /

=

J

.

D'(x, s, t,

is a formal power series in (x, s, t, with coefficients that are Hence, the product CZD'(x, s, is a formal power differential operators in series in (x, s, t, with coefficients which are differential operators, each of the form ZP for some differential operator P. Since Z(po) =0, we obtain, as formal power series in (x, s, t, where

(3.4.55)

0,

s, I.

for every smooth function f(u) defined in

This completes the proof of the

0

lemma, in view of (3.4.54).

We are now in a position to complete the proof of Theorem 3.4.3. We claim the following. ClAIM 3.4.56. If C(x, s, t) is the commutator appearing in the right hand side of (3.4. 19), then

(3.4.57)

Exp(x . X + tXK +

Si + C(x,s, r))

= Exp(h'(x, s, :)Z1). . . Exp(hk(x, s, t)Zk) Exp(F(x, 5, t) + O(mh + 1)), where the Zj are smooth vector fields such that Z, (p0) = h(x, s, t) are weighted homogeneous polynomials, and

(3.4.58) F(x, s, t) = (x+g(x, s, t)).X

0,

for i =

1,

. . .

k, the

(x, s,

t) appearing in (3.4.58) is an RT-valued polynomial which

is 0(2), the

s, t), j = 1,

. . .

h,

are weighted homogeneous polynomials which

polynomials of degree m,, and the fi(x, s, t) are are + 1).

We first complete the proof of Theorem 3.4.7 assuming that Claim 3.436 has been proved. Recall that the Taylor series of the coefficients (a(x, s), b(x, s)) =

p3.4. CANONICAL FORMS FOR REAL VECTOR FIELDS OF FINITE

(a(x, s), bi(x, s),...

s)) of X are given by (3.4.18). By (3.4.19), Claim 3.4.56, and repeated use of Lemma 3.4.52, we obtain ,

(Exp(F(x, s, t) + O(mh +

(3.4.59) (a(x, s), b(x, s)) --

where means, as before, that both sides are equal as formal power series. It follows immediately from the definition (3.4.16) that (3.4.60)

(Exp(F(x, s, r) + O(mh +

= (Exp(F(x, s, t))®')(po) + O(mh + I) and, hence, from (3.4.17) that (3.4.61)

(Exp(F(x, s, t) + O(mh + l))€Yj(po) —

®'(exp(F(x,s,t))

.

po)

+ O(mh + 1).

Now, in view of the definition (3.4.11) of® and the form (3.4.58) of F(x, s, t), we deduce that (3.4.62)

c(x, s, t) + O(mh + 1),

(Exp(F(x, s, r) + O(mh +

where c(x, s, 1) is the Re-valued polynomial (3.4.63)

c(x,s,t) = (x,

Here, 81K denotes the Kronecker delta, i.e. the symbol which is zero if i

K

and

one if I = K. Equation (3.4.59) yields (3.4.64)

(a(x, s), b1 (x, s), .

. .

,

s))

+ O(mh). This completes the proof of Theorem 3.4.7 by taking

s) =

S))1
and

(3.4.65)

p7(x,s) = -f!L(x,s,O).

j = 1,...

(recall that K has been fixed throughout this proof). The fact that p7(x, s) is weighted homogeneous of degree rn —1 is immediate since (x, s, 1) is weighted

III. VECTOR FIELDS: COMMUTATORS, ORBITS. AND HOMOGENEITY

86

homogeneous of degree m1. To complete the proof of Theorem 3.4.7, it remains to prove Claim 3.4.56. PROOF OF CIAm4 3.4.56. We denote by D(x, s, t) the expression inside the exponential on the right hand side of (3.4.19), i.e.

D(x, s, 1) = x X + tXK +

(3.4.66)

S1

S1 + C(x, s, t).

We shall show that D(x, s, t) satisfies the hypotheses of Lemma 3.4.35 with

2.

Observe that #1(D) = 0. Recall that each commutator Y in C(x, s, t) is of the form (3.4.20). We write such a commutator in the form (3.4.21). It suffices to show that I + O(yjp) m1 for each yjp(U) in (3.4.21), which follows immediately from Lemma 3.4.22. Thus, we may apply Lemma 3.4.35 to D(x, s, t) repeatedly until the resulting D'(x, s, t) satisfies = 0 fort This completes the proof of Claim 3.4.56. 0

The proof of Theorem 3.4.7 in the smooth case is complete, in view of the arguments preceding the proof of the claim. The real-analytic case follows immediately from the fact that the change of coordinates (3.4.11) is real-analytic when the vector fields in V are. 0 REMARK 3.4.67. Theorem 3.4.7 has a simpler proof for some systems of vector fields which form a Lie algebra; the proof given above was inspired by this case.

We indicate now how the proof simplifies in this special case. Let g be a finite dimensional real Lie algebra of smooth vector fields on an open neighborhood of 0 in such that dimg(p) = dimg = n for every p where dimg is the dimension of g as a real vector space. Suppose that g has a vector space direct sum decomposition (3.4.68)

g=

ga...

with

= [g1, g], J

? 1, 9h+2

=

0.

11(3.4.68) holds, we say that g is a graded nilpotent Lie algebra. We introduce new coordinates on near the origin as follows. Let X = (X1,... , Xr) be a basis of 9i and S1 = a basis of for j = 1, . . h. Then the mapping .

.

is

.

i-÷ u

= 8(x,s) = exp(x . X +s1

+ . . +Sh .

.0

ThesetV = {X1,... , =h+1

of finite type atO, with Hörmandernumbersm1 = 2, m2 = 3,...,

and multiplicities , 4. By Theorem 3.4.7 and its proof (see 3.4.11), in the (x, s) coordinates one has (3.4.69)

Xk = Xk._i + 0(0), k = 1,

. . .

§3.5. CANONICAL FORMS FOR REAL VECTOR FIELDS OF INFINITE TYPE

87

where Xk._I is weighted homogeneous of degree —1. However, in this case one can prove (3.4.69) more easily directly and also show that the error term 0(0) vanishes of infinite order as follows. From (3.4.68) and the Baker—Campbell— Hausdorif formula (Theorem 3.4.9), one can check that (3.4.70)

Exp(x . X

+ Exp

(x . x + tXK +

+ qj(t, x, s)) . si).

where qj(t, x, s) is a weighted homogeneous polynomial of degree j

t,x given weight! (3.4.71)

given weightj + 1. Indeed, if2

i

+ 1, with

h + 1, then

=a

. . .

for some constant real vector a (and the left hand side of (3.4.71) vanishes if i > h + 1). Thus the proof of Theorem 3.4.7 is greatly simplified in this case, since Lemmas 3.4.22, 3.4.35, and 3.4.52 are not needed. One can then finish the proof of (3.4.69) with 0(0) replaced by O(oo) using (3.4.6 1) to (3.4.65) as before.

§3.5. CanonIcal Forms for Real Vector Fields of Infinite Type In this section, we shall consider the case where the collection of (pointwise) linearly independent real vector fields V = (X1,... , X,) in c W are of infinite type at P0 E i.e. the values at p0 of the vector fields in V together with their commutators of all lengths span a proper subspace E The Hörmander numbers 2 m1 < ... their multiplicities , 4, and the subspaces E0 C can be defined as in §3.4 with the only difference being that C ... C Eh = E We denote by the codimension of E in i.e. h

(3.5.1)

:= n

—

(r +

We will introduce smooth coordinates (x, s, X

E Rr,Sj E

forj = 1,... ,h,s'

E

s') = (x, si,... , s') E R", with Weassigntheweightonetox,

the weights m1 to Sj, and the weight m to s'. In what follows we shall vary the weight m, always requiring m > mh. With these weights, we define the notion

of weighted homogeneity for polynomials p(x, s, s') and differential operators

Q(x, s,s',

D?) as in §3.4.

The main result in this section is the following, which extends Theorem 3.4.7 to the infinite type case.

III. VECTOR FIELDS: COMMUTATORS, ORBITS. AND HOMOGENEITY

88

TIIE0RBM3.5.2. Let 1 bean open subsetofW, POE = {X11... , Xr) a collection of real vectorfields with smooth coefficients in c that are linearly independent and of infinite type at po• Then there are smooth coordinates (x, s, s') = (x, s1,... 5h. s') near P0. vanishing at P0. such that

k = 1,... ,r,

Xk =

(3.5.3)

jq

j.q

where Pq (x, s1,... , i) are weighted homogeneous polynomials of degree — and is a vector field which is O(O)for any choice of weight m > mh assigned —

1

to s' as above. Moreover if the vector fields in V have real-analytic coefficients, then (x, s, s') may be chosen to be real-analytic coordinates. following Theorem 3.4.7, we note that Theorem 3.5.2 implies that the expansion of Xk in weighted homogeneous vector fields is is given by given by Xk = Xk._I + Xk.o + Xk.I + . . ., where REMARK. As in the remark

Xk._I =

a

a k

jq

j.q

regardless of the weight m assigned to s'.

We have the following corollary which is a reformulation of Theorem 3.5.2. We leave the proof to the reader.

po. V be as in Theorem 3.5.2. In the coordinates given by Theorem 3.5.2, we have, fork = 1,... , r, COROLLARY 3.5.4. Let

Xk = ak(x, s, s').

(3.5.5)

s')•

+

+

s')•

where the components of the vector valued functions a'(x, s, s'), b(x, s, s'), and

s, s') are given by

(3.5.6)

s, s') = =

+

s, s')

s) + s'

s')

with is the Kronecker symbol, 0,0) = 0, are as in si,... , Theorem 3.5.2, s) is0(m1), s) isasmoothfunction which vanishes at 0 together with its derivatives of all orders, and where (x, s') and (x, s, s')

§3.5. CANONICAL FORMS FOR REAL VECTOR FIELDS OF INFINITE TYPE

89

jf the vector fields in D are realare smooth vector valued functions. analytic, then all the functions in (3.5.6) are real-analytic and, in

As in §3.4, we shall refer to coordinates in which the collection of vector fields V takes the form (3.5.5) as canonical coordinates for V at P0. We also refer to vector fields in the form (3.5.5) as being in canonical form.

3.5.2. The proof follows very closely that of Theorem

PROOF OF

3.4.7. However, we need a slightly modified version of Lemma 3.4.22. We of length m1 such that their values at P0 choose commutators Sjq' q = 1,... , modulo E1...1 as in the proof of Theorem 3.4.7. We also choose smooth span such that, after shrinking if necessary, the values at p vector fields Si,... , of the collection of vector fields

(3.5.7)

(Xj)i<j
. .

.

i

The order of vanishing at po relative to V of for all p a smooth function y(u) is defined as in §3.4. Note that in the infinite type case, o(y) = oo means that every term in the Taylor series at P0 of y(u) vanishes when s' = 0. form a basis of

LEMMA 3.5.8.

Z=

Let m > mh be a fixed integer assigned as weight of s'. Let ... ] .. .1] be a commutator of vector fields in V of

length i and write (3.5.9)

Z = a(u) X + >2 yj(u) Sj + y'(u)

SI,

Yj(U) =

wherea(u) = are smooth in

.

=

Then

0(Yjq) m—

REMARK. Let

1,...

Z be as in Lemma 3.5.8, and Z = [z,

for

some £ =

If we expand 2 as in (3.5.9) then the corresponding coefficients and necessarily satisfy (3.5.10) with i replaced by i + m. Indeed, the right hand sides in (3.5.10) become negative, so that the condition is vacuous. Thus, we may extend the result of Lemma 3.5.8 to commutators that involve both the and the provided we regard the S as if it were a commutator of length m. ,

90

III. VECTOR FIELDS: COMMUTATORS, ORBITS, AND HOMOGENEITY

The proof of Lemma 3.5.8 is very similar to that of Lemma 3.4.22 and we shall not give it. As in the proof of Theorem 3.4.7, we introduce the smooth change of coordinates

(3.5.11)

u

Fix m >

mh,

= ø(x, Si,...

,

s') = exp (x . X +

sj Sj + s'

5')

the weight assigned to s'. Proceeding as in the proof of Theorem 3.4.7 (leaving the details to the reader), we obtain for k = 1,... , r (3.5.12) k

j.q

jq

si,... , s1...I) are weighted homogeneous polynomials of degree q(x, s) is a weighted homogeneous polynomial of degree m — 1, and is — 1, a vector field which is 0(0), relative to the weights I
(3.5.13)

+s'.

+s' .

where the formal series on the right hand side of(3.5. 13) is the decomposition of the

s) Taylor series of (x, s) into its weighted homogeneous terms; here, is weighted homogeneous of degree v. It follows from (3.5.12) and (3.5.13) that (3.5.14) Since (3.5.15)

(x, s) is weighted homogeneous of degree rn —

1,

we deduce that

v=O,... ,m—2.

(x, s) is identically s) together with its derivatives of all orders vanish at po. In particular, 0, i.e. for any choice of rn > mh, we have q(x, s) 0 in (3.5.12). Observe that the vector fields .. . 5 can be chosen real-analytic, even constant, in the u coordinates if we choose sufficiently small. Hence, if the vector fields in V are real-analytic, the change of variables given by (3.5.11) is real-analytic. l'his concludes the proof 0 of Theorem 3.5.2. Since m was arbitrary, we conclude that the Taylor series of

§3.6. WEIGHTED HOMOGENEOUS REAL VECTOR FIELDS

91

We would like to point out that Corollary 3.5.4, applied in the real-analytic case, implies the local Nagano Theorem (Theorem 3.1.4). Indeed, we may take for V a collection of vector fields in g which span g(po), where g is as in the statement of Theorem 3.1.4. Here, the set of commutators Sjq is empty and the local Nagano leaf at p0 is given by the manifold {s' = O}. On the other hand, we would like to point out that Theorem 3.4.7 in the real-analytic case together with Theorem 3.1.4 imply Theorem 3.5.2. Indeed, the collection of vector fields in V (see §3.3) and are of finite type at are tangent to the local Nagano leaf at p0 of po in that manifold. Choosing local coordinates so that the local Nagano leaf at po becomes (s' = O} and applying Theorem 3.4.7 to V in the manifold (s' = 0) yields Corollary 3.5.4. In the smooth case, Theorem 3.5.2 does not seem to follow from the finite type result, Theorem 3.4.7, in any simple manner, since the local Sussman orbit of p0 (see Theorem 3.2.1) is not determined by the Taylor series of the vector fields in V at P0.

§3.6. Weighted Homogeneous Real Vector Fields where Let (x, s, s') = (x, s1,... , s') be coordinates vanishing at 0 E R', We assign the weight one to x E Rt' for j = 1,... , h, s' E x, the weight m1 to for j = 1, . . . h, and the weight m to s'. We assume 1
< ... THEOREM 3.6.1. Let (x,s,s') = (x,s1,... ,sh,S'), 1 < m1 < fork = I,... ,r, the be as above. rnh < m, and following weighted homogeneous vector fields of degree —1

(3.6.2)

X&._i =

—f--

axk

+

s1_i).

s1

h

a

k k

where

(x, s1,...

, s1 - i)

Li

j=I q=I

a

jq

are weighted homogeneous polynomials of degree

III. VECTOR FIELDS: COMMUTATORS, ORBITS, AND HOMOGENEITY

92

Also consider, fork = 1,... , r, (3.6.3)

J = 1,...

,

h

and Q = 1,... ,Lj,

Xk.I = 1=1 Q

a

,Sj_j)—. asjq

q=I

For k = 1,...

,

r, let X* be a real smooth vector field defined near 0 in

such

that Xk = Xk._I + 0(0).

(3.6.4)

Then, the following are equivalent,

as Hör(i) The collection of vector fields (X1,... X,.) has m1,... , mander numbers with multiplicities Li,... , at 0. (ii) The collection of vectorfields { X1 } has m1,... , m,, as Hör, ,

.

mander numbers with multiplicities Li,... , at 0. (iii) For any integers J, Q as in (3.6.3), the collection

has .tj_1, QatO. ,

Hörmandernumbersm1,... ,m1 withmultiplicitiesti,

. .

PROOF. First, observe that if X and X' are weighted homogeneous vector fields

of degree d and d', respectively, then [x, Y] is weighted homogeneous of degree

d + d'. We shall start by proving that (i) and (ii) are equivalent. Denote by

thespanofa/axkato,fork E0 C a commutator of length t, (3.6.5)

= 1,... ,r. Leti

andconsider

. . .

=

. . .

+ 0(—L + 1).

Note that any vector field of the form c a/asjq or c a/as:, for some constant c 0, or —m, respectively, and, hence, is not is weighted homogeneous of degree It follows that 0(—l + 1), since L (3.6.6)

.. .

=

..

.

[Xvg_1._1,

mod Eo.

It follows from (3.6.6) that the first Hörmander number of the collection of vector fields (X1,... ,X,.) is m1 with multiplicity if and only if the same is true for to be the Xr._i }. Assuming this, we define the subspace E1 C span of E0 and the right hand side of (3.6.6) for such commutators of length m1.

93

NOTES FOR CHAPTER III

Repeating this procedure and defining the subspaces E1 inductively, we find that, for every t (3.6.7)

[xv,,

.. .

=

.

. .

mod E1,

which proves that the collection of vector fields in (1) has the Hörmander number with multiplicity £1+i if and only if the same holds for the collection of vector fields in (ii). This proves the equivalence of (i) and (ii). The fact that (iii) implies (ii) is trivial. The converse is easy to prove using a (x, Si,... , — i) of d/dSjq similar argument if one observes that the coefficient 0 is independent of St for £ j. We leave the details to the reader.

Notes for Chapter III Theorem 3.1.4 is due to Nagano [1], and Theorem 3.2.1 is due to Sussmann [1]. The importance of the notion of finite type for real vector fields was discovered by Hörmander [1] in his fundamental work on the hypoellipticity of second order

differential operators formed by the sum of squares of such vector fields; his methods of proof involved the use of the Baker-Campbell—Hausdorff formula. Folland and Stein [11 proved Theorem 3.4.7 for the special case of sets of real vector

fields arising from Levi-nondegenerate hypersurfaces. The "lifting theorem" of Rothschild and Stein [1] used exponential coordinates to approximate arbitrary sets of real vector fields of finite type by left invariant vector fields on a Lie group of higher dimension. Heifer and Nourrigat [1] established Theorem 3.4.7 using the representation theory of nilpotent Lie groups. (See also the book HeIfer and Nourngat [2].) The generalization given in Theorem 3.5.2 for vector fields of infinite type can be found in Baouendi and Rothschild [I]. The proofs of Theorem 3.4.7 and Theorem 3.5.2 given here are somewhat different from those which have previously appeared in the literature. For a different approach to these proofs see also the book of Treves [21. The reader can also consult the notes of Chapter IV for references concerning canonical coordinates for generic submanifolds in CN.

CHAPTER IV

COORI)INATES FOR GENERIC SUBMANIFOLDS

Summary In this chapter, we return to the study of generic submamfolds of CN. We begin by defining the CR orbit of a point in a CR submanifold, and relating it to minimality and finite type. Next, we introduce so-called normal coordinates, and prove their existence, for real-analytic generic submanifolds. Their importance will become apparent in Chapter IX for the proof of the reflection principle. We also introduce a notion of approximately normal coordinates for a smooth generic submanifold M. In §4.3, we use the results of Chapter III (applied to the real and imaginary parts of the CR vector fields on M) to prove the existence of canonical coordinates forM. These will be used in connection with the Segre sets in Chapter X. We conclude the chapter by proving the existence of canonical coordinates that are also normal.

§4.1. CR Orbits, Minimality, and Finite In this section, we shall define the notion of a local CR orbit of a point in a generic submanifold and relate this notion to that of minimality and finite type, as introduced in Chapter I. We start with the definition of CR orbit.

DEfIrsJmoN4.l.l. LetM c

CN

beasmoothCRsubmanifoldandp EM.

We

define the local CR orbit of p to be the local Sussman orbit of p in M, as defined in §3.2, relative to the set of all smooth sections of TCM. the complex tangent bundle of M, defined by (1.2.7). We denote the Local CR orbit of p by Orb1,. Hence Orb, is a germ at p of a smooth submanifold of M whose tangent space

at each point q contains TM. In defining Orb,,, one could replace M by any open neighborhood of p in M (see the remark following Theorem 3.2.1). Now, with a submanifold that represents let po M. We will identify the germ it. Recall that dimR TM equals 2n, where n denotes the CR dimension of M; it follows that (4.1.2)

2n

dima Orb,,0 94

dime M.

§4.2. NORMAL COORDINATES FOR GENERIC SUBMANIFOLDS

95

It follows from the definition that the local CR orbit is a CR submanifold of Also, as a = T;M for every p CR dimension n. In fact, consequence of Theorem 3.2.1 (1), if W' C M is any CR submanifold containing fl V C of CR dimension n, then there is a neighborhood V of po such that

W,nV. LetM beasmooth CRsubmamfoldin(Y' and po EM. The following are equivalent

(i) M is minimal at po; (ii) dimR = dlmR M.

If in addition, M

is real-analytic, then (1) and (ii) are equivalent to

(iii) M is of finite type at p0. It should be noted here, in view of Theorem 3.3.1, that when M is real-analytic, then the CR orbit coincides with the Nagano leaf at of the real-analytic sections of Tc(U), where U is a sufficiently small neighborhood of p0 in M. PROOF OF THEOREM 4.1.3. Recall from Definition 1.5.12 that M is minimal at po if and only if there is no CR submanifold W' of M through P0 of CR dimension

n such that dimR W'
§4.2. Normal Coordinates for Generic Submanifolds Let M be a smooth generic submanifold in CN, and po E M. It is sometimes useful to choose holomorphic coordinates which satisfy special conditions. In this section, we shall define and prove the existence of "normal" coordinates for a real-analytic generic manifold M with p0 E M, and "approximately normal" coordinates when M is assumed to be only smooth. We start with the real-analytic case, i.e. we let M be a real-analytic generic submanifold in Ce'. We denote by n the CR dimension of M and by d the real codimension of M in C". Since M is generic, we have n + d = N. By a transversal coordinate for M we shall mean a holomorphic function w, valued in Cd defined near for which

IV. COORDINATES FOR GENERIC SUBMANIFOLDS

96

defined near

there exists a holomorphic function z, valued in following properties:

satisfying the

(1) The functions (z, w) form a system of holomorphic coordinates near P0 with (z. w)(po) = 0. s), defined in a (ii) There exists an We-valued real analytic function neighborhood of 0 in R2'4' such that in a neighborhood of P0 in M is given by 1mw

(4.2.1)

where the function (4.2.2)

s) satisfies

0, s)

s)

0.

In this case the coordinates (z, w) for which w is transversal are called normal. We note that by the reality of the second identity in (4.2.2) implies the first. The notion of normal coordinates can also be made independent of the choice of a real analytic defining function as follows. If u(z, w, is a real-analytic function defined near 0 in then it can be extended as a holomorphic function by replacing by x and th by rin u(z, w, T) in a neighborhood of 0 in the power series expansion of u. We have the following characterization. be a real analytic generic subman (fold PROPOSITION 4.2.3. Let M C through Oand (z, w) any holomorphic coordinates near 0. Then (z. w)are normal

coordinates for M if and only if there exists a real analytic R&valued defining function p for M near 0 such that (4.2.4) Furthermore, if (4.2.4) holds for some real analytic defining function, then it holds for any such function.

PROOF. If (z. w) are normal coordinates, it follows from (4.2.1) that condition (4.2.4) is satisfied for p = Im w — Re w). For the converse, note that condition (4.2.4) is equivalent to

(4.2.5)

p(z, 0, w,

= K(z, w, ti')(w —

where K (z. w, t) is ad x d-matrix valued holomorphic function in a neighborhood A A. . . A 8pd(O) 0 (see Definition 1.3.3), it follows Since from (4.2.5) that K (0) is an invertible matrix. We can therefore use the implicit function theorem to solve for Im w in the equation p = 0 to obtain satisfying

of 0 in

14.2. NORMAL COORDINATES FOR GENERIC SUBMANIFOLDS

97

(4.2.2); this proves the normality of (z, w). The last statement follows from the fact that if p' is another real analytic defining function, then p' = ap where a is a

0

d x d-matrix valued real analytic function.

The next theorem shows that one can always find normal coordinates for any real analytic generic submanifold.

If M C

is a real analytic generic subman(fold and po E M, then there exist normal coordinates for M at po• THEOREM 4.2.6.

Although Theorem 4.2.6 gives the existence of normal coordinates, it should be noted here that normal coordinates are not unique. be regular coordinates at PROOF. Let such that M is given by (4.2.7)

WW

i.e. holomorphic coordinates

th±W)

= 0. a real-analytic We-valued function with 4(0) = 0 and with We shall choose new coordinates (z, w) satisfying = z, ii' = w + h(z, w) with h holomorphic near the origin, h(0) = 0, and dh(0) = 0. We claim that h can be chosen so that (z, w) are normal coordinates. In the new coordinates, M is given by

(4.2.8) w— th+h(z,w) —h(z,w)

I

-

w+th+h(z,w)+h(z,w)\)=0. 2

Then, by Proposition 4.2.3, (z, w) are normal coordinates if and only if the left hand side of (4.2.8) is identically zero when = 0 and th = w, i.e.

(4.2.9)

h(z, w) — h(0, w) —

0,

+

h(z, w) ± h(0, w))

= 0,

where w) is the holomorphic function obtained from h(0, w) by replacing the coefficients in the Taylor expansion by their complex conjugates. To solve (4.2.9), we seek a solution h(z, w) satisfying also h(0, w) = —h(0, w), i.e. the coefficients of h(O, w) are pure imaginaiy. For this, we substitute for h(0, w) in (4.2.9) to obtain

(4.2.10) h(z, w)—(—h(O,

h(z, w)-i-(—h(0, w)))

98

IV. COORDINAThS FOR GENERIC SUBMANIFOLDS

Putting z = 0 in (4.2.10) we obtain h(0, w) = i4(0, 0, w). Substituting this for h(0, w) in (4.2.10) we obtain (4.2.11)

h(z, w) —

w))

h(z, w) +

0, w)) 2

By the implicit function theorem (4.2.11) has a unique solution h(z, w), which 0 satisfies h(0, w) = i4(0, 0, w) and dh(0) =0. This proves Theorem 4.2.6.

PRoposmoN 4.2.12. If(z, w) are normal coordinates for M, then there is a C" -valued holomorphic frnction Q(z, r) defined in a neighborhood of 0 in satisfying Q(z, 0, r) Q(0, r) r and such that M is given by (4.2.13)

w = Q(z,

ü').

PROOF. If M is given by Im w = 4(z, s), then by the implicit function theorem for holomorphic functions, one can solve for w to find a unique holomorphic function Q(z, x1 r) so that M is also given by w = Q(z, ü). Since (z, 0, s) 0 by the nonnality of the coordinates, the unique solution for = 0 is

Similarly,wehaveQ(O,x,r) w = üi,whichisequivalenttoQ(z,O,r) The proof of Proposition 4.2.12 is complete.

0

We now turn to the smooth case. Let M be a smooth generic submanifold of CN of CR dimension n and codimension d, and let p0 E M. Although we cannot prove as strong a result as Theorem 4.2.6, we can find holomorphic coordinates near po that are approximately normal of an arbitrarily given order K. By this we x C'1, vanishing at mean holomorphic coordinates (z, w) E such that M is given by (4.2.2) near po = (0,0) with some smooth function s) satisfying (4.2.14) where (4.2.15)

mod K,

mod K,

0, s) denotes the following part of the Taylor series of

at (0,0,0)

aaap

and similarly for s). Moreover, denotes equality as formal power series, and equality mod K means that the lowest order terms in the difference are of degree at least K.

§4.2. NORMAL COORDINATES FOR GENERIC SUBMANIFOLDS

99

THEOREM 4.2.16. If M is a smooth generic submanifold and po E M, then, given a positive integer K, there exist approximately normal coordinates of order

K forMat p0. PRoOF. The proof follows very closely that of Theorem 4.2.6. First, observe

that Proposition 4.2.3 holds in the smooth case if identity, written as "a", in (4.2.4) is replaced by equality of Taylor series mod K, written "-- mod K", the conclusion then being that the coordinates (z. w) are approximately normal of order K. The proof is then completed by solving (4.2.9) for h(z, w) as a formal power series, and then making the polynomial change of coordinates = z,

=

+

w), where hK(z, w) denotes the series h(z, w) truncated at level K. The details are left to the reader. 0 w

We now give the analogue of Proposition 4.2.12 in the smooth case. PROPOSITION 4.2.17. If(z, w) are approximately normal coordinates of order then there exists a C" -valued snzooth function Q(z. w, th) defined in a neighborhood of 0 in such that, for each j = 1,... , d, and each multiindex a,

Kfor M at

Da i—j(z,z,w,w)=O, (z,w)EM, a w1

(4.2.18)

/

where D = (D1,...

represents the gradient in all the (real) variables,

,

and such that M is given by (4.2.19)

w=

with Q(z,0,w,th)--

mod K.

PRoOF. We assume that M is given by (4.2.2) near P0 = (0, 0), where (z, w)

x are the given approximately normal coordinates of order K. We may further assume that #(z, s) is defined in some product neighborhood x where

and are open balls centered at the origin in and R", respectively. We let #(z. s, s') be an almost holomorphic extension ins s) to x V5 x Rd. By this we mean that s, s') is smooth in x V3 x Rd. s, 0) 4(z, s), and, for all j = 1,... , d and all multi-indices a,

+

(4.2.20)

as,

s,0)

/

0,

where D is as in the statement of the proposition. The construction of the function 'b(z, s, s') is fairly simple and standard (see e.g. Nirenberg [1]). Indeed, it

suffices to take -

s,s') =

i a.

s

Z,

IV. COORDINAThS FOR GENERIC SUBMANIFOLDS

100

is a > 2, and = 1 for < 1, = 0 for where A E sequence of positive numbers tending to infinity sufficiently fast. The reader can check that (4.2.20) implies e.g. that

for any multi-index fi. As a consequence, we have 4'(z, 0, s, s')

(4.2.21)

s,

as formal power series. Denote by (4.2.22) defined

s')

-I

w-th\ 2

for z E

mod K,

0

'

2i

)'

This function satisfies

and w

s,

s)

s). Moreover, (4.2.20) becomes

(4.2.23)

\ d and all multi-indices a, and (4.2.21) becomes

for all j = 1

0, w, ü)

(4.2.24)

0(0,

w, ü,)

mod K.

0

We claim that we can solve, for z, u in a neighborhood of the origin in respectively, the equation

and

(4.2.25)

for w, and obtain w = Q(z. ü, u)with Q(0, 0,0,0) = 0. To prove the claim, we make use of the implicit function theorem in the following setup (whose derivation is left as an exercise to the reader). LEMMA 4.2.26.

Let F(X, X, Y, Y) be a smooth C" -valued function in a neigh-

borhood of the origin in CN x Cd. If F(0, 0, 0, 0) = (4.2.27)

Idet Fy(0,0,0, 0)12 — Idet

and

0, 0, 0)12

then there exists a unique smooth solution Y =

a neighborhood of(0, 0, 0, 0) with f(0, 0) =

0

0.

f(X. X) of F(X, X, Y. Y) = 0 in

0.

The claim above now follows from Lemma 4.2.26 and (4.2.23). Observe that M is defined by (4.2.2), which is equivalent to (4.2.25) with u = th. Hence, M is also defined by w = Q(z, w, i,). The fact that Q(z, 0, w, th) Q(0, w, th)

follows immediately from (4.2.24), (4.2.25), and the definition of the function Q. The property (4.2.18) follows by implicit differentiation in (4.2.25) and the property (4.2.23). 0 An inspection of the proof of Proposition 4.2.17, shows that we also have the following slightly sharper result.

§4.3. CANONICAL COORDINATES FOR GENERIC SUBMANIFOLDS

101

PROPOSITION 4.2.28. Let M C

be a smooth generic subman (fold of codimension d, N = n + d, and po E M. Then there exist formal power series Q(z, x, t) = (Qe(z, x r))I<,
Q(z,

w, th)

Q(z,

mod K.

th),

We conclude this section with the following remark.

4.2.30. By Proposition 4.2.12, if M is a real-analytic generic submanifold of C'1' of codimension d and CR dimension n, then there are coordinates Z = (z, w) E C" x near any given point po E M such that M is given near po by (4.2.13). In this remark, we shall address the converse. Assume that Q(z, r) is a holomorphic C'1-valued function defined near 0 in C" x C" x C'1 such that

Q(z,0,r)

Q(0,x,r)

Q(z, x. Q(x, z, w))

(4.2.3 1)

w.

Then, w = Q(z, th) defines a generic submanifold M of codimension d near 0 in Indeed, by the implicit function theorem, there exists a unique valued holomorphic function (z, x, u) such that u

+

x. u)

Q(z,

x' u —

x. u)).

Thus, if we can prove that s) is W1-valued when s E R", then the equation w = Q(z, ü) is equivalent to Im w = Re w), which defines a generic

submanifold of codimension d near the origin. This is done by using the second identity in (4.2.31) to deduce that both s) and s) satisfy the equation

forX,

s+iX =

—iX).

Since the latter equation has a unique solution near the origin, the result follows.

§4.3. Canonical Coordinates for Generic Submanifolds In this section, we shall present another system of special coordinates for generic submanifolds corresponding to the canonical forms for real vector fields given in §3.4—3.5.

DEfiNiTION 4.3.1. If M is a CR manifold and P0 E M, then the Hörmander numbers of M, and their multiplicities, at Po are defined to be the Hörmander

IV. COORDINATES FOR GENERIC SUBMANIFOLDS

102

numbers and multiplicities of the collection of real and imaginary parts of a basis for the CR vector fields of M near p0.

The Härmander numbers of a collection of linearly independent real vector fields were defined in § 3.4. Clearly, the Hörmander numbers of a CR manifold do not depend on the choice of basis for the CR vector fields.

M c CN be a smooth generic submanifold of CR dimensionnandpo €M. Let2 Sm1 < ... < mhbetheHc'rmandernumand their multiplicities. Let d = bers of M at po and , THEOREM 4.3.2. Let

d' =

Then, for every integer m > mh, there exist local vanishing at near with z E C", Cd with w1 E C", and w' E Cd', such that, in these w = (WI,... Wh) coordinates, M is given near P0 = 0 by dimR M — 2n

—

d.

holomorphic coordinates (z, w, w') E ,

hflWk =Pk(Z,Z,sI,... —

1mw =B(z,z,s,s)s I

F

,sk._I)+Ak(z,z,s,s)s+rk(z,z,s)

+r(z,z,s), —

is a w, s'=Re w', andk=l...h. Here, have wei ghted homogeneous vector- valued polynomial of degree mk, where z weight one, s1 has weight m1for j = I, . . . h, rk(Z, s) is a smooth vector-valued function which is 0 (mk + 1), r' (z, s) is a smooth vector-valued function which is 0(m), Ak(z, s, s') and B(z, s, s') are smooth matrix valued functions with

wheres=Re

Ak(O, 0,0,0) = 0 and B(0, 0,0,0) =0. If M is real-analytic, then ailfunctions in (4.3.3) are real-analytic and one can take r'(z, s) 0. If M is of finite type at po. then d' = 0 and M is given only by the first set of equations in (4.3.3) with Ak 0. We shall refer to coordinates in which the submanifold M takes the form (4.3.3), with a choice of the integer m > as in Theorem 4.3.2, as canonical coordinates (of level m) for M at and the form (4.3.3) as a canonical form (of level m) for M at po. In the real analytic case we take r'(z, s) 0 and hence in this case the integer m is irrelevant; we then call these simply canonical coordinates and canonical form.

PROOF. We choose a basis L = X1 + i Ye,, j = 1, . . n, for the CR vector fields on M near po. We may assume that the L1 commute with each other (see Proposition 1.6.9). We shall apply Theorem 3.5.2 to the collection of 2n smooth real linearly independent vector fields {X1,... , X,,, Y1 Y,, }. It follows that there are smooth coordinates (x, y, s, s') = (x, y, s') € , .

§4.3. CANONICAL COORDINATES FOR GENERIC SUBMANIFOLDS

such that in these

which are local coordinates on M near p0, vanishing at coordinates (4.3.4)

Lk

103

= Lk._I + 0(0),

where

(4.3.5)

=

Lk._I

where z

a —— +

h

a

,

—,

azk

= x+iy. Here, as in §3.5, we have assigned the weight one to x andy (and, = 1,... ,h,andmtos',wheremisthe

integergiven in the statement of the theorem. The coefficient q(z, s1,... , is a complex vector valued weighted homogeneous polynomial of degree m1, for

j=1,... ,h. t: M -+ C's' be the smooth local embedding of M

P0. We let (z. s, s'), as introduced above, be the local coordinates on M near P0. The components of t(z, s, s') satisfy the system Lku = O.k = 1,... , n. Hence, the Let

linear part of each component is independent of write (4.3.6)

i(z,

.c,

s') = (x(z,

s, s'), t(z,

near

in view of (4.3.4—4.3.5). We

s, s'), t'(z,

S.

where x. r, and r' are defined near 0 in

and valued in C", and C' respectively. After a linear change of coordinates in we may assume that the components of i are of the form

x(z,z,s,s') =z+ 0(2) (4.3.7)

r'(z, where

ft(z, SI,...

= Sk s, s') = s' + f'(z, ,

,Sk_I)+ O(mk) s)

+ 0(m),

is a polynomial, without linear terms, which is a sum

of weighted homogeneous polynomials of degrees between 2 and mk — I, for k = 1,... , h. Similarly, f'(z, s) is a polynomial, without linear terms, which is the sum of weighted homogeneous polynomials of degrees between 2 and m — I. The remainders 0(2), O(mk), and 0(m) do not have any linear terms. In what

follows, we shall use the notation 0(v) for a remainder that is 0(v), v ? 2, and that, in addition, does not have any linear terms. Of course, if v > m, then a term which is of weighted degree v cannot have any linear terms and, hence, the two notions 0(v) and 0(v) coincide. It is sometimes convenient to denote r' by Th+I and m by mh+I. The following lemma will allow us to eliminate the polynomials 1k and f' in (4.3.7).

IV. COORDINATES FOR GENERIC SIJBMANIFOLDS

104

h + 1, that a local embedding Assume, for some k, I k intoCN is given of theform Th+I)ofM

LEMMA 4.3.8.

=

,

=z+ 0(2) si,... , si_i) + + I), j= 1,... ,k—1

r1(z,

s,

s') = + i/.imj(Z,

rk(Z,

s,

s') = s& + r,L(z, Z, Si,••.

(4.3.9)

where u/rn1 (z,

'5k—i) +

0(i + 1)

s1,... , si_i) is a vector valued weighted homogeneous polyno-

Sk_ i)i5 a vector valued weighted homogeneous (z. polynomial, without linear terms, of degree /.L, and /L < mk. Then there is a without linear terms, of weighted homogeneous polynomial , degree such that, after the following polynomial change of coordinates in

mial of degree

4310 the imbedding t,

,

X=X ,Tk_i), in

the new coordinates

with the corresponding F,L

b,...

is

,

of the form (4.3.9)

0.

PROOF. First, we claim that the vector fields for j = 1,... , n, commute are assumed to commute, we have with each other. Indeed, since the

(4.3.11)

0=

EL1,

=

+ O(—l).

[L1._1,

By identifying the terms of weighted degree —2, the claim follows. Since the components of the embedding i satisfy the system L1u = 0, j = 1,... , = 0 for it follows, by taking the lowest weighted homogeneous term in

t=1,...,k,thatforj=1,...,n

.. . ,St—i))

0

,5k—i)O. We introduce the truncated vector fields M1._1 obtained by replacing the sum in

the right side of (4.3.5) with a sum of only the first k —

(4.3.12)

1

terms, i.e.

j=1,...,n.

§4.3. CANONICAL COORDINATES FOR GENERIC SUBMANIFOLDS

Since the

commute with each other, it is clear that the

105

also commute.

...Sk_i)and,

...St....i) = hence, (4.3.13)

Mj._i(Sg

+

,Sti)) = 0,

Z,

s1,...

,

—1,

£ = 1,.

= 0.

are also annihilated by the vector It follows that the components of the complex vector-valued funcfields £ = 1,... , k — 1, form a basis of linearly tions z, St + 'J'mt(Z, Si,... = 0, j = 1,... , n (in the restricted independent solutions of the system

Note that the coordinate functions zr,... ,

,k—l). as We replace by n independent complex variables and consider the — Since the vector fields MJ._I are linearly holomorphic vector fields is a independent and commute with each other, and since 51,••• , = 0, j = 1,... , n, we may apply polynomial solution of the system the holomorphic Frobenius theorem (Theorem 2.1.12) to conclude that there is a holomorphic function r1,... , Tk_I) such that

(4.3.14)

,Sk—l)

= P(z,si +

. . .

+

. . .

Using the weighted homogeneity of

and the fact that r1,1 has no linear terms, one can easily check that P is a weighted homogeneous polynomial without linear tenns. With this choice of in (4.3.10), the reader can verify r1,... , that the conclusion of the lemma follows. 0

By repeatedly using Lemma 4.3.8, we may assume (4.3.7) holds with ft and f' 0. We write

0

= z+ 0(2) (4.3.15)

Tk(Z,Z,S,5')=Sk+*rnt(Z,Z,SI,...,sk_i)+O(mk+l)

r'(Z,z,s,s')=s'+t/Irn(Z,z,s)+O(m+ where

I),

and 1//rn are weighted homogeneous polynomials of degree mk and m, , h. We need the following lemma.

respectively, and k = 1,...

LEMMA 4.3.16. Let t = (t1,... , be a weighted coordinate system near 0 E Rk in which :j is assigned the weight 1, j = 1,... , k. Consider the following change of coordinates near 0 (4.3.17)

IRk

=:j + f'(t), j = 1,... ,k,

IV. COORDINATES FOR GENERIC SUBMANIFOLDS

106

where each fJ(t) is a smooth function which is O(v1),for some v1 inverse change of coordinates is also of the form

2.

Then, the

j=1,... ,k,

(4.3.18)

where each fi(i) is a smooth function which is 0(v1), provided that we assign ,k. the weight iefor€=

The fact that there is an inverse change of coordinates of the form (4.3.18), with fi(O) = 0 and dfi(O) = 0, follows immediately from the inverse function theorem. The fact that fi (1) is also 0(v1), if we assign the weight can be proved by contradiction as follows. Assume not, and define v0 to each to be the smallest integer such that there is Jo E (1,... , k} with the property that PRooF.

(1) denotes the weighted homogeneous part of 0, where degree vo of the Taylor series of /10(1). Substituting (4.3.18) in equation (4.3.17) and identifying weighted homogeneous terms of degree vo on both sides in the equation with J = Jo yields the desired contradiction. The proof is complete. 0 vO

and

(1)

We can now complete the proof of Theorem 4.3.2 in the smooth case. It follows from (4.3.15) and Lemma 4.3.16 that we can solve for z, 5k and s' in terms of x' Re tk, and Re r' and we obtain

z = x + 0(2) (4.3.19)

,Rerk_1)+6(mk+ 1)

5k

0(m+

s'

1),

and 'I'm(X, Ret1,... ,Re Re t) are weighted homogewhere 'frm(X, neous of degree mk and m, respectively. The conclusion of Theorem 4.3.2, in the smooth case, follows by taking the imaginary part of (4.3.15) and substituting for z, s and s' using (4.3.19). We shall complete the proof in the case where M is a real-analytic generic submanifold, that is, we need to show that we can make a further holomorphic change of coordinates such that the term corresponding to r'(z, s) in (4.3.3) is identically zero. Proceeding as in the smooth case, observing that all coordinate changes on M may be taken to be real-analytic (and also beginning with a real-

analytic embedding i: M -+ CN), we take as a starting point an embedding L(Z, s, s') = (x(z, s, s'), r(z, s, s'), r'(z, s, s')) satisfying

x(z,z,s,s') =z+ (4.3.20)

0(2)

Tk(Z,Z,51S)=5k+*mt(Z,Z,51,... ,sk_I)+O(mk+l) r'(z,

s, s')

= s' + g'(z,

s) + B(z,

s,

107

CANONICAL COORDINATES FOR GENERIC SUBMANIFOLDS

are vector-valued weighted homogeneous polywhere (z, SI,... , nomials of degree mk, g'(z, s) is a real-analytic vector-valued function, and B(z, s, s') is a real-analytic d' x d' mathx-valued function with B(O, 0,0,0) = 0. Observe that, by Corollary 3.5.4, the vector fields L*, k = 1,... , n, are tangent to the local Nagano leaf Orb0 = {s' = 0). Hence, we may take their restrictions Rk, k = 1,... n, to Orb0. The components of x (z, s, 0), r (z, s, 0) form a basis of linearly independent solutions of R*u = O.k = 1,... , n. Since g'(z, s) is a real-analytic solution, it follows that there exists a holomorphic function G'(X, t) such that ,

g'(z,

(4.3.21)

s) = G'(X(z,

s,O),

r(z,

s,0)).

The reader can check that the holomorphic change of coordinates in which r' is r) yields the following form of the embedding replaced by r' —

=z+O(2) (4.3.22)

where erty

Tk(Z,Z,S,S')=Sk+tfrmk(Z,Z,SI,... ,Sk_I)+O(mk+l) r'(z, s, s') = s' + B(z, s, s')s',

B(z,

s, s') is a real-analytic d' x d' matrix-valued function with the prop= 0. The remainder of the proof is the same as in the smooth

that B(0, 0,0,0)

0

case.

We relate the Hörmander numbers to the Levi map, discussed in §2.2. We have the following proposition, whose proof is easy and left to the reader. PROPOSITION 4.3.23. Let (M, V) be a smooth CR manifold of CR codimension

d, with V its CR bundle. Then, the Levi map of M at p. defined by (2.2.4), is surjective if and only if M is of finite type a: p with m1 = 2 being the only Hörmander number (with multiplicity d). In the embedded case, when M is

number (4.3.24)

where z E

C"

result.

of finite type

at 0 and the only HOnnander

is 2, the equation (4.3.3) becomes

Im w

=

p(z,

+ r(z,

s),

p is an R'1-valued quadratic polynomial. The remainder r is 0(3), is of weight one and

s

E

R" of weight

where

two. We have the following related

IV. COORDINATES FOR GENERIC SUBMANIFOLDS

108

PRoPosrrioN 4.3.25. Let M C be a smooth generic subman (fold of CR dimension n given near 0 by (4.3.24), where p = (p',... , p") is an R'1-valued quadratic polynomial and the remainder r is 0(3) with z E of weight one and s E R" of weight two. Then M is of finite type at the origin with 2 the only Hörmander number (fand only (fthe Hermitian matrices L= 1,... , d, are linearly independent over R. PROOF. By Proposition 4.3.23, it suffices to show that the Levi map of M at £ = 1,... , d, 0 is surjective if and only if the Hermitian matrices are linearly independent over R. Also, recall, by Proposition 2.2.10, that the Levi map at 0 can be represented by these Hermitian matrices. The proposition is then a consequence of the following two simple facts from linear algebra. LEMMA 4.3.26.

Let A1,...

,

Ad

be n x n complex matrices and consider the

mapping (4.3.27)

£:

x

—*

C",

£(X, Y) = ((AiX,

with the usual notation (X, Y) = Y for the Hermitian inner product Then C is surjective and only if the matrices A1, j = 1,... , d, are in linearly independent over C. In addition, if A1,... , Ad are Hermitian, then they are linearly independent over C if and only (fthey are linearly independent over R.

The proof of the first part of Lemma 4.3.26 is an easy consequence of the fact that the image of C, given by (4.3.27), is a linear subspace of C". The proof of one of the implications in the second part is immediate. The other is obtained by assuming that the A1,... , Ad are linearly dependent over C, taking the Hermitian adjoint, and using the fact that A = for j = 1,... , d. As mentioned above, this completes the proof of Proposition 4.3.25. 0

§4.4. Weighted Homogeneous Generic Submflnifolds < ... In this section, we shall relate the Hörmander numbers 2 m1 of a generic submanifold M C of mh and their multiplicities Li, £2,... , finite type at P0 E M to those of certain weighted homogeneous submanifolds that "approximate" M near P0. The result is similar in spirit to Theorem 3.6.1 for real vector fields and the proof essentially reduces to an application of that theorem. m1 < m2 < ... < mh and t1,t2,... ,L,, be Thus, let positive integers2 given. Let (z, w) = (z. Wi,... , Wh) be local coordinates in CN, vanishing at C" andw1 E Ct' forj = 1,... ,h;hence,n+Et1 = N. poE We assign the weight one to z and the weight m1 to w1. In what follows, we shall also use the notation w1 = (Wjr)i
109

§4.4. WEIGHTED HOMOGENEOUS GENERIC SUBMANIFOLDS

THEOREM4.4.1.

Let(z,w)=(z,wi,...

< ... be as above. Consider the following generic submanifold, 2

and £2,... CN, defined nearOby M0 c

mh,

(4.4.2)

r=

= Pjr(Z,Z,51,... ,s1_1),

l,...t1,

j=

1,...h,

where = Re w1 and each Pit (z, s1,... , sj_i) is a weighted homogeneous polynomial ofdegreem1. ForJ E(1,... ,h)andRE (1,... ,tj},consideralso

MgR c CN(J.R), N(J,

R) =n+ti +

Ifli Wjr =

.

..

+tj_i + R, defined by r=

. .

.

1,..

i = 1,...

(4.4.3)

Im Wj, Let

= pjr(Z, Z,

.. .

,Sj_i),

r

=

1, . . .

M c CN be a smooth generic submanifold with 0 E

,

J—

1,

R.

M such that

M is defined

near 0 by

r=

(4.4.4)

l,...t1,

j=

where

,s1_1)+ O(m1 + 1).

(4.4.5)

Then, the following are equivalent.

M is of finite type at 0 with Hörmander numbers 2

(1) The

m1 < ...

<me, and multiplicities

,

£h.

(ii) The submanjfold M0 is offinite type at 0 with Hörmander numbers 2 m1 < ... < mh and multiplicities Li,... ,

(iii) For any J, R as in

(4.4.3),

c

the

type at 0 with Hörmander numbers 2

is of finite

m1 <<mj and multiplicities

Li,... ,Lj_1,R. s) = (z, si,... sh) be local coordinates on M near 0. Let a basis for the CR vector fields on M near 0 such that Lk, for

PROOF. Let (z, be

I, (4.4.6)

...

, n, is of the form

a Lk = —:— aZk

h

+

for some smooth vector valued functions a1 (z, in § 1.6). We shall prove that

(4.4.7)

a

s). —, s) (a formula for which is given

Lk = L*._i + 0(0),

IV. COORDINATES FOR GENERIC SUBMANIFOLDS

110

where (4.4.8)

a

Lk._1 =

h

,sJ_I)

Ztc

= weighted

a

j

,Sj—i))i
homogeneous polynomial of degree mj — 1. Moreover, we shall show

si,...

that each polynomial

,

is completely determined by the poly-

= 1,...

,j

— landz = 1,...

1,... ,r— 1. to be local coordinates on M0 near 0 as well, it follows that there is a basis for the CR vector fields on M0 that has the same form (4.4.7) with L*._1 as in (4.4.8). 1) to be local Similarly, for each J, R as in the theorem, taking (z, s1,... coordinates on MOJR near 0, where = (SJr)I
= Lk._I + 0(0),

(4.4.9) where

(4.4.10)

i—I

a

=

a /

k

R

a

,sj_i)• as1,. —,

r=1

s1,...

as above. The proof of Theorem 4.4.1 will follow by applying Theorem 3.6.1 to the collection of real and imaginary parts of the L*,..i, and Lk._). To apply Theorem 3.6.1, we need to show that the CR vector given by (4.4.6) are of the form (4.4.7) with Lk._I as in (4.4.8). fields L1,... , We write with

(4.4.11)

,

Lk =

4aZk

j=1

as1

+ 0(0),

s) is a vector-valued polynomial consisting of terms of weighted Since Lk annihilates + s), for v = 1,... h and degree less than where

CICJ

,

k=1,... (4.4.12)

,n,wefindthat

Ill

WEIGHTED HOMOGENEOUS GENERIC SUBMANIFOLDS

where a = 1,...

s) denotes the

s),

x L, matrix with matrix elements

= 1,... ,Lj.

homogeneousofdegreem1 —1, fork = 1,... ,n andj = 1,...

,h.

Toseethis,

the smallest integer such that the weighted homogeneous part of denote by 0. We observe that, for is not identically zero, of degree (z, s) and k E { 1,... ,n}, the jth term in the sum on the right side fixed v (1,... , h} of (4.4.12) contributes only with terms of weighted degree at least

ifj
+

(4.4.13)

1ff > v s). Fix k E (1,... ,n}

and assume, in of the form (4.4.5) of < — 1. Denote to reach a contradiction, that there is v such that denote by vo the largest v such that and the smallest such ps" =

because order

if .j = v

1,

by in (4.4.12) with v = v0 using (4.4.13), we find that Identifying terms of degree and, hence, the claim above is 0. This contradicts the choice of s) proved. — 1 in (4.4.12) To complete the proof, we identify terms of weighted degree

(1,... ,n}andv E (1,... ,h}. Startingwithv = landusing(4.4.13),

fork

we obtain

=

(4.4.14)

Forv> l,weobtain (4.4.15)

Z,Si,...

S) =

,

v—i —

—

— %_,,IC

. . .

i=I a = 1,... , Lv and where Pv.s, denotes the Li x Lv matrix with elements = 1,... , A simple inductive argument using (4.4.14) and (4.4.15) shows that (4.4.16) where each q(z,z,si,... ,s1...1) = valued

is a vector-

weighted homogeneous polynomial of degree

—1

and each q., is com-

pletely determined by the polynomials

(44 17) This

. .. Z, Si,

. . . ,

V

Sj..i),

= 1,... ,f —

IL = 1,

.

.

.

,r

1,

= 1,...

— 1.

is the desired conclusion; hence the proof of Theorem 4.4.1 is complete. 0

IV. COORDINATES FOR GENERIC SUBMANIFOLDS

112

§4.5. Normal Canonical Coordinates In this section, we shall show that for a real-analytic generic submanifold M one may choose canonical coordinates (as defined in §4.3) at p0 M that are also normal (as defined in §4.2). In the smooth case, we shall show that one may choose canonical coordinates that are "approximately normal", up to any order. We stait by formulating the result in the real-analytic case. THEOREM 4.5.1. Let M C Ce" be a real-analytic generic submanifold and p0 M. Then there exist canonical coordinates (z, w, w') = (z, w1,... , Wh, w'), vanishing at Po (as given by Theorem 4.3.2) that are also normal, i.e. in these

coordinates, M is given near Po by

Im w =

(4.5.2)

1mw

where s = (

.

.

Re

w, s' =

Re

s,

,

-

,

s'),

w', and where

)

0, s, s')

s,

s')

0.

The existence of approximately normal canonical cordinates in the smooth case is given in the following.

THEOREM 4.5.4. Let M C C" be a smooth generic submanifold and P0 E M. Let 2 m1 < ... be the Hörmander numbers of M at P0 and Li,... , their multiplicities. For any integers m, K, K m > mh, there exist canonical coordinates (z, w, w') of level mfor M at P0 (as given by Theorem 4.3.2) satisfying the following. The subman (fold M, in these coordinates, is given by

k=l,...,h, (4.5.5)

where,

Im w' = Ø'(z,

s = Re w, s' =

Re w',

s,

)Ia

s') s,

s') and s) and

s,

valued functions defined near 0, geneous vector valued polynomials of degree j, and, fork 0, Vj, a,

s') are smooth vectors) are weighted homo-

= 1,... ,h,

j

+ mIaI < K.

(4.5.6)

0, s)

The equivalence

in (4.5.5) means equality between Taylor series.

0,

0, s)

In the notation of Theorem 4.5.4, the coordinates (z, ui, w') being canonical of level m is equivalent to the following (4.5.7)

s)

0, Vj <mk,

and

s)

0, Vj
§4.5. NORMAL CANONICAL COORDINATES

113

We start by proving Theorem 4.5.4, since part of this result will be used in the proof of Theorem 4.5.1.

For the proof it is convenient to write wh+i = Wh+i), and s = (Si,... ,Sh,sh+I). w = (WI,... , Sh+i Recallthatweassigntheweightonetoz,theweightmj tosj,j = 1,... ,h + 1; (thus, s' = sh+i is assigned the weight m = mh÷i). With this notation, it suffices to prove the first identities in (4.5.5) and (4.5.6) for k = I,... , h + 1. That is, we must find canonical coordinates (z. w) (of level m) and smooth functions s) such that PROOF OF THEOREM 4.5.4.

= 5,

=

m,

k=l,...,h+1,

(4.5.8) j=mt

with

(4.5.9)

weighted homogeneous polynomial of degree j and satisfying

Vk=l,...,h+1,Vj:m&<j
begin by dealing with the case j = mk in (4.5.9). We first claim that we can find canonical coordinates (z, w) such that M is given by (4.5.8) and We

(4.5.10)

k= I,... ,h+l.

We shall prove the claim by induction on k. For k = I we begin by taking any canonical coordinates th) of level m = mh+i, with th = (ü1,... , given by Theorem 4.3.2. We take the weighted homogeneous change of variables z = fork = 2,... ,h + I, and = th1 — (Note that by = homogeneity, (z, s) is independent of s.) The reader can easily check that (z, w) are still canonical coordinates of level m and that (4.5.10) holds fork = I, i.e. the corresponding 0. (z, 0) (z, satisfies We assume now that we have found canonical coordinates, denoted by ü,) such that M is given by an equation of the form (4.5.8) for which (4.5.10) holds for 1 k £ — 1. We define new coordinates (z. w) by the following weighted homogeneous change of coordinates (4.5.11)

ZZ, WkWk, k=1,...,h+1, ,th,.1).

Using a simple homogeneity argument, the reader can check that the new coordinates (z, W) are canonical of level m and that identity (4.5.10) now holds for k = 1,... , £. The proof of the claim now follows by induction.

IV. COORDINATES FOR GENERIC SUBMANIFOLDS

114

We complete the proof of Theorem 4.5.4 by another induction. Assume that that, for some choice of canonical coordinates ü) of level m for M, where M is given by

k=1,...,h+1,

(4.5.12) j=mt the

following holds. For some j

(4.5.13)

E

fm1, m1 + 1,...), Vk=1,...h+1,

Vt=mk,...

,j.

We interpret the condition in (4.5.13) as being vacuous for those j and k that satisfy Observe that we have proved (4.5.13) for j = ml. We claim that there

j

exist canonical coordinates (z, w) of level m for M at 0 such that M is given by (4.5.8) with (4.5.14) Assuming the claim, the proof of Theorem 4.5.4 follows by induction. To prove theclaim,letthe integerko E (1,... , Ii + 1}bedeflnedasthelargestk forwhich

mk<j+l,

(4.5.15)

and consider the following change of coordinates in CN near 0 given by

ZZ, (4.5.16)

Wk

=

t=k0+1,...h+1, -

Wk

0, tip)

—

+

-

0, ü),

k = 1,... , ko.

that, by (4.5.15), the change of coordinates equals the identity modulo terms that are weighted homogeneous of degree at least j + 1. Lemma 4.3.16 implies that the inverse change of coordinates is also of this form, i.e. Note

il=Wf,t=kO+1,...,h+1, (4.5.17)

Wk=Wk+O(j+l), k=1,...,k0,

where we have assigned the weight one to z and mk to w*, k = I,.. .h + 1. The submanifold M in the new coordinates (z, w) (with s = Re w) is given by (4.5.8),

where

(4.5.18)

=

—

v=O

+

115

§4.5. NORMAL CANONICAL COORDINATES

fork=1,... ,ko,and (4.5.19) v=.r41

for e = k0 + 1,... , h + 1. Substituting the inverse change of coordinates for on the right hand side of (4.5.18)1 identifying weighted homogeneous terms in (4.5.18), and using (4.5.17), we obtain

k= 1,...k0,

(4.5.20)

(4.5.21)

v=mk,... ,f,

—

+

fork=1,... ,k0,and ,j+1.

£=k0+1,...h+1,

(4.5.22)

In view of the induction hypothesis, equations (4.5.20) and (4.5.22), it suffices to 0, s) 0, fork = 1,... , Ico, to prove the claim (4.5.14). This check that is easy to verify, by using the reality of (0,0, s) for real vector-valued s. This 0 completes the proof of Theorem 4.5.4.

ui,, si') be a set of canonical coordinates for M at po as described by Theorem 4.3.2. We may write the defining equations of M as follows, with = Re th and s' = Re W, PRooF OF THEoREM 4.5.1. Let

si,,

=

,

=

Imwk (4.5.23)

k= 1,... ,h, =

where

j,

is a vector valued weighted homogeneous polynomial of degree i, F) are real-analytic matrix-valued functions with i, 1') and

B(O,0,0,0) = Oand Ak(0,0,0,0) = 0. AninspectionoftheproofofTheorem 4.5.4 shows that we may assume that any finite number of the 0. We shall assume the following 0, (4.5.24)

0,

0,

k = 1,...

,

h.

.i) satifies

IV. COORDINATES FOR GENERIC SUBMANIFOLDS

116

According to the proof of Theorem 4.2.6, there are normal coordinates (z, w, with

=

(4.5.25)

ii' =

z,

w

ill =

+ h(z, w, w'),

w'

+ h'(z, w,

where h(O) = 0, dh(0) = 0, h'(O) = 0, dh'(O) = 0, and where h(z, w, W') and

h'(z,

w, w') satisfy

(4.5.26)

hk(z, w, W') —k

+

0, w,

f

w')

h(z, w, w') —

0,

0, 2

w')

0,

w') 0, w,

w,

w')

w, w,

ui')

is

W,

some matrix-valued holomorphic function, and that

hk(z,w,0)=O(mk+1),

(4.5.28)

k=1,... ,h.

To prove (4.5.27), it suffices to verify that A = h'(z, W, 0)

following equation, since h(z, (4.5.26), (4.5.29)

A+

A

by (4.5.23).

w')

w, w') and

h'(z, w,

w') are

0 is a solution of the unique determined by

0, w, 0)

0 is a solution of (4.5.29) is immediate since Ø'(z, 0, w, 0)

0

§4.5. NORMAL CANONICAL COORDINATES

117

To prove (4.5.28), we substitute w' = 0 in the first equation of (4.5.26) and use

(4.5.23), (4.5.24), and (4.5.27) to obtain (4.5.30)

hk(z, w, 0) —

j=mt+I

j=mt+I

The proof of (4.5.28) is now very similar to the proof of Lemma 4.3.16. As-

sume, in order to reach a contradiction, that (4.5.28) does not hold. Define t'0 (1,... , h } with the property to be the smallest integer such that there is k w, w') denotes the weighted 0 and v0 that w, 0) mk, where homogeneous part of degree v in the Taylor series of h"(z, w, w'). Also, denote {1,... , h} the largest integer for which w, 0) 0. The desired by k0 contradiction follows, as in the proof of Lemma 4.3.16, by identifying terms of weighted degree vo on both sides of (4.5.30) with k = k0. Thus, we have proved the claims (4.5.27) and (4.5.28). Performing the change to normal coordinates given by (4.5.24), we find that M in the new coordinates is defined by Im (Wk

+ hk(z, w, w')) =

s

+ Re h(z, w, w'),

s' + Re h'(z, w, w')) (4.5.31)

Im (w' + h'(z, w, w')) =

s

+ Re h(z, w, w'),

s' + Re h'(z, w,

wheres = Re w,s' = Rew',andk = 1,... ,h. Solvingforhnwandlm w'in (4.5.3 1) we obtain Im Wk

=

Im w' =

s, s') s,

where k = 1,...

, h. The proof of the theorem will be complete if we show that l,k(z, s,0) is O(mk), k = 1, ... , h, and that s, s') is of the form B(z, L s, s')s'. To check the latter it suffices to verify that Im w' = 0 satisfies the second equation of (4.5.3 1) when s' = 0. This is immediate from (4.5.23) and (4.5.27). To check the former, we set s' = 0 in the first equation of (4.5.31), use the previously established fact that Im tv' = 0 on M when s' =0, and obtain

(4.5.32)

Im Wk

Since h'(z, w, 0) is

+ Im hk(z, w, 0) = + 1) for j = 1,...

is O(m&), by the assumption that

s

+ Re h(z, W, 0), 0).

s,s') th, th') were canonical coordinates to begin ,

h, by (4.5.28), and

118

IV. COORDINATES FOR GENERIC SUBMANIFOLDS

with, a similar argument to the one used to prove (4.5.28) above shows that the s,0) of (4.5.32) is O(m*) for each k = 1,... , h. We solution Im Wk = leave the details to the reader. The proof of Theorem 4.5.1 is complete. 0

Notes for Chapter P/ Normal coordinates for real analytic hypersurfaces, as given in §4.2, were given in Chern and Moser [1]. For generic manifolds of higher codimension, the proof of Theorem 4.2.6 was given in Baouendi, Jacobowitz, and Treves [1]. The results in the smooth case, in particular Proposition 4.2.17 and 4.2.28, are new. Canonical coordinates, as introduced in §4.3, for generic manifolds of finite type were given by Bloom and Graham [1]. We also refer to Baouendi and Rothschild [1], where the infinite type case and uniqueness are treated. A different proof for the existence of canonical coordinates can be found in the book Treves [2].

CHAPTER V

RINGS OF POWER SERIES AND POLYNOMIAL EQUATIONS

Summery Chapter V, which is again independent of the preceding ones, covers a number of methods and results from commutative algebra and complex analytic geometry, some of which are hard to locate in standard texts. In §5.1, we review some basic

results concerning ideals of finite codimension in rings of power series. These will be used in connection with the notion of essential finiteness, discussed first in Chapter IX for hypersurfaces and then in Chapter XI for generic submanifolds of higher codimension. Some fundamentals from analytic geometry, including the Weierstrass Preparation Theorem, are treated in §5.2—5.3. Basic definitions of algebraic function, manifolds, and varieties are given in §5.4; mappings of algebraic manifolds and varieties will be discussed in Chapter XIII. The chapter concludes with some results on roots of polynomial equations with holomorphic coefficients. In particular, it is proved that a smooth function that satisfies such an equation is necessarily real-analytic. This result is important for the proof of the reflection principle given Chapter IX.

§5.1. Thute Codimensional Ideals of Power Series Rings In this section, we introduce some basic definitions and results from commutative algebra as well as local analytic geometry. We shall denote by C(z11... , Zn } = C{z} the ring of germs of convergent power series in n complex variables, and by

the ring of fonnal power series in n indeterminates. Zn]] = In what follows, denotes either of the rings C{zi,... , We note here the standard fact that is a unique factorization domain and a Noetherian ring, i.e. all ideals in are finitely generated. (See e.g. Hörmander [2, Chapter VII. These results use the Weierstrass preparation theorem, which is stated without proof in §5.3.)

DEfiNITION 5.1.1. LetI be an ideal in

We shall say that I is of finite

codimension if the quotient ring II is finite dimensional as a vector space over C. The codimension of I, written codiml, is defined as the dimension of 119

120

V. RINGS OF POWER SERIES AND POLYNOMIAL EQUATIONS

For J1,... the Since this form.

,

i,,

the ideal in generated by we denote by I(J1,... , is a Noetherian ring, any ideal is finitely generated and hence of

EXAMPLE 5.1.2. In any nonzero ideal is of finite codimension. In 1Z2 the of finite codimension, while the ideal z1Z2) is not. ideal is + Z2)

We recall a basic result from commutative algebra which we shall need. LetI be an ideal in 1Z,, andm the maximal

THEOREM 5.1.3 (KRULL'S

ideal, i.e. m = I(zi,... ,Zn). Then

1= can be derived from The proof of Krull's Lemma, as stated above for the following more general statement: If R is a Noetherian ring with a unique maximal ideal iii, then flN th" = (0). (See e.g. Zariski and Samuel [11 for the proof.) Theorem 5.1.3 then follows by taking R = 1?,,/I and th = rn/I. aaZa E C(z}, we shall a sequence in R.,, anda(z) = If a'(z) = converges simply to a(z) if say that —÷ aa as j —÷ oo for all a. The following is an easy consequnce of KrulI's Lemma. PROPOSITION 5.1.4. Eveiy ideal in

is closed under simple convergence of

and

sequences. That is, if us an ideal, converges simply to a(z) =

then a(z) E

E I for all j. Then for any positive integer N,

PROOF. Assume

EI+m". IakN

By Krull's Lemma, to show that a(z)

EI+rn".

I, it suffices to show that for each N,

ForN fixed, letIN =

which Since

may be regarded as a subset of the finite dimensional vector space I is an ideal in 1Z,1, 'N is a vector subspace of /m" and therefore closed. On is in IN for j = 1, 2,..., it then follows that the other hand, since EIaI
cI

The following is a useful characterization for an ideal to be of finite codimension.

PRoPosmoN 5.1.5. Let I be an ideal in

and rn the maximal ideaL Then,

the following are equivalent:

(i) I is of finite codimension. (ii) There exists a positive integer N such that zf E I, j = 1,... (iii) There exists a positive integer N' such that me"

ci.

,

n.

FINITE CODIMENSIONAL IDEALS OF POWER SERIES RINGS

PROOF. Clearly (iii) (i)

,'

j, I

(1),

,'

since mi"

has

121

finite codimension. To prove that

(ii), we observe that if I is of finite codimension, say d, then for any are linearly dependent modulo I, i.e. there exist n, z1, , zff'

j

a11... ,ad+I E C, not all vanishing, such that

apzj' E I. If P0 is the

0, we conclude by factoring that E I. smallest integer with (iii). If z7 E I for all j, then To complete the proof, we must show (ii) C I, which completes the proof of the Za, with Ial > nN, is in I. Thus, 0 proposition.

Recall that if I is a proper ideal, then its radical (5.1.6)

:= ff E

:

is defined as

fP E I for some positive integer p).

The following is an immediate consequence of Proposition 5.1.5. COROLLARY 5.1.7. LetI be an ideal in and m the maximal ideal. Then, I = in. is offinire codimension if and only if

PRoPosmoN 5.1.8. Let the ideal generated by the (5.1.9)

E 1Z,, and let 1(f) Assume that

1"...

,

1k

dimc

= I(f11...

,

fk) be

<00.

Then there exists an integer N', depending only on 1(f), such that the following holds. Ifg1 ... ,g,, E —gj E m1V, j = 1,... ,k, then 1(f) =1(g), where I(g)=1(g1,... , g,). More precisely, if C 1(f), then the conclusion above holds with N' = N + 1. PROOF. By Proposition 5.1.5, there exists an integer N such that C 1(f). We shall prove the conclusion of the proposition with N' = N + 1. We first show that if —g1 E then

1(f) C 1(g).

(5.1.10)

In particular, this will show that 1(g) has finite codimension. Then, by symmetry, it will follow that 1(g) c 1(f) and, hence, that 1(f) = 1(g). To prove (5.1.10), we observe that f, — gj E implies (5.1.11) Since

(5.1.12)

m

C 1(f), we obtain

CI(g)+m".m cI(g)+I(f).mcl(g)+(I(g)+mN+I).m C 1(g) + mN+2.

V. RINGS OF POWER SERIES AND POLYNOMIAL EQUATIONS

122

Continuing in this way, we obtain, for any £> 1,

1(f)

(5.1.13)

Now, the inclusion (5.1.10) follows from (5.1.13) and Krull's Lemma (Lemma 0 5.1.3). This completes the proof, in view of the remarks above. The following two results are useful for determining the codimension of an ideal.

be the ideal PRoposrrIoNS.1.14. Letf1,... and le:I(f) C generated by the f1. Then, for any positive integer p, the following are equivalent.

(i) codim 1(f)

(ii) codiml(fi,...

p; ,

,zr')

p.

Furthermore, if (i) or (ii) holds, thenl(f)

,

,zr').

PROOF.

Clearly, we have I C I'. Assume (i) holds. As in the proof of (i) (ii) in Proposition 5.1.5, we have z7 I, for j = 1, . . n. Hence, I' C I, which proves ,'

.

I = I' and, hence, (ii) holds.

C I'. To see this, pick any multi..< ..< =cr asequenceof and -< y means yj for if I As in the proof of(i) (ii) in j.

Now, assume (ii) holds. We claim that

indexcr with al = p. DenotebyO= multi-indices with = J — 1, where

j=

divides Observe that Proposition 5.1.5, we deduce that 1

. .

. n.

..<

,

are

linearly dependent (mod

I) and hence E I, for some j. This implies that Za E I, which proves the claim. By Proposition 5.1.8, we have I = I', proving (i). The proof is done. 0 The following can be used to show that a small perturbation of the generators of an ideal I of finite codimension generates an ideal of finite codimension not exceeding that of I. It also gives an algorithm to determine whether a given ideal has a certain codimension.

THEOREM 5.1.15. Given positive integers n, p and k, there exist a positive integer N1 and polynomials K1, ... KN1 in k(p + 1)" variables such that, for any

fit... ,fkElZn, (5.1.16)

3jL,

Here we have used the following notation. For (5.1.17)

j=1,... ,k,

§5.1. FINITE CODIMENSIONAL IDEALS OF POWER SERIES RINGS

123

we denote by (5.1.18)

1

PROOF. Let n, p and k be fixed positive integers. Denote by 1',, the vector caza, where p for some = 0 if a space of polynomials of the form and denote I = 1,... , n. Denote by N the dimension of 1's,, i.e. N = (p + consisting of the monomials Za E 2,,. Let natural basis of the by (e1,... eN} Denote by -÷ 2,, the obvious truncation operator. For 11, . . . 1k E In view of Proposition we denote by If = E C , 5.1.14, we have ,

codiml(f1,... (5.1.19)

and if either condition holds we also have

,f*)=I(fi,...

(5.1.20)

Since, by the definition of If' above, we have (5.1.21)

,zr')=I(fp,...

,fk,Zi,...

we then obtain, if either condition of (5.1.19) holds, (5.1.22)

It follows from the above that codim 1(f)

p if and only if there exist

gi'... ,g,,Efei,... such that for any r E

2,,, *

p

+

r(z) =

(5.1.23)

,eN}

mod

,

j=I

forsomea=(aI,...,ap)ECPandh=(hI,...,hk)E(Pp)k.Foreachg= (gi,...

,g,,) E {e1,...

2,,

defined by

/k

p

(5.1.24)

+ tcp

a)(z) = j=I

he(z)f/'(z)),

I

V. RINGS OF POWER SERIES AND POLYNOMIAL EQUATIONS

124

where tc, is as defined above. Thus, codim 1(f) p if and only if there exists g = (gi,... , g,,) E {e1,... , eN)" such that Ag is surjective, i.e. there is an which is not 0. Observe any N x N determinant formed from the matrix of such determinant is a polynomial in the coefficients of the If. Take K1, ... KN, to be the collection of all such determinants, corresponding to the finitely many different choices of g E {e1,... , eN)". This completes the proof of Theorem

0

5.1.15.

5.1.25.

(1) As stated in the theorem, the polynomials K1,... , Kpj1 depend only on or the three integers k, p, and n (and not on whether equals C(z}). Hence, the theorem gives a constructive procedure to determine if

codim(f)
the truncation of the Taylor series as defined in the proof. A consequence of Theorem 5.1.15 is that, for any positive integers k, p, q, with q, there is an algebraic subvariety (see §5.4) Vk.p.q C (PqY' p generates an ideal in of E such that f = (1,,... , codimension> p if and only if tcq(f) E Vk.p.q. As an immediate consequence of Theorem 5.1.15 (see also Remark 5.1.25 (1)), we obtain the following. COROLLARY 5.1.26. Let

f"...

,

E C{z}. Denote by 1(f) the ideal gener-

ated in Cf z} by the f1 and by 1(f) the ideal generated in

Then,

dimC{z)/I(f).

(5.1.27)

5.1.28. The reader should note that, in contrast to Corollarly 5.1.26, E C[z], where C[z) denotes the ring of polynomials, then the ideal if f',... generated by the fJs in C{z) may be of finite codimension whereas the ideal generated in C[z] is of infinite codimension. The following representation of formal power series in terms of the generators of a finite codimensional ideal will be useful in the proof of Theorem 5.1.37 below, and is also interesting in its own right.

PROPOSITION 5.1.29. Let ft,... , and assume that the ideal 1(f) E generated by fi,... , has finite (positive) codimension in Then, there can be written in the form exists an integer K such that any h E (5.1.30)

h(z) =

§5.1. FINITE CODIMENSIONAL IDEALS OF POWER SERIES RINGS

where aa(z) are polynomials of degree at most K — I and

125

=

Equivalently, such h (z) can be written in the form

h(z) =

(5.1.31) where

E

Proposition 5.1.29 can reformulated as follows. If the ideal is a finitely generated module 1(f) is finite codimensional, then the ring and the generators can be chosen to be polynomials over of degree K — 1. The reader can easily check that the converse also holds. REMARK 5.1.32.

may choose K such that mK C 1(f). Clearly, the representation (5.1.31) follows from the representation (5.1.30). Also, the representation (5.1.30) is obvious for a polynomial of can be written in degree K — 1. Thus, it suffices to show that any h(z) E the form (5.1.30). Since c 1(f), we have PROOF OF PROPOSITION 5.1.29. By Proposition 5.1.5 we

h(z) =

(5.1.33)

for some E j = 1, . . n. Let us decompose c1(z) as c1(z) = c(z) + c7(z), where c (z) is a polynomial of degree at most K — and c7 E mK. Again such that using mK C 1(f), we find E .

1

c'(z) =

(5.1.34)

Substituting (5.1.34) in (5.1.33), we obtain

= (5.1.35)

such that

k

+

h(z)

Iterating this process, we deduce, for any integer m (5.1.36) where the ep

h(z)

=

1, that we have

aa(z)(f(z))a + IPI=m+I

are polynomials of degree at most K — 1 (independent of m) and the The desired conclusion (5.1.30) follows. 0

The following shows that finite codimensionality of an ideal implies the nonvanishing of some Jacobian determinant of a given set of generators.

V. RINGS OF POWER SERIES AND POLYNOMIAL EQUATIONS

126

THE0REM5.1.37. Letfi,...

erated by Ii,...

,

E

and assume tharthe ideall(f)gen-

has finite codimension in 1Z,,. Then, there exists 1

(5.1.38)

(z))i

i1

<

0.

PRooF. In view of Corollary 5.1.26, it suffices to prove Theorem 5.1.37 in the Hence, we prove Theorem 5.1.37 only in this case. Denote by F case = Also, denote the field of fractions of the subnng C Observe that F and S have characteristic 0. by S the field of fractions of F[T1,... , -+ S, given by Consider the ring homomorphism

:= E F[TJ, i.e. the sum is finite and

where

(5.1.39) with image of

da(f(Z)) ea (1(z))

Ca(f(Z))

and ea(f(z))

ea(f(z))

0. Denote by R C S the

Observe that S is an (infinite dimensional) vector space over the field

F. We claim that R is a finite dimensional subspace (over F) of S. Since is F-linear, R is a linear subspace. To prove that S is finite dimensional, we shall use Proposition 5.1.29 to show that the monomials za = K— 1 for lal (where K is given by Proposition 5.1.29), span R. Indeed, for fi with IflI ? K, we have (5.1.40)

= IYI$K—I

Thus, for any element

where (5.1.41)

E R,

we may substitute the right hand side of (5.1.40) for any fl I

in (5.1.40) with K. This proves the claim, since the sum on the right hand side of (5.1.41)

is finite.

Now, let J c F[T1 be the kernel (5.1.42)

R

so that

§5.1. FINITE CODIMENSIONAL IDEALS OF POWER SERIES RINGS

127

Observe that 3 is a prime ideal, since R, being a subring of a field, is an integral domain. We claim that there exist nontrivial polynomials H, 3, j = 1,... , = !11(T1). Indeed, for any i.e. H,(T1,... , such that H1 depends only on 7)2 integerj {1,... , n), considerthe infinite sequence of monomials 1, Since F[TJ/3 is a finite dimensional vector space over F, in view of (5.1.42), there must be nontrivial linear relations between these monomials that are in the ideal, i.e. there exist ao,... , E F, not all 0, such that

Eael)tE3.

(5.1.43)

This proves the claim. Now, take H1(T1),...

3, as above, such

,

that 111(T,) has the lowest possible degree (with the property H1(T1) E 3). By clearing the denominators, we may assume that the H,(T,) have coefficients in the ring Now, since the H1 were chosen of lowest possible degree, it follows that

j=1,... ,n.

(5.1.44) Hence, since 3 is prime, we have

(5.1.45)

det

= [1

g 3. I

Since 3 is the kernel of 4', we deduce that 4,(H1) 0, forj = 1,... , n, and also 4, (det(H,j)) 0, where det( H,,) denotes the determinant in (5.1.45). Since the coefficients of H1 are formal power series in the 1(z), we shall write H, (1(z), T),

for Hj(T) with the obvious abuse of notation. In this notation, we have e.g. çb(Hj)(z) = Hj(f(z), z). Thus, the above can be written

1=1'... ,n,

(5.1.46)

Differentiating the first equations in (5.1.46) with respect to zt, we obtain by the chain rule (5.1.47)

0

af

B(z)1—(z) +

au

z),

where B(z) is an n x k matrix with matrix elements in Here, al/az and ÔH/8T are the Jacobian matrices of f and H with respect to z and T, respectively. Since 0, it follows from (5.1.47) that there must be an n x n minor of the Jacobian matrix of f with a determinant 0. 0

V. RINGS OF POWER SERIES AND POLYNOMIAL EQUATIONS

128

§5.2. Analytic Subvarieties

In this section we will give some basic definitions for real and complex analytic subvarieties. We will pay special attention to the complex analytic ones and state a few fundamental results for these (e.g the Nullstellensatz) which will be useful later in the book.

Let K denote either of the fields R or C. Let c be an open subset of A subset V C is called a (K)-analytic subvariery of if, for every x E there exist an open neighborhood cv C of x and functions fi.... , fic, real-analytic in cv if K = R and holomorphic in cv if K = C, such that (5.2.1)

Vflcv=(xEcv: fI(x)=...=fK(x)=O).

It follows, in particular, that an analytic subvariety V C is a closed subset We recall some basic notation and properties of analytic subvarieties; the of reader is referred to e.g. Gunning and Rossi [1], Gunning [1] (in the case K = C), Malgrange [1], and Guaraldo, Macr'i, and Tancredi [1] (in the case K = R) for details and further properties. An analytic subvariety V of is called irreducible if it cannot be written as a union V1 U V2 of two distinct (meaning also that neither is contained in the other) nonempty analytic subvarieties of A point x E V is called regular of dimension k if there is a neighborhood cv of x in such that V flcu is a k-dimensional submanifold of cv. The set of regular points of V is the set of regular points of any

dimension. If a point x E V is not regular (i.e. regular of any dimension), then it is called singular. The dimension of V at a point x V which is regular of dimension k, is of course said to be k. The dimension at an arbitrary point x V is the maximum of positive integers k such that every neighborhood of x contains a regular point of V of dimension k. The dimension of V is the maximal dimension of V at any point.

In the case 1K = C. the set of singular points of V

again a C-analytic subvariety, called the singular variety of V, which has a dimension strictly less than V. If V is also irreducible, then the set of regular points is a connected submanifold of dimension dim V. The reader should be warned that when K = the set of singular points need not be an R-analytic subvariety, and even if V is irreducible, the set of regular points need not be connected and could consist of several different dimensional pieces. However, in both cases it is true that the set of regular points of V is dense in V. There are other fundamental differences between IR- and C-analytic subvarieties. (Most notably perhaps, the analogue of the Nullstellensatz below does not hold for R-analytic subvarieties.) For the most part, we shall not need any properties of IR-analytic subvarieties beyond the fact that the complement of a proper R-analytic subvariety of an open set is dense is

§5.2. ANALYTIC SUB VARIETIES

129

Thus, we shall confine the remainder of this section to the case 1K = C. Also, for the remainder of this book, a C-analytic subvariety will be referred to as a holomorphic subvarieiy and an R-analytic one as a real-analytic subvariety. Let us remark, as the reader may already have noticed, that in this book real-analytic subvarieties (e.g. submanifolds) of are also considered, in which case C" is tacitly identified with

in

The following theorem (see e.g. Gunning and Rossi [I, Theorem ll.E.3], or Gunning [I, Corollary II.H. 10]) will be useful in this work. We use here the notation

for the space of holomorphic functions in an open subset

be an open subset of C", and let F C

THEOREM 5.2.2. Let

(5.2.3)

V

:= (z

E

of C". Then

f(z) = 0, Vf E F)

is a holomorphic subvariely of

5.2.4. In the real case it is easy to see that if {f1} is a sequence of (real-valued) real-analyic functions defined in an open set 2 C R", then the set of points V :={x E Vj) a real-analytic subvariety of which is defined in any relatively compact open subset U C by the vanishing of a single function fu (x). Indeed (as in the proof of Theorem 1.5.10) one may take is

fu(x) .—

21

1f(y)12

for a defining function of V fl U. We shall mostly be concerned with local properties of holomorphic subvarieties.

Given two holomorphic subvarieties, V c and V' C and p E V fl V' we say that V and V' are equivalent at p, denoted by V --,, V'. if there exists an open set W C clflc2',withp E Wsuch that VflW = V'nW. Theequivalenceclass, under the relation of the holomorphic subvariety V C will be called the germ of V at p. and denoted by V,,. For p C", we denote by 0(p) the ring of germs of holomorphic functions at p. This ring is isomorphic to the ring C{z} of convergent power series in z. Given a germ at p of a holomorphic subvariety we denote by I( C O(p) the ideal of germs vanishing at i.e. those germs f E 0(p) for which there is an open neighborhood of p, a holomorphic subvariety V C representing V,,, representing f such that f(z) = 0 for z E V. and f

V. RINGS OF POWER SERIES AND POLYNOMIAL EQUATIONS

130

Next, supposelis an ideal in 0(p). We denote V(I) the germ ofaholomorphic I be a finite set subvariety at p defined by I as follows. Let f',... , of generators of I, an open neighborhood of p, and f',... , 1K E representatives

of ft,...

,

1K. The germ V (1) is the germ at p of the holomorphic

subvariety of

Clearly, V (I) is independent of the choice of Q and generators ft,... , IK. (The reader should note that analogous definitions could be made for real-analytic subvarieties.) is called irreducible if it A germ V,, of a holomorphic subvariety at p E of two distinct nonempty germs of cannot be written as the union (V1 U is irreducible if and only if its ideal holomorphic subvarieties at p. The germ can be decomposed uniquely as a finite union of is prime. Any germ irreducible germs of holomorphic subvarieties at p, called the irreducible compo(The analogous statements are also true for real-analytic subvarieties. nents of However, the analogue of the following theorem does not hold for real-analytic subvarieties.)

C's". ff1 is an ideal in 0(p) THEOREM 5.2.5 (NULLSTELLENSATZ). Letp is a germ of a holomorphic subvarieiy at p. then following hold: and (a) (b)

=

I(V(I)) =

are easy to prove. A proof PROOF. Part (a), and the inclusion I(V(T)) C of the opposite inclusion in part (b) can be found in Gunning [1, Theorem II.E.2] (see also Gunning and Rossi [1], Theorem ffl.7). 0 As an immediate consequence of the Nullstellensatz and Corollary 5.1.7, we have the following corollary. COROLLARY 5.2.6. A proper ideal I is of finite codimension

and only if

V(I)=(O}. The following notion will be used in Chapter IX. Recall the notation either of the rings or C(z}, where z = (Zi,... , Zn).

for

with DEFINITION 5.2.7. An element 1(z) = (fi(z),... , f(O) = 0, is a finite (holomorphic or formally holomorphic) map if the ideal 1(f) has finite codiinension in

fff = (fi,... that f(0) =

0.

f : (Ca, 0) —+ (Ca, 0) to indicate , The following is an immediate consequence of Corollary 5.2.6.

§5.2. ANALYTIC SUB VARIETIES

131

THEOREM 5.2.8. If f(z) is a holomorphic mapping (C",O) —+ (C",O), then (0). 1(z) is finite if and only 0 is an isolated point of We shall also use the following well—known property (see e.g. Rudin [1, Theorem 15.1.6]) of finite holomorphic mappings.

5.2.9. If f(z) is afinite holomorphic mapping (C", 0) -+ (C", 0). then f(z) is an open mapping. One can find a precise local description of holomorphic subvarieties in e.g. Gunning [1, Chapter DJ. We shall not need such a precise description in this book and, hence, we do not give it here. In the next section (Theorem 5.3.9), we shall prove a weaker result that suffices for our purposes. However, we will need the

following easily stated consequence of the precise local description. Since we have not found a reference for this exact statement, we provide a proof, based on the local description of a holomorphic subvariety given in Gunning [1], for completeness and for the reader's convenience. 5.2.10. Let V be a holomorphic subvariely of an open set U C C" and p0 E V. Assume that the dimension of V at Po is positive. Then, there exists a nonconstant analytic disc A: A —+ C", i.e. a C"-valued function, holomorphic in which extends continuously to the closure A, such that A(0) = p0 the unit disc andA(A) C V. PROOF. Let V' be a holomorphic subvariety in a neighborhood of po such that is irreducible. (We may take V'to be a representative of an irreducible component, of positive dimension, of the germ V,,0.) Let us denote the dimension of V' at p0 by k > 0. By the local parametrization theorem (Gunning [1, Theorem II.D.10]), we may assume, after a translation and a linear change of coordinates if necessary, that p0=O and that there is an open neighborhood U = Ui x U2 C CA x CnA of 0 such that the projection n: C" —+ CA, taking z = , Zn) to the first k components z' = . -. , z,), exhibits V' fl U as a finite branched holomorphic covering of pure order r, for some integer r > 0. (Since we shall not use the notion of finite branched holomorphic covering further in this book, we refer the reader to Gunning [1, Chapter II.C], for the definition and basic properties of such.) This implies that there exists a proper holomorphic subvariety X C such that fl V' every point z' U1 \ X has a neighborhood w with the property that consists of r disjoint k dimensional submanifolds and it restricted to each of these is biholomorphic. We may assume that X is nonempty and goes through the origin in CA; otherwise, the restriction of it to V' does not branch at 0 so that V' is a k dimensional submanifold at 0, in which case the desired result is trivial. Denote the dimension

po E V' c V. V' has positive dimension at po. and the germ

V. RINGS OF POWER SERIES AND POLYNOMIAL EQUATIONS

132

of X at 0 by t, 0

£
shrinking U, if necessary) a linear subspace S C Ck of dimension k — £ > 0 such that SnX = (0). Chooset EU1 EU1 E U, \ X, we may define r holomorphic mappings in a neighborhood Since t/2 woft/2in U,, cv -+ forj = 1,... ,r, such that each H, parametrizes fl V'. Consider the nonconstant mappings one component of ir — '(cv)

j = 1,

:=

(5.2.11)

. .

. r,

from a neighborhood of = 1/2 in the unit disc A into V'. Since S n X = (0), each mapping can be continued holomorphically along any curve in A \ (0). By (5.2.11), the continuation of the A1 along e.g. the circle = 1/2 results in a permutation of the A'. Hence, there exists a positive integer m r such that the repeated continuation of A' around that circle m times yields A' again. The monodromy theorem implies that the mapping A = A' is holomorphic obviously remains bounded as -÷ 0, we deduce that in A \ (0). Since V A: A —+ is a nonconstant holomorphic mapping. Since the mapping clearly extends continuously to the closure (indeed, it extends holomorphically across -.+ V1 is an analytic disc with the desired properties. This the unit circle), A: completes the proof of Theorem 5.2.10. 0

§ 5.3. Weierstrass Preparation Theorem and Consequences The Weierstrass Preparation Theorem is a fundamental tool in the theory of several complex variables. In this section, we state the version of this theorem that we shall need and derive from it some important consequences. We refer the reader to [Hörmander [2], §6.11 for a proof and further discussion. Recall as in

or C(z,,... , zn).

§5.1 that 7Z,, denotes either

THEOREM 5.3.1 (WEIERSTRASS PREPARATION THEOREM). Let K E

K (0) = (5.3.2)

0.

Assume there exists a positive integer N satisfying

—(0) =0,

. 1

N

Then the following holds:

(5.3.3)

K(z)

+

—

I,

dNK 0.

with

§5.3. WEIERSTRASS PREPARATION THEOREM AND CONSEQUENCES

with U E

= (Zj, .

= 0, and with the notation

0, and a1 E 1Z,,_1,

U(O)

133

,Zn—i).

. .

We shall also need a well-known result concerning symmetric convergent power

series. Forz = (Zi,...

= 1,... ,n,betheelemen-

E

tary symmetric functions, i.e. the coefficients of the polynomial

fl(X_ZJ)=Xn A convergent or formal power series S(z1,...

S(zi,

. . .

,

S(z,r(I),

is called symmetric if , Zir(n)),

. . .

for any pennutation of the set (1, 2,... , n}. The following is a generalization of Newton's Theorem for symmetric polynomials. THEOREM 5.3.4.

For every symmetric for,nal power series S(z1,... such that ,

there

,

exists a unique formal power series K (wi,... S(z1, where

ai(z)

K(a1(z),

. . .

. . .

are the elementary symmetric functions of

,

as

above. Furthermore, if S is convergent, so is K. PROOF. We decompose S(z) into its homogeneous parts S(z)

where each Sj(Z) is a homogeneous polynomial of degree j. Clearly, each Sj(z) is symmetric. By the fundamental theorem on symmetric functions (Newton's theorem; see e.g. Van der Waerden [1, section 261), there exist, for j = 0, 1,. polynomials K1(ci, such that , K1(ai(z), ..

.

To complete the proof it suffices to show that if S is convergent, the formal series converges in some neighborhood of 0 E C". For this, choose 8 > 0 such that the series S(z) converges for Izi < 8. Observe that for each E there exist Z E C" such that = a1(z). Indeed, one may = (fl,... take Z = as the n roots of the monic polynomial equation , ,

V. RINGS OF POWER SERIES AND POLYNOMIAL EQUATIONS

134

as is easy to see from the definition of oi(z),...

By using Lemma 5.5.1 , (stated and proved in §5.5), we deduce that if belongs to the open neighborhood < then JzI < 8. Hence, the series U= U, which completes the proof of Theorem 5.3.4. 0 converges for

As an application of the Weierstrass Preparation Theorem, we shall prove the following proposition which will be used in connection with the reflection principle in Chapter IX. PROPOSITION 5.3.5. Let F1(z),... satisfying

det as

azk

aformal power series. If K E

necessarily K

with no constant term, and

E

,

satisfies K(Fi(z),...

0, then

,

0.

PROOF. We reason by contradiction. Suppose K(Fi(z),... K 'p" 0. We shall show that det

0, with

,

0, contrary to the assumption.

After a change of holomorphic coordinates, we may assume (5.3.2) is satisfied. Therefore, by the Weierstrass Preparation Theorem we have (5.3.6)

—0,

with fr(z) = (F1(z),... 1,

(5.3.7)

...

We differentiate (5.3.6) with respect to Zt,

,

, n, to obtain

(NFn(z)"' + N—I n—i

=—

t

j=o p=i

We consider now two cases. Suppose first (5.3.8)

Multiplying det

+

0.

by the left hand side of (5.3.8), which is the same

as multiplying the nth row of the determinant by the same quantity, we conclude

§5.3. WEIERSTRASS PREPARATION THEOREM AND CONSEQUENCES

135

from (5.3.7) that

Jai(F'(z))Fn(zY') det

+ contradicting

the

assumption. Hence (5.3.8) does

polynomial equation by induction on N. respect

to

not

— 0,

hold, i.e. F(z)

satisfies a

similar (0(5.3.6), but of degree N — 1. We may then continue

Indeed, if (5.3.6)

holds for N

=

1, then differentiating it with

(z))

would yield the vanishing of det

,

again contradicting

0

the assumption.

the following more technical result, which may be viewed as a partial generalization of the Weierstrass Preparation Theorem for the zeros of a set of r functions (see Remark 5.3.13). We shall give two proofs of this result. The first is based on theorems in analytic geometry not included in this book; the second proof we give is self-contained. In Chapter IX we shall need

THEOREM 5.3.9. I..et E and suppose Z E z),... z) are holomorphic functions in some neighborhood U x V of(0, 0) E C C(z} the ideal generated by F1(0, z),... , F,.(0, z) and by E C U x V the common zero locus ,

E :=

= ... =

EU x V:

o}.

Assume that I has finite codimension, i.e.

dimc (C(z}/I) <00.

(5.3.10)

Then there exist S > 0 and, for j in

= 1,...

,

q, Weierstrass polynomials

z1)

i.e. -1

= z7' +

(5.3.11) ak

(5.3.12)

are holomorphic for I I <S and ak (0) = 0, such that

E fl (Be(s) X Bz(S))

c

E

x

= ... =

Here, we use the notation B1(S) = < 5) with similarly Be(S) = (IzI <5) with IzI = SUPI<j
=

0}. sup1<1<,,

and

5.3.13. Note that with q = r = 1 the conclusion of the lemma follows from the Weierstrass Preparation Theorem. Indeed,

the conclusion holds

V. RINGS OF POWER SERIES AND POLYNOMIAL EQUATIONS

136

with equality in (5.3.12). However, it is not possible in general to find P1, ..., 2. such that equality holds when q 5.3.9. By choosing FIRST PROOF OF in view of Corollary 5.2.6 assume that (z E

> 0 sufficiently small, we may

F1(O, z) =...

,

z) =

0)

consists of only the point z = 0. We then choose 82 > 0 such that the cylinder x does not intersect E. This means that, for each fixed F= E B1(82), the fiber

E

x is a compact (and thus finite) subset of the projection 1,... , q, by it1:

/2). Denote, for each j =

:=

(5.3.14)

It follows from the above that the restriction of it1 to the variety

Efl(B1(82) x is proper (indeed, finite to one). Thus, yrj (E) is a holomorphic subvariety in /2) by Remmert's Proper Mapping Theorem (see e.g. Gunning (82) x [1] or Gnffiths and Hams [1]). Moreover, since the fibers are finite, there is x for some sufficiently small a holomorphic function z1) in 8 > 0, such that (5.3.15)

ir1(E) n

x B21(8)) C ((i, z1):

z1) = 0)

and

$0.

(5.3.16)

would contain ((c,

Indeed, if such a function did not exist then

= 0),

for sufficiently small z1, which would contradict the finiteness of E0. By the Weierstrass Preparation Theorem (l'heorem 5.3.1), we deduce the existence of a Weierstrass polynomial (5.3.17)

z1) in zj such that

7r1(E)fl(Bc(8) x

C

=0),

by shrinking 8 > 0 further if necessary. This completes the first proof.

0

The above proof of Theorem 5.3.9 used Remmert's Proper Mapping Theorem (although in a very simple form). Since we have the ambition to make this text

§5.3. WEIERSTRASS PREPARATION THEOREM AND CONSEQUENCES

137

self-contained and since giving a proof of the proper mapping theorem seems outside the scope of this book, we shall present an alternative proof of Theorem 5.3.9 that does not use analytic geometry. ALTERNATIVE PROOF OF THEOREM 5.3.9. We first reduce the situation to a simpler one. By Proposition 5.1.5, there are holomorphic functions a1k (z) near and an integer N such that 0E

zf =

(5.3.18)

for j = 1,... , q. Define holomorphic functions

z) near (0,0) such that

=

(5.3.19)

fork= 1,... ,r. Itfollowsthat (5.3.20)

zf + K1(C,

z)

=

z)

z), k=l 1=1

for j = 1,... , q. Note that the variety E is contained in the variety E' defined by (5.3.22)

Now, Theorem 5.3.9 is a consequence of the following. LEMMA5.3.23. €C", z = 1,... morphic functions defined in a neighborhood U x V of (0,0) E

j

K1 (0, z)

0.

j

= 1,...

,

,q.beholo-

satisfying q. Given positive integers N1,... , Ne,, define

(5.3.24) Then, there existS > 0 and Weierstrass polynomials P1 (see 5.3.11) such that (5.3.25)

E' n

zj) in Zj. J = 1,... , q,

x C

f(c,z)

Bc(S) x Be(S):

... =

0),

V. RINGS OF POWER SERIES AND POLYNOMIAL EQUATIONS

138

where

(8) are the polydics defined in Theorem 5.3.9.

(8) and

We reason by induction on q. Forq = 1, we obtain (5.3.25) by applying the Weierstrass preparation theorem to the function z= We shall now show how to reduce the case of q to that of q — 1. We apply the Weierstrass Preparation Theorem to z7' + K1 z). Hence there 0,... , N1 — satisfying exist holomorphic functions ct(C, Z2,... , zr,), PROOF.

1

ct(0, Z2,

,

0 is equivalent to

z)

N1

(5.3.26)

.

Z71

-

+

= 0.

Z2,... ,

k = 1,...

be the roots (counted with multiplicity) k = 1,... , N1, in of (5.3.26). Then replace Zi by z) for in + Taking products over k we obtain the equations = 2,... , q. j 12t Pk(C, Z2,...

(5.3.27)

,

Zq),

,

N1,

fl[z7' + K1(C, pk(Z2,...

,

Zq,

Z2,... , zq)]

=0.

Since the left hand side of (5.3.27) is a symmetric holomorphic function of p1.... PN1. by Theorem 5.3.4, it is a holomorphic function of the elementary symmethc in (5.3.26). Therefore we may rewrite functions of the Pk i.e. of the coefficients (5.3.27) in the form (5.3.28)

J-Ij(C,z2,... ,Zq)O,

j=2,... ,q,

where H1 is holomorphic near 0 and by the hypothesis Z2,... , zr,) on Therefore, (5.3.28) is a system of the same form as the one defining E' in (5.3.22) with q replaced by q —1. By induction, we obtain Weierstrass polynomials

,q,suchthat (5.3.29)

E" = {(C,z') EU' x V': H2(C,z') = ... = Hq(C,Z')

0},

where z' = (z2,... , z,,) and U' x V1 is a sufficiently small neighborhood of (0, 0)

(5.3.30)

x C", satisfies x

c {(C,z'): P2(C,z2) = ... = P,.,(C,z,) =01,

for sufficiently small 8 > 0. To find the missing Weierstrass polynomial Pi(C, zi),

we start with (5.3.26) and replace each Zj. j

= 2,...

,

q, by one of the roots

§5.4. ALGEBRAIC FUNCTIONS, MANIFOLDS. AND VARIETIES

139

(counted with multiplicity) of z1) = 0. Taking a product, similar to (5.3.27), over all possible expression so obtained and again using symmetry and Theorem H1 is holomorphic 5.3.4, we obtain an equation of the form Applying once more the Weierstrass for some integer and H1(O, Preparation Theorem, we obtain a Weierstrass polynomial Zi) in zi and it is 0 straightforward to check that the inclusion (5.3.25) holds.

§5.4. Algebraic Functions, Manifolds, and Varieties In this section, we shall recall some classical material on algebraic functions, manifolds, and varieties which will be used later in the book. We let K denote either of the fields R or C (although most of the material presented here is valid the ring of convergent power over any field of characteristic 0), and denote by series in n variables over K. We shall denote the variables by x = (x1,... , so that Sn = K{x). We will identify with the ring of germs of holomorphic functions at the origin in C and with the ring of real-valued, realanalytic functions at the origin in if K = R. For the remainder of this section, we will refer to the germs of functions in Sn simply as being analytic. We shall also abuse the notation in the usual way and identify a germ of a function with some representative of it.

DEFINmON 5.4.1. A function f E is (K-)analytic algebraic if there exists a nontrivial polynomial P E K[x1,... , Xn, t] = K[x, t] such that

P(x, f(x))

0,

for x in a neighborhood of 0 E K". 5.4.2.

(i) We will also refer to C-analytic algebraic functions as holomorphic algebraic and to R-analytic algebraic functions as real-analytic algebraic. The latter functions are also called Nash functions. (ii) A complex-valued real-analytic function near 0 R" is called realanalytic algebraic if its real and imaginary parts are real-analytic. (iii) Although we only defined, for the sake of convenience, what it means for a germ of a function at 0 to be analytic algebraic, it should also be clear what it means for a function to be analytic algebraic at a point x E K",

We denote by An the subset Of Sn consisting of the analytic algebraic functions. The following basic properties will be useful.

V. RINGS OF POWER SERIES AND POLYNOMIAL EQUATIONS

140

PROPOSITION 5.4.3. The following properties hold:

(a) (b)

is a subring of Sn. 1ff E then

(c) 1ff E

E

for j = 1,...

, n.

satisfies a nontrivial equation P(x, f(x))

0, where P(x, t)

is a polynomial in t with coefficients that are analytic algebraic functions of x (which we can write P E then f E (d) 1ff is a holomorphic algebraic function near 0 then its restriction to is real-analytic algebraic. Conversely, is a real-analytic algebraic function near 0 then its holomorphic extension to is holomorphic algebraic.

(e) lfg = (gi

withg(0)=Oandf

E

thenh = fog

is in .A,. PROOF. The properties (a)—(c) are standard results about algebraic extensions. We refer the reader to the book by Hodge and Pedoe [1, Volume I, Chapter III] or

Bochner and Martin [1J. The statement (d) also follows easily from the material in Hodge and Pedoe [1, Volume I, Chapter III] by observing that C is an algebraic field extension of IR. We leave the details to the reader.

To prove (e), we first observe that h(y) = f(g(y)) is in 8,. Now, let P(x, t) be a polynomial such that P(x, f(x)) 0. We write

P(x,t) =

(5.4.4) where ak

(5.4.5)

By (a), each

K[x] fork = 1,... 0

,

p. We then have, with h(y) = f(g(y)),

P(g(y), h(y)) = is in A,. Hence, h E

by (5.4.5) and (c).

0

The next result states that functions defined implicitly by algebraic equations are algebraic. This theorem, henceforth referred to as the algebraic implicit function theorem, will be important in Chapter XIII. THEOREM 5.4.6 (ALGEBRAIC IMPLICIT FUNCTION THEOREM). Let x E

and

y E Ktm. Assume that A(x,y) = (Ai(x,y),... is in A (0,0) = 0, and that the Jacobian matrix (aA/ay)(0, 0) is invertible. Then, there exists a unique Ktm -valued function f(x) = (f1(x),... , f,,,(x)), with f(0) = 0 such that A(x, f(x)) 0. and f Before we prove Theorem 5.4.6, we shall present some basic material on algebraic subvarieties of We refer the reader to e.g. the books by Hodge and

§5.4. ALGEBRAIC FUNCTIONS. MANIFOLDS, AND VARIE1IES

141

Pedoe [1, Volumes I—if] and Zariski and Samuel [1] for further information and proofs of the facts stated here. The reader can also consult the book by Bochnak, Coste, and Roy [1] for material that is specific to algebraic geometry over R. is an algebraic subvariety if there exists such that a collection of polynomials (Pa) in K[x1,... , DEfiNITION 5.4.7. A subset V C

V=(x€Km:Pa(x)=O,

(5.4.8)

Va).

When K = R (resp. K = C) we say that V is a real-algebraic (resp. complex algebraic) subvariety.

Clearly, every polynomial in the ideal I C K[x] generated by the collection (Pa) vanishes on the algebraic subvariety V and hence, since K[x] is noetherian, V can be presented as the common zero locus of a finite collection of polynomials We shall use the same notation as in §5.2. Given an algebraic P1,... , subvariety V we denote by 1(V) the ideal of polynomials that vanish on V. and given an ideal I we denote by V(I) the common zero locus of the polynomials in I. The fact that the same notation was used for the corresponding holomorphic objects in §5.2 should cause no confusion. An algebraic subvariety V is called irreducible (over 1K) if it cannot be written as a union Vi U V2 of two distinct (meaning also that neither is contained in the other) nonempty algebraic subvarieties. A subvariety V is irreducible if and only if the ideal 1(V) is prime. Moreover, if V is reducible, i.e. not irreducible, then it can be decomposed as a finite union of irreducible algebraic subvarieties. For this reason, it often suffices to consider only irreducible subvarieties. be an irreducible algebraic subvariety and let p = 1(V) be the Let V C

prime ideal of all polynomials vanishing on V. The integral domain R(V) =

K[x1,...

is called the coordinate ring of V and the quotient field F(V) of R(V) is called the field of rational functions on V. The field F(V) is a finitely generated extension of K and its transcendence degree over K, i.e. the maximum number of algebraically independent elements over 1K, is called the dimension of the irreducible algebraic subvariety V. We shall denote the dimension of V by dim V. An irreducible subvariety of dimension 0 consists of a single point and one of dimension n must be the whole space If V1 and V2 are irreducible algebraic subvarieties and V1 C V2. then dim dim V2. Moreover, if V1 is a proper subset of V2, then dim V1
,

dx

=

142

V. RINGS OF POWER SERIES AND POLYNOMIAL EQUATIONS

does not exceed n — d. The set Vmg of points x E V at which the rank of (5.4.9) equals n — d is a nonempty subset of V called the set of regular points of V. The of (5.4.9) is strictly less than n — d is V set an algebraic subvariety of dimension strictly less than d called the set of singular points.

is a d-dimensional submanifold of K". If is connected and dense in V. This is not true in general K = C, then when K = R. (See e.g. Example 13.3.4 in Chapter XIII.) Another fundamental difference between R and C (cf. also §5.2) is that the analogue of part (b) of the The reader should note that

Nullstellensatz (Theorem 5.2.5) in the algebraic setting holds in the case K = C but not when K = R. Indeed, it is easy to construct e.g. prime ideals I in REx1, x21

such that even the dimensions of I and I(V(T)) are different. Part (a) of the Nullstellensatz, on the other hand, holds over both R and C. The reader should be warned that even though an algebraic subvariety is also an analytic subvariety (see §5.2), the definition of regular (and hence singular) points differs in the two cases. A regular point on an irreducible algebraic subvariety V. in the algebraic sense, is a regular point of dimension dim V in the analytic sense. For K = C, the two notions coincide since the set of regular points of an irreducible

holomorphic subvariety is a connected submanifold of dimension dim V. For K = R, the converse is not true, as is illustrated by the following example taken from Benedetti and Risler [1]. Let V be the real-algebraic subvariety of R2 given by the vanishing of the irreducible polynomial P(x, y) = x4 + 2x2y — y3. It is easily checked that the point (0, 0)is a regular point in the real-analytic sense, but not in the real-algebraic sense. In fact, the point (0, 0) is a singular point in the complexification of V. We shall need the following fundamental fact.

5.4.10. LetS C K" be a connected analytic sion d through the origin. The following are equivalent: (i) S has m = n — d analytic algebraic defining functions A1,... , Am near the origin, i.e. A = (A1,... , Am) E (An)'" hasaiacobianmatrixaA/ax of maximal rank (m = n — d) in some neighborhood U C K" of the origin

andSflU={xEU: A(x)=0). (ii) There exists an irreducible algebraic subvariety V C K" of dimension d

suchthatSc V. PROOF. We begin by proving that (i) (ii). The proof is by induction on m. Thus, suppose first that m = iso that S is a (smooth) analytic hypersurface through the origin in K" defined by a single analytic algebraic equation A1 (x) = 0. Since is algebraic, there exists a nontrivial polynomial P E K[x, tJ such A(x) = ,'

§5.4. ALGEBRAIC FUNCTIONS. MANIFOLDS, AND VARIETIES

143

that P(x, A(x)) 0. We may assume that the polynomial P has minimal degree in t, so that Q(x) = P(x, 0) is nontrivial. The zero locus of Q(x) is an algebraic subvariety that contains the analytic submanifold S fl U. Indeed, if A(x) = 0, then Q(x) = P(x, 0) = P(x, A(x)) = 0. Since S fl U is a connected analytic submanifold on which Q(x) vanishes, it is clear that S fl U is also contained in the zero locus of some irreducible polynomial R(x) in the prime factorization of Q(x) over K. If we denote by V the zero locus of R(x), then V is an irreducible algebraic subvariety that contains S fl U. Moreover, V has dimension n — 1 since it contains S and is not the whole space. This completes the proof in the case

m=l. mo. We may Assume that the implication (i) ; (ii) has been proved for m assume that mo < n (otherwise, there is nothing more to prove). We shall prove the implication form = mo +1. Denote by d0 = n — m0, by S1 the d0 dimensional analytic submanifold defined near 0 by the m0 first equations

Ai(x)... ,Amo(X)0,

(5.4.11)

and by 52 the d0 dimensional analytic submanifold defined by them0 last equations

(5.4.12)

A2(x)=... ,Amo+i(X)0.

By the hypothesis on the rank of the Jacobian matrix aA/ax(o), the submanifolds S1 and S2 intersect nontangentially at the origin and their intersection is, of course, S.

By the induction hypothesis, there are irreducible algebraic subvaneties V1 and C V1 and 52 C V2. The submanifold S is contained V2 of dimension such that in the algebraic subvariety V1 fl V2. (The intersection of two algebraic subvarieties is again an algebraic subvariety.) Now, there are two possibilities. Either V1 V2

is a proper subset of V1 or V1 fl V2 = V1. In the former case, each irreducible component of V1fl V2 is a proper subset of V1 and, hence, has dimension strictly less than dim V1 = d0. Since S is a connected analytic submanifold which is contained in V1 fl V2. it must be contained in an irreducible component V of V1 fl V2. Thus, V is an irreducible algebraic subvariety of dimension strictly less than d0 such that

Sc V.

Since

Shasdimension4— 1,wemusthavedimV =d0— 1 = dimS.

This completes the induction step under the assumption that V1 fl V2 is a proper subset of If V1 flV2 = V1,then V1 C V2. Since V1 and V2areirreducibleandofthesame dimension, we must have V1 = V2. We claim that S fl U, for some neighborhood U of the origin, must be contained in (V1 Indeed, near each point x E S fl U, S is defined as the intersection between Si and 52. If x were a regular point of V1,

V.

144

RINGS OF POWER SERIES AND POLYNOMIAL EQUATIONS

dimensional submanifold in a neighborhood of x. then V1 would be a smooth This is impossible, since V1 must contain both of the d0 dimensional submanifolds S1 and S2 which, if we choose U sufficiently small, intersect nontangentially at x. Hence, S fl U C (Vi )sing and, since each irreducible component of (Vi )sing has dimension strictly less than 'h), the induction step, under the assumption that V1 fl V2 = V1. is completed as above. This completes the proof of the implication (i)

(ii).

(i), let V c K" be an irreducible alTo prove the converse, i.e. (ii) ; gebraic subvariety of dimension d containing S. By renumbering the coordinates if necessary, we may assume that S is described near the origin as a graph x" = f(x'), where x = (x', x") with x' E Kd and x" E K'" and where

f = (Ii,... , fm) E

(Se)'". We claim that for every j E (1,... , m} there exists a nontrivial polynomial p1(x', x7) in the prime ideal p = 1(V). Indeed, if no x7 such polynomial existed, then the image of the d + 1 monomials in the quotient field of K[x', x"]/p would be algebraically independent over K. This would, by definition, contradict the fact that the dimension of V is d. (See also Hodge and Pedoe [1, Volume II, Chapter X.5].) Since S is contained in V near the origin, we must have p3(x', f3(x')) 0. Hence, each analytic function m. This is algebraic and we may take A1(x) = x7 — f1(x') for j = I completes the proof of Theorem 5.4.10.

0

It follows from Theorem 5.4.10 that a connected analytic submanifold S C K" of dimension d is contained in an irreducible algebraic subvariety V c K" of the same dimension if and only if it is defined near some point x0 E S (and hence at every x E 5) by analytic algebraic equations. We make the following definition.

DEfiNITION 5.4.13. A connected analytic submanifold S C K" is algebraic (or K-algebraic) if it is contained in an irreducible algebraic subvariety V C K"

with dimS =dimV. An analytic mapping f: (K", 0) —+ (K'", 0) is said to be analytic algebraic if j = 1,... , ni, for each component of f. By Theorems 5.4.6 and E 5.4.10, the mapping f is analytic algebraic if and only if its graph in K" x K'" is an algebraic submanifold. We now prove the algebraic implicit function theorem, which, at this point, is very easy. PROOF OF THEOREM 5.4.6. The implicit function theorem for holomorphic or

real-analytic functions asserts that there is a unique solution y = f(x) of the equation A(x, y) = 0 such that f(0) = 0 and f E (Sn)'". We must show that

§5.5. ROOTS OF POLYNOMIAL EQUATIONS

145

Denote by S the analytic n-dimensional submanifold through the origin defined by the equations A(x, y) = 0. By Theorem 5.4.10, there is an irreducible algebraic variety V of the same dimension d such that S C V. The fact that the functions fj are algebraic now follows from the same argument used to conclude the proof 0 (i) in Theorem 5.4.10 above. of the implication (ii) We conclude this section with a remark. In Chapter Xffl, we shall consider realalgebraic submanifolds M of C's'. This simply means that we identify CN with R2N and that M is a real-algebraic submanifold of R2N. In view of Proposition 5.4.3(d), a real-analytic function f(Z, Z) is algebraic if and only if it is an algebraic function of (Z, Z). Hence, a connected submanifold M c C's' is real-algebraic if and only if it is defined locally near some point po E M (and hence, by Theorem 5.4.10, at eveiy point) by real-analytic functions that are algebraic in (Z, Z). §5.5.

Roots of Polynomial Equations

Holomorphic Coefficients

This section deals with functions which are roots of polynomials with holomorphic or real-analytic coefficents. The main result (Theorem 5.5.32) is that a smooth function which is a root of a monic polynomial with real-analytic coefficients is itself a real-analytic function. The first lemma gives a crude bound for the size of the roots of a polynomial in tenns of the size of its coefficients. LEMMA 5.5.1. Let z' +aj_1

+. . . +ao be a monic polynomial with complex j = 0,... , J—lforsomeC >0. Then

coefficients. Suppose that 1a11 any root p of the polynomial satisfies p I

C.

PROOF. Assume by contradiction = — ...

contrary to the assumption. This proves the lemma.

I < I/C' for all j > 0. wehave

0

Let V be a domain in the complex plane C and denote by 0(V) the ring of holomorphic functions in V. We consider the polynomial ring 0(V)(z], i.e. the ring of polynomials of the form

p(w,z) =

(5.5.2) where

ao,... , aj

0(V) with aj $ 0. Note that, in contrast with the ring

of convergent power series C{w}, the ring 0(V) is not a unique factorization

V. RINGS OF POWER SERIES AND POLYNOMIAL EQUATIONS

146

domain (UFD), and hence 0(V)[z] is not a UFD. Nevertheless, we need to factor polynomials in 0(V)[z] and we shall factor such polynomials into irreducible polynomials over a slightly different ring. Recall that an element a in a ring R is called irreducible if given a factorization a = bc with b, C E R, either b or c is a unit. First, we prove the following.

5.5.3. Let V C C be a domain and let p(w, z) be a monic polynomial in 0(V)[z], i.e. p(w, z) = z' +

(5.5.4)

where ak E 0(V)for k = 0,... , J p(w,z)

— 1.

p(w, z) = pi(w,

(5.5.5)

Then there is a unique factorization of

.

zY',

. .

where p',... p E 0(V)[z] are distinct, irreducible monic polynomials. PROOF. Denote by M(V) the field of fractions of 0(V), i.e. the field of meromorphic functions in V. Since M(V) is a field, the polynomial ring M(V)[z] is a UFD (see e.g. Zariski and Samuel [1]). Thus, the monic polynomial p(w, z) has a unique factorization in M(V)[zl

p(w,z) = pi(w,z)" ..

(5.5.6)

Pie... ,

are distinct, irreducible polynomials in M(V)(zJ. Since unique here means modulo multiplication by elements in the field M ( V) and since p(w, z) is monic, we may assume that the distinct, irreducible factors pj (w, z), 1 are monic i.e. where

j

—I

k= 1,... ,t,

(5.5.7)

j=o

where a E M(V). We claim that the coefficients a(w), for j = 1,... , — 1 and k = 1,... , £, are in fact holomorphic in V. The conclusion of Lemma 5.5.3 follows from this claim; if a monic polynomial in 0(V)[z] is irreducible in M ( V)[z], then it is irreducible in 0( V)[z], and the uniqueness of (5.5.6) (assuming p1.... , are monic) follows easily from the uniqueness of factorizations in

M(V)[z]. V To prove the claim, denote by the poles of all the coefficients a(w), for j

V

= 1,...

,

—1 and k =

1,... , t. Let

§5.5. ROOTS OF POLYNOMIAL EQUATIONS

147

K be an arbitrary compact subset of V. Since the coefficients ao(w),... , a1...1 (w) of p(w, z) are holomorphic in V, there is a constant C > 0, depending on the compact K, such that

j=O,... ,f—i,

(5.5.8)

for all w E K. By Lemma 5.5.1, the roots P'.... , p., (counted with multiplicity) of p(wo, z), at any given WO E K, are bounded by C. Resthct now further to

wo E KflV.

z)

,

in (5.5.6) are formed as elementary symmetric polynomials of a subcollection of the roots Pie... , pj, it follows that there is a constant C' > 0, depending only on C, such that

k=I,... ,t.

j=O,...

(5.5.9)

was arbitrary, the coefficients (w) are bounded meromorphic Since w0 E K fl functions in the interior of K and thus holomorphic there. Since K was arbitrary,

we deduce that the a(w) are holomorphic in V. proving the claim and hence

0

completing the proof of Lemma 5.5.3. The following is a corollary of Lemma 5.5.3.

COROLLARY 5.5.10. Let V C C beadomain and let p(w, z)beapolynomialin O(V)[z]. Denote by J the degree of p(w, z) and bya(w) = aj(w) the coefficient of in p(w, z) (cf (5.5.2)). Then, IE C V denotes the (possibly empty) discrete set of points where a(w) = 0 and V = V \ E, there is a unique factorization of p(w, z) of the form

(5.5.11)

p(w,z)

where the factors Pj (w, z), 1 <j

...pe(w,z)re,


in O(V)[z]. PROOF. Factor a(w) and apply Lemma 5.5.3 to the polynomial p(w, z)/a(w), whose coefficients are holomorphic in 0 The following is the well-known fact that the roots of a polynomial with holomorphic coefficients are holomorphic away from a small set.

LEMMA 5.5.12. Let V C C be a domain and let p(w, z) be a polynomial in O(V)[z]. Denote by J the degree of p(w, z). Then there is a (possibly empty) discrete set of points E c V, and an integer k J, such that for any wO E

V. RINGS OF POWER SERIES AND POLYNOMIAL EQUATIONS

148

= V\E there exists a neighborhood V0 of w0 and k holomorphic functions in Vo, with distinct values at every point in Vo, such that for all w E V0 Pie... the roots of p(w, z) are given by p1(w),...

,

in p(w, z) by a(w), and by E0 c V the discrete set of zeroes of a(w). By Corollary 5.5.10, we may factor p(w, z) as follows PRooF. Denote, as in Corollary 5.5.10, the coefficient of

(5.5.13)

p(w, z) = a(w)pi(w, z)"

. . .

pe(w,

z)",

where P1 (w, z),... , (w, z) are irreducible monic polynomials in O(V \ Eo)[z], which are pairwise distinct. Let (5.5.14)

q(w,z) =a(w)pi(w,z).. .pe(w,z).

Note that for any w0 E V \ E0 every root of p(wo, z) is also a root of q(w, z) (possibly of smaller multiplicity). Recall (see e.g. Van der Waerden [1]) that the discriminant d(w) of the polynomial q (w, z) is the resultant of q and its derivative and hence is a holomorphic function in V \ E0. Since the discriminant of a polynomial vanishes identically if and only if the polynomial has a repeated factor, it follows from the definition of q(w, z) that d(w) 0. We let E1 be the set of zeroes in V \ E0 of d(w). By the implicit function theorem, the roots in z of q(w, z) are all distinct and are given by holomorphic functions in w in a neighborhood of any point w0 E V\E, where E = E0 U E1. This completes the proof of Lemma 0 5.5.12. The following proposition, which deals with monic polynomials with holomorphic coefficients, is the basis of the classical Puiseux expansion of the roots.

PROPOSITION 5.5.15. Let V C C be a neighborhood of0, and let p(w, z) be a monic polynomial in O(V)[z]. Then there exists an integer N anda neighborhood V' of 0 in Cfor which all the roots of z) are holomorphic functions of w in V'. In fact N can be taken to be I!. PRooF. Without loss of generality, we may assume that V is a disc centered at

the origin of radius r > 0. If r is sufficiently small, by Lemma 5.5.12, we may V, w assume that there exists k so that near every point w 0, the roots of p(w, z) are given by k holomorphic functions near w. We let N = k! and we choose V' to be a disc centered at the origin of radius sufficiently small so that its image under the map w i-÷ wN is contained in V. We shall define k holomorphic functions in V' as follows. By Lemma 5.5.12 and the monodromy theorem, in any simply connected open subset U of V \0 there exist k holomorphic functions

149

§5.5. ROOTS OF POLYNOMIAL EQUATIONS

in U such that at each w E U the roots of p(w, z) are given by the values of these functions at w. We shall take U = V\(R± U (0)), where R4 denotes the positive real numbers. It also follows from Lemma 5.5.12 that each Pj has continuous limits as w -+ R+ from above and below. These limits, which need not be the same, will be denoted Pj+ and Pj— respectively, j = 1,... , k. By the uniqueness of the roots, there is a permuation a of (1,... , k } so that the Pa(j)+(S) for s E (0, r). continuous functions Pj+. Pj— on (0, r) satisfy We may now define functions j = 1,... , k, in W' = V'\(E' U (0)), are line segments with and the E R+) = where E' = (w E V1 : with opening of endpoint 0. Then W' is the disjoint union of N open sectors are numbered by starting with The separated by the line segments angle and continuing counterclockwise. We define = (w E W1 : 0 < arg w <

b1(w),j=1,... ,k,forw€W'asfollows.Put ifw E

Paji(W), (5.5.16)

if w E

b1(w) := PCN_I(J)(WN),

if w E

at is taken in the sense of composition. Note that aN = I, the identity permutation of the set (1,... , k }. Hence by the property of the I)j stated above, where

away from 0, and the functions b1 extend continuously through the segments therefore the are holomorphic in V'\(O}. Since all the roots of p are bounded in V (see Lemma 5.5.1), so are the functions b1, which implies that the extend holomorphically to all of V1. It is clear that the functions b1(w), given by (5.5.16), z). The proof of Proposition j = 1,... , k, are the roots of the polynomial 0 5.5.15 is now complete. The following is an immediate consequence of Proposition 5.5.15. COROLLARY 5.5.17. Let p(w, z) be as in Pmposition 5.5.15. Then there exist

positive integers N, Li,...

,

and holomorphic functions b1(w),...

,

in

a neighborhood V' of 0, with the b1 (w) distinct at all w E V1 except (possibly) at

w =Oandsuchthat k

(5.5.18)

z)

—

w E V1.

The following proposition shows that a smooth function of one variable which

satisfies a monic polynomial equation with real-analytic coefficients is itself realanalytic; in Theorem 5.5.32 this will be generalized to several variables.

V. RINGS OF POWER SERIES AND POLYNOMIAL EQUATIONS

150

PRoiosmoN 5.5.19. Let p(s, z) = z' +. . +ao(s) be a monic polynomial with real-analytic coefficients in an open interval I of R, and u(s) a smooth function defined on I satisfying p(s, u(s)) 0. Then u is real-analytic in I. .

PROOF. It suffices to show that u is real-analytic in a neighborhood of every point in I. We fix such a point in I and, by translation, we may assume that point is 0. Without loss of generality we may also assume I is a connected, symmetric interval around 0. Since p(s, u(s)) 0, s E I, we have u(w")) = 0, for w E C with I. We take N given by Corollary 5.5.17. By that corollary (and after shrinking I if necessary), on each connected segment in the preimage of I\{0} under the mapping w i-+ w", the function u(w") is equal to the restriction of one of the holomorphic functions b1 (w) to that segment since u is continuous and the functions (w) are pointwise distinct on these segments. We restrict to the segments of arguments 0 and respectively, and we note that the images of these by the map w i-+ cover 1\(0). Let b1, and be the corresponding = holomorphicfunctions,sothatb11(s) = >0 with E I. We claim that = J2, and the coefficients cq of the common Taylor series of and bh at 0 satisfy Cq = 0, for q pN. For this, we define new smooth functions v1, v2 on an open interval containing 0 by v1(s) = u(s") and = v2(S) = By writing the Taylor series of the in terms of that of u, we easily obtain (5.5.20)

=

= 0, q

pN,

and

=

b11 (s) = v1(s) and b11 = v2(s) for s > 0 sufficiently small, the claim easily follows from (5.5.20). It follows from the claim that u is given on I by the Since

same convergent series for s > 0 and for s < 0 and hence is real-analytic in a

0

neighborhood of 0, which completes the proof of the proposition.

In order to prove the generalization of Proposition 5.5.19 to the case of a realanalytic function of several variables, we shall need the following version of a classical inequality for polynomials. THEOREM 5.5.21 (BERNSTEIN5 INEQUALITY). Given a cube Q C

with

nonempty interior and a real number R, with R > 1, there is an open neighborhood V of Q in such that for any polynomial P (zi, complex coefficients the following holds: (5.5.22)

sup IP(z)I zEV

. . .

of degree k with

RksuplP(x)I. xEQ

The proof of Theorem 5.5.21 is based on a reduction to the classical inequality

in one variable. For this we need the following lemmas.

OF POLYNOMIAL EQUATIONS

§5.5.

151

LEMMA 5.5.23. Let P(z) be a polynomial of degree k in one variable with complex coefficients, and R a real number with R> 1. Then

sup IP(z)I

(5.5.24)

Rk

sup IP(z)l. IzI=I

PROOF.

Consider the polynonial A(z) := zkP(1/Z). Then it is easy to check

that

(5.5.25)

sup IA(z)I

=

(zI=1/R

sup I'P(z)I,

sup IA(z)I = sup IP(z)I.

IzI=R

IzI=I

IzI=I

On the other hand, by applying the maximum principle to the polynomial A and

then to P we have (5.5.26)

sup IA(z)I IzI=I/R

sup IP(z)I

sup IA(z)I, IzI=I

sup

IP(z)I.

zI=R

Combining (5.5.25) and (5.5.26) we obtain the desired inequality (5.5.24) which

0

completes the proof of the lemma.

LEMMA 5.5.27 (ClAssicAL BERNsTEIN INEQUALITY). Let P(z) and R be as in

Lemma 5.5.23. Then (5.5.28)

sup IP(z)I zEE*

Rk

IP(x)I,

sup xE(—IJ)

where ER is the ellipse given by x2/a2 + y2/b2

1, with

PROOF. Note first that the mapping takes the circle I + onto the interval [—1, 1] and the circle k I = R onto the boundary of the ellipse ER. Now apply (5.5.24) to the polynomial of degree 2k given by + I/i)) and the maximum principle to obtain (5.5.28). 0

Theorem 5.5.21 may now be proved by iterating (5.5.28). First, by dilation and translation, for any interval [a, b] C R and any R' > 1 there is a neighborhood VR'

of[a, bJ inC such that

(5.5.29)

sup IP(z)I

IP(x)I.

sup xE(a.b)

Without loss of generality we may assume Q

R' =

R"

= [a, b] x ... x [a, b]. Taking

and applying (5.5.29) successively for each variable, we obtain the theorem with V = (VR')". 0 The next result is that a formal power series which converges on any real line passing through the origin is convergent.

V. RINGS OF POWER SERIES AND POLYNOMIAL EQUATIONS

152

THEOREM 5.5.30. Let a(z) = a formal power series in n indeterminates with complex coefficients. Assume that the restriction of a(z) to any line passing through the origin is convergent, i.e. for every w E the unit in (tW)G converges for all t in some open sphere in R", the power series t (depending on w) containing 0. Then a(z) is a convergent power series.

PROOF. For every integer k, let ak (z) be the homogeneous polynomial of degree Za. The assumption implies that for every w E k defined by ak (z) = there exists a positive constant for all k 1. For such that Iak (w) By the above remarks, : Iak(w)I j, k I let Ejk = fw E

j

k

Since EJk is closed, so is flk EJk. Hence, by the Baire Category Theorem there

exists Jo such that flk EJOk has nonempty interior. Therefore, there is an open subset U C and a positive constant C so that Iak(cv)I Ck for all w E U 1. By homogeneity, we conclude that lak(x)t and all k Ck for all x E Q, a closed cube in with nonempty interior. We may now apply Theorem 5.5.21 with R = 2 to conclude that there exists a complex nonempty open set V C cn such that sup Iak(z)I

(2C)k,

Vk >

1.

zE V

is constant and equals &aa for Icri = k, we conclude by Cauchy's Chat for all inequalities that there exists a positive constant C' such that Iaa I multi-indices a, Ia I > 0. This proves the desired convergence of the power series a(z). 0 Since

The following are important corollaries of Theorem 5.5.30. THEOREM 5.5.31. Let u (x) be a smooth function in an open neighborhood U of 0 in R't such that the restriction of u to every line £ through 0 is real-analytic in U fl £. Then there exists an open neighborhood U' of 0 with U' C U such that the restriction of u to U' is real-analytic.

PROOF. Let a(x) be the Taylor series of u at the origin. By assumption, a(x) satisfies the hypothesis of Theorem 5.5.30, and hence is a convergent power series

in some neighborhood U' of 0, which after shrinking, may be assumed to be contained in U. Since the restriction of u to any line £ passing through the origin agrees with its Taylor series, we conclude that ulu'nc = alunc. Hence u is realanalytic in U'. 0 From Theorem 5.5.31 we easily obtain the following generalization of Proposition 5.5.19.

§5.5. ROOTS OF POLYNOMIAL EQUATIONS

153

a monic THEOREM 5.5.32. Let p(s, z) = z' + and u(s) a smooth polynomial with coefficients real-analytic in an open set U C function defined in U and satisfying p(s, u(s)) 0. Then u is real-analytic in U. PROOF.

It suffices to show that u is real-analytic in a neighborhood of every

in U. Without loss of generality, we may assume s0 = 0 and U is

point starshaped, so that every line through 0 intersects U in an open interval. Applying Proposition 5.5.19 to the restriction of u to every line passing through the origin, and then making use of Theorem 5.5.3 1, we conclude that u is real-analytic in a neighborhood of 0 in U. This completes the proof of the theorem. 0 RErbIAIU 5.5.33. In contrast to a basic property of holomorphic functions of several complex variables (see e.g. Hörmander [2, Chapter II]), a smooth function in several real variables which is separately real-analytic in each variable need

not be real-analytic, as is shown by the following example. Let u(x1, x2) = It is clear that for each XI fixed, the function x2 u(xI,x2) is real-analytic, and similarly for each x2 fixed. However, u is not real-analytic in any neighborhood of the origin. The reader should compare this observation with the result in Theorem 5.5.3 1. REMARK 5.5.34. It is possible to find a smooth function u in a neighborhood of 0 in R2 such that the restriction of u to every line through the origin is real-analytic in a neighborhood of 0 in that line, without u being real-analytic in any neighborhood of 0 in R2. Let be a smooth function defined on R such that çt(t) 0 fort 1 and 4(t) > 0 otherwise. Let u(x1, x2) = + (x2 — + (x2 + 1)2). Note that u is smooth in R2 and vanishes exactly on the line x2 = 0 and inside the circles or radius one with centers (0, 1) and (0, —1). Hence u vanishes on some

open interval containing 0 on any line through 0, but is not real-analytic in any neighborhood of the origin in R2. Hence the hypothesis of Theorem 5.5.3 1 cannot be weakened to allow the size of the neighborhood on each line to depend on that line.

The following can be viewed as a generalization of Theorem 5.5.32 and is a corollary of that result. THEOREM 5.5.35. Let R1(w, z),... , RN(W, z) be germs at 0 of holomorphic functions in Ck x with = 0, j = 1,... , N. Assume that the ideal z)) (generated by the R1(0, z)) in C(z) is of finite codimension. If u(s) = (u1(s),... ,uk(s)) are smooth functions defined in a neighborhood of 0 in Rk and satisfy the equations R1(s, u(s)) = 0, j = 1,... , N, then u1(s) is real-analytic in some neighborhood of Ofor j = 1,... , k.

I=

154

V.

RINGS OF POWER SERIES AND POLYNOMIAL EQUATIONS

PROOF. By Proposition 5.1.5, since I is of finite codimension, there exists a positive integer J such that 4 E I, j = 1, . . n, i.e.

j = 1,... ,n.

zJ = Hence we have

(5.5.36)

4=

z) +

wqHjq(w, z).

where the holomorphic functions Hjq (w, z) are determined by the relation

=

—

It follows from (5.5.36) that for (w, z)

R1(w,z)=0, j=1,... ,N,wehave

C" sufficiently small and satisfying

Zj =

(5.5.37)

This in turn implies by Lemma 5.3.23 that the following relations also hold: N1

(5.5.38)

-I

z7' +

0,

j = I,... ,N,

p=I

are convergent power series vanishing at 0. Since by assumption u satisfies R, (s, u(s)) = 0, j = 1,... , N, we conclude from the above that we also have where the N1 are positive integers and the

N1 -

u1(s)Ni

I

+

= 0,

j = 1,...

,

N,

p=I

for s real, sufficiently small. The desired result is then an immediate consequence of Theorem 5.5.32. 0

Notes for Chapter V This chapter contains mostly classical and well-known material. We shall not attempt (nor would we succeed even if we tried) to give complete and proper credit for each result.

NOTES FOR CHAPTER V

155

Most results in §5.1 are undoubtedly known in one form or another. Many results can be found in standard texts on commutative algebra such as e.g. Zariski

and Samuel [1]. Other results, such as e.g. Theorem 5.1.15, are hard to find in the literature. (The latter theorem can be found in e.g. Baouendi and Rothschild [51.) Theorem 5.1.37, in the case k = n, can be found in Eisenbud and Levine [1]. In the case = C{z}, Theorem 5.1.37 is a standard result in complex analytic geometry. The authors are grateful to Joseph Lipman for his help with the proof of this theorem in the general case given here. The material in §5.2.45.4 is classical and most results are stated without proofs. Each section contains references to standard texts, where much of this material and further results can be found. However, the algebraic implicit function theorem (Theorem 5.4.6), although well known, is not easily found in the literature, and since this result is crucial in Chapter XIII, we include a proof of it. A more general version of Theorem 5.5.31 is due to Abhyankar and Moh [1). The main results of Section 5.5, Theorems 5.5.32 and 5.5.35, are due to Malgrange

[1), although the proofs given here are different. These, and related results in a more algebraic setting, can also be found in Artin [1] and [2].

CHAPTER VI

GEOMETRY OF ANALYTIC DISCS

Summary In Chapter VI, we study the set of analytic discs attached to a generic submanifold M of CN. We begin with a review of the classical theory of the Hubert and Poisson transforms on the unit circle St. In particular, we provide a proof of the boundedness of the Hubert transform on the Holder space Cka (S'). In §6.2, we define the notion of an analytic disc attached to M, and introduce the Bishop equation for the construction of such discs. We also prove, by means of the approximation theorem of Chapter H, that any CR function on M extends holomorphically into any open set in CN which is a union of small discs attached to M. The concept of an infinite dimensional submanifold of a Banach space is reviewed in §6.3. Applications of the implicit function theorem, including the construction of the tangent space to such a submanifold, are also given. The chapter concludes with the important result that the set of small analytic discs (of class to the generic submanifold M of forms a smooth submanifold of the Banach space of all analytic discs with values in The results of §6.3 are used to give a description of the tangent space of this submanifold at a constant disc, which is then applied to prove the existence of a unique solution of the Bishop equation.

The results of this chapter are used to study the holomorphic extension of CR functions from generic submanifolds in Chapter VII.

§6J. Hilbert and Poisson Transforms on the Unit Circle Let its closure. By an analytic disc in C C be the open unit disc and CN we shall mean a continuous mapping from to CN whose restriction to is

holomorphic. As a preliminary to studying analytic discs in N > 1, in this section we shall review some classical results for discs in C. In the sequel, we shall use in particular the boundedness of the Hilbert transform (Theorem 6.1.20). Let u(e'°) be a continuous, real valued function defined on the unit circle S' and write = S°. We define the function Cu, the Cauchy transform of u, in a by (6.1.1)

Cu(z):=—!-f 156

157

§6.1. HILBERT AND POISSON TRANSFORMS ON THE UNIT CIRCLE

THEOREM 6.1.2. If u (e10) is continuous and real-valued on 51, then its Cauchy The real and imaginary parts of transform Cu(z) is a holomorphic function on Cu(z) are given by

Cu(z) = Pu(z,

(6.1.3)

where Pu(z,

+ iTu(z,

(6.1.4)

u(C)Re

=

(6.1.5)

given by

are harmonic functions in

and Tu(z,

-

f

2ir —

u(e'°)

o

ZZ

Ie' —zi

2d0,

(C +z) dC 2iri

C

111=1

,9ze u(e).

1

2ir j0

—19—19 ze —zi

do.

We observe that = idO. The formulas (6.1.4) and (6.1.5) are immediate identities. Since all the integrands and their derivatives in z and are all absolutely integrable for z E the fact that Cu (z) is holomorphic and that 'Pu(z, and Tu(z, are harmonic, follows by differentiating under the integrals PROOF.

in(6.1.1),(6.1.4)and(6.1.5). The function Pu (z. defined by (6.1.4) is the Poisson transform of u. We shall show that the Poisson transform of a continuous function extends continuously to PROPOSITION 6.1.6. ff0

r < I and u a continuous, real valued function on

SI,

(6.1.7)

1

Pu(re", re1') = — I 2ir j0

\l+r

u(e'9) 1

Furthermore, Pu extends continuously to (6.1.8)

lim Pu(z,

— r2 2

—2rcos(t—O)j

do.

and

=

PROOF. The formula (6.1.7) follows immediately from (6.1.4) by setting z = reit. To prove the continuity and (6.1.8) we observe that 1

1—r2

2ir

l+r2—2rcosO

P(r,O)= —___________

VI. GEOMETRY OF ANALYTIC DISCS

158

r <

satisfies the following for 0

1:

(i) P(r,9) >0, (ii) P(r, 9)d9 = 1, (iii) P(r, 9) converges uniformly to 0 as r

I

on any interval of the form

[S,2ir —8]for0 <8 By (ii),

'Pu(re", re")

(6.1.9)

—

u(eito)

1 = — 2,rj0I

(u(e'°) — u(e"o))

l—r2

l+r —2rcos(t—9) 2

dO.

To complete the proof of the proposition, we break up the integral on the right hand side of (6.1.9) into the regions (I) 19 — tol < 28, (II) I2jr — (9 — t0)) <28, and (HI) 245 — 28. To estimate the integral on (I) and (II), we 19 — tol use (1), (ii), and the continuity of u; for the integral over (III), we use (iii) and let

0

t—÷t0andr--+1.

In general, in contrast to the situation for Pu, the function Tu does not extend if u is assumed only to be continuous. For this, we need to continuously to assume further regularity on u. Recall that if 0 < a < 1, the Holder spaces Ca(SI) are defined as follows: u E C such that (6.1.10)

—

for all 9 E [0, 2,r]. Then

u(e'°)I

becomes a Banach space with norm —

V

PROPOSITION 6.1.12. If u E Ca(Sl)for some a, 0
1,

IIuIIa:=suplu(eie )I+sup

(6.1.11)

/

V

then Tu extends

continuously to z

= reit so that (6.1.5) becomes

(6.1.13) Tu(re", re_it) =

!

rsifl(t 9) 1+r 2rcos(t

= JoI where

K(r, 9) :=

Since

—

9)

K(r,t—9)u(e'°)dO,

the kernel K(r, 9) satisfies K(r, —9) =

—K(r, 0) and is periodic of period 2,r, we obtain

(6.1.14)

Tu(re', re_it)

=

f

u(e'°)dO

K(r,

—

§6.1. HILBERT AND POISSON TRANSFORMS ON ThE UNIT CIRCLE

K(r, 0)1 C/I0I

159

the integrand in (6.1.14) is continuous in r and r, except at0 = 0, and bounded by which is integrable, the continuity is proved. 0 Since

and u

It follows from Proposition 6.1.12 that if u E Cu, then the restriction Tu of Tu to S' is a continuous function, called the Hubert transform of u. (Some authors call Tu the conjugate frnction of u.) For u E it follows from (6.1.14) that we have

(6.1.15)

Tu(e") =

2n

sinO

f

—

1—cos8

o

dO

or (6.1.16)

Tu(e") =

f

cot(9/2)dO.

—

We also have 1

(6117)

Tu(e") = tim — E-.o2fl Je

=

f

—

u(eit)) cot(0/2)dO,

since

(6.1.18)

urn I

cot(O/2)dO

= 0.

We shall write

Tu(e't) =

u(e )cot

2

do,

where PV stands for principal value. 6.1.20 (PRlwa.ov's ff0 T is a bounded operator from Ca(SI) into itself PROOF. Dropping PV for convenience, we have

(6.1.21)

Tu(e') =


1,

the Hubert transform

VI. GEOMETRY OF ANALYTIC DISCS

160

It suffices to assume 8 > 0. By using (6.1.18), for any small positive 8, we may rewrite the right hand side of (6.1.21) as (J1 — J2)/2ir, where ,38

J1 :=

(6.1.22)

+

+

J0

—

2

and

J2 :=

(6.1.23)

j +

o

+J

j

— u(e")]cot—dO. 2

which gives Note that the integrand in J1 is bounded in absolute value by and the first for the first integral in J1. The second integral in the estimate 0 (8a) two in J2 are similarly estimated. After a change of variables, the third integral in

ii becomes p2,r—28

j

(6.1.24)

do.

—

2

Subtracting the third integral in J2 from the integral in (6.1.24), we obtain

(6.1.25)

—

Tu(e") = O(c5a) +

2,r

+ '2),

where

:=

(6.1.26)

f

—

u(e")) [cot

0+8 —

cot

do,

and

(6.1.27)

'2

:= J

We estimate

by writing

(6.1.28)

(u(e(t — 0))

Ii:

(u(e")

= —

0 —

+ u(e"))[cot

+6 2

do.

For the first we obtain 8 —

.;-

cot

OdO
After translation, a similar estimate holds for the other integral in only to estimate 12. We write (6.1.29)

12 = (u(e")

—

cot

o

+6 2

do.

It remains

§6.1. HILBERT AND POISSON TRANSFORMS ON THE UNIT CIRCLE

161

Using the cancellation property (6.1.18) we obtain 12 = —(u(e1') —

J

e cot —dO, 2

which proves 12 = O(&), and completes the proof of Theorem 6.1.20.

0

If u is defined on S'. we define differentiation by associating to u the periodic

function on the line ü(O) = u(e'°). We write u'(e'°) = dii/dO(O). For k a nonnegative integer and 0 < a < 1 we define the more general Holder spaces

C'" (51) as the space of functions u

Ck (S'), k times continuously differentiable is belongs to C"(S1). functions on S1, for which the kth derivative

a Banach space with the norm (6.1.30)

j We have the following corollary of Theorem 6.1.20.


COROLLARY 6.1.3 1. Fork a nonnegative integer and 0 into itself transform T is a bounded linear operator from

1,

the Hubert

It follows from equation (6.1.17), by differentiating under the integral sign, that = for j = 1,... , k. It suffices to then apply Proposition PROOF.

0

6.1.20.

If u is a complex-valued function defined on S1. we define Pu, Tu, and Tu by the formulas (6.1.4), (6.1.5) and (6.1.18) respectively. It is often convenient to use the Fourier expansion of u to define the operators C, 7', T, and the Hilbert transform T. If u is continuous on 51, we write

(6.1.32)

with

u(e18) =

=

_!_

f

u (e'0) and its Fourier series is understood in the sense of

distributions.

PRoPosmoN 6.1.33. For u(e'°) = (6.1.34)

Cu(z)

(6.1.35)

Pu(re'°, re'°)

we have

=

VI. GEOMETRY OF ANALYTIC DISCS

162

Tu(re'°, re'°) = —i

(6.1.36)

where sgn(O) = 0 and

sgn(n) = n/In for a nonzero integer n, and

(6.1.37)

Tu(e'°) =

I

—i

The proof of Proposition 6.1.33 is left to the reader.

PRoPosmoN 6.1.38. The following hold for u E C"(S1), 0
(ii) PTu = Tu. (iii) u is the restriction of a holomorphic function in

continuous on

if and

only if Tu + iu isa constant function on S'. PROoF. The proof of (1) follows immediately from (6.1.37). The proof of (ii) follows from (6.1.35), (6.1.36), and (6.1.37). For (iii), one can easily check, by Proposition 6.1.33 again that Tu = —iu + c, with c constant, if and only if the Fourier coefficients u,, of u vanish for n <0. The latter condition is equivalent to Cu = 2Pu — uo. Since Cu is holomorphic in and continuous up to the boundary, by Propositions 6.1.6 and 6.1.12, the same is true of Pu, whose boundary

value is u. This proves that u extends holomorphically if Tu + iu is constant. Conversely, if u extends holomorphically, by the Cauchy integral formula, we have

=

0,

n=

0,

1, ..., which implies that u1 = 0 for all n <0 and,

hence, proves the converse, by the above remarks.

0

§6.2. Analytic Discs Attached to a Generic Submanifold In this section, we shall define the notion of an analytic disc attached to a generic

submanifold M and explore the relationship of such discs to the holomorphic extension of CR functions on M. As in §6.1, we denote by the open unit disc < 1) in C, by S1 its DEfiNITION 6.2.1. An analytic disc in CN is a continuous mapping A: -+ An analytic disc A is said to be attached to a

CN which is holomorphic in

subsetM C C CN be a generic submanifoid. We denote by .4(M) the set of all

analytic discs attached to M. We shall show later in this chapter that the subset

§6.2. ANALYTIC DISCS ATTACHED TO A GENERIC SUBMANIFOLD

163

has a natural structure as an infinite dimensional manifold, and we shall calculate its tangent space. The interest in analytic discs in the study of holomorphic extension of CR functions is indicated by the following proposition, which shows that locally every CR function on M has a natural extension to the union of the images of all sufficiently small analytic discs attached to M.

of A(M) consisting of those analytic discs that are also

PRoIosmoN6.2.2. LetM an of P0 such that for every C' CR function I on U there exists a continuous function F defined on W = UAEAW A(A) with the property that, for any A E A(U), the following hold.

(i) F o A is holomorphic in A. (ii) The restriction of F to U coincides with f. (iii) F is holonzorphic in W°, the interior of W. PROOF. If U is sufficiently small, by the Approximation Theorem (Theorem

2.4.1), for every CR function f in U there exists a sequence of holomorphic polynomials

whose uniform limit in U is f. We note that U C W =

UAEA(U) A(A), sinceanypointp E

p. We define F as follows. For any Z E W, choose A E .4(U) such that Z is in A(A). o A converges uniformly to f o A on S', by the maximum principle for Since o A is a Cauchy holomorphic functions in the unit disc, we conclude that and hence converges to a function FA E C°(4&) which is sequence in We claim that F is holomorphic in A. We put F(Z) = FA(C), where Z = independent of the choices of A and Indeed, the previous argument shows that isaCauchy sequence for the uniform norm on W,and F(Z) = l'his proves the claim. Using a similar argument, the reader can also check that the function F is defined independently of the choice of the approximating sequence Pv The properties (i) and (ii) follow from the construction. The holomorphy of F in W° follows from the fact that a uniform limit of holomorphic functions is 0 holomorphic. For the remainder of this section we study analytic discs attached to a generic submanifold M in terms of the regular coordinates of § 1.3. Suppose that M is of codimension d and CR dimension n in C", and let P0 E M. By Proposition 1.3.6 and we may locally choose holomorphic coordinates (z, w) near P0 , with z E 4(z, s) so that, near M is given by Im w = wE = 0. By writing w = s + it with is smooth and satisfies = 0 and

VI. GEOMETRY OF ANALYTIC DISCS

164

s, t

E

we may write the defining equations in the form

(6.2.3) Let A:

t

CN be an analytic disc. In the local coordinates above, with

=

=

E

A,

=

If A is sufficiently small, it is attached to M if and only if

=

(6.2.4)

We shall show that given any analytic disc

A —+ C's, which is sufficiently small (in a sense to be made precise), there is an analytic disc = attached to M, i.e. satisfying (6.2.4). Let T be the Hilbert transform defined in §6.1, and T the operator defined by E Ca(SI), forsomeO
(6.2.5)

= Tu(e'°) —

:= Tu(e'°) —

CO),

last equality following from Proposition 6.1.38. Note that 7'0 = T. The following properties are easily checked by using the results of §6.1. the

= 0,

(6.2.6)

=

(6.2.7)

E S',

—u +

In particular, we have

(6.2.8)

=

—u +u(C°),

ES'.

Using (6.1.37) and (6.2.5) we also have for, n E Z,

(6.2.9)

=

—

I — I.

PROPOSITION 6.2.10. For every

the mod jfied Hubert transform info itself

-

ifn > 0; — ifn <0.

E A, nonnegative integerk,

and0
is a bounded linear operator from

1,

(S')

PROOF. The boundedness follows from the definition (6.2.5), Corollary 6.1.31 and Proposition 6.1.6. 0

165

A GENERIC SUBMANIFOLD

ANALYTIC

We shall often consider the modified Hubert transform T,, obtained by taking

=

1

e S'. The following formula, which holds for every

CU(Sl), is easily proved from (6.1.5), (6.2.5), and the fact that purely imaginary for E S'.

E

S' and u E

+

—

is

17

(6.2.11)

T,u(17)

= PV

I

f

— q) (

—

fl(

—

We have the following important result. THEOREM 6.2.12. Let M be a smooth generic submanifold through 0, given by (6.2.3), and let 0 < a < 1. Then there exists e > 0 such that for any analytic disc —+ C" in with IIZIII.a <E, andfor any c E R" with id <E, there z: attached of class exists a unique (small) analytic disc =

to M such that Re w( 1) =

c.

The proof of Theorem 6.2.12 will be given in §6.5, by using the Banach space methods developed in §6.3 and §6.4. We close this section with some comments and motivation for this proof. In the hypotheses of Theorem 6.2.12 we are given a small analytic disc with values in C"; the conclusion states that there is a unique small analytic disc A A is attached to M and Re w(1) = c. of the form with valued in That is, we need to find a (unique and small) analytic disc for e satisfying s(l) = c = + and t(C) = S'. Since s+it is holomorphic, so is —t+is. Therefore, = for some a E Rd and E S', by Proposition 6.1.28 (iii). Since (T,u)(1) = 0, we must have a = c. We consider the following equation for

(6.2.13)

ES'.

=

We shall refer to (6.2.13) as the mod jfied Bishop equation for M at 0. We note here is a function valued in Rd , and T, acts on each component of the vectorthat If the modified Bishop equation (6.2.13) has valued function then we may define on S' by a unique solution

(6.2.14)

:=

ES'.

By (6.2.13) and Proposition 6.1.38, we have = + c', where c' is a is holomorphic in A. We put = constant. Hence + + is an analytic disc attached to M. By (6.2.13) and so that = satisfying (6.2.13) the definition of T1, s(1) = c as desired. The existence of s will be an easy consequence of the Banach space results proved in §6.4 and 6.5.

VI. GEOMETRY OF ANALYTIC DISCS

166

§6.3. Submanifolds of a Banach Space In this section we will give some background material on infinite dimensional submanifolds of a real Banach space, which will be used to study the set of analytic discs attached to a generic submanifold in Ce". IfU c X isanopensubsetofaBanachspace X andu: U -÷ Y isamapping of class Ck, k 1, into another Banach space Y, we shall denote by u'(x) the Frechet derivative of u at x U. Recall in particular that u'(x): X -÷ Y is a continuous linear mapping. (See e.g. Abramson, Marsden, and Ratiu [1] for basic results concerning calculus on Banach spaces.) If a: X —÷ Y isa continuous linear mapping, we shall say that a has a right inverse if there exists a continuous linear mapping Y —÷ X such that a o fi = Jy, where Jy is the identity mapping on Y. Observe that if a has a right inverse, then kera and im are closed subspaces of X and X = kera im $. Ifa and are as above, we shall also say that a is a left inverse Throughout this section, E and F will denote fixed (real) Banach spaces. We now define submanifolds of E modeled on F.

DEIINmON 6.3.1. A subset B C E is a Banach subman(fold modeled on F of class Ck if for every Yo E B there exist an open neighborhood V of Yo in E, an open neighborhood U of 0 in F, and a mapping U -+ E of class Ck satisfying the following.

(i) The mapping U -÷ is a homeomorphism with the topology induced by E.

is equipped

(ii)

(iii) The linear mapping 4"(O): F -+ E has a left inverse a: E —* F. As noted above, condition (iii) in the definition implies that E has the Banach space direct sum decomposition (6.3.2)

E=

im

® kera.

As in the finite dimensional case, a submanifold of E can also be locally characterized by defining equations in E. More precisely, we have the following. PROPOSITION 6.3.3. A subset B C E is a Banach submanifold modeled on F of class (fond only (ffor every Yo B there exist an open neighborhood V of Yo in E, a real Banach space G, and a mapping R: V -÷ G of class Ck such that (6.3.4)

B fl V

=

the linear mapping R'(yo): E -+ (6.3.5)

{y

G

V: R(y) = 0},

has a right inverse fi: G -+ E, and

ker R'(yo) is isomorphic to F.

§6.3. SUBMANIFOLDS OF A BANACH SPACE

167

We shall call the mapping of Definition 6.3.1 a local parametrization of B near Yo• We shall say that R given by Proposition 6.3.3 is a local defining function has a left inverse if of B near Yo• Note that in the finite dimensional case, and only if it is injective. Similarly, R'(yo) has a right inverse if and only if it is surjective. As in the finite dimensional case, the proof of Proposition 6.3.3, which will be given below, is based on the Implicit Function Theorem and the Inverse Mapping

Theorem. For the convenience of the reader, we shall state here without proof these theorems in the case of Banach spaces in the form in which they will be used. Let X and Y be Banach spaces

6.3.6 (INVERSE

and U C X an open neighborhood of x0. If U: U -+ Y is a mapping of class Ck, k 1, such that U'(XO) is an isomorphism from X to Y, then u has a local inverse near x0. That is, there exist open neighborhoods U0 and V0 of xO and u(xo) in X and Y respectively and a mapping v: V0 -+ X of class such that (6.3.7)

u(v(y)) = y, v(u(x)) = x,

Vx E U0, y E V0.

If X, Y, and Z are Banach spaces, U and V open subsets of X and Y containing

xoandyorespectively,andu: U xV —÷ ZofclassC*,k 1,thenthecontinuous linear mapping du/ax(xo, Yo): X —÷ Z is defined by au/ax(xo, yo)(x) = u'(xo, yo)(x, 0). Similar notation is used for au/ay(xo, yo). THEOREM 6.3.8 (IMPLICIT FUNCTION THEOREM). Let X, Y, and Z be Banach

spaces, U and V open neighborhoods of X0 and Yo in X and Y, and u: U x V —÷ Z of class C', k 1, with u(xo, Yo) = 0. Suppose that 9u/9y(xo, Yo) is an isomorphism from Y onto Z. Then there is a neighborhood Uo C U of x0 and a unique map v: -+ V of class C' satisfying (6.3.9)

u(x, v(x)) =

0 Yx E U0.

PROOF OF PROPOSITION 6.3.3. Suppose first that B C E is a Banach subman-

ifold modeled on F, and let yo B. We shall construct a local defining function R. Since by assumption has a left inverse a, we write the decomposition (6.3.2) in the form (6.3.10)

E = im

kera =

F0

G,

with F0 and G closed Banach subspaces of E. For y E E we write the unique decomposition y = y' + with y1 F0, and y2 E G. Similarly, for x E U we

VI. GEOMETRY OF ANALYTIC DISCS

168

with is an = E F0 and E G. Then write + isomorphism from F onto F0. By the Inverse Function Theorem (Theorem 6.3.6) of exists and is of class U —+ F0, the local inverse applied to the map Ck. For y sufficiently close to ye,, y = Yt + y2 as above, we define the G valued mapping

R(y) :=

(6.3.11)

If we write Yo =

Y2 —

then we have = The reader can easily check, by using (ii) of yo.2 — Definition 6.3.1 that (6.3.5) holds, possibly after shrinking V around and that R satisfies the rest of the conditions of Proposition 6.3.3. Yo.i

+ Yo.2' i.e. Yo.i = øi(O) and Y02

=

R(yo) =

0.

It remains to prove the converse. We assume now that R is a defining function for B around yc, satisfying the the conditions of Proposition 6.3.3. We shall construct a map as in Definition 6.3.1. It follows from the assumptions that we have the Banach space splitting

(6.3.12)

=

E

P that is to G. Again for y E E, we write the direct sum decomposition in the form y = + Y2' y' E F,

Similarly we write yo = Yo.i + yo.2. The mapping Y2) R(yi, Y2) = R(y1 + y2)' defined in an open neighborhood of (yo.i, yo.2) in F x G and valued in G, is of class C' and satisfies k(yo.1, yo.2) = 0, and the linear mapping y2 E O.

-

OR

C

—+ G

an isomorphism. Therefore we can make use of Theorem 6.3.8 and conclude that there is a unique function x of class Ck defined in a neighborhood of yo. i in P and valued in O satisfying is

x(yi))

R(yi + x(yi))

0,

Differentiating this identity and evaluating at y' =

X(yo.i) = yo.

Y0.2•

we obtain

0

0= —(yo.I,yo.2)+ —(yo.I,yo.2)X (Yo.i). 0Yi

0Y2

Since OR/0y1(yo.1, yo.2) = 0 (by the definition of P) and invertible, it follows that (6.3.13)

x'(yo.i) = 0.

yo.2) is

§6.3. SUBMANIFOLDS OF A BANACH SPACE

169

Recall that by assumption fr = ker R'(yo) is isomorphic to F. Let i: F -+ F be

such an isomorphism and define

0(x) := i(x) + yo.i + x(i(x) + yo.i).

(6.3.14)

It follows from the properties of x that 4) is of class Ck, defined in an open 0. The reader neighborhood of 0 E F, valued in E, and satisfies R(Ø(x)) can now easily check that 4) satisfies all the conditions of Definition 6.3.1. This completes the proof of the proposition. 0 6.3.15.

(i) Let 4) and R be as in Definition 6.3.1 and Proposition 6.3.3 respectively. Since 4)'(O) has a left inverse a, i.e. a o 4)'(O) = 'F' follows that for x close to 0 in F, the mapping a o 4)'(x) is an isomorphism of F onto itself, and hence, if a1 = (a o 4)'(x)y' o a, we have

a1 o4)'(x) = 'F'

(6.3.16)

4)'(x) has a left inverse. A similar argument shows that R'(y) o is an isomorphism of G for y close to Yo in E, and hence R'(y) has a right i.e.

inverse

R'(y) o

(6.3.17)

=

=

(R'(y) o (ii) It follows from (6.3.14) that B near Yo is a graph of a function over a neighborhood of 0 in F with values in O, where and O are as in with

o

(6.3.12).

In order to define and calculate the tangent space to a Banach submanifold of E, it is useful to show that a local parametrization is unique modulo a local diffeomorphism of F. We have the following.

PRoPosmoN 6.3.18. Let 4): U C F -+ E and U C F —+ E be two local parametrizations of a Banach manifold B modeled on F (both satisfying the conditions of Definition 6.3.1.) Then there exists an invertible Ck mapping v defined in a neighborhood 010 in F and valued in F, with v(0) =0, such that (6.3.19)

4)(x)

çb(v(x))

for all x in a sufficiently small neighborhood of 0 in F. PROOF. As in the proof of Proposition 6.3.3 we consider the direct sum decomposition given by (6.3.10). Consider the mapping J(x, Y2) := 4)(x) + Y2 defined

170

VI. GEOMETRY OF ANALYTIC DISCS

in U x G and valued in E. It follows immediately from the assumptions on that J'(O, 0) is an isomorphism from F x G onto E. Hence by the Inverse Mapping Theorem (Theorem 6.3.6), 1 admits a local inverse of class C' defined in an open in E. Let be the mapping defined in a neighborhood neighborhood of yo = of Yo in E by

fr(y) := pruf'(y)

(6.3.20)

where pru denotes the projection on U from U x G. It immediately follows that i.e. *(Yo) = 0 and is a local left inverse of class Ck

VxEF closetoO.

(6.3.21)

We also have that

is a local right inverse of

on B, i.e.

y, Vy E B close to yo.

(6.3.22)

We claim that there is an open neighborhood U1 C U fl U of 0 in F such that (6.3.23)

=

o

o

(x), Vx

U1.

B and is close to Indeed, for x E U sufficiently close to 0, we have Thus, the claim (6.3.23) follows from (6.3.22). Define v(x) = for U1, where U1 C U is a sufficiently small open neighborhood of 0. Then v x is of class C' valued in F, and v(0) = 0. It follows from (6.3.23) that (6.3.19) holds for x U1. To show that v is a local diffeomorphism of F near 0, we repeat the argument above, interchanging and We find a mapping defined in a neighborhood of 0 in F and satisfying (6.3.24)

in a small neighborhood of 0 in F. Differentiating (6.3.19) and (6.3.24) and using the chain rule, we obtain (6.3.25)

=

=

Substituting the first equality of (6.3.25) into the second (and conversely) and multiplying the result by the left inverse of (or we obtain v'(O)IY(O) = ii'(O)v'(O)

=

Ill

§6.3. SUBMANIFOLDS OF A BANACH SPACE

which shows that v'(O) is an isomorphism on F and hence v is a local diffeomor0 phism of F near 0. This completes the proof of the proposition.

We shall define the notion of tangent space to a Banach submanifold of E. If the subset of E obtained B c E is a subset and y B, we shall denote by as follows. For any curve y: (—€, e) -.÷ E, e > 0, of class C', we let y(t) = dy/dt(t). The set C E is defined as the set of all vectors y(O) such that is a cone, i.e. invariant by y(—E, E) C B and y(O) = y. We observe that multiplication by real numbers. When B is a Banach submanifold of E, we have the following description of TB in terms of local parametrizations and defining functions of B (see Definition 6.3.1 and Proposition 6.3.3).

PRoPosrrIoN 6.3.26. Let B c E be a Banach

modeled on F. Then for every y E B, is a closed subspace of E isomorphic to F. Moreover, if 4 isa local parametri zation of B near Yo and R is a local defining function of B near then for all y E B sufficiently close to yo, (6.3.27)

TB = im

= ker R'(y),

wherex E Fandy =qS(x). For B as in the proposition we shall call

the tangent space of B at y.

PRooF OF PRoPosmoN 6.3.26. Observe that the first part of the proposition

follows from (6.3.27). To prove (6.3.27), it suffices to take x = 0 and y = yo. (Indeed, for a point x near 0 and y near it is easy to reduce to this case by translation and making use of Remark 6.3.15.) We first show (6.3.28)

T,bB = im

For this, let x E F. Then the curve y(t) = /.(tx) is of class Ck with image in B (for t sufficiently small). Since y(0) = Yo and y(0) = we obtain the inclusion im C To prove the opposite inclusion, we shall use the map defined by (6.3.20) in the proof of Proposition 6.3.18. Lety E and y (t) a curve with image contained in

Bsuchthaty(0) = y. Definethecurveo(t)withvaluesin Fbyo(:)= We have, by the properties (6.3.21) and (6.3.22) of 4(a(t)) y(t). Differentiating the latter, we obtain which completes the proof of (6.3.28).

a(t) and

= y(0) =

y,

To prove the second equality in (6.3.27), we note that by (6.3.28) (or by Propo-

sition 6.3.18) the subspace im is in fact independent of the choice of the parametrization Hence it suffices to assume that R is given and that is obtained from R by (6.3.14). Differentiating (6.3.14), taking x = 0, and using

VI. GEOMETRY OF ANALYTIC DISCS

172

(6.3.13), we obtain that = e, and hence its image is F = ker R'(yo) which yields the desired equality. The proof of Proposition 6.3.26 is complete. 0

of

The following gives a linear parametrization of the tangent space T%.13 in terms for y sufficiently close to yo.

PRoPosmoN 6.3.29. Let B C E be a Banach modeled on F and a local defining function of B near satisfying the properties of Proposition 6.3.3. Then for all y 13 sufficiently close toYo R

(6.3.30) where

= =

(6.3.31)

o

(R'(y) o $)'.

(JE

—

o

In addition, the mapping

z i-+ z —

E

TB

is infective.

Recall, by Remark 6.3.15, that the mapping G -÷ E is well-defined for y close to Yo and is a right inverse of the linear mapping R'(y): E —+ G. Note that o R'(y) = 'E — fi o R'(yo) is the projection of the mapping 'E — for y = E onto F = kerR'(yo).

PRooF OF PR0P0sm0N 6.3.29. For y E E close to yo, let E -+ E be the continuous linear operator R'(y). Recall that is a right inverse of = JE — R'(y). We claim that is a projection onto ker R'(y). (The latter coincides with for y B by Proposition 6.3.18.) Indeed, since o R'(y)) o o R'(y)) = a R'(y), it easily follows that o = Ps., i.e. is a projection. Since = 0 the claim is proved. = z if z ker R'(y), and R'(y) a To complete the proof, we use the direct sum decomposition (6.3.12). We first show that (6.3.32)

We observe that by using the expression for we have G C ker since Also, since fib. is a right inverse for R'(y), we have the direct sum G = im decomposition E = ker R'(y) Im = ker R'(y) G. Hence if z E ker F,, we write z = zi + Z2 with Zi E kerR'(y) and Z2 Since = zi and = = 0, it follows that Zi = 0, i.e. z E O, which completes the proof of (6.3.32).

By the decomposition (6.3.12), if z E we may write z = + with and Z2 E C. Hence, by using (6.3.32), for z kerR'(y), we have

§6.3. SUBMANIFOLDS OF A BANACH SPACE

173

B close to Yo• This and therefore T,.B = for y completes the proof of (6.3.30). The injectivity of the mapping (6.3.31) follows from (6.3.32) and the decomposition (6.3.12).

z=

=

We now introduce the derivative of mappings between Banach submanifolds and then give a version of the rank theorem when the target manifold is finite dimensional. Since the definition is very similar to that in the case of finite dimensional manifolds, we will be very brief. Let E1, E2, F1, F2 be real Banach spaces and assume that C E1 and C E2 are Banach submanifolds of class Ck, k 1, modeled on F1 and F2 respectively. A mapping f: is said to beofClasSCk U CF1 —+ parametrization of as in Definition 6.3.1. For such a mapping f and any Yo Bi we define a continuous linear mapping f'(yo): —÷ Tf(%b)132 as follows. If we claim that yi: (—E, E) —+ E1 is of class C' with values in 13, and yi(0) = the map t i-+ f(y,(t)) E E2 is of class C' in a neighborhood of 0. For this, we let (U, be a local parametrization of 13, near as in Definition 6.3.1 and given by (6.3.20). It follows from (6.3.22) that f(yi (1)) (1 (1)) for a close toO. The claim follows since f o and are of class Ck. Observe that the curve t >'2(t) = f(y1(t)) is valued in B2. Put z = We set E (6.3.33)

=

f'(yo)z =

The reader can easily check that f'(yo) is a continuous linear mapping from to Tf0.0)82.

When the image of the mapping f'(yo):

i.e. dim im f'(yo) = r
the

is finite dimensional

—+

rank of the mapping f at yo.

THEOREM 6.3.34 (Ti-re RANK THEOREM). Let

f:

-+

B2

be of class Ck as

above, and let r be a positive integer such that the rank off does not exceed r at any point. Iffor some yo E has rank r at Yo. then there exists a neighborhood U of Yo E B, such that f(U) is a subnianiold of 132 of class Ck and dimension r. Furthermore, if U is sufficiently small, then the set

f

(6.3.35)

(y E U:

f(y) =

is a Banach submanifold of 13, modeled on ker f'(yo).

Since the statement of the theorem is local, by taking local parametrizations of B, and B2, it suffices to assume that B, = U, where U is an open subset of a Banach space F,. We may also assume that B2 = F2, another Banach space, and that yo = 0 and = 0. The subspace F21 of F2 given by F2, = f'(O)F, PROOF.

VI. GEOMETRY OF ANALYTIC DISCS

174

is of dimension r by assumption. By the Hahn-Banach Theorem, we may find a closed subspace F22 of F2 such that

F2 =

(6.3.36)

F21

F22.

We may now write the corresponding decomposition for f, i.e

f(x) =

F22, x E U.

+ f2(x) E F21

By assumption, we have f(0)F1

=

=

F21.

Let F11 = kerf(0),

and

note that F1/F11 is isomorphic to the finite dimensional space F21. Hence we can find a closed subspace F12 of F1 so that we have the direct sum decomposition

=

(6.3.37)

Forx E

put f =

we write x = /1 + /2. Since

x1

F11

+ x2 E F11 e

(6.3.38)

F12. F12.

F12 -÷

is an isomorphism, we

Let J(x1, x2) =

f(xi

+ x2), and

F21

function theorem (Theorem 6.3.8) find a (unique) function g(t, x2) defined in a neighborhood of 0 in F21 x F12 satisfying may apply the implicit

to

fi(g(:,x2),x2) — t

(6.3.39)

0, g(0,0) = 0.

We claim that the function

(t, x2) i-* f2(g(t, x2), x2) E

(6.3.40)

is independent of x2. Assuming this claim, we see that, after shrinking U if necessaiy, the image f(U) is parametrized by V t i-+ t + a(t) E F21 +

where V isaneighborhoodof0in F21 anda(t) = f2(g(t,0),0). It remains to prove the claim that the function (6.3.40) is independent of x2. We first observe that since the rank off is maximal at 0 and that f'(O) F1 = f11(0) F1 = F21, it follows that (6.3.41)

for

kerf1'(x) C

x in a neighborhood of 0 in F.

function

(6.3.40)

with respect to X2

C F1

Differentiating the identity (6.3.39) and the will then yield the desired independence of

§6.3. SUBMANIFOLDS OF A BANACH SPACE

175

(6.3.40) with respect to x2. The proof of the first statement in Theorem 6.3.34 is now complete. To prove the second statement, using the first part of the theorem, it suffices to assume that = U, an open subset of F1, and 82 = R'. We may also assume yo = 0 and I (Yo) = 0. Since f'(O): F1 -÷ R' is suijective, and hence has a right continuous inverse, it suffices to apply Proposition 6.3.3 to conclude that the modeled on ker f'(O). This set given by (6.3.35) is a Banach submanifold of completes the proof of Theorem 6.3.34. We close this section with another application of the implicit function theorem (l'heorem 6.3.8) for mappings into finite dimensional manifolds.

6.3.42. Let f: —÷ B2 be of class Ck as in Theorem 6.3.34, and assume that 82 is a finite dimensional man (fokL Suppose that I is a submersion ata point yo E B1, i.e. the rank off at yo equals the dimension of 82. Then there is a neighborhood U of in B2 and a Ck mapping g: U —* B1 such that U, and = yo. Moreover, g(U) is a subman (fold of f(g(z)) = z, for all z of dimension r of class Ck. PROOF. Since the statement of the theorem is local, we may assume, without loss of generality, that = U, an open subset of a Banach space F1 and 82 = R'. -+ R' is surjective, by the finite dimensional ity of R' there exists Since f'(yo):

a continuous linear right inverse of f'(yo). i.e. there exists a map R" -+ with f'(yo) o = Iw. As in the proof of Proposition 6.3.3 we may write

=

(6.3.43) Now

JR'

—+

F12

F1 we write y =

im fi =

Fl2.

is an isomorphism whose inverse is f'(yo)1F12. For y + y2 with y' E F11 and Y2 F12. We introduce the

function f(yl, Y2) = f(yI + y2).

We

have !(yoi, Yo2) = f(Yo) = Zo, and

-÷

8f/8y2(yol, y02): F12 JR' is an isomorphism. Therefore, by Theorem 6.3.8 we may find a function h of class Ck defined in a neighborhood of (yoi. Zo) in F11 x JR'

(6.3.44)

f(yi,

z))

—z

0, Y(y, z) E

Zo) = y02.

To prove the first statement of the theorem it suffices to take g(z) = Yoi +h(yoi, z).

The second statement then follows from the first part of Theorem 6.3.34 since g is of rank r. This completes the proof of Theorem 6.3.42. 0

VI. GEOMETRY OF ANALYTIC DISCS

176

§6.4. Mappings of the Banach Space In order to apply the theory of §6.3 to describe the set of analytic discs attached to a generic submanifold of CN, we shall prove a result concerning the composition

of analytic discs with smooth mappings. We shall write here C' .a (5', RP) for [CIa (S' , R)]P for any positive integer p. The main result of this section is the following. THEOREM 6.4.1.

-+

Let g: U C

Re

be of class

with g(O) =

0.

Then

the mapping given by

C'"(S', defined for u

u i-+ g o u

C'"(S', Rt),

RP) with sufficiently small norm, is of class

For the proof of Theorem 6.4.1 we shall need the following two lemmas. LEMMA 6.4.2. For scalar valued functions u, v E

the product uv

CIa(SI) and (6.4.3)

IIut'IIi.c,

CIIUU1.aflVIIi.a,

where C depends only on a.

PROOF. The inequality follows in a straightforward manner by using the Leibnitz The details are

formula for differentiation and the definition of the norm of C' left to the reader.

0

LEMMA 6.4.4. Let g: U C -÷ R, where U is a neighborhood of 0, be of class with g(O) = 0. Then there are constants C and EO > 0 such that with E <En, j = 1,... p, then (6.4.5)

llgo(u,,...

,Up)IIia

Wewriteu = (u,,...

We shall firstestimate where we use the notation g(u) for g o u. By abuse of notation, we shall identify u(9) with PROOF.

u(e'°). By definition, (6.4.6)

IIg(u)tIa = sup lg(u(O))J + sup

g (0) = 0, by the mean value theorem, we may estimate sup9 Ig (u (0)) I by

C sup9 Iu(0)I, where C =

SUPXEU.

Ig'(x)I and U' is a relatively compact open

§6.4. MAPPINGS OF THE BANACH SPACE

177

convex neighborhood of 0 in U such that u(e) E U' for DUDa <EO. We also have by the mean value theorem, + 8)) — g(u(O))I + 8) — u(O)I. By multiplying both sides by sa and taking suprema we obtain

IIg(u,,... ,Up)IIa

(6.4.7)

1114) ha.

I

Next, we observe that dg/dB(u(8)) = g'(u(O))u'(O). We first use the inequality sup Ig'(u(O))u'(O)l 9

and then write

(6.4.8) g'(u(O + 8))u'(O + 8) — g'(u(8))u'(8)

= [g'(u(O + 8)) — g'(u(0))]u'(6 + 8) + g'(u(8))[u'(O + 8) —u'(8)].

Multiplying both sides of (6.4.8) by sa and taking suprema yields the desired

0

results.

PRoOF OF THEOREM 6.4.1. Since the norm is measured on each component, it

suffices to consider only the case £ = 1. By Taylor's formula we may write for U, V E

CIdt(Sl, RP),

(6.4.9)

g(u + v) — g(u)

—

djg(u)v1 =

f(l

We

to

a13g(u + fv)u13dt.

— 1)

0

131=2

first show that the mapping a i-+ g(u) = gou is of class C from R) with derivative at u given by the linear mapping

(6.4.10) Let uO

RP)

u=(v1,... RP) be fixed. By (6.4.9) and Lemma 6.4.2, it suffices to show

there is a constant C > 0 such that for all u in a neighborhood U, of uO in RP) and a neighborhood U2 of 0 in CIa(SI, RP) (6.4.11)

jld13g(u +1V)IIi.a

C,

Vu

EU,, V

U2, t

By applying Lemma 6.4.4 with g replaced by 813g,

(6.4.12)

11813g(u

+ tv)hI

C'IJu

+

E [0,1],

= 2.

= 2, we obtain l.a

for some constant C' depending only on g. The desired conclusion (6.4.11) then follows for u in a fixed neighborhood of u0 and v in a fixed neighborhood of 0 in

0

VI. GEOMETRY OF ANALYTIC DISCS

178

§6.5. Banach Subm2nifolds

of Analytic Discs

In this section we shall apply the definitions and results of §6.3 and §6.4 to show that the set of analytic discs attached to a generic manifold has a natural structure as a Banach submanifold. with denote the set of all analytic discs valued in CN of class Let

0
1,

i.e.

ri

:= (A: & -+ CN, A E

(6.5.1)

is the Holder space introduced in §6.1 and 0(A) is the space of holomorphic functions in the unit disc & if A = (A11... , AN), we define its where

norm as

IN (6.5.2)

(

\j=I

HA1

)

/

(A) as given in §6.1. Since a will be denotes the norm of u in fixed throughout this section, we have omitted it in the notation above. Also, for the remainder of this section, by an analytic disc we shall mean an analytic disc which is also of class equipped with the norm given by (6.5.2) is a real It is easy to check that Banach space. (It is also a complex Banach space, but we will consider it only a we denote by (p) the constant disc with value p i.e. the real space.) For p E analytic disc p for E A. If M c CN is a subset, we shall denote by where lu fi

(M) C

theset

(6.5.3)

(M) := ((p): p E M).

With this notation, if M c Ce" is an rn-dimensional submanifold of class Ck, then it is easy to check that (M) C D" is a Banach submamfold of class Ck modeled on Rm (as defined in §6.3) and that the mapping M p i-÷ (p) E (M) is injective and of class Ck. (Indeed the inclusion p (p) is a linear mapping from Cr1' to

We may now give a Banach manifold structure to the set of analytic discs attached to M and sufficiently close to (M). THEOREM 6.5.4. Let M C CN be a (smooth) generic subman (fold of CR di-

mension n and CR codimension d. There exists an open neighborhood U of (M)

in D" such that the set ..4(M, U) of all analytic discs in U attached to M is a Banach submani fold of

modeled on 1)" x R", of class C°°.

We note in particular that (M) C A(M, U). It will follow from the proof that if M is of class Ck, k 2, then the manifold A(M, U) is of class Ck_I.

§6.5. BANACH SUBMANIFOLDS OF ANALYTIC DISCS

6.5.4. It suffices to show that for every po

PROOF OF

179

M there

exists e = e(po) > 0 such that the set of analytic discs (A EDTM: IA — (po)II <E, A attached to M}

(6.5.5)

is a Banach submanifold of DN modeled on iY' x R". Indeed, if this is the case it suffices to take

U= poEM

where for Ao

DN we write

B(Ao, e) := (A

(6.5.6)

DN:

QA — A0II

<E}.

Then A(M, U) is the union of the sets defined by (6.5.5) taken over all P0 in M. Now fix po E M and let p = (pi,... , Pd) be defining functions of M near po. For e > 0 sufficiently small, we consider the mapping R: B((po), E) —÷

R"), defined by (6.5.7) R is of class C°°.

u (e'°) = A (&°) R'(A0) for A0 (6.5.8)

Indeed, we may apply Theorem 6.4.1 with It follows also from Theorem 6.4.1 that the derivative — Po• B((po), e) is given by

+

=

where Pz is considered as an d x N matrix, and A E is regarded as an N x 1 matrix. Next, we shall prove that the continuous linear mapping (6.5.9)

R'((po)):

—+

CIa(SI,R(I)

has a continuous right inverse. For this, recall that the d x N matrix pz(po, is of rank d. Hence there exists a constant N x d matrix P, with complex coefficients, such that (6.5.10)

=

Now we shall define the right inverse of (6.5.9),

follows. For!

R)d) ...+

as

R"), we denote by T1f the real vector valued function obtained from I by the action of the modified Hilbert transform T1 given by (6.2.5)

VI. GEOMETRY OF ANALYTIC DISCS

180

and (6.2.11) on the components of f. It follows from Corollary 6.2.10 that the linear mapping (6.5.11) iT1 f is the boundary value of (a unique) analytic disc in V" (see Proposition 6.1.38), we may define fif D" by

ES'.

:=

(6.5.12)

Since P. defined by (6.5.10), is a constant matrix, the continuity of the map

fi: C(S',RY") defined by (6.5.12), follows from that of the map (6.5.11). It follows immediately from (6.5.8), (6.5.10), and (6.5.12) that isa right inverse of R'((po)), i.e.

R'((po))(flf) =

(6.5.13)

f

Vf

We next determine the kernel of the map R'((po)). Since the dimension of the kernel of the constant matrix Po) is n = N — d, we may choose n linearly

independent vectors ei,... 6.5.8 that for A

,

E Ce" spanning kerpz(po, jo). It follows from

we have

A E ker R'((po)) and hence, since (6.5.14)

+

Pz(po,

0,

we have

A E kerR'((po))

ic, C

pz(po,

E

R".

Hence

(6.5.15)

kerR'((po)) = V'e,

...

(iPR")

denotes the obvious Banach space isomorphism. We shall now make use of Proposition 6.3.3 with E = V", G =

<

where

R")

and F = V" x Rd. Observe that by Remark 6.3.15, we may choose0 eso> small that R'(A) also has a left inverse for A B((po), E). It then follows from Proposition 6.3.3 that the set given by (6.5.5) is a Banach submanifold of x R". This completes the proof of Theorem 6.5.4. modeled on F = 0

§6.5. BANACH SUBMANIFOLDS OF ANALYTIC DISCS

181

We shall now show how Theorem 6.2.12, which gives the existence of the solution of the modified Bishop's equation (6.2.13), is a consequence of Theorem 6.5.4.

6.2.12. We assume that (z. w)

regular holomorphic coordinates vanishing at po and that Mis given by (1.3.7). We take p(z, w, th) = Im w — Re w). Hence the d x N matrix pz(O, 0) takes the form PROOF OF

are

Pz(O,O) =

(6.5.16)

the d x n zero matrix and 'dxd the d x d identity matrix. Hence we can take the N x d matrix P (6.5.10) to be where °dxn denotes

P=

(6.5.17)

/ °nxd (

It follows from (6.5.15), (6.5.16) and (6.5.17) A

that

for A E DN

E kerR'((O))

(6.5.18) ,Cd),

The linear mapping

with

AJED1andCJER.

given by (6.5.12) becomes

=

(6.5.19)

(0,

if for A E VN we write = V". the decomposition (6.3.12) becomes

=

(6.5.20)

with z(•)

Re w(1)) + (0,

77' and w(.)

— Re w(l)),

with

(z(.),(Rew(1))) /

P=kerR'(O),

(O,w(.)— (Rew(1))) E G

Since by Theorem 6.5.4, .4(M, U) is a manifold, if follows from Proposition 6.3.3 (and Remark 6.3.15(u)) that near (0), ..4(M,U) is the graph of a function defined near Oon ker R'(O) and valued in O. Theorem 6.2.12 is then an immediate consequence of the decomposition (6.5.20) and (6.5.21). 0

We close this section by showing that the subset of discs attached to M passing through a fixed point in M is again a submanifold. We shall say that an analytic disc A is attached to M :hmugh p M if A is attached to M with A(1) = p.

For p M we write

(6.5.22) We have the following.

:= (A E ..4(M,U): A(1) = p).

VI. GEOMETRY OF ANALYTIC DISCS

182

PROPOSITION 6.5.23. For p E M, if U, given by Theorem 6.5.4, is sufficiently U) is a Banach submanifold of ..4(M, U) of class C°° modeled small, then

on

:={A ED": A(l)=O).

(6.5.24)

= ..4(M, U), 82 = M, Yo = PROOF. We shalt apply Theorem 6.3.34 with (p), and given by f(A) = A(1), for A E ..4(M, U). It is easily checked that for T(p)A(M,U), we have f'((p))B = B(l). Making use of the description of B T(p)A(M, U) given by (6.5.15), it follows that

f

kerf'((p)) =

.. .

where, as in (6.5.15), e1,... , e,, span kerpz(p, kö) in C". The proposition is then 0 a consequence of Theorem 6.3.34.

Notes for Chapter VI The material in §6.1 on the Cauchy, Hubert, and Poisson transfonns is classical and can be found e.g. in Zygmund [1]. The technique of analytic discs attached to has a long history, beginning perhaps with the work of real submanifolds of Lewy Li], L2] who used discs attached to hypersurfaces to study boundary values of holomorphic functions and to give the first example of a nonsoivable first order

differential equation. Other early fundamental work is due to Bishop [1] who considered discs attached to submanifolds of higher codimension. Since then, analytic discs attached to real submanifoids have been used in many contexts. We mention here only a few. For references concerning the use of analytic discs to prove extension of functions from CR submanifolds, we refer the reader to the notes of Chapter VII. Analytic discs have also been used to study hulls of holomorphy of manifolds with CR singular points, e.g. in the work of Kenig and Webster [1], Bedford and Gaveau [1], and Huang and Krantz [1]. We also mention here the important work on the Kobayashi metric by Lempert [1]. In addition, filling with analytic discs has been very fruitful in topology, and we refer the reader e.g. to the survey article by Eliashberg [1]; see also Alexander [1] and Gromov [I]. For recent work on the Riemann-Hiibert problem and embeddability of large discs, For the use of analytic discs see Globevnik [1] and in H°° control theory, we refer the reader to Heiton and Moreno [1]. The concept of manifolds modeled on a Banach space and the implicit function theorem in Banach spaces, reviewed in §6.3, are standard. We refer the reader e.g. to the book of Abraham, Marsden, and Ratiu [1]. In §6.5, we follow the approach

NOTES FOR CHAPTER VI

183

of Baouendi, Rothschild, and Trépreau [1) in studying small discs attached to generic submanifolds. In particular, the use of the implicit function theorem in a Banach space setup to solve the Bishop equation and the proof of Theorem 6.5.4 is taken from that work. The approach of studying infinite dimensional families of discs was inspired by the work of Tumanov [2].

CHAPTER VII

BOUNDARY VALUES OF HOLOMORPHIC FUNCTIONS IN WEDGES

Summary In this chapter, we define the notion of a wedge with edge on a generic submanifold. We show that a holomorphic function in an open wedge with slow growth near the edge admits a distribution boundary value on the edge. In §7.3, we show that if the boundary value is a continuous function on the edge, then the holomorphic function extends continuously up to the edge. We also show, in §7.4, that if the distribution boundary value is zero, then the holomorphic function must vanish identically in the wedge. To prove the latter result, we construct a family of analytic discs partly attached to the edge whose images fill an open subset of the wedge. We conclude by stating, without proof, a version of the "edge-of-the-wedge" theorem for generic edges.

§7.1. Wedges with Generic Edges in In this section, we shall describe an open wedge with a generic edge in CN. of codimension d and CR dimension n (so Let M be a generic submanifold of that N = n + d) and p = (ps,... , Pd) be a defining function of M near po and 0 a small neighborhood of p0 in CN in which p is defined. If r is an

open convex cone with vertex at the origin in R", we set

(7.1.1)

W(0,p,r) :=(Z E 0 :p(Z,Z)E

I').

The set defined by (7.1.1) is an open subset of CN whose boundary contains M Such a set is called a wedge of edge M in the direction r centered at p0. When M is a hypersurface, i.e. d = 1, a wedge with edge M centered at po is simply a one-sided neighborhood of i.e. an open set of the form Ofl (Z: p(Z, Z) > 0)

orOfl(Z: p(Z,2) <0). The following shows that the set defined in (7.1.1) is in a sense independent of the choice of p, which will allow us to change defining functions freely.

PRoPosmoN 7.1.2. Let p and p' be two defining functions for M near po, where M and P0 are as above. Then there is a d x d real invertible matrix B such 184

§7.2. HOLOMORPHIC FUNCTIONS OF SLOW GROWTH IN WEDGES

185

that for every 0 and r as above the following holds. For any open convex cone relatively compact in r n Sd_I (where S"' denotes fl r1 C R" with such an open neighborhood of p0 in there exists the unit sphere in R"), that

W(01, p', F1) c W(0, p, r).

(7.1.3)

PROOF. As indicated in § 1.2, there exists ad x d smooth invertible matrix a defined in a neighborhood of po such that p' = ap near p0.We put B = a(po, which If p'(Z, Z) E F11thena(p0, f,oY1a(Z, Z)p(Z, Z) a(po, we obtain

(7.1.4)

(1+ O(Z, 2))p(Z, Z) E a(po,

0 where! is thed x d identity matrix and O(Z, Z) is a matrix with IIO(Z, Z)II 5d-I is relatively compact in r n The hypothesis that B!'1 n as Z then implies that p(Z, 2) is in r for Z sufficiently small. This proves (7.1.3) for 0 01 sufficiently small.

§7.2. Holomorphic Functions of Slow Growth in Wedges Let M c Ci" be a generic smooth submanifold and p0 E M. As before, we denote by p = (Pi,... , Pd), where d is the codimension of M, a defining function for M near p0. In this section, we shall consider a wedge W = W(0, p, r') of the form (7.1.1) with edge M. We say that a holomorphic function f(Z) in W' is of slow growth if there exists a constant C > 0 and an integer k 0 such that

If(Z)I

(7.2.1)

<

VZ E W'.

,

—

If dist (Z, M) denotes the distance from a point Z

to the submanifold M, then

(7.2.1) is equivalent to (7.2.2)

If(Z)$

Idist(Z,MW'

YZ E W,

where C > 0 is a constant, possibly different from the one in (7.2.1). We shall

show that a holomorphic function of slow growth on W' has a distribution boundary value on the edge M. We

at

choose regular coordinates Z = (z, w) E C" x Cd (see (1.3.7)), vanishing so that M is given near Po = (0,0) by

(7.2.3)

Im w =

Re w),

VII. BOUNDARY VALUES OF HOLOMORPHIC FUNCTIONS IN WEDGES

186

= 0. By Proposition 7.1.2, there exist E > 0 and a convex open cone F C R" such that if

= 0 and

where

(7.2.4) W :=

+iv,

w): w = s

IzI

<

< E, lvi < E, V E

E,

thenWCW'. We may assume, without loss of generality, that 0 is not in the open convex cone F. It follows that if f(z, w) is holomorphic in W, given by (7.2.4), then it has slow growth if and only if for some C > 0

If(z, s +

(7.2.5)

s)

+ iv)I

S E IR", and v E 1. for all sufficiently small z E If F1 c F is an open cone such that F1 fl S"_', where

5d—I

as in §7.1 denotes

the unit sphere in W', is a relatively compact subset of F fl S"', then we write -< F. Similarly, if W1 is a wedge of the form (7.2.4) with and F' replaced by Ei and F1 with El < E and r'1 -< 1, then we write W1 -< W. We have the following result. THEOREM 7.2.6. Let f(z, w) be a holomorphic function of slow growth in W. Then, there exists a CR distribution u = bvf defined in a neighborhood of P0 in

M by (7.2.7)

(u,

x)

:= urn

I

f(z, 5 + iØ(z,

s)

+ iv)x(x, y, s)dxdyds,

with sufficiently small compact where z = x + iy, for smooth functions x in support near the origin. Furthermore, u is continuous, then for any wedge W, with W -< W, the restriction off to W extends continuously up to a neighborhood of po in M. In addition, uniqueness holds, i.e. if u 0, then f 0. The boundary value u = bvf is independent of the choice of regular coordinates. The reader should recall (see Hörmander [3, Definition 6.3.3]) that a distribution } is a covering u on a manifold M is defined in terms of an atlas (Us, Xa)' where {

of M by coordinate neighborhoods and (Xa} is the corresponding collection of coordinate functions xa: Ua —+ R"', as follows: for each a there is a distribution ua

Xa(Ucr) such that

(7.2.8)

o

°

xj'

= up,

§7.2. HOLOMORPHIC FUNCTIONS OF SLOW GROWTH IN WEDGES

187

whenever the composition Xa o xj' is defined. Thus, the statement in Theorem 7.2.6 claiming that the boundary value bvf is independent of the regular coordinates (z, w) means that if we define the boundary value using different regular coordinates (z'. w'), then the two boundary values are related by (7.2.8) with Xa and Xp being the corresponding coordinate maps. The proof of Theorem 7.2.6 will be given below. An immediate corollary of Theorem 7.2.6 is the following coordinate free formulation.

COROLLARY 7.2.9. Let M C CN be a generic submamfold and p0 E M. Assume that p = (pie... Pd) 15 a defining function for M near p0 and let and F is an W = W(O, p. F), where 0 is an open neighborhood of P0 in ,

open convex cone in R", be a connected wedge of edge M in the direction F centered at p0. 1ff is a holomorphic function of slow growth in W, then it has a boundary value bvf E D'(M fl 0). bvf vanishes in an open subset of M fl 0, then f vanishes identically in W. For the proof of Theorem 7.2.6, we shall need the following results. PROPOSITION 7.2.10. Let v E F and x(x, y, s) be as in (7.2.7). Set, for small

o >0, (7.2.11)

:= f

Then, the limit lim

f(z, s +

s)

+ iav)X(x, y, s)dxdyds.

exists.

It is clear that the function a i-÷ is in some open interval (0, r), r > 0. We shall show that extends continuously to a = 0. For this we need the following lemma, whose proof is elementary and left to the reader. PROOF.

LEwIA 7.2.12. Let F(o) be a smooth function in the open interval (0, r), for some r > 0, and let k be a nonnegative integer If for every nonnegative integer j, there exists a constant C1 > 0 such that (7.2.13)

0


then F(o) extends as a smooth function in [0, r).

It follows from (7.2.5) that I for j 1, we differentiate (7.2.14)

= e=i

J

)I

C/ak, for some C > 0. To prove (7.2.13)

s+iØ(z,

s)+iav)(ivt)x(x, y, s)dxdyds.

188

VII. BOUNDARY VALUES OF HOLOMORPHIC FUNCTIONS IN WEDGES

We use the identity (7.2.15)

—f—.

asm

(f(z, s +

s)

=

+ lay))

s + i4(z,

+

s) + 10

s)),

where Sme denotes the Kronecker symbol. It follows from (7.2.15) and the fact that =0 that (7.2.16)

s+

s)

+ jay)

=

asm

m=1

where a,m(z,

+ lay)),

(f(z,s +

s) are smooth functions near the origin with a,m(O)

=

81n,• Sub-

stituting (7.2.16) in (7.2.14) we obtain, after an integration by parts, (7.2.17) —

f

= f(z, s+i4'(z,

y, s))dxdyds. Sm

Thus, we have, by (7.2.5), (7.2.18)

for some C1 > 0, i.e. (7.2.13) holds for j =

1. Repeating the differentiation and subsequent integration by parts yields (7.2.13) for all j 0. The proof of Proposition 7.2.10 is complete by Lemma 7.2.12. 0

It is not yet clear that the limit urn

is independent of the choice of v E r.

This will be a consequence of the following more general result. Let T(Z, s, a) be a smooth C'-valued function defined in such that Im r(z, s, a) is valued in a compact subset of the open cone r. Define (7.2.19)

:=

f(z, s +

s) +

ar(z,

a))x(x, y, s)dxdyds.

We claim that the point (z,s is in the wedge W for (z, s) sufficiently close to (0,0) and sufficiently small 0 > 0. To see this, note

that, with t = t(z,

s,a),

(7.2.20) s + aRe t) + aIm t).

§7.2. HOLOMORPHIC FUNCTIONS OF SLOW GROWI'H IN WEDGES

=

189

and Im r stays in a compact subset of the open cone I', there (z,s) = (0,0)suchthatfor(z,s) €U existsanopenneighborhoodU c Since

0

and a > 0 sufficiently small

€r.

(7.2.21)

This proves the claim, since (z, s + oRe r + iØ(z, s + aRe r)) E M. Hence, the integral in (7.2.19) is defined for all sufficiently small a > 0, provided that x has sufficiently small support near the origin. PROPOSITION 7.2.22. For any v E F and r(z,

(7.2.23)

—

= 0(a),

s,

a) as above, we have

a -+

PROOF. The difference f(z, s + i4 + at) — f(z,s + s) and v = r(z, s, a), can be written (7.2.24)

+ jay), with

=

f(z,s

=0)

d

—r)v))(r1

—iv1)dr.

0

By writings + 1(/ + a(rr + i(1 — r)v) as in (7.2.20), we observe that it is of the form

(7.2.25) where

s + i4 + a(rt + 1(1 — r)v) = w + ia(ra + (1 —

(z, w) M and a, b

F'. Since the open cone I' is convex, we deduce that (z, s + içb + a(rv + i(l — r)v)) E W. Consequently, we have (7.2.26) a

ff

—

= s+içb+a(rr+i(l —r)v))(r1 —ivj)x(x, y, s)dxdydsdr.

As in the proof of Proposition 7.2.10 (see (7.2.15) and (7.2.16)) we find (7.2.27)

+1(1 — r)v)) —r)v))),

VII. BOUNDARY VALUES OF HOLOMORPHIC FUNCTIONS IN WEDGES

190

=

where s, a, r) are smooth functions near the origin with integration by parts in (7.2.26) using (7.2.27) yields

(7.2.28)

An

—

d

I Jo

f(z,s+iØ+a(rr+i(l—r)v))

I

x

—

We denote by Hjg (a) the integral (7.2.29)

Hj1(a):= 11 Jo

f(z,s+iØ+a(rr+i(1—r)v)) s,a,

x

— i Vi)7,(X, y,

s))dxdydsdr.

By using Lemma 7.2.13, Proposition 7.2.22 follows from (7.2.28) if we can show

that for each nonnegative integer m there exists a constant Cm > 0 such that (7.2.30)

C

dm

4, a

do"

£ = 1,... , d. We shall show that (7.2.30) holds for m = 0. The proof of (7.2.30) for larger m follows by differentiation and integration by parts as in the proof of Proposition 7.2.10. To prove (7.2.30) for m = 0 it suffices to prove, in view of (7.2.5), that

where j,

(7.2.31)

s)+a(rr(z,

s,

a)+i(l —r)v) = w+ioa(z,

s,a, r),

where (z, w) E M and a(z, s, a, r) F is a smooth function bounded away from 0. This is done by using an analogue of (7.2.20) and the fact that F is an open convex cone that does not contain 0. We leave the details to the reader. This completes the proof of Proposition 7.2.22. 0

Proposition 7.2.22 implies that lim GT(a) exists and is equal to lim

a-+O

any

choice of v

F. In particular, lim

a-+O

is independent of the choice of the

vector v F. It is not yet clear that the distribution u defined in a neighborhood of 0 by

(7.2.32)

(u,

x) := alim

for

§7.2. HOLOMORPHIC FUNCTIONS OF SLOW GROWTH IN WEDGES

191

for some choice of v E r, defines a distribution in a neighborhood of po in M independent of the choice of regular coordinates (z, w). However, if the function f extends continuously to W U M, then

(7.2.33) and

urn

=

a -+0

f f(z, s +

s))x (x, y, s)dxdyds

hence in this case u, being equal to the restriction of f to M, transforms

according to (7.2.8) under a change of local coordinates on M. We shall show that this is also true in the general case by approximating a general function f, satisfying the assumptions of Theorem 7.2.6, by functions that also extend continuously up to M. Let Ek(W) denote the Banach space of holomorphic functions f(z, w) in W (defined by (7.2.4)) which satisfy the estimate (7.2.5) with the norm (7.2.34)

1111k

:=

sup (z.w)EW. uEr

with w =

s

+

(7.2.35)

s) + iv. We fix V0

fm(Z, w) :=

E

1', and we define the sequence

f(z. w +

If we let be a wedge with W, i.e. W is defined by (7.2.4) with e' > 0 ffl, then fm(Z, w) is and r' instead of E and r with E' < e and r' -< r (see §7.1), holomorphic in W, form sufficiently large, and w) extends holomorphically across M. In particular, fm extends continuously up to the edge M. In what follows we also assume that r' has been chosen so that v0 E I" and so that

VvEr',

(7.2.36) with a

>0.

PROPOSITION 7.2.37. For any f PROOF. We have,

with 4 =

Ek(W), the sequence

converges to f in

s) and v E 1", in view of (7.2.35)

(7.2.38)

E Wfor(z,s) E

xRdandv

sufficiently small, and m sufficiently large. We leave it to the reader to verify

192

VII. BOUNDARY VALUES OF HOLOMORPHIC FUNCTIONS IN WEDGES

(e.g. using the Cauchy estimates) that (7.2.38) that (7.2.39)

lfm(Z, S + içb

+ iv)

—

is in Ek+1 (W). Hence, it follows from

f(z, s + içb + iv)l

iv +

To estimate the integral on the right hand side of (7.2.39), we decompose v as avo + v', where v' is orthogonal to vo and a > 0. Note that a lvi by (7.2.36).

Using the fact that lv + rvo/mI

ivol Ia + r/mi, a straightforward calculation

shows that (7.2.40)

0

This completes the proof of Proposition 7.2.37.

An inspection of the proof of Proposition 7.2.10 shows that we have the following estimate (7.2.41)

I

(u, x) I = I tim

C11lf tie.

for any £ k, where C1 > 0 and k is as in (7.2.5). Hence, Proposition 7.2.37 and (7.2.4 1) with £ = 2k imply that the distributions Urn defined by the functions frn converge, in the sense of distributions, to u. Since Urn coincides with the restriction of frn to M (and hence, as mentioned above, is independent of the choice of regular coordinates), it follows that the same holds for U.

Thus, u given by (7.2.7) defines a distribution on the submanifold M. This distribution will be denoted by bvf and referred to as the boundary value of the function f. Since bvf is the limit of the CR functions frn restricted to M, it follows that by! is a CR distribution on M. This completes the proof of the first statement in Theorem 7.2.6. The proof of the continuity and uniqueness stated in the theorem will be given in the next sections.

§7.3. Continuity of Boundary Values In this section, we shall prove the second statement of Theorem 7.2.6 by showing that if the boundary value by! of a holomorphic function f in the wedge W, given by (7.2.4), of slow growth, i.e. satisfying the estimate (7.2.5), is continuous, then f extends to be continuous up to M, with by! = JIM. Let s, t) be an almost holomorphic extension ins of in the

193

*7.3. CONTINUITY OF BOUNDARY VALUES

and satisfies (4.2.20) (with s' replaced hereby:). Consider the cuty, s, t) defined in the following way. Let 8(x, y, s) be a smooth

origin in off function

<e/2, si <€/2}, with 0 1 function with compact support in (lxi <e/2, in a neighborhood of the origin. Take 9(x, y, s,:) to be an almost holomorphic 1 in a neighborhood of the extension in S of 0(x, y, s) such that 0(x, y, s,:) and vanishes for (x, y, s) outside the support of 0(x, y, s). We origin in let y, s,:) = Ô(x, y, s, :)A(:), where A(:) is a smooth function with compact 1 in a neighborhood of the origin in Rd. support in (1:1 < E/2} such that A(:) 1) is a smooth function of compact support which is one Observe that *(x, y, s, near the origin and which is almost holomorphic in s and hence satisfies (4.2.20). Let f(z, w) be a holomorphic function in W (given by (7.2.4)) satisfying (7.2.5). Consider the function

g(x, y, s,t) := f(z, s + it +

(7.3.1)

s,

t))*(x, y, s, t).

We let, for a smooth function X(x, y, s) as in Theorem 7.2.6 and V E 1',

G(a)

(7.3.2)

Recall that x is allowed to have support in some fixed, but sufficiently small,

neighborhood U of the origin in We may choose *(x, y, s, 0) is contained in U. Writing G(o) as (7.3.3)

f(z, s + joy +

s,

ov))*(x,

y, s, av)X(x, y, s)dxdyds,

reader can check that tim G(o) exists and is equal to

by expanding the almost holomorphic extension Proposition 7.2.22. The mapping (7.3.4)

that the support of

G(a)

=f the

so

•, •, 0)bvf, x)

s,t) in t and applying

(x,y,s,t)

is a diffeomorphism near the origin in and maps the flat edge ft = 0) to the submanifold M C We denote by r' an open convex cone with -< 1' as in Proposition 7.2.37. The mapping (7.3.4) maps a flat wedge W in the direction I" with edge on ft = 0) into the wedge W. Hence g, given by (7.3.1), is defined in the wedge W and has slow growth up to the edge ft = 0). In fact, by the assumption on the support of g, it is defined in the whole wedge x r'.

VII. BOUNDARY VALUES OF HOLOMORPHIC FUNCTIONS IN WEDGES

194

Moreover, by using the slow growth of f up to M and the almost holomorphic property of and the reader can easily check that there exists a constant C > 0 such that (7.3.5)

C,

j = 1,... ,d,

for (x, y, s,t) E R2n+d x F'. Assume that the CR distribution bvf on M is a continuous function in a neigh-

borhood of 0 in M. We denote by u the corresponding distribution in a neighborhood of 0 in the flat edge ft = 0) By shrinking the support of if necessary, we may, and we will, assume that uifr(., •, 0) is a continuous function (with compact support) in Clearly, f extends continuously up to M if and only if g extends continuously up to the flat wedge ft = 0) in a neighborhood of 0 in Hence, the desired continuity off up to M, near 0, is a consequence of the following proposition.

PR0P0sm0N 7.3.6. Letg(x, y, s, t), u(x, y, s), andG(a)be as above. Assume that the distribution u is a continuous function near 0 in R2n4.d. Then, g extends continuously up to the edge ft = 0) and the restriction of its extension to ft = 0) coincides with *(.' ., ., 0)u.

PROOF. Weshallusethenotationr = s+it and writeg(x, y, r)forg(x, y, s, t). (The reader should keep in mind that g (x, y, r) is not holomorphic in r, but satisfies the bound (7.3.5).) Let y, s) be a smooth function with compact support in such that = 1. Let po(x,y,s) = yS—'s8—') for 8 > 0, and put

(7.3.7)

g5(x, y, s, r) :=

J g(x', y', s',

— x', y — y', s — s')dxdyds.

We also write y, r) y, s, t). Observe that lyl, isi, and itt large, and satisfies the bound

(7.3.8)

sup

at1

s.C,

j=I

y, s, t) = 0, for lxi,

d,

for (x, y, s, t) E

x F', where C > 0 is the same constant as in (7.3.5) and, hence, independent of 8. We claim that, for fixed 8 > 0 sufficiently small, V. r) is continuous up to the edge ft = 0) for (x, y, s) sufficiently small. To see this, we observe that (7.3.9)

y, s, t) = (g(., ., t), pj,(x —

., y — ., ç —

§7.3. CONTINUITY OF BOUNDARY VALUES

195

and

(7.3.10)

where (.,.) denotes the pairing between distributions and test functions in y, s, t) and u is the distribution corresponding to bvf. The continuity of up to the edge {t = 0} is a consequence of the following facts: (1) The limit of •, ., r) in the sense of distributions as t —÷ 0 in r' coincides with u*(., •, •, 0) (see (7.3.2), (7.3.3), and the in a sufficiently small neighborhood of 0 in observation following these). (ii) The test functions (x —•, y —•, s — .) converge to — •, y' — •, s' — •) as (x, y, s) —÷ (x', y', s'). (This is a in standard fact in the theory of distributions.) (iii) The pairing (.,.) is continuous x —+ C. (See e.g. Schwartz [11.) as a bilinear mapping Clearly, (i), (ii), and (iii) imply the claim. x Our next claim is that, for each (x, y, s, t) E

Iimg5(x,y,s,0) = *(x,y,s,0)u(x,y,s), (7.3.11)

Iim——(x,y,s,t)=——-(x,y,s,t),

j1,...,d.

The first equality in (7.3.11) is a standard fact on regularization of continuous functions. The second one follows from the same argument after a differentiation in the convolution. We need the following lemma, whose proof is a direct consequence of Stoke's Theorem (see e.g. Hörmander [2, Theorem 1.2.11) and is left to the reader. LEMMA 7.3.12. Let = + i,1, > 0} and a E C: = smooth function in fl + that extends continuously to the closure 11+. Assume that h = Ofor large I and that dh is bounded in Then, the following holds:

(7.3.13)

h(I, 1)=

We apply Lemma 7.3.12 to the function go(x, y, s + iCt), which satisfies the hypotheses of the lemma for each (x, y, s, t) E R2n+d x r'. We obtain (7.3.14)

=

y, s + it) —

+2

ff

196

VII. BOUNDARY VALUES OF HOLOMORPHIC FUNCTIONS IN WEDGES

Using the fact that 1

(7.3.15)

'°°

1

—

+1

2

,r

dJ7

=

1,

we obtain (7.3.16)

y, s

+ it) — 1

y', s') =

(00

y, s —

17:) — go(x',

y',

172+1

+

6 -÷

using the second claim (7.3.11) established above, the bound (7.3.8), and the fact that Letting

0,

sup Iu(x,y,s)*(x,y,s,O)I,

sup lgs(x,y,s)I

(7.3.17)

we deduce that

g(x, y, s + it) — u(x', y', 1

y', SI, 0)

(00 u(x, y, s — q:)iJ'(x, y, s

in-00 +

—

— u(x', y', s')ifr (x', 772+1

y,s + it:)

desired continuity of g up to the edge {t = 0) with limit u*(•, •, •, 0) follows, by letting (x, y, S.:) •-÷ (x', y', s', 0) with 1 r', from the continuity of The

., ., 0)

and the estimate (7.3.5).

0

§7.4. Uniqueness of Boundary Values

In this section, we shall complete the proof of Theorem 7.2.6 by showing that 0 in the whole wedge W. In view of the previous section, we may assume that f is continuous up to the edge M. We reduce to the case where the edge M c C" is maximally totally real (see §1.3) as follows, If M C Ce" is a generic submanifold near po E M, given in x regular coordinates (z, w) vanishing at po by (7.2.3), then consider the

if bvf E 0, then f

197

§7.4. UNIQUENESS OF BOUNDARY VALUES

maximally

totally real submanifold U C M given by y = 0, i.e.

M

is given near

po = (0,0) by (7.4.1)

If W

Imz=O,

is an open wedge of the form (7.2.4) with edge M, then we define an open

wedge

W with edge M

as follows.

Pick v0

E

F' sufficiently small and observe that

the point (7.4.2) (x, s)

is

E

sufficiently close to the origin. Let f' C W x Rd such that

be

an open

convex cone containing the vector (0, vO)

W,

(7.4.3) and (v1, V2) E f' sufficiently for (x, s) E of such a cone follows from writing

(7.4.4)

s

+

close to the origin.

The existence

x, s) + iv2

so that

(7.4.5)

(x +iv1,s +i#(x,x,s) +iu2) E W

if E

(7.4.6)

and

F',

using the fact that d#(0) = 0 as in §7.2. Define the wedge W with edge U

by

(7.4.7) W := {(x

+iy,s

lxi <E',

< E',

IsI <E', (ti <E', (y,t) E i}

with e' > 0. If e' < e is sufficiently small, then W C W. Since W is an open subset of the connected open set W, it suffices to show that f vanishes in W. Since we may assume, as mentioned above, that f is continuous up to the edge M, so that restriction off to the wedge W commutes with taking boundary values

on the edge, it is clear that it suffices to prove the uniqueness for the maximally totally real submanifold M. Thus, the proof of Theorem 7.2.6 will be completed by the following result.

VII. BOUNDARY VALUES OF HOLOMORPHIC FUNCTIONS IN WEDGES

198

PROPOSITION

7.4.8. Let M C Cd be a maximally total!)' real

with

P0 E M and W a connected wedge with edge M. 1ff is holomorphic in W and continuous up to the edge M with I IM 0, then f 0 in W.

For the remainder of this section, we shall assume that M is a maximally totally real submanifold of C" given in a neighborhood of 0 E C" by (7.4.9)

Im w =

w)

where is a smooth function near the origin with = 0 and = 0. To motivate the method of the proof of Proposition 7.4.8, we consider first the flat case M = R" c Cd. We may assume that the wedge W is of the form

(7.4.10)

W = (w = s +it E

isI <

E,

itt < E, t

EF),

for some open convex cone r C Rd. Consider the family of half discs parametrized by so E Rd and t0 E r, defined by

(7.4.11)

:=

+

E

where & := = + E > 0); here, Et is the open unit disc as in Chapter VI. Observe that for Isol <€12, ltoI <€/2, C W and A((—i, i)) c M (where (—i, 1) is the open interval from —ito i on the imaginary axis of the plane). Also, as and t0 vary, satisfying the conditions above, the images of the half discs fill an open subset of the wedge W. (Indeed, for instance the points = + ito/2 fill an open subset of W as and to vary as above.) For o fixed so, to, consider the function h defined by (c). Clearly, —+

f

U (—i, i) and vanishes on the line segment (—i, i). h extends continuously to Hence, h vanishes identically in From the observation above, it follows that f vanishes identically in an open subset of W and, hence, by connectedness of W, in all of the wedge W. This proves Proposition 7.4.8 in the flat case. In the general case where the edge M is not necessarily flat, the half discs given by (7.4.10) are no longer attached to M along the line segment (—i, i). Hence, we need to construct a family of discs, partly attached to M, whose interiors fill an open subset of the wedge W. The proof of Proposition 7.4.8 will be concluded by the same uniqueness argument as in the flat case. Thus, Proposition 7.4.8 is a consequence of the following two results. The reader should recall the definition of the Banach space V' of C' .G analytic discs valued in given by (6.5.1).

§7.4. UNIQUENESS OF BOUNDARY VALUES

199

i)", defined Rd, satisfying the following pmperties in a neighborhood U x V of(O, 0) in Rd x THEOREM 7.4.12. There exists a smooth mapping (s, v) -÷

forall(s,v)E U xV. (i) A00 = (0), where (0) denotes the constant disc as in §6.5. = s + içb(s). where M is given by (7.4.9). M, for 0

(ii) (iii) (iv)

=

1.' +

içb'(s) .

V.

We shall use the infinite dimensional implicit function theorem as in §6.5 to prove Proposition 7.4.12. LetS: V" —÷ be the mapping given by PROOF.

(7.4.13) where p(w,

<

S(A)(O) = p (A(ebo),

= Im w — (Re w), which is defined for A

IY' with I IA II l.a
for r > 0 sufficiently small (i.e. A B((0), r) using the notation of §6.5). The reader can check, using the arguments in §6.5, that S is a smooth mapping and its differential S'((O)): Vd ir/2], R") is given by (7.4.14)

(S'((O))A)(O) = Im A(e18),

0

C' .a We shall construct a continuous right inverse of r/2], R") -+ the linear mapping S'((O)) as follows. We first define a continuous linear mapping

(7.4.15)

y:

—+

such that for h ir/21, R"), (yh)(0) = h(0) for 0 ir/2. 0 be a smooth function with compact support in the open interval Let *(0) (—sr, which is one on ir/2] and define

h(0),

(yh)(0) :=

(o

—

i)),

Define (nh) restricted to S' by —T,(yh) + iyh, where T1 denotes the modified Hilbert transform as in §6.2. This defines the desired right inverse of S'((O)). Set (7.4.16)

B:=(A

S(A)=0}.

VII. BOUNDARY VALUES OF HOLOMORPHIC FUNCTIONS IN WEDGES

200

Since S'((O)) has a right inverse, Proposition 6.3.3 implies that B is a Banach submanifold of 7)' modeled on its tangent space at the origin, ker S'((O)). It is clear from (7.4.14) that 7)': Im A(e10 =

kerS'((O)) = (A

(7.4.17)

such that g(e'°)

= 0 for

R") x R" (7.4.18)

the subspace of g E

C

We denote by by

9

We may then identify ker S'((O)) with

0

the following

ker S'((O))

= (A €D": AIs' = —Tig+s+ig, (g,s)e Consider

the smooth mapping c:

xR"j.

7)' -+ TC" defined by

Q(A) := (A(1)1

(7.4.19)

ir/2}, it E B is attached to M along the half circle {—ir/2 9 follows that the restriction of Q to B is a smooth map Q: B TM. By Theorem 6.3.42, the proof of Theorem 7.4.12 will follow if we show that the mapping C: B -÷ TM is a submersion at the constant disc (0), i.e. the corresponding tangent map Q'((O)): T(o)13 To(TM) is surjective. We parametrize M by an open neighborhood of 0 in Rd by s i-+ s + E M. The tangent space TM is then parametrized by x Rd and T(o)(TM) can be identified with Rd x Rd. Using this identification we compute Q'((O))A. We have, with the identification T(o)B = kerS'((O)) given by (7.4.18), A

(7.4.20)

C'((ODA

=

The reader can easily check, using (6.2.11), that if u(0) is a function of class

defined on 5' with u(9) =0 near 9 =0, then d

1

—T,u(9)Ieo——

(7.4.21)

dO

Observe

that if u(O)

0 and u $ 0,

the tangent mapping C'((O)): T(o)B

and

(7.4.20).

The

u(0)

<0. The surjectivity

then

-+ To(TM) is

2d9.

immediate

now

of

from (7.4.18)

The proof of Theorem 7.4.12 is complete.

second result needed to

following.

—fl

complete the proof of Proposition 7.4.8 is

0 the

UNIQUENESS OF BOUNDARY VALUES

201

Let (s, v) i-÷ E 7Y',for (s, v) U x V, be given by Theorem 7.4.12. Then, for every compact subset K C I' fl V, there exists an open neighborhood U' C U 010 in R" and S > 0 such that PROPOSITION 7.4.22.

fl

(7.4.23)

E

U' x K,

is a disc of radius S centered at 1 E C. Moreover, if the compact set K where <8, such that the image of with 0 has nonempty interior, then there exists the mapping E W, U' x K (s, v) i-+ —

has nonempty interior in W. PROOF. We Taylor expand

we

=

at

e0. Using the analyticity of

obtain, for —ir/2 <9
=

(7.4.24) We

i-÷

+

—

write s' = s'(9, s, v) = Re

and

v' =

—

Ce'°) +

v'(O,

s, v) =

Re

By Theorem 7.4.12 (iii), we have

= s' + i4(s'),

(7.4.25)

= v' + hf/(s')u'.

Thus, we may write

= s' + i4(s') + + i(Re (1 —

(7.4.26)

+

— Im (1 —

—

1)2),

where

(7.4.27)

:=

s', v',

:= Re

—

l)Ø'(s')v' + un (Ce'° — l)v'.

vary in an open set wi'., C as follows. For each 9 with 191 <8', for some 5' > 0, we let vary along a portion of the inner normal to the unit circle at e'°, i.e. such that Im (1 — = 0 and 0 < Re (1 — 0 (independent of 9). The reader can easily verify that if we choose 8 > 0 fl C where sufficiently small, then is the disc of radius S centered at 1 E C. For such and 9, (7.4.26) can be rewritten We let

(7.4.28)

= s' —

+ + iØ(s' + *)+

4(s' + if') + Re (1 —

+ O((Re (1 —

202

VII. BOUNDARY VALUES OF HOLOMORPHIC FUNCTIONS IN WEDGES

where

(7.4.29)

:=

VI, Re

=

(Ce'° — Re

1))

+ O((Re

(Ce'° —

—

1))2).

Observe, by Theorem 7.4.12 (ii) and (iv), that s'(O, 0, v) = 0 and v'(O, s, v) = v. Since d4(0) = 0, we may for any E' > 0 choose 8' > 0, r > 0, and U' c U sufficiently small so that (7.4.30)

—

+

+ O((Re (I

<E'Re

—

(1 —

(with 8 related to as above), s E U', V E K. By choosing E' > 0 for sufficiently small, depending only on K, the first part of Proposition 7.4.22 follows from (7.4.28). U' x K —÷ W, for To complete the proof, we consider the mapping 0 < < 8, defined by To prove the second part of v) = — Proposition 7.4.22, we shall show that the tangent map of at (0, vO) has full rank for any Vo in the interior of K, provided that we choose sufficiently small. at = 0, This fact follows immediately from Taylor expanding — (7.4.31)

and using

—

= s+

+

+

+

= 0. This completes the proof of Proposition 7.4.22.

The proof of Proposition 7.4.8 is now complete, in view of the remarks preceding Theorem 7.4.12.

§7.5. Additional Smoothness up to the Edge The aim of this section is to generalize the results of §7.3 to smooth boundary values. We shall show that if the boundary value of a holomorphic function of slow growth in a wedge is smooth, then the holomorphic function is smooth up to the edge. More precisely, we have the following.

THEOREM 7.5.1. Let M C CN be a generic submanifold and P0 E M. Assume that p = (pie... , Pd) is a defining function for M near P0 and let W = W(O, p, F') be a connected wedge of edge M in the direction F' centered at where 0 is an open neighborhood of po in and F is an open convex cone in R". 1ff is a holomorphic function of slow growth in Wand its boundary value on M, then, for any wedge W -< W, the restriction off to W bvf is of class is Ctm up to the edge M.

We use the notation introduced in the previous sections. For the proof of Theorem 7.5.1, we shall need the following lemma.

§7.5. ADDITIONAL SMOOTHNESS UP TO THE EDGE

203

LEMMA 7.5.2. Let f(z, w) be as in Theorem 7.2.6 and bvf the CR distribution

given by (7.2.7). There

exist vector fields R1

d

of the form a

(7.5.3) (=1

where the a1e are smooth functions defined in a neighborhood of 0 in

such

that

=

(7.5.4)

R1bvf,

=

wherej=l,...,d,and€=l,...,n. PROOF. By Proposition 7.2.37 and (7.2.4 1), it suffices to prove the lemma for

functions f that extend smoothly up to M (cf. the proof of Theorem 7.2.6). We take vector fields R1, of the form (7.5.3), so that (7.5.5)

R1(se

+

Z,

s)) =

where the denote the Kronecker symbols. To prove the existence of the R1 satisfying (7.5.5), we fix j and find solutions of the linear system (7.5.6)

ajq(z,

s)

+i

s)

8jq'

I

q

d.

We note that for fixed j, with j d, the system (7.5.6) has a unique solution defined near the origin, by the fact that = 0. The lemma follows by differentiating the boundary value f(z, s + s)) 1

firstwithrespecttos1,

j = 1,...

£ = 1,... ,n.

0

The details are left to the reader. PROOF OF THEOREM 7.5.1.

By Theorem 7.2.6, the conclusion of Theorem 7.5.1 holds for m = 0. We shall give the proof for nz = 1 and leave the induction for

m > I to the reader. Suppose then that u = bvf is of class C1. Let W be a wedge with W -< W. It suffices to prove that each derivative

and

where

j = I,... ,d, £ = 1,... ,n, extends continuously (in W) up to the edge M. We make use of the identities (7.5.4) of Lemma 7.5.2. The continuity of and (in W) up to the edge M follows from Theorem 7.2.6, since the right hand sides of the identities in (7.5.4) are continuous, and are of slow growth in any wedge (with edge M centered at 0) with W -< W1 -< W. The proof of Theorem 7.5.1 is complete form = 1. 0

CHAFFER VIII

HOLOMORPHIC EXTENSION OF CR FUNCTIONS

Summ2ry The main result of this chapter is that minimality is a necessary and sufficient condition for holomorphic extension of all CR functions from a generic submanifold M in C" into an open wedge; this result is stated in §8.1 In §8.2, we give sufficient conditions, in terms of disc derivatives, for filling a wedge, with edge M, with analytic discs attached to M. A precise description of the tangent space of the manifold of analytic discs attached to M at any disc is given in §8.3. The definition of the defect of an analytic disc (which is a nonnegative integer) is presented in §8.4, first in geometric terms and then in local coordinates. In §8.5 the rank of the evaluation and derivative maps at a disc is given in terms of the defect of that disc. The proof of the sufficiency in the main result is given in §8.6. The necessity is addressed in §8.7. Further results on wedge extendibility of CR functions are mentioned without proof in §8.8.

§8.1. Criteria for Wedge Extendability of CR Functions Recall from § 1.5 that a CR submanifold of CN is minimal at po if it contains no proper (i.e. of smaller dimension) CR submanifold of the same CR dimension through P0. We may now state the main result of this chapter: Minimality is a necessary and sufficient condition for holomorphic wedge extension of all CR functions. THEOREM 8.1.1. Let M be a generic submanifold of

and p0

of codimension d

M. if M is minimal at po, then for every open neighborhood U of po in

M there exists a wedge W with edge M centered at P0 such that every continuous

CR function in U extends holomorphically to the wedge W. Conversely, if M is not minimal at then there exists a continuous CR function defined in a neighborhood of P0 in M which does not extend holomorphically to any wedge of edge M centered at The sufficiency of minimality (Tumanov's Theorem) will be proved in §8.6 by using the Banach space methods developed in Chapter VI. The proof of the necessity will be given in §8.7. 205

VIII. HOLOMORPHIC EXTENSION OF CR FUNCTIONS

206

§8.2 Sufficient Conditions for Filling Open Sets with Discs

We shall use the notation of §6.5 and give a sufficient condition at a disc U) (see (6.5.22)) to guarantee that the images of all discs close to A

A

in A(M, U) fill an open wedge with edge M centered at p0.

If A E ..4(M, U), we let A(9) :=

A(eae). Since the curve 9 i-+ A(9) lies in M,

Let We also write A'(O) := we have for every 9, E U) for which A'(O) P0 E M. We shall be interested in finding discs A E the complex tangent space to M defined by (1.2.7). The significance of this condition is contained in the following observation. Let = + h1. If A is as i-+ E Ci" does not necessarily lie in M. above then the curve [—1, 1] However we have: LEMMA

8.2.1. If A E

then E

E

T;0M.

PROOF. By the definition of A, we have A'(O) = in

A

in

we then obtain, using Propo-

sition 2.3.7,

JA'(O) = JA.

(8.2.2)

=

= —Ad,

where J is as defined in § 1.2. This proves the lemma by the definition (1.2.7) of

0

TAM.

For motivation, we shall deal first with the case of a hypersurface, and then proceed to the more general case of a generic submanifold of higher codimension. Let M is a hypersurface in M, p a defining function for M near P0. p0 and 0 a small neighborhood of po in We denote by (resp. 0) the set (Z 0 : p(Z, 2) > 01 (resp. {Z 0 : p(Z, 2) <0}). We shall refer to and

as the two sides of M in 0. THEOREM 8.2.3. Let M be a hypersurface in

0

such that

and let P0 E M. There

if there exists an analytic disc A0 E

U) with hA0 — g then the images of all the analytic discs (P0)11 <E and satisfying A E .A(M, U) close to A0 fill (at least) one side of M near po.

is

PROOF. We may assume that Po = 0 and (z, w) are regular coordinates given is small enough, we may write =

by (1.1.7). Then if

§8.2 SUFFICIENT CONDITIONS FOR FIWNO OPEN SETS WITH DISCS

where z°(C) and

ci.a,

vanishing at

are

=

1.

207

analytic discs valued respectively in C of class = We write Note that since Ao +

is attached to M we have

ES',

=

(8.2.4)

where T1 is the modified Hubert transform defined by (6.2.5) and (6.2.11). Let yE and a E R, both sufficiently small. We define a family of discs attached to M and satisfying a)), with A00 = A0. = (y, a + For this we write where = + y. Then = = s ) + it and hence the family of analytic discs A are uniquely determined by Theorem 6.2.12. Let U be a small neighborhood of(y, a) = (0,0) in x R. We shall need the following lemma. The mapping U x [—1, 1] maximum rank (= 2n + 1) at y = 0, a = 0, LEMMA 8.2.5.

PROOF. It is easy to check that = s°(e'°). For clarity we write

w(y, a; = s(y, a; + it(y, of z and w with respect to y, and

0.

(y, a, = 1, jf and only

is of

E

g

g

if and only if 0, where a; and w(y, a; = (z(y, a; We must compute the partial derivatives a, at the point (0, 0, 1). Since zQ', a; = z with respect to y, written is the identity

We claim that 0, 1) = 0. Indeed, we have s(y, 0; 1) = 0 0) from which the claim immediately follows. Similarly,

and t(y, 0; 1) = Ø(y,

l)=0. It is clear that *(0, 0; 1) =

0 and that

By the Cauchy-Riemann equations, 0; 1) = 1). Since is tangent to 5' at = for

hence 4ui(0, 0; 1) = (y, a,

i—÷

Lemma 8.2.5.

at

E S'. we have

0; 1) = 0; 1) and

0; 1) =

1.

0; 1) =

= 0, and t(0,0,C) = 1) = and

We conclude that the Jacobian of the map (0,0, 1) is a nonzero multiple of which proves

0

We may now complete the proof of Theorem 8.2.3. It follows from Lemma 8.2.5 that the mapping U x [—1, 1] (y, a, i-+ E is a 1-sided local diffeomorphism near (0, 0, 1). (This can be seen by extending the map to U x ]R in a C' fashion.) Note that the image of U x (1) is an open neighborhood of 0 in M. This proves Theorem 8.2.3. 0

EXAMPLE 8.2.6. We take M to be the Lewy hypersurface in C2 given by {(z, w) : Im w = 1z12) and po = (0, 0). Let E be positive. We use the notation of

208

VIII. HOLOMORPHIC EXTENSION OF CR FUNCTIONS — 1). Using (6.2.11) the disc = —iE2(e'° — e10). Since 10(e'°) = Hence the = 2iE2(1 — 1), we obtain

Theorem 8.2.3 and its proof. We take for

and (8.2.4), we obtain s°(O) = Iz°(e10)12 = €2((e'°

—

1)(e_iO

—

sO(eiO)

disc

=

— 1),

2ie2(l

—

and we have = 2E2 0. Again using attached to M, with Ao(1) = (6.2.11) and (8.2.4), the reader can verify by direct computation that is

l)+y, —1 so that the can be easily checked.

1)+a+iIyI2),

1, fill the region above the hypersurface near 0, as

8.2.7. It follows from the proof of Theorem 8.2.3 that the sign of is positive, then determines the side filled by the analytic discs. if is negative, Re w) > 0 near 0. if the discs fill the side Im w — w) <0 near 0. In Example 8.2.6 the Re then the side is given by Im w — = 2E2 is positive. constructed discs fill the side Im w — 1z12 > 0 since

We return now to the general case of a generic submanifold of codimension 1. The following result states that the existence of a single disc A0 E ATh(M, U) with certain properties is a sufficient condition for the images of small discs attached to M to fill an open wedge with edge on M.

d with d

8.2.8. Let M be a generic subman (fold CN and p0 E M. Then, disc A0 E A,,0(M, U), there exists E > 0 with the following property. If there is a -+ defined with lAo — (po)ll <E, such that the mapping Q: by

(8.2.9)

Q(A) :=

a submersion at A = A0, i.e. its tangent map TAO(ATh(M, U)) -+ surjective, then for any 8 > 0 sufficiently small, the set is

(8.2.10)

U

is

A([—1, 1]),

AE.4(U.U)

contains an open wedge W with edge M centered at P0. In particular, the images

of the discs A E ..4,,0(M, U) with hA — Aohl <8flllthe open wedge W. g then W is a wedge in the direction r, where r is an open convex if

§8.2 SUFFICIENT CONDITIONS FOR FILLING OPEN SETS WITH DISCS

209

If A(O) E TAM, then W is

cone containing the vector an open neighborhood of po in CN.

PROOF. We assume that there exists a disc A0 as in the statement of the theorem.

We choose regular coordinates (z, w) E C" x

vanishing at po, so that M is

given by

1mw

(8.2.11)

= 0 and d4'(O) =

where

0.

In these coordinates, we write

=

(8.2.12) We also identify the tangent space Thus, we may write

with the space C" x Rd in the usual way.

EC" 8.2.13. Let A0 be as above. Then, there exist r > 0 and a smooth mapping v A,, E A,,0(M, U), defined for v E Rd with lvl
= (uo,vo+ v),

v E R", lvi
PROOF. Recall that the mapping g: U) —+ T,0M C" x Re', defined by (8.2.9), is a submersion at A0. By Theorem 6.3.42, there exists a smooth right inverse from a neighborhood of (u0, v0) in C" x Rd. The desired mapping is obtained by (8.2.15)

v —÷ A,,

:= fl(uo, vo + v),

0

which is defined in a sufficiently small neighborhood of 0 E Rd.

We now construct a smooth mapping C" x Rd x Rd

(y, a, v) -÷

Ay.a.,, E

..4(M, U), defined for (y, a) in a sufficiently small neigithorhood U of 0 E C" x Rd and lvi 0, as follows. We write

(8.2.16) A,,

is as in Lemma 8.2.13. We define, for(y,o)

C" x

and V E Rd sufficiently small,

(8.2.17)

:= (z(v;

+ y, w(y, a, v;

(y,a)€ U, lvi
E

where w(y, a, v; .) is the unique C" valued analytic disc with w(y, a, v; 1) = a associated to the C" valued disc z(v; + y given by Theorem 6.2.12. We shall complete the proof of Theorem 8.2.8 by proving the following lemma.

VIII. HOLOMORPHIC EXTENSION OF CR FUNCTIONS

210

0, then the set

With the notation above, if v0

LEMMA 8.2.18.

(8.2.19)

1))

(,J

contains an open wedge W with edge M centered at p0 in the direction r, where r is an open cone in containing v0. If vO = 0, then (8.2.19) contains an open neighborhood of Po in CN. We replace by I — in (8.2.17) and expand in > 0) using (8.2.14). We obtain PROOF.

(8.2.20)

—

U = (y + a(v,

a)+i(vo+

a +h/(y,

= 0 (for

around

+b(y, a, v,

+c(y, a, v,

+

wherea(v,

b(y, a, v, are C'1 valued C'" functions and c(y, a, u, isa function, all defined near 0 in C'1 x x x R. (For convenience, we have extended the disc in a C'" fashion across = 1; the expression a CIa denotes function such that is bounded.) We must show that there exists a neighborhood w of 0 C'1 x Rd and a conic neighborhood F' of V0 E Rd and a neighborhood ofO x R" x Rd such that if(z, s) w, t E F' sufficiently small, then there exist (y, a, v) E and 0 < <2 such that Cd valued

I

(8.2.21)

=

We start by considering the equations for y and a (8.2.22)

as

functions of v,

z,

s,

=z a + Re (b(y, a, v)y + c(y, a,

+

= s.

The first equation gives

y =z

(8.2.23)

Substituting this into the second equation and applying the C' implicit function theorem we obtain a solution for a of the form (8.2.24)

a

is an R" valued C'

where z, s, (8.2.20), we obtain

(8.2.25)

—

U=

+ i(v0 +

s

+ i4(z,

+ urn

function.

Substituting for y and a in

s) +

(b(y, a, v)y + c(y, a,

a)

s)}

—

+

§8.2 SUFFICIENT CONDITIONS FOR FILLING OPEN SETS WITH DISCS

and a = a(v, z, s,

where y = y(v, z, s,

211

Comparing (8.2.25) with (8.2.21),

we obtain the equation (8.2.26)

t= +

+ Im (b(y, a, v)y + c(y, a, We write

(8.2.27)

f(v, z, s,

:=

a) — 4'(z,

s)

+

+ Im (b(y, a, v)>' + cQ', a, where y

y(u, z, s,

and a = a(v, z, s, f(v, z, s,

(8.2.28)

=

as above. Thus, writing z, s,

the equation (8.2.15) becomes (8.2.29)

t

Observing, making use of (8.2.23) and (8.2.24), that (8.2.30)

s) =

a) —

s+ and ö = à(v, z, s, Using (8.2.27) and (8.2.30), the reader can check that there exist a neighborhood cv of 0 E C" x R" and a 8 > 0 such that where a = a(v,

(8.2.31)

r0

—,

ae

1

—,

2

4

(z, s)

cv, 0

8,

lvi
where e(v, z, s, is given by (8.2.28) and r0 is as in (8.2.17). (For this, note that e(v, 0, 0, 0) = 0 and 0, 0, 0) = 0 for lvi
for each (z, s) E cv, 0

8, and U' E

(8.2.32)

v + e(v, z, s,

=

v'.

We claim that (8.2.32) has a unique solution v = v(v', z, s, for each iv'i ro/4. Indeed for such v', the mapping v —+ v'—e(v, z, s, is a contraction on the closed

212

VIII. HOLOMORP}HC EXTENSION OF CR FUNCTIONS

ro/2 by (8.2.3 1). Hence, it has a unique fixed point v = v(v', z, s, ball lvi which gives the desired solution of (8.2.32). This proves the claim. 0. To complete the proof of the first part of Lemma 8.2.18, we assume that vO be an open convex cone generated by B(vo, ro/4), the ball centered Let r C at vo E Rd of radius ro/4. (After shrinking r0 if necessary, we may assume, 0, that ro/4 < ivol.) We claim that there exists 8' > 0 such that, for since 00

every (z.s) E wand t

F with Ill < 8', (8.2.29) can be solved for v and 8. To prove the claim, we pick A > 0 such that ro/2 and 0 with lvi At E B(vo, ro/4). We choose 8' > 0 such that if iti < 6' and At E B(vo, r0/4), <8 by the choice of 8'. To then A> 1/8. Choose = 1/A. Observe that 0 solve (8.2.29), we must then solve (8.2.33)

At =

00

+ v + e(v. z, s,

ro/2. Since At B(vo, ro/4), there is a unique such v with for v, with lvi ro/2 by the previous claim. This proves the claim and, hence, the first part lvi of Lemma 8.2.18. To prove the second part of Lemma 8.2.18, we assume 00 = 0. The equation (8.2.29) becomes (8.2.34)

t=

+ e(v, z, s,

We claim that there exists 8' > 0 such that, for every (z, s) E w and t E Rd with 1:1 < 6', (8.2.34) can be solved for v and with lvi ro/2 and 0 6. The proof of this claim, with the choice 6' < 6r0/4, is completely analogous to the proof above and is left to the reader. The proof of Lemma 8.2.18, and hence Theorem 8.2.8, is now complete. 0

§8.3. Thngent Space to the Manifold of Discs In A0 E

we shall prove that if M is minimal at po there exists a disc

(M, U) satisfying the conditions of Theorem 8.2.8, which is the main step in the proof of the sufficiency in Theorem 8.1.1. We begin by giving a more explicit description of the tangent space to the Banach manifold ..4(M, U) of analytic

discs attached to M introduced in §6.5. We shall first prove an auxiliary result, Proposition 8.3.1 below, concerning general matrix-valued families of analytic discs.

We denote by M(d, C) the space of d x d matrices with entries in C and by GL(d, C) the open subset of invertible matrices. We use similar notation M(d, R)

*8.3. TANGENT SPACE TO THE MANIFOLD OF DISCS

213

and GL(d, R) for matrices with real entries. We shall also consider the Banach M(d, R)), where C' a denotes the Holder spaces C'"(S', M(d, C)) and space as defined in Chapter VI and S' is the unit circle. For a E M(d, C) we = a for 5', M(d, C)) the constant function denote by (a) E so that (GL(d, C)) C C'°(S', M(d, C)) denotes the set of constant invertible functions with values in GL(d, C). PR0IosmoN 8.3.1. Let 0
<

1

and d be fixed. There exist R., an open M(d, C)) anda (unique) smooth map-

ping (8.3.2)

M(d, R)),

: R.

the following. i-* (I) For each m E R.., the mapping S' E M(d, C) extends holomorphically to a mapping g on the open unit disc with

E GL(d, C)for all in ó. (ii) [v(m)J(1) = 'd. where 'd is the d x d identity matrix. PROOF. We shall make use of the implicit function theorem in the Banach space setting (Theorem 6.3.8). Let X = C1 a(SI,M(d, C)) and Y the subset of X consisting of those matrix valued functions whose components all extend

holomorphically to & Observe that Y is a closed vector subspace of X since Y is defined by the vanishing of the negative Fourier coefficients of the entries of the

M(d, R)). Consider the

matrix function (see e.g. (6.1.32)). Let Z = bilinear (and hence smooth) mapping

x M(d, R) B(n, h) := (Im hn, [Re hn](l)). B : X x Y -± Z

(8.3.3)

Note that B((Id). (Id)) = (0, Id), where 0 is the zero function in Z. We claim that (Id)) is an invertible linear mapping from Y to Z x M(d, R). First, the reader can easily check that (8.3.4)

(Id))i = (lmi,Reiz(l)), vh

y•

To prove the claim, we must show that for any f E Z and any a E M(d, R) there is a unique h = fl(f, a) E Y such that Im 1 = and Re h(1) = a. For this we

f

take

(8.3.5)

a) =

(a)

—

T, f

+ if,

VIII. HOLOMORPHIC EXTENSION OF CR FUNCTIONS

214

where T1 is the modified Hubert transform defined by (6.2.5). The continuity of the inverse mapping defined by (8.3.5) follows from that of T1 (see Proposition 6.2.10). Finally, the uniqueness follows from the fact that a holomorphic function with vanishing imaginary part must be constant. Hence the mapping defined by (8.3.4) is invertible, which proves the claim. By the implicit function theorem, there exists a unique smooth mapping n i—* h(n) E Y defined in a neighborhood U of (id) in X such that

Lmh(n)n =

(8.3.6)

0,

Viz €U.

IRe h(n)n](1) = Id.

We may now define v as follows. Consider the smooth mapping (8.3.7)

m

i-÷ m(l)m' E X

defined in an open neighborhood of (GL(d, C)) in X. To prove the proposition, it suffices to take

v(m) =

(8.3.8)

e(m),

where U is the domain of definition of h as above. To show that m( 1) has the desired property, it suffices to apply the maximum principle to the determinant of [h which is close to 1 on 5' and hence nonvanishing on A. This completes the proof of Proposition 8.3.1. El with liZ. =

=

COROLLARY 8.3.9. LetlZandv(•), be given by Proposition 8.3.1. If

d x I vector of real valued functions in C°(S') andm E

)f

E

Cd extends holomorphically to

ES'.

extends holomorphically to

PROOF. Since

(8.3.11)

liZ such that S'

then

= T[v(m)](C)f(l),

(8.3.10)

i-+

isa

so

does

=

by the properties of v. Since the latter is real valued, it must be constant, and hence

(8.3.10) follows since [v(m)](1) = I,. The proof of Corollary 8.3.9 is done.

0

Before giving the description of the tangent space to ..4(M, U), as mentioned above, we shall introduce some necessary notation. Recall that for any positive integer k, Dk is the space of all analytic discs valued in Ck defined by (6.5.1). As in

§8.3. TANGENT SPACE TO THE MANIFOLD OF DISCS

215

M, let p(Z, Z) be a local defining function If A B((po), E) (defined by (6.5.6)) for M in a neighborhood of po in and E > 0 sufficiently small, then by using (6.5.10), the matrix-valued function is close to (Id) in M(d,C)). Hence, we S' may make use of Proposition 8.3.1 and write the proof of Theorem 6.5.4, for p0

:=

(8.3.12)

with

=

For convenience we define the d x N matrix-valued function rA on S' by

:=

(8.3.13) so that

=

(8.3.14)

S'

With this notation we have i-+

E

extends holomorphically to

For a constant k1 x k2 matrix Q with complex coefficients, we extend Q to act as in the obvious way. a linear operator, denoted also by Q, from to

M andA E A(M,U) fl B((po), E). Then for PROPOSITION 8.3.15. Let P0 E > 0 sufficiently small, an analytic disc B E D's' is in TAA(M, U) and only if there exists C DN satisfying (8.3.16)

150)C = 0,

and c E Rd such that

B=C+PD,

(8.3.17)

where P is the constant N x d matrix given by (6.5.10), and D E its restriction to S' is given by

(8.3.18) —2b = (vArA PY'

+ VArAC + i T1

is such that

+ VArAC) + ic),

where VA is given by (8.3.12), rA by (8.3.13), and T1 the modified Hubert transform given by (6.2.5). C and c are unique.

We observe that if A =

then (8.3.17) reduces to the description of the tangent space at (P0) given by (6.5.15). For the proof of Proposition 8.3.15 we need the following. (P0)

VIII. HOLOMORPHIC EXTENSION OF CR FUNCTIONS

216

8.3.19. Let A E B( (p0). €). If E > 0 is sufficiently small, then for any f, g E C1 (S', R") the following are equivalent. where R'(A) is given by (6.5.8) and by (6.5.12). (i) ,g = (R'(A) o (ii) VA g = Re [(VArA P)(f + IT1 f)], with VA, rA, P and T1 as in Proposition 8.3.15.

PROOF. Assume (i). Then we have by the definition of R'(A) and fi, (8.3.20)

(rAP(f +iTif)

g=

—

iTif)).

Multiplying both sides by the real-valued matrix we obtain (ii). The converse is proved similarly since VA is real valued and invertible. 0 PROOF OF PROPOSITION 8.3.15. We shall make use of the parametrization of the tangent space TAA(M, U) in terms of the tangent space U) given by Proposition 6.3.29 for A sufficiently close to (P0). On the other hand, it follows from the description of U) given by (6.5.15) that

(8.3.21)

CEVN, Hence (8.3.22)

D's' isin

cElRd}.

TAA(M,U) ifandonly ifitisoftheform

B=

(iPc)

(iPc)).

We now make use of Lemma 8.3.19 to calculate

(8.3.23)

f = (R'(A)fi)'R'(A)(à + (iPc))

by taking (8.3.24)

g=

We first consider the case

= 0 in (8.3.22—24). We have by (ii) of Lemma 8.3.19

(8.3.25)

VA R'(A)(iPc) =

+ (iPc)).

Re

[(UATAP)(f + iT1f)],

and replacing for R'(A) we have (8.3.26)

i(VArAP

—

= Re [(VArAP)(f + iT,f)].

217

§8.3. TANGENT SPACE TO THE MANIFOLD OF DISCS

Since the function

P1(e) extends holomorphically to

i-+

2j(VATA P)c

(8.3.27)

(8.3.26) yields

= (VATA P)(f + iTif) + ic1,

R" uniquely determined from c. Multiplying both sides of (8.3.27) by P(VArA Py' (which extends holomorphically to by the choice of VA given by Proposition 8.3.1) we have with

c1

=

(8.3.28) and

hence (8.3.22), with

= iPc—iP(VATAPY'cl, = 0 becomes

B = iP(VATAPY'cl.

(8.3.29)

By using the injectivity of the map (6.3.31), we observe that Cl must take all values in Rd as c varies in R". We next consider the case c = 0 in (8.3.22—24). With the corresponding choices

for f and g we obtain in this case (8.3.30)

vAg = VATAC + VATAC = Re [(VATAP)(f

Hence since

extends holomorphically to

i—*

+ iTif)]. and so does

i-+ [VATA? + VArAC + IT, (VATAC +

VATAC

+ VArAC + iTl(VArAC + VATAC)

= (VArAP)(f + iTif) + 1C2, where

C2

(8.3.32)

R" is uniquely determined by Hence we have

P(f+iT1f)= P(VArAPY'[vArAC + VArAC + iTI(VArAC + VArAC) — 1C2],

so

that

(8.3.33)

in this case (C = 0) is given by

=

(vArAa+vArAe)—1c2].

The proof of (8.3.17) and (8.3.18) now follows from (8.3.33) and (8.3.29) (corresponding to the case = 0) since, as mentioned above, Cl in (8.3.29) is an arbitrary vector in Rd. The uniqueness of and C follow from the injectivity of

(6.3.3 1). This completes the proof of Proposition 8.3.15.

0

The following corollary gives a description of the tangent space to the submanifold (given by (6.5.22)) and is an immediate consequence of Proposition 8.3.15 as well as Proposition 6.5.23 and its proof.

VIII. HOLOMORPHIC EXTENSION OF CR FUNCTIONS

218

COROLLARY 8.3.34. Let p0 E M and A E A,,0(M,U)

fl B((po),

E). Then for

E > 0 sufficiently small, an analytic disc A E DN is in TA A,,0 (M, U) if and only if there exists C E V's' satisfying

(8.3.35)

= 0,

pz(po,

C(1) = 0,

such that (8.3.36)

A

+ pb,

=

where P is the constant N x d matrix given by (6.5.10) and b E

is such that

its restriction to S' is given by

(8.3.37) VA

—2b =

+ VATAC

+

+ i

is given by (8.3.12), rA by (8.3.13), and T1 the modified Hilbert transform

C is unique.

given by (6.2.5). §8.4.

Defect of an Analytic Disc Attached to a Manifold

In this section we shall give a geometric definition of the notion of defect of an analytic disc, which is crucial in the proof of Theorem 8.1.1.

As usual, we denote by TCN the real tangent bundle of C" and CTCN its and CT*CN complexification. Let T*CN be the (real) cotangent bundle to CN, can be a covector a in the complexified cotangent bundle. For Z written in the form (8.4.1) consisting of covectors of the form N. The bundle T*I.OCN is then a complex mani= 0. j = (8.4.1)with fold which may be identified with C2N by coordinates (Z1 AN). ZN, A1 We denote by T" 0C" the subbundle I

We

0

described as follows. Given

consider the mapping S : T*CN .÷

=

(8.4.2)

+

E

S(6)

define S(O) by

a = 2i

E

so that (8.4.3)

(6, X) = Im (S(O), X),

VX

TZCN.

§8.4. DEFECT OF AN ANALYTIC DISC A1TACHED TO A MANIFOLD

219

For each Z E CN, the mapping S is then a real linear isomorphism from onto

Let M be a smooth real submanifold of CN and denote by TM C TCN, the real tangent bundle of M and by CTM C CTCN its complexification. We also denote by T*M and CTM the real and complexified cotangent bundles of M, respectively. Let L M -+ be the embedding of M into For p E M, the pullback of real covectors is given by -÷ M such that :

X) = (0, X),

(8.4.4)

X E TAM,

0E

(,) on the left hand side denotes duality between T,,'M and and (,) the right hand side denotes duality between TCN and Similarly, we have the pullback of complex covectors where on

:

C

TC" ofMisdefinedforp := {e

(8.4.5)

T;c":

Mby

(0, X) =

0, VX

Note that is the kernel of the map defined above. If ps,... Pd valued smooth local defining functions for M we have for p M

=

(8.4.6)

a1

=

are

real

RI.

Similarly we have

= f >bjdpj(p), b1 E C}.

(8.4.7)

It easily follows that the restriction of

cT;M.

(8.4.8)

is injective for all p T*CN

(8.4.9)

M if and only if M is generic. Since the mapping S

T;I.OCN defined by (8.4.2) is injective, it follows that o

5:

T*CN -+

CT;M is injective

We have the following proposition.

M is generic.

220

VIII HOLOMORPHIC EXTENSION OF CR FUNCTIONS

then the PROPOSmON 8.4.10. Let M C Ci" be a generic smooth o is a smooth submaiufoldof T*M, and for every p E M S)(EM) set

(8.4.11)

= (T;M)1.

o

where TCM is the complex bundle of M (as defined in

1.2)), and where orthog-

and TM.

onaliiy is taken in the sense of duality between

o PROOF. The fact that is contained in T;M easily follows from (8.4.2), (8.4.3) and (8.4.5). To prove the equality (8.4.11) it is convenient and instructive to take local regular coordinates Z = (z, w) E x C" vanishing at p and assume that M is given by (1.3.7). In these coordinates, is spanned by

the following tangent vectors at 0: (8.4.12)

dz1

+ 2-, — aWk

—--,

I

aWk

<j
1
d.

It follows from (8.4.12) that (8.4.13)

ER),

j

and hence (8.4.14)

= { ibjdWj(O).

b1

E

Rj.

We take z1,

and 5k = Re Wk to be local coordinates for M near p. Since on M s) with we have wk = sk + = 0, we conclude that = dsk(0). Then (8.4.14) implies

b1 ER).

(8.4.15)

Since TM is spanned by

and

8.4.16. Since

=

ectors {idpi(p),... for p E M

,

the desired equality (8.4.11) follows.

+ ap1, it follows from (8.4.6) that the coyover R. Hence we have

form a basis of

tiER), (8.4.17) o

0

={i

E

Rj.

§8.4. DEFECT OF AN ANALYTIC DISC ATrACHEI) TO A MANIFOLD

221

REMARK 8.4.18. Let p = (pi,... ,Pd) be a real-valued, smooth, defining function forM near po E M, and P an N x d complex matrix satisfying (6.5.10). the embedding of M in CN. The reader can As usual, we denote by L : M —÷ easily check that for p E M sufficiently close to p0. any covector 9 TM can be uniquely written as

9=

(8.4.19)

with tPa

0, where a is the N x 1 matrix whose coefficients are the complex numbers a1 and the are real numbers. It will sometimes be convenient to write (8.4.19) in the more compact form (8.4.20)

+ T&dZ + i Tt8p(p)), a E CN, TPa

9=

= 0, t ER'1,

with the obvious matrix notation.

Since S(EM) is a real submanifold of dimension 2N contained in the complex manifold of complex dimension 2N, we may consider continu—* ous analytic discs B attached to S(EM). The base projection of such a disc will then be an analytic disc in attached to M. Taking (Z1,... , Zn,, A1,... , for global holomorphic coordinates in we have

=

(8.4.21)

with where each A1(.) is a scalar continuous analytic = (Xi(C),... , disc. The disc B given by (8.4.21) is attached to EM if and only if

(8.4.22)

A(S') C M and

E

E S'.

For a continuous analytic disc A attached to M, we denote by VA the set of all discs of the form (8.4.21) which are attached to S(EM). Note that by (8.4.22), VA is equipped with a natural real vector space structure. For E S'. let be the subspaoe defined by

C

(8.4.23)

:=

DEfiNITION 8.4.24. If

define the defect of A at defined by (8.4.23).

y=

(A(.),A()) E VA}.

E S' and A is an analytic disc attached to M, we deft A, as the dimension of the real vector space VA

VIfl. HOLOMORPHIC EXTENSION OF CR FUNCTIONS

222

By Proposition 8.4.10, we have, for every

S'.

C

Recall also that is injective on S(EM) and hence def1A is also the dimension For convenience of notation we shall define the of the real subspace real subspace C T(1)M by

=

(8.4.25)

The following shows that for a disc A

disc, its defect at

C

C

A(M, U) sufficiently close to a constant

S' is independent of

We have:

THEOREM 8.4.26. If U is a sufficiently small neighborhood of (M) in Dr', then is independent forA E ..4(M,U), €S'.

If U is as in the theorem and A A(M,U), we denote by def A the defect of Theorem 8.4.26 will be a consequence of the following. A at any PRoPosmoN 8.4.27. Let P0 M and p a defining function of M near Po• If A(M, U)is sufficiently close to (Po) S', is in VA(CO) if and only A

=i

(8.4.28)

for some b

Rd andvA given by (8.3.12), and such that the mapping 5' extends holomorphically to &

The reader should note that in (8.4.28) is ad x 1 matrix with values in is ad x d matrix of real numbers, and Tb, the transpose of b, is a 1 x d matrix of real numbers. PROOF OF PROPOSITION 8.4.27. We shall make use of Remark 8.4.16 and, in particular, of the first equation in (8.4.17). If A A(M, U) is sufficiently close to

(po),ananalyticdisc EM if and only if

oftheform(8.4.21)with A

=i

(8.4.29)

A(M,U) isattachedto

S',

with continuous in and holomorphic in i-+ and 5' class C°. This implies (using the notation (8.3.13)) that the map SI

€

E Rd of

§8.4. DEFECT OF AN ANALYTIC DISC ATrACHED TO A MANIFOLD

223

extends holomorphically. In particular, if P is the matrix given by (6.5.10), then the map given by C" S' 3 l-+ given by also extends holomorphically. By applying Corollary 8.3.9 to where is given by (8.3.12). (8.3.12), we conclude that = This completes the proof of the The characterization (8.4.28) easily follows.

0

proposition.

sI the covectors

PROOF OF THEOREM 8.4.26. Since for every

linearly independent and the matrix sition 8.4.27 that are

is

invertible, it follows from Propo-

= dimR fb

(8.4.30)

S'

extends holomorphically to

i-÷

This completes the proof of the theorem, since the right hand side of (8.4.30) is

independent of

E

0

S'.

REMARK 8.4.31. It follows from the proof of Theorem 8.4.26 that if

is

a continuous analytic disc attached to S(EM) of the form (8.4.21) with A E A(M,U) sufficiently close to (Po) in DN, then in fact B is of class C'°. COROLLARY 8.4.32. flu is as in Theorem 8.4.26, then the mapping

A(M,U)

A i—p def A

is upper semicontinuous. PROOF. Let

denote the space of infinite sequences valued in

and

regarded as a real vector space. For A E A(M, U) sufficiently close to (Po) as in the proof of Proposition 8.4.27, consider the linear mapping UA Rd .+ (CN)Z+ defined as follows. For b E Rd. write the Fourier series expansion

=

rbvA(eiO)pz(A(eio)

A) E

(8.4.33)

and define Rd

b i-* UAb

UAb = (a_i (b, A), a_2(b,

(C's')"- by

A),...

,

A),...).

224

VIII. HOLOMORPHIC EXTENSION OF CR FUNCTIONS

Clearly UA is a linear mapping, and each component

A) depends continuously on A E A(M,U). By (8.4.30), we have def A = dimRkerUA. Since d — rank UA, it suffices to show that the mapping A(M, U) is lower semicontinuous. If A A(M, U) is sufficiently A i-+ rank UA E close to (p0) with rank UA = r, we claim that rank UN > r for A' A(M, U) sufficiently close to A. Indeed, to prove the claim, it suffices to observe that the rank UA = r implies the nonvanishing of an r x r determinant extractCd from the components of a_j(e*, A), j = I,... , and I
of the Evaluation and Derivative Maps

§8.5.

Throughout this section we fix po E M and we assume U sufficiently small so U), the set of discs attached to M through P0 defined by (6.5.22), is that a manifold. (See Proposition 6.5.23.) S' we consider the evaluation map defined by For (8.5.1)

DN

:=

A i-÷

We also define the derivative map

(8.5.2)

by

:=

A

TA(I)C" C TCN.

We note that the evaluation map (for

S' fixed) and the derivative map are both smooth. (The former is linear.) We shall restrict both mappings to the fixed submanifold C 1Y'. Hence we have the smooth mappings between manifolds (8.5.3) (8.5.4)

-+ M,

:

: ATh(M,U) -+

For convenience of notation we shall write A for ATh(M,

U). For

A

A, we

consider the derivatives 10(A) and Q'(A) which are real linear mappings (8.5.5)

TAA —÷ TA(CO)M,

(8.5.6)

TAA

T,,0M.

Note that since the target manifold

in (8.5.4) is a vector space, we have identified, as usual, its tangent space to the space itself in (8.5.6). The main theorem of this section is the following, which relates the rank of the mappings (8.5.5) and (8.5.6) to the defect of the disc A.

225

§8.5. RANKS OF THE EVALUATION AND DERIVATIVE MAPS

E S',

THEOREM 8.5.7. Let

and

AE

C

1 and P0 E M. flu is sufficiently small, defi ned by (8.4.25), then

=

(i)

(ii) Q'(A)TAA = where orthogonality is taken in the sense of duality between TM and T*M. By using the inclusion in (8.4.25), we obtain the following immediate corollary of Theorem 8.5.7. COROLLARY 8.5.8. Under the assumptions of Theorem 8.5.7, the codimension coincide and the codimension of g'(A)TA.A in in and are equal to the defect of A. Furthermore the following inclusions hold:

of

c

(8.5.9)

C g'(A)TA..4.

PROOF OF THEoREM 8.5.7. We shall first prove (i). For this, we shall begin by

showing the inclusion (8.5.10)

.F'CO(A)TAA c

Note that since the map given by (8.5.1) is linear, we have

for all A

=

Corollary 8.3.34 we can write A = C satisfying (8.3.35) and D given by (8.3.37). To prove the

TAA. Using

inclusion (8.5.10) it suffices, by Proposition 8.4.27, to prove that

=0,

(8.5.11)

t-÷ such that S' extends holomorphically to for all b E We replace D in (8.5.11) by using (8.3.37). The left hand side in (8.5.11) becomes TbvA

_.! [rbvA

(

T1

Using Proposition 6.1.38 (iii) we note that T1 f = —if

(to)].

if f is a function on S'

which extends holomorphically to A and vanishes at 1. Now (8.5.11) follows immediately, since both TbvArA and C extend holomorphically to This completes the proof of the inclusion (8.5.10). To show that the opposite inclusion holds, we shall prove

j

(8.5.12)

[710(A)TA.A11 C

VIII. HOLOMORPHtC EXTENSION OF CR FUNCTIONS

226

such that

For this, let (8.5.13)

VA E TAA.

By Remark 8.4.18 (see (8.4.20)),

We must show that decomposition in the form

=

(8.5.14)

has a unique

+ T&dZ + i ¶t

with a E CN, TPci = 0 and t E W'. We shall need the following two lemmas. LEMMA 8.5.15.

Let

be of the form (8.5.14) and satisfying (8.5.13).

E

Then 5' be of the form (8.5.14).

LEMMA 8.5.16.

then necessarily a =

satisfies (8.5.13)

0 in (8.5.14).

We now complete the proof of the inclusion (8.5. 12) assuming Lemmas 8.5.15 and 8.5.16. By Proposition 8.4.27, using Lemma 8.5.16, it suffices to show that such that there exists b e

= TbvA

(8.5.17) and

S'

(8.5.18)

i-+

extends holomorphically to

By Lemma 8.5.15, it suffices to take

=

(8.5.19)

which completes the proof of the inclusion (8.5.12) (and hence part (i) of Theorem

8.5.7). It remains to prove Lemmas 8.5.15 and 8.5.16. PROOF OF LEMMA 8.5.15. We assume the hypotheses of the Lemma. We shall show that if u CN satisfies rA(l)v = 0 then

(8.5.20)

S'

E C,

i—+

extends holomorphically to Suppose this is true. Then we claim that (8.5.20) extends holomorphically for all u CN, which will prove Lemma 8.5.15. Indeed, we write for any v E (8.5.21)

u=

v1

+ u2,

U,

kerrA(l),

v2

Image P,

§8.5. bANKS OF THE EVALUATION AND DERNATIVE MAPS

227

by (6.5.10). If it can shown that (8.5.20) extends holomorphically for v =

kerrA(1), then it suffices to prove it extends holomorphically for v = Image P. The latter statement follows from the fact that '-÷

v2

E

(see (8.3.12) and (8.3.13)). To prove the lemma, it suffices to prove (8.5.20) extends holomorphically for

extends holomorphically by the definition of v

kerrA(1). We begin by choosing A in (8.5.13) of the form (8.3.36) with DIV satisfying (8.3.35) and = 0. We shall show first that with this

choice we have (8.5.22)

= 0.

T1[Tt

and

For this, we note since A(co) = (8.5.23)

(TcidZ + T&dZ,

= 0 it foLlows that

= 0.

Hence it follows from (8.5.13) that (8.5.24)

= 0.

Tt

By replacing i5 in (8.5.24), using (8.3.37) and the fact that

= 0, we obtain

(8.5.25)

VArAC +

= 0.

—

0.

Replacing

by

we also have

(8.5.26)

Now (8.5.22) follows from (8.5.25) and (8.5.26). Recall (see (6.2.11)) that for any function u on S' of class ca we have

PV f

(8.5.27)

For v E kerrA(l) and a nonnegative integer k we let in (8.5.22). It follows from (8.5.22) and (8.5.27) that

(8.5.28)

j

—

—

= 0.

Since (8.5.28) holds for all nonnegative integers k, we conclude that (8.5.20) extends holomorphically for v ker rA (1). This completes the proof of Lemma 8.5.15.

0

228

VIII. HOLOMORPHIC EXTENSION OF CR FUNCTIONS

PROOF OF LEMMA 8.5.16. Let be as in the assumptions of Lemma 8.5.16. We first observe, using Lemma 8.5.15 and Proposition 8.4.27, that E Hence by the inclusion (8.5.10), we have

(8.5.29)

(i

A of the form (8.3.36). Since TPcx = 0, (8.5.13), (8.5.14) A = 0 for all C satisfying (8.3.35). Replacing and (8.5.29) imply that Re the disc C by iC, we conclude that = 0. Since can be taken to be an arbitrary vector in kerrA(1) (and Ci" = kerrA(1) Image P), it follows that 0 a = 0. This completes the proof of Lemma 8.5.16.

To complete the proof of Theorem 8.5.7 we must prove (ii). As in the proof of (i), we begin by showing the inclusion (8.5.30)

Q'(A)TAA D WA(1)1.

Note that Q'(A)A = hA(e18)Ioo for A E TA..4. By Proposition 8.4.27 and Corollary 8.3.34, it suffices to show that

= 0,

+

(8.5.31)

extends holomorphically, with C and for b E Rd such that D as in (8.3.35) and (8.3.37). Recall that VA(l) = rA(l)P = 'dxd. Since 0, we also have 0. Therefore the left hand side rA(1)C(e'°)

of (8.5.31) becomes

Tbd[b(eie)JI

(8.5.32)

Since C(1) =

0, by using the expression for i3 given by (8.3.37) we obtain the following equivalent expression for (8.5.32):

(8.5.33)

—

(T1

+

)

Since TbVATAà extends holomorphically to ó, (8.5.33) becomes

(8.5.34)

—

+

Both terms in the expression (8.5.34) vanish since, as mentioned above, C(l) = 0 0. This proves the inclusion (8.5.30). and

§83. RANKS OF THE EVALUATION AND DERIVATIVE MAPS

229

To show that the opposite inclusion Q'(A)TAA D WA (1)1 holds, we shall prove

[Q'(A)TAA]1 C

(8.5.35) For this, let

E

such that

(8.5.36)

E TAA. We must show that E WA(l). As before, has a unique decomposition in the form (8.5.14) with a E CN, TPa = 0 and t E Rd. We first = 0. Then since rA(l)P = 1, (8.5.36) choose in (8.3.36) so that A

becomes

= 0.

(8.5.37)

Substituting the expression for b from (8.3.37) in (8.5.37) (and using again ë( 1) = = 0) yields

(8.5.38)

TVATAC

+ TtVArAC](ebo)I

= 0.

After replacing C by iC in (8.5.38) we conclude that (8.5.39)

T1 [

] (e'°)

Tt

= 0.

The reader can easily check, using (8.5.27), that if u(C) is a function of class defined on S' with u(1) = = 0, then

=

(8.5.40)

f

V E with rA(l)v = 0 and k a nonnegative integer, we take 1)2ev. Then it follows from (8.5.39) and (8.5.40) that

For —

=

(8.5.41)

This, together with the fact that VA holomorphically, implies that extends holomorphically.

(8.5.42)

i

)rA P extends

E VA(I).

A similar proof to that of Lemma 8.5.16 now shows that a = 0 in (8.5.14). It then follows from (8.5.42) that E WA (1), which completes the proof of the inclusion (8.5.35). The proof of Theorem 8.5.7 is now complete. 0

VIII. HOLOMORPHIC EXTENSION OF CR FUNCTIONS

230

§8.6.

and Extension of CR Functions

In this section we shall use the results of §5.2 to §5.5 to show that minimality implies extension of CR functions, i.e. the sufficiency of minimality in Theorem 8.1.1. This is given by Theorems 8.6.1 and 8.6.2 below. As in the previous section, we let M C CN be a generic submanifold and we fix p0 E M. We shall again use the notation A = U) for the set of discs attached to M through p0 in A(M, U) with U, a sufficiently small neighborhood of the set of constant discs (M). THEOREM 8.6.1.

Let M C be a generic submanlo Id and p0 M. Suppose there that for every E > 0 exists A E A of defect 0 and VA — (po)II <e. Then every continuous CR function defined in a neighborhood of P0 extends holomorphically to some wedge in CN with edge M and centered at THEOREM

8.6.2. If M is minimal at p0. then there exist analytic discs A

A

of defect 0 with II A — (po)II arbitrarily small.

It is clear that Theorems 8.6.1 and 8.6.2 imply the first conclusion of Theorem 8.1.1. (The converse, i.e. the second conclusion of Theorem 8.1.1 will be proved in §8.7.) PROOF OF THEOREM 8.6.1. Let f be a continuous CR function defined in an open neighborhood w of P0 in M. By Proposition 6.2.2 (with M = w) there exists

a neighborhood U of P0 in wand a continuous function F defined in W(U), where W(U) is as in Proposition 6.2.2, satisfying (i), (ii), and (iii) of that proposition. By the hypothesis of Theorem 8.6.1, we may choose a disc A0 of defect 0 such that Ao(S') C U. We may also assume that VAo — (Po)II is smaller than the given by Theorem 8.2.8 and small enough to satisfy the hypothesis of Theorem 8.5.7. By Theorem 8.5.7 (ii), the mapping : A,,0(M,U) -÷ T,,0M is a submersion

at A0. We may then apply Theorem 8.2.8 with 8 > 0 such that A(S') C U for all A A(M,U) with IA — A0II < 8. Let W be the wedge centered at P0 with edge M given by Theorem 8.2.8 with this choice of 8. We therefore have W c W(U)°. (Here, as in Proposition 6.2.2, W(U)° is the interior of W(U) in C".) The restriction F to W gives the desired holomorphic extension of f to the wedge W. This completes the proof of Theorem 8.6.1.

IJ

REMARK 8.6.3. An inspection of the proof of Theorem 8.6.1 shows that the holomorphic extension F of the continuous CR function f given in the proof is in fact continuous up to M. This fact would also follow from Theorem 7.2.6. PROOF OF THEOREM 8.6.2. We shall argue by contradiction. Suppose that there exists E > 0 such that any A E A with hA — (po>II < E has positive

§8.7. NECESSITY OF MINIMALITY FOR HOLOMORPHIC EXTENSION

231

defect. Choose such a disc A0 such that def A0 is as small as possible. Since A0(1) = po, the following lemma shows that without loss of generality we may assume Ao(—l) = po. LEMMA 8.6.4. There exists E

> 0 such that

— (Po)II < E and A0 E .4,

then def A0 = def A1, where A1 is the analytic disc given by A1

=

PROOF OF LEMMA 8.6.4. We note first that by the definition of the matrix The lemma is then an immediate conse=

(see (8.3.12)), we have VA1(C)

quence of (8.4.30), since a function u(e'°) defined on S' extends holomorphically 0 does. if and only if to

We may now complete the proof of Theorem 8.6.2. As mentioned above, we assume that A0(—1) = p0. By Theorem 8.5.7 (i), the mapping F....1 : A -÷ M has maximal rank r < dim M in a neighborhood l..Io of A0 in A. Hence by the rank theorem (Theorem 6.3.34) after shrinking U0 if necessary, the set F... is a smooth submanifold M1 C M with dim M1 = r and Po E M1. By the first inclusion of (8.5.9) it follows that M1 is a CR submanifold with the same CR dimension as that of M. This contradicts the minimality of M at Po. which 0 completes the proof of Theorem 8.6.2

§8.7. Necessity of Minimality for Holomorphic Extension to a Wedge In this section we shall consider the second statement in Theorem 8.1.1, i.e. that minimality is a necessary condition for the extension of all CR functions into a wedge. We start with the case where the generic submanifold M is realanalytic, since the proof of the necessity in this case is much simpler than in the general (smooth) case. (By contrast, we should mention here that the proof of the sufficiency of the minimality condition in Theorem 8.1.1 does not seem to be any simpler when M is real-analytic.) PROPOSITION 8.7.1. Let M C C" be a generic real-analytic submanifold which is no: minimal a: Po M. Then, there exists a smooth CR function h defined in a neighborhood of P0 in M that does not extend holomorphically to any wedge with

edge M centered at PROOF. Recall that, in the real-analytic case, minimality at P0 is equivalent to finite type at Po (see Theorem 4.1.3). Thus, we may assume that M is of infinite type at p0. We choose canonical normal coordinates (z, w, w') E C" x x C". with d' > 0, as given by Theorem 4.5.1. In these coordinates, M is given near

VIII. HOLOMORPHIC EXTENSION OF CR FUNCTIONS

232

po = (0,0,0) by un w (8.7.2)

Im w' = B(z,

Re w, Re w')Re WI,

0,5, s') s, s') 0 and B, as in Theorem 4.3.2, is a d' x d' matrix-valued real-analytic function with B(z, 0,s, s') B(O, s, s') 0. Note that the scalar function is an R"-valuedreal-analytic function with

(s' + iB(z,

s, s')s').

(s' + iB(z,

s, s')s')

has real part cls'12, for (z, s, s') sufficiently close to 0 E c > 0. We claim that the continuously differentiable function (8.7.3)

g(z,

s, s')

= ((s' + iB(z,

s,

s')s') (s' + iB(z,

for some

s,

where the cube root is the usual one defined in the right half plane Re > 0, is CR and does not extend holomorphically to any wedge with edge on M centered at po = (0,0,0). That g is CR follows from the fact that g is C' and coincides with the restriction toM of a branch of (w'•w')213 for all s' 0. To see that g does not extend to any wedge, we argue by contradiction. Thus, assume that g extends to a wedge W with edge on M in the direction r centered at 0. Using a linear transformation, we may assume, without loss of generality, that the open convex cone r contains x R" with:' = (1,0,... , 0). Since g is continuous, its holomorphic (0, :') E

extension f to W is continuous up to the edge M by Theorem 7.2.6. Hence we

mayrestrictftothesetWfl((z,w,w'): z=0, W=0, ,0)). bethecoordinateonthelinet =((z,w,W'):z =0, uj =0, w' = (Wi, 0,... , 0)). Observe that the intersection of M with the line £ contains an open neighborhood of 0 on the real axis in the -plane, and the intersection of W with £ contains a piece of the upper half plane in t. The restriction of g to the real axis of £ is = )4/3• The contradiction will follow by observing that the C' function does not extend holomorphically to the upper half plane in any neighborhood of 0. To construct the desired smooth counterexample h, it suffices to take h(z, s, s') = exp(—1/g(z, s, s')). We leave the details to the reader. This completes the proof of Proposition 8.7.1. 0 The method of the proof of Proposition 8.7.1 does not extend to the case where M is merely smooth. The obstruction to carrying out the same construction is due to the error term r'(z, s) (which vanishes in the real-analytic case) that appears in the defining equations in canonical coordinates given by Theorem 4.3.2. In the

§8.7. NECESSITY OF MINIMALITY FOR HOLOMORPHIC EXTENSION

233

smooth case, we will first construct a CR distribution in a neighborhood of P0 in M which is not the boundary value of a holomorphic function in any wedge. We will then give some indication of how to conclude the existence of a smooth CR function which is not wedge extendible, as stated in Theorem 8.1.1.

In view of Corollary 7.2.9, the existence of a CR distribution which is not the boundary value of a holomorphic function in any wedge centered at p0 is a consequence of the following result, which is of independent interest. THEOREM 8.7.4. Let M C C's' be a generic manifold which is no: minimal a: of M containing po, with dimRMI
PROOF. We begin by choosing regular coordinates (z, w) E C" x Cd, vanishing as in Proposition 1.3.6, so that M is given by at

A basis ofCR vector fields L1, j = 1,... , n, can then be written in the form

LJ=2—+

(8.7.6)

with aJk smooth and (Xjk(O) = 0. We shall use (z, s) as local coordinates on M

Po =0. Assume that M1 c Mis given by

j

d2, and the = 0, 1 differentials do1 are linearly independent. Since the CR dimension of M1 and M are are equal, we must have I d2 d and we write d = d1 + d2. Since the LJOk on M1 since the a1 are real valued. tangent to M1, we must have LJO& = =0 Hence, by (8.7.6), (8.7.7)

j=1,... ,d2, k=1,... ,n.

ax,,

We conclude that rank

=d2.

(8.7.8) k

a linear change of variables and using the implicit function theorem, we can assume that M1 is given as a subset of M by After

(8.7.9) Put Uj

=

(8.7.10) for

5a1

— if'1 (z,

s),

j

,sa1),

1

= 1,...

,

d*1(O)=O. d2,

(x,y,s',u) = (x,y,si,...

and take ,5d1,U1,... ,ud2),

new coordinates on M, so that M1 is given (as a subset of M) by u = 0 near 0. Theorem 8.7.4 will be a consequence of the following proposition.

VIII. HOLOMORPHIC EXTENSION OF CR FUNCTIONS

234

PRoPosmoN 8.7.11. Let M, M1, and p0 be as in Theorem 8.7.4, and let (z, s', u) be the local coordinates on M near P0 as defined above. Then there is a V(O) 0, smooth function V(z, s') defined in a neighborhood of 0 in such that the distribution T defined by 8(u) =ö(u))Ø...08(ud2),

T=

(8.7.12)

where 8 is the Dirac measure at the origin in R, is a CR distribution on M.

We shall need the following general lemma (which is analogous to Proposition 2.5.4) for the proof of Proposition 8.7.11 LEMMA 8.7.13.

Let M

c C" be a generic subman:fold of codimension d given

be a basis of CR vector fields of the near p0 E M by (8.7.5), and let L1, . form (8.7.6). Define the smooth functions f1 on M by .

(8.7.14)

.

j = 1,...

f3

,d.

Then there exist d vector fields R3 of the form (8.7.15) where the

R1

are smooth complex valued functions defined near 0, such that the

following holcL

(i) RkfJ

Sj,, I

j, k

d, where 8jk denotes the Kronecker symbol. n, 1 k d.

(ii) [L1, Rk] 0, 1 j (iii) 1 (iv) The set (L1,... , L1,... , R1, . . . Rj) forms a basis for the sections of the complex tangent space to M near 0.

We claim first that (ii), (iii) and (iv) are all consequences of(i). For this, note first that since all the EL3, and [Ri, are of the it suffices to show form E1 b1(z, PROOF OF LEMMA 8.7.13.

(8.7.16)

[Li, Rk]fq

0,

IRk, Re]fq

0,

for all indices j, k, t, q, which will follow immediately from (1) by expanding the commutators. It is also clear from (8.7.6) and (8.7.15) that (i) implies (iv). This proves the claim. For (i), it suffices to take the vector fields R1, j = I d, given by Lemma 7.6.2, satisfying (8.7.17)

+

s))

=

§8.7. NECESSITY OF MINIMALITY FOR HOLOMORPHIC EXTENSION

235

U

The proof of Lemma 8.7.13 is complete. PROOF OF PROPOSITION 8.7.11. In the coordinates the U may be written

(8.7.18)

=

?, u), given by (8.7.10),

+

+

s', t) near 0. We introduce the vector are functions of (z, obtained from L1 by setting u = 0:

where fijk and

fields

(8.7.19) The 1
s', u) = +

(8.7.20)

5', u) = Uk + *k(Z,

gk(z, where the L1g*

=

are

s', u + *(x, y, s')) s') + (z, s, u + '/'(z,

s')),

given by (8.7.5) and *k by (8.7.9). Note that we have LJfk =

0.

For a smooth function h(z, i,?, u)

near 0 we write d2

(8.7.21)

h(z,

s, U) = h°(z,

hk(z,

s) +

s) Uk + O(1u12).

1 Wenote j that M1 may be regarded as a generic submanifold of codimension d1 of by = + the embedding z' = z, s', 0). Hence we may apply Lemma 8.7.13 above to obtain d1 vector fields of the form

SinceL)fk

d3

=

(8.7.22)

k=I

satisfying the following relations:

(8.7.23)

0,

ERJ°,

0,

VIII. HOLOMORPHIC EXTENSION OF CR FUNCTIONS

236

wherek = 1,... ,n and j,p = 1,... ,d1. Itfollowsfrom(8.7.23)andthefact that

is a CR function on M1 that we have

=

(8.7.24) Since the

d1

a —

zi

*=i

are linearly independent we obtain, for j = 1,... ,n,

+

L1 =

(8.7.25)

+

uk

Uk

Ur

I<*s42

where the coefficients in (8.7.25) are smooth near 0.

s') satisfying the conditions of the

We shall now construct the function V(z, proposition. Using the relation (8.7.26)

u&

E '8kr 8(u),

au,

k,r = 1,...

we conclude that V must satisfy the equations

1<j
(8.7.27)

Since we require V(O, 0,0) (8.7.28)

0, equation (8.7.27) is equivalent to

V) —

s')

0,

1

j

n.

The proof of Proposition 8.7.11 will then be completed by the following lemma.

8.7.29. Let

be the snzoo:hfrnczions determined by (8.7.25). Then isin the range of U nearthe origin.

PROOF. We begin with the calculation of the coefficients vj',.,. It follows from (8.7.25), that if L1h = 0 then

(8.7.30)

Ur

= where we have used the notation h = h°-f

hku&

O(1u12),

+ O(1u12) as in (8.7.21). Setting the coefficient of u* equal to 0 in (8.7.30) we obtain

(8.7.31)

Lyhk+

Taking for h relations (8.7.32)

the

d1, q in (8.7.23), we obtain,

functions fq'

=

1

(8.7.31) we now replace h by gq. 1 q its expression given in (8.7.32). We obtain

+

defined

(—

Lyf;

d2, defined in (8.7.20) and

+

—

Rewriting (8.7.33) by collecting the coefficients of

(8.7.34) Let B be the matrix s') 8rq Set D(z, obtain

in (8.7.20), and using the

—

In

(8.7.33)

237

FOR HOLOMORPHIC EXTENSION

OF

§8.7.

= 0.

we have

= —Lyg +

—

and note that Brq(O, 0,0)

—

= det B(z,

by

=

s'). By Cramer's rule applied to (8.7.34) we

1<j
(8.7.35)

where Djt,. is the determinant of the matrix obtained by substituting the rth column Summing (8.7.35) in B by the vector (— + over r we obtain

(8.7.36)

Vp.,. =

=

= —14 log D.

Here we have used the formula for differentiating a determinant and the identity = = 0(cf. (8.7.23)) and = 0. Since D(0) 0, it follows from (8.7.28) and (8.7.36) that a desired solution of (8.7.27) is given by V = This completes the proof of Lemma 8.7.29 0 This completes the proof of Proposition 8.7.11 and that of Theorem 8.7.4.

0

As mentioned in the observation preceding Theorem 8.7.4, we have shown that if a smooth generic manifold M C is not minimal at then there exists

VII! HOLOMORPHIC EXTENSION OF CR FUNCTIONS

238

a CR distribution which does not extend holomorphically to any wedge of edge M centered at p0. We have also shown (Proposition 8.7.1) that if, in addition, M is real analytic then there is a smooth CR function which does not extend holomorphically to any such wedge. In the smooth case, in order to show the existence of a continuous CR function which is not wedge extendible, we use the following representation of CR distributions.

Let T be a CR distribution defined in a neighborhood of P0 in M. Then there exists a nonnegative integer p and a continuous CR function u defined in a neighborhood of p0 in M such that LEMMA 8.7.37.

(8.7.38)

T

=(

where the R1 are the vector fields given by Lemma 7.6.2.

We omit the proof of this lemma, which is based on techniques of elliptic partial differential operators; we refer the reader to Treves [1] for details. We now apply Lemma 8.7.37 to the CR distribution T given by Theorem 8.7.4. We claim that the continuous CR function u, satisfying (8.7.38), is not the boundary value of a holomorphic function in any wedge of edge M centered at p0. We reason by contradiction. Assume that u = bvf, where f is a holomorphic function with slow growth defined in a wedge W of edge M centered at By (8.7.38) and repeated use of the first identity in (7.6.4) of Lemma 7.6.2 we conclude that

(8.7.39)

T=

It would then follow from (8.7.39) that Tis the boundary value of a holomorphic function with slow growth in a wedge W -< W, contradicting the fact that the distribution T is not wedge extendible. The proof of the necessity in Theorem 8.1.1 is now complete. REMARK 8.7.40. Using a Baire Category argument, one can show that if M is a smooth generic submanifold of not minimal at p0. then there exists a smooth CR function defined in a neighborhood of Po which is not wedge extendible. We refer the reader to Baouendi and Rothschild [6] for the details.

§8.8. Further Results on Wedge Extendabiity of CR Functions We shall conclude this chapter by stating, without proof, another condition which guarantees that a given continuous CR function h on a generic submanifold Mc extends holomorphically to a wedge with edge M. As we have seen, for

NOTES FOR CHAPTER VIII

239

a point P0 M, a necessary and sufficient condition for all CR functions defined near po on M to extend to a wedge with edge M centered at p0 is that M is minimal at Po• Thus, if we wish to remove the minimality condition on M, we must impose some condition on the CR function h in order for it to extend into a wedge. The condition given here is inspired by that of the edge-of-the-wedge theorem in §7.6.

We shall need some notation. Let M C CN be a generic submanifold of M. Letp(Z, Z) = 0, where p = (p1,... . Pd), be A ... A apd a defining equation for M near po so that P(po. = 0 and 0 CN,k near Define the generic submanifolds Mk c = 1,.. .d,ofcodimension

codimension d> land p0

d—

1

by

Mk := {Z:

(8.8.1)

=0, j = 1,... ,d, j

We also define the relatively open submanifolds Mk C Mk by

Mk := (Z E Mk:

(8.8.2)

fork =

I

Z) > 0},

d. Observe thatM =

each Mk. The following result gives a sufficient condition for a CR function h on M to extend to a wedge with edge M.

[I]). Let M, M1,...

and P0 be as above. For any sufficiently small open neighborhood U of po in C's' and any open convex cone F' such that F' -< (x E R": x1 > 0, j = I,... ,d}, there THEOREM 8.8.3 (AIRAPETJAN AND HENKIN

,

exists a wedge W with edge M centered at P0 in the direction F' with the following pmperzy. Any continuous CR function h on M fl U for which there are continuous CR functions hk on Mk 1) U, fork = 1,... , d, that extend continuously to M fl U and coincide there with h, extends holomorphically into the wedge W.

A proof of Theorem 8.8.3, different from the original one by Airapetjan and Henkin [1], using a construction of attached analytic discs similar to the one in Chapter VI is given in Vityaev [1]. The proofs do not use the edge-of-the-wedge theorem in §7.6, and it is not difficult to see that the latter theorem is in fact a consequence of Theorem 8.8.3.

Notes for Chapter VIII The sufficiency in the main result of this chapter, Theorem 8.1.1, is due to Tumanov [2), and the necessity was proved in Baouendi and Rothschild [6]. The proof of the sufficiency given here, using infinite dimensional manifolds, is an adaptation of the approach in Baouendi, Rothschild, and Trépreau [I]. We also mention that, for hypersurfaces, Theorem 8.1.1 was proved earlier by Trépreau

240

VIII. J4OLOMORPHIC EXTENSION OF CR FUNCTIONS

[1]. A proof for type finite type hypersurfaces was first given by Baouendi and Treves [2]. As mentioned in the notes of Chapter VI, holomorphic extension of CR functions goes back to the work of Lewy [1], who considered the case of Levi nondegencrate hypersurfaces. Subsequently, a number of mathematicians have contributed to the understanding of this problem. We mention here the work of Hill and Taiani

[1], Boggess and Polking [1], Bedford [1], Fomaess and Rca [1], Boggess and Pius [1], all of whom used the method of analytic discs. Work on this problem using the FBI (Fourier-Bros-Iagolnitzer) transform was done by Baouendi, Chang and Treves [1], Baouendi and Treves [2], Baouendi, Rothschild, and freves [1], and Baouendi and Rothschild [1]. For further discussion of these methods, we [2]. refer the reader to the books by Boggess [1] and As mentioned in the text, Theorem 8.8.3 is due to Airapetjan and Henkin [1]. In the case where the generic submanifolds Mk are hypersurfaces the corresponding result goes back to Chirka [1]. Tumanov [3] gives a sharper version of Theorem 8.8.3 where the wedge W is of the form (Z E U' : p1(Z, Z) > 0) for some small open neighborhood U' of po in CN.

Propagation of wedge extendibility, although not discussed in this book, is closely related to the methods and results of this chapter. For such results we refer the reader to the work of Hanges and Treves [11, [2,3], Baouendi, Rothschild, and ftépreau [1], Tumanov [4], Jöricke [1], and Merker [1].

CHAFFER IX

HOLOMORPHIC EXTENSION OF MAPPINGS OF HYPERSURFACES

Summary This chapter discusses generalizations to CN of the classical Schwarz reflection principle in one complex variable. More precisely, we study holomorphic mappings defined on one side of a real-analytic hypersurface M, smooth up to M, and mapping M into another real-analytic hypersurface M'. We address the question of when such a mapping extends holomorphically across M. In §9.1, we review the Schwarz reflection principle in one complex variable and also give some examples showing that the analogue of the Schwarz reflection principle fails to hold in higher dimensions without additional assumptions. The aim of this chapter is to give sufficient conditions for such a reflection principle to hold. in §9.3 we illustrate some of the techniques that will be used by first giving a reflection principle for CR diffeomorphisms between Levi nondegenerate hypersurfaces. We then define and discuss, in §9.4, the notion of essential finiteness for real-analytic hypersurfaces (a notion that will be given for more general submanifoids in Chapter XI). The notion of a not totally degenerate mapping is introduced in §9.5. The main resuit (Theorem 9.6.1) of this chapter is that a holomorphic mapping H defined on one side of a real-analytic essentially finite hypersurface, M, smooth up to the boundary and taking M to M', extends holomorphically to a full neighborhood of M provided that H is not totally degenerate. This result is stated and proved in §9.6—9.8. In §9.9, we prove some geometric properties of CR mappings between real-analytic hypersurfaces, which lead to some reformulations of Theorem 9.6.1. We then apply these local results to proper mappings between bounded domains with real-analytic boundaries in §9.10. In §9.11, we state, without proofs, a few results on holomorphic extension of CR mappings between real-analytic hypersurfaces under different hypotheses than those of Theorem 9.6.1. In that section, we also state, again without proof, a reflection principle for CR diffeomorphisms between higher codimensional real-analytic generic submanifolds.

241

IX. HOLOMORPHIC EXTENSION OF MAPPINGS OF HYPERSURFACES

242

§9.1. Reflection Principle in the Complex Plane The Schwarz reflection principle in one complex variable can be stated as

follows. PROPOSITION 9.1.1.

Let M and M' be two real-analytic curves in C and H a

holomorphic function defined on one side of M, extending continuously to M, and mapping M into M'. Then H has a holomorphic extension across M.

PROOF. We reduce to the case of the usual Schwarz reflection principle by showing that M and M' are locally biholomorphically equivalent to R. More precisely, we may assume that M and M' pass through the origin and that M is given near 0 by y = 4)(x), where 4) is a real-analytic, real-valued function of x in an open neighborhood of 0 in R satisfying 4)(O) = = 0. We shall construct a holomorphic function x defined near the origin such that x (x) E M for x real.

Since M isparametrizedbyx+i4)(x), x E R, wemaytakex(x) = x+i4)(x)and extend x (x) to C by complexifying 4) near 0 since 4) is real-analytic; we obtain x(z) = z + i4)(z), z E C for Izi sufficiently small. Since 4)'(O) = 0, we have x'(O) = I and hence x is a biholomorphism near the origin in C. It follows that near 0 x maps the upper half plane to one side of M and, after replacing x (z) by x (—z) if needed, we may assume that x maps the upper half plane to the side of M on which H is defined. Similarly, one can find a local biholomorphic function x' mapping M' into R in a neighborhood of the origin. Then J = x' o H o is defined in the upper half plane near 0, is smooth up the boundary, and maps the real line into itself. By the usual Schwarz reflection principle, J extends holomorphically in a neighborhood of 0. Since x and x' are invertible, H also extends holomorphically to a neighborhood of 0. 0

For the remainder of this chapter we shall address the problem of extending this classical theorem to higher complex dimensions. To make this precise, we first introduce some notation. Let M be a real-analytic hypersurface in containing the origin, with local defining function p. If is a sufficiently small open neighborhood of 0 in C"4', we denote by the set = (Z Z) > 0}. We consider a mapping H, holomorphic in

smooth in

valued in

C M', where M' containing the origin. We shall

is another real-analytic hypersurface in always assume H(0) = 0. We shall say that H extends holomorphically at 0 if there exists a smaller neighborhood of 0 in to which H has a holomorphic

extension. We discuss conditions on M, M' and H which guarantee that H extends holomorphically at 0. Let us remark that we shall not address the question of finding optimal a priori regularity of the holomorphic mapping H up toM. We

§9.2 REFLECTION PRINCIPLE: PREUMINARIES

243

shall always assume that H extends smoothly to M. The following simple example shows that the conclusion of the Schwarz reflection principle does not hold for "flat" hypersurfaces in for n 1.

=

Let M be the real hyperplane defined by Im put M' = M. Let H be the mapping defined by EXAMPLE 9.1.2.

0, and

111(Z)=Z1, j=2,...,n+1, where h is a holomorphic function defined in the upper half plane in C, smooth up to the boundary, but which does not extend holomorphically across 0 (such as that given in Example 1.4.9). If we take h to satisfy h(0) = h'(O) = 0, then the mapping H is a local diffeomorphism near 0 in for a sufficiently small We have H(M) C M', and H does not extend holomorphically at 0. The reflection principle can also fail to hold even if M is not flat, if the mapping H is in some sense too degenerate, as shown by the following. EXAMPLE 9.1.3.

Let M be the hypersurface in C2 given by

ImZ2 —(ReZ2)IZ,12 =0,

and take M' = M. Take H

to

be the mapping given by H1(Z) = f(Z2), and

112(Z) = 0, where f is holomorphic in the complex plane with a cut in the negative imaginary axis and does not extend holomorphically in a neighborhood of 0. It is easily checked that H is holomorphic above M, maps M into M', and does not extend holomorphically to any neighborhood of 0 in C2. We shall show in the rest of this chapter that under some finiteness conditions on M, M' and H, the reflection principle also holds in higher dimensions.

§9.2 Reflection Principle: Preliminaries We start by giving a criterion for a holomorphic function defined on one side of a real-analytic hypersurface to extend to the other side. First, we recall the following (nonholomorphic) change of coordinates in from § 1.7. Let (z, w) be regular coordinates for M at 0, with M given by

where 4 is real-analytic. We consider the mapping '4' : U C by

(9.2.1)

i—+ (z,w)=

...+

given

IX. HOLOMORPHIC EXTENSION OF MAPPINGS OF HYPERSURFACES

244

and s,t E zE real-analytic, we have extended it by complexifying s. s) is R; also, since of the parametrizing map for M given The mapping 'P is an extension (to by (1.6.3), also denoted 'P. Note that if U is sufficiently small, then the mapping '4' is a real-analytic diffeomorphism mapping the hyperplane ft =0) onto M and (See also § 1.7.) Here U is an open neighborhood of 0 in

the side t > 0 (resp. t <0) onto the side of the hypersurface defined by Im w> one Re w)). In fact, for every fixed z E Im w (s, fr(z, s)) given may view the mapping 'P as the flattening of the curve s in the proof of Proposition 9.1.1. We may now give a criterion for a function holomorphic on one one side of a hypersurface to extend holomorphically to the other side. As in §9.1, we let be and let a neighborhood of 0 in fr(z,

Re w) (resp.

(9.2.2)

ci'

denote the space of functions holomorphic in The following We let is an immediate consequence of Proposition 1.7.5 for the case of a hypersurface. be as above. PR0I'osmoN 9.2.3. Let M be a real-analytic hypersurface and let u(z, s) be the CR function defined by

For h E C' () fl u(z,

s)

= h(z, s +

s))

= h('P(z, s,0)).

(land only Then h extends holomorphically to a full neighborhood of 0 in if there exists E > 0 such that such that for every z E Izi < E, the function s ,-+ u(z, si <E, ill <E) + it) is a bounded, measurable function such a way that the extension u(z, s in of all its variables. The results of this section will be used in the following context. We begin with a mapping H, holomorphic in smooth in (2k, valued in and satisfying H(M) C M', where M' is another real-analytic hypersurface in containing the origin and is one side of M as above. We shall apply Proposition 9.2.3 to the components H, of H. In order to prove that the mapping extends holomorphically to a neighborhood of 0, it then suffices to show that the functions s i-+ u1(z, s) extend holomorphically in s for z fixed (as in Proposition 9.2.3)z, where uj is the to M. CR function obtained by restricting We shall also need some further preliminary results for extension of mappings. We let M and M' be real-analytic hypersurfaces through 0 in with (z. w) and

(z', w') normal coordinates for M and M' respectively (see §4.2). We let M be given by

(9.2.4)

245

§9.2 REFLECTION PRINCIPLE: PRELIMINARIES

and M' by

Im w' = *(z',

(9.2.5)

Re

w').

the holomorphic mappings defined in

Let us denote by and valued in

satisfies H(M) = M' fl If H E with H defined and H(O) = 0, we write H = (F, G), where F = (F1,... , w). The components F, G are called normal by z' = F(z, w) and w' = G(z, components of H, and G is called a transversal component of H. We let f = and G to M, and g be the CR functions obtained by restricting (fr,... s)). Let s) s)) G(z, s + s) i.e. and g(z, Fj(z, s + i4(z,

j = 1,...

,

n, be the basis of CR vector fields on M given by (1.6.5).

PROPOSiTiON 9.2.6. Let H = (F, G) be as above. Then the following condi.. tions are equivalent.

(I) HIM is a local djffeomorphism in a neighborhood of 0 in M onto a neighborhood of 0 in M'. Oanddet

(ii)

(iii)

0.

(Lk1)I<Jk
5&0.

PROOF. First note that since f, and g are CR functions on M and since Lk lo

it follows that follows from (9.2.5) that

*(0) =

0,

1 < j, k

n. Also, since H(M) c M'

it

(9.2.7)

Applying L1 to both sides of (9.2.7) and noting that d*(O) = 0, we obtain also = 0, 1 j n. The equivalence = 0 from which it follows that of (1) and (ii) easily follows. Finally, since = 0 it follows that L1 f*(0) = (0). Hence (iii) is equivalent to (ii). The proof of the (0) and (0) = proposition is complete.

For E >

0

we denote by R( and L the rectangles in the upper half of the

complex plane given by (9.2.8)

RE

JsI<E,0
t <e).

< €,

=

(IzI

sI

<e,ItI <e}.

We also write V7

={(z,s,t) =

{(z, s, t)

(zf <E, Isi < E,

t<

Izi <e, Isi <e, —E < t

eJ,

0}.

IX. HOLOMORPHIC EXThNSION OF MAPPINGS OF HYPERSURFACES

246

If u(z,

s) is a function on M (of at least class C'), we shall say, to be brief, u(z, z, s) extends up (resp. down) differentiably if there exists an E > 0 such that, for each z E Izi < €, the functions i—* u(z, s) extends holomorphically to Rf (resp. —RE) in such a way that the extension u(z, s + it) is a C' function of all its variables in (resp. V). Similarly, we shall say that u(z, s) extends up (resp. down) continuously if the extension u (z, s + it) is only continuous in (resp. VE). We shall use the analogous terminology for u extending up or down smoothly to mean that the extension u(z, s + it) is smooth in all its variables in (resp. If no smoothness is indicated, it will be assumed that the extended function is bounded and measurable in (z. s, t) and that for each fixed z the function (s, t) i-+ u(z, s + it) is holomorphic in R( (resp. —RE) and continuous on (resp. We shall need the following observations, which will be used in the proof of the reflection principles in the subsequence sections.

9.2.9. If u (z, s) is afunction on M (of class at least C') which extends up, then its complex conjugate, ü(z, s), extends down with the same smoothness. If L is a vector field on M with coefficients which extend up continuously (resp. smoothly), andv(z, s) extends updifferentiably (resp. smoothly) then Lv extends up continuously (resp. smoothly). If u(z, s) is a CR function (of class C')

which extends up and down, then there is a holomorphic function h(z, w) in a neighborhood of 0 in such that u(z, s) = h(z, s + icti(z, s)) near 0. PROOF. The first and second statements are obvious from the reflection property in one complex variable and the definitions of extending up and down given above. The third statement is an immediate consequence of Proposition 1.7.5. 0

§9.3. ReflectIon Principle for Levi Nondegenerate Hypersurfaces The following illustrates, in a simple example, the methods that will be used in the remaining part of this chapter.

PROPOSITION 9.3.1. Let M be a hypersurface in C2, given by (9.2.4), and fl H as above with H(0) = 0, H(M) c M', where W is the Lewy hypersurface given by Im w' = If the restriction of H to M is a local diffeomorphism near 0 in M, then H extends to a holomorphic mapping in an open neighborhood of 0 in C2. PROOF. Let H = (F, G) as above, and let (f, g) be the CR functions obtained by restriction to M. Since H(M) C M' we have

(9.3.2)

§9.3. LEVI NONDEGENERATE HYPERSURFACES

247

Since f and g are restrictions of holomorphic functions defined on one side of M, it follows that f and g extend up (say) differentiably. Hence by Proposition 9.2.3 it suffices to show that f and g extend down continuously. Let L = — i a CR vector field on M. Applying L to both sides of (9.3.2) we obtain

= 2ifLf.

(9.3.3)

By Proposition 9.2.6, since the restriction of H to M isa diffeomeorphism, we

have Lf(0)

0. Dividing both sides of (9.3.3) by Lf, we obtain f =

and f extend down differentiably (by Lemma 9.2.9), and the coefficients of L, being real-analytic, extend up and down, it follows that and U extend down continuously. Hence f extends down continuously. Using (9.3.2), we also Since

have that g =

+ 2iff extends down continuously. This proves Proposition

0

9.3.1.

We shall now generalize Proposition 9.3.1 to prove a reflection principle for an important class of hypersurfaces. Recall from §2.2 that a hypersurface M is Levi nondegeneraze at 0 if its Levi form at 0 is nondegenerate. By Proposition 2.2.10, if M is given by (9.2.4), then M is Levi nondegenerate at 0 if and only if the matrix )(0) is invertible. If M is real-analytic and given in normal coordinates by as in (4.2.13), it is easy to check that w = Q(z, (0), where (0) =

A is an n x n constant invertible matrix. Hence M is Levi nondegenerate at 0 if and only if (0) is invertible. THEOREM 9.3.4. Let M be a real-analytic hypersurface in given by (9.2.4) = 0, H(M) CM', H(0) = 0, where M' is a Levi nondegenerare real-analytic hypersurface in If HIM is

andH e

a local djffeomo,phism near 0 in M, then H extends to a holomorphic mapping in an open neighborhood of 0 in Hence M and M' are biholomorphically equivalent. PROOF. We shall assume that M' is given by ü/ = z', w'). If H = (F, G) as above, we have the identity for the corresponding CR functions on M:

(9.3.5)

= Q(f, f, g).

Applying

to (9.3.5) and then replacing g by Q(f, f,

(9.3.6)

+

we obtain

=0, j = 1,...

f, Q(f, f,

Let B1(f, f, each

, n.

, denote the left hand side of (9.3.6). Then is a power series which conververges near the point

qo = (0,... ,0, Ljfi(0),...

,

E

248

IX. HOLOMORPHIC EXTENSION OF MAPPINGS OF HYPERSURFACES

Let By,, denote the derivative

(qo) (where

is considered as a variable). Since

= 0, a direct calculation shows that B1,, = we can express the matrix (B11,) as a product of matrices (9.3.7)

Hence

=

(0)) is invertible by the assumption of Levi nondegeneracy of M' and the remarks preceding the theorem. The matrix (L, fk (0)) is invertible by Proposition 9.2.6. It follows from (9.3.7) that the matrix (B31,) is also invertible. The matrix

Hence we may apply the implicit function theorem for systems of holomorphic equations to (9.3.6) to solve for the functions f3 as convergent power series in the other functions, i.e. for j = 1,... , n, we have (9.3.8)

fj = K1(f,

,

with K3 holomorphic in all its arguments. It follows from the assumptions on

H that the components of f and g extend up differentiably. Hence by making use of Lemma 9.2.9 the right hand side of (9.3.8) extends down continuously. We conclude from (9.3.8) that the f3, j = 1,... , n, extend down continously. we see that g also extends down Rewriting (9.3.5) in the form g = Q(f, f, continously. This show that H extends holomorphically to a full neighborhood of and completes the proof of Theorem 9.3.4. 0 o in REMARK 9.3.9. Fefferman [1) proved that a biholomorphism between two with smooth boundaries extends as a strongly pseudoconvex domains in smooth diffeomorphism between their closures. If one combines Theorem 9.3.4 with Fefferman's theorem, it follows that any biholomorphism between strongly pseudoconvex domains D and D' with real-analytic boundaries extends as a biholomorphism between neighborhoods of their closures D and D'. § 9.4. Essential Finiteness for Real-analytic Hypersurfaces In the remainder of this chapter, we shall prove more general reflection principles, in which we will no longer assume that the hypersurfaces are Levi nondegen-

crate nor that the mapping is a diffeomorphism. The following simple example shows that if the defining function of a real-analytic hypersurface is independent of one of the complex variables, there may be holomorphic mappings on one side of the hypersurface which do not extend. EXAMPLE 9.4.1. Let M C C3 be given in normal coordinates (zt, Z2, w) by 1mw = IziI2, and let H = (F1, F2, G), where w) = Zi, G(z, w) = wand where F2 is a function holomorphic on one side of M, smooth up to M, which does

§9.4. ESSENTIAL FINITENESS FOR REAL-ANALYTIC HYPERSURFACES

249

not extend holomorphically in any neighborhood of 0. (Such a function exists by an immediate modification of Example 1.4.9.) Then 11(M) C M, but 11 does not extend holomorphically to a full neighborhood of 0 in C3. Note that F2 may be chosen so that HIM is a diffeomorphism.

We shall first introduce an algebraic condition on a hypersurface M which Let implies that the defining function of M depends on all the variables of M be a real-analytic hypersurface through 0 given in normal coordinates near 0 by the defining equation Im w = write

Re

w). Since

is real-analytic, we may

(9.4.2) where

E

Cf z), the ring of convergent power series in n

indeterminates

Z=(Zi,... The hypersurface M is essentially finite at 0 if the ideal :aE is of finite codimension in C{z} (see Definition 5.1.1). Its codimension is called the essential type of M at 0. DEFINITION 9.4.3.

We will show (see Proposition 9.4.7) that Definition 9.4.3 is independent of the choice of normal coordinates (z, w). We have the following. LEMMA 9.4.4. Let (z, w) be normal coordinates at Ofor a real-analytic hypersurface M, and let z'(z, w), w'(z, w) be another set of normal coordinates at 0

obtained by a local holomorphism in (9.4.5)

w'(z,

Then

w) = k(z, w)w,

k E C(z, w),

k(0,

0)

0,

and the mapping z i-÷ z'(z, 0) is a local biholomorphism in C". PROOF. Let p'(z', w', üi') be a real-analytic defining function for M near 0 in the coordinates (z'. w'). The function

p(z,

w, tii) := p'(z'(z, w), z'(z, w), w'(z, w), w'(z, w)J

is then a defining function for M near 0 in the coordinates (z, w). Since (z, w) and (z', w') are normal coordinates (4.2.5) implies p'(z', 0, WI, 0) = K(z', W')w', with K(0) 0 and p'(z'(z, W), 0, w'(z, w), 0) = R(z, w)W, with R(0) 0, from which (9.4.5) follows. The rest is immediate, since (9.4.5) implies that

0 PROPOSITION 9.4.6. Let a(z, x) with z

E C" and x

C" (with

= Iz

I

aa(z)x" be a convergent power series I small) and let I be the ideal in C(z

250

IX. }IOLOMORPHIC EXTENSION OF MAPPINGS OF HYPERSURFACES

generated by {aa (z)

I=

where Z

a

There exists €0

}.

> 0 such that if 0 <

is the ideal in C(z} generated by the power series z

€o. then a(z, x)

for ally E C", lxi PROOF.

Choose €o so that

Since write

is absolutely convergent for Jz I,

E I for all a, for any N, aa(z)Xa

=

Ix I <€0.

aa(z)Xa E I. For Ixi < €0, we Since for and a(z, x) =

bp(x) as N —+ oo, by Proposition 5.1.4 we €o. For the converse, since for lxi < E, a(z, x)1 E 'El so are all its x derivatives, by Proposition 5.1.4; indeed, such (z This proves derivatives can be expressed as limits. Hence for each a, aa (z) 0 C Z and completes the proof of Proposition 9.4.6.

each fixed fi, x' we have conclude that Z C I for

0<

PRoPosmoN 9.4.7. The definitions of essential finiteness and essential type given in Definition 9.4.3 are independent of the choice of normal coordinates

(z,w). PROOF. Let (z'(z. w), w'(z, w)) be another set of normal coordinates at 0, as in Lemma 9.4.4. Assume that in these coordinates M is given by

Im

w' =

Re

w').

It follows from Lemma 9.4.4, since any two defining functions are nonvanishing multiples of each other, that

(9.4.8)

where A(0)

x1 0)

= A(z',

x', 0),

0 and (z, x) are related to (z', x') by z' = z'(z, 0) and x' =

0). By differentiating (9.4.8), we deduce that the ideal generated in C{z} by the derivatives

(9.4.9)

—(z,0,O), axa

is contained in the ideal generated in C(z} by the series

(9.4.10)

a biholomorphism (by Lemma 9.4.4), the above shows that if the ideal generated by the derivatives (9.4.9) has finite codimension, then so does that generated in Clz'} by (9.4.11)

a

ax' a

(z',O,O),

aE

§9.4. ESSENTIAL FINITENESS FOR REAL-ANALYTIC HYPERSURFACES

251

The opposite implication follows by dividing (9.4.8) by A (z', x') (which is possible since A (0,0) 0) and repeating the argument. The symmetry also shows that the codimensions of these ideals are the same. This completes the proof of Proposition 0 9.4.7.

Using Proposition 9.4.6 and the Nullstellensatz (Theorem 5.2.5), we obtain the following.

PROPOSITION 9.4.12. if M is a real-analytic hypersurface through 0 in defined near 0 by or

with (z, w) normal coordinates at 0, then the following are equivalent.

(I) M is essentially finite a: 0.

(ii) There exists Eo > 0 such that for every €, 0 < E < e0, and for every 0. Z E C'2, 0< lzl <e, there exists x E C'2, lxi <e, with and for every (ii') There exists eo > 0 such that for every e, 0 < e 0. z E C'2, 0< Izi <E, there exists X E C", ixI <E, with Q(z, = Ofor all a, and (iii) There exists > Osuch that if z E C", Izi then z = 0. Here the (z) are as defined in (9.4.2). < eoandQa(z) =Ofor all a, (iii') There exists e0 > Osuchthatifz E then z = 0. Here the (z) are defined by (9.4.13)

Q(z, x' 0) =

Qa(z)x".

PROOF. The equivalence of(i), (ii), and (iii) follows from Proposition 9.4.6 and Corollary 5.2.6. To show that (ii) and (ii') are equivalent and that (iii) and (iii') are equivalent, we note that

(9.4.14)

Q(z,

0) = a(z, x)/'(z, x' 0),

0 since any two defining functions are nonvanishing multiples of where a (0) each other. The proof of Proposition 9.4.12 is complete. 0

:aE and V = V(I) the germ at 0 of Let I C C{z) be the ideal the subvariety in C" defined by the ideal I as in §5.2. We claim that the germ at is contained 0 of the holomorphic subvariety given by WM = V x {0} C in M. Indeed, if we take w = 0, z E V then (9.2.4) holds. Note that by Lemma 9.4.4 and (9.4.8), WM is well defined up to a biholomorphic change of normal

252

IX. HOLOMORPHIC EXTENSION OF MAPPINGS OF HYPERSURFAcES

coordinates. We shall call WM the essential variety of M. With this terminology M is essentially finite if and only if its essential variety is trivial, i.e. WM = (0). DEfiNITION 9.4.15. A real-analytic hypersurface M in through 0 is of D-fini:e type at 0 if there is no germ W at 0 of a nontrivial holomorphic subvariety

withWCM. The following proposition gives a relationship between the three notions of nondegeneracy: essential finiteness, finite type (or minimality), and D-finite type for a real-analytic hypersurface. PRoPosmoN 9.4.16. If M is a real-analytic hypersurface in consider the following properties:

with 0 e M,

(i) M is of D-flnite type at 0. (ii) M is essentially finite at 0. (iii) M is offinite type at 0.

Then (I) implies (ii) implies (iii). Furthermore, in C2 the three properties are equivalent. PROOF. If M is not of finite type then by Theorem 1.6.7, for normal coordinates

(z, w) at 0 and M given by (4.2.2), we have 4'(z,

0)

0. Hence M is not

essentially finite at 0, which proves (ii) implies (iii). If M is not essentially finite at 0, then the essential variety of M, WM is a nontrivial holomorphic subvariety

contained in M through 0. Hence M is not of D-finite type, which proves (i) implies (ii). For the claim concerning C2, we shall prove by contradiction that (iii) implies

(i) in this case. Suppose that M C C2 is of finite type at 0 but not of D-finite type at 0. Then there exists a nontrivial holomorphic subvanety W through 0 with W C M. Necessarily, W is a (smooth) complex curve near every regular point p of W. At such point p, M is not minimal, since near p. W is a CR submanifold of the same CR dimension (= 1) contained in M through p. Hence M is not of finite type at p since finite type implies minimality (and the two notions are equivalent in the real-analytic case). However, since the set of points of finite type is open in M (e.g. by Theorem 1.5.10) and since p can be taken arbitraily close to 0, we reach the desired contradiction. The proof of Proposition 9.4.16 is complete. (J

§9.5. Formal Power Series of CR Mappings Let M be a real-analytic hypersurface in through 0 defined by (9.2.4), and u(z, s) a smooth CR function on M defined in a neighborhood of 0. By

§9.5. FORMAL POWER SERIES OF CR MAPPINGS

Proposition 1.7.14 we associate to u

a

formal power series U(z, w) capzawP,

U(z, w)

(9.5.1)

253

pEZ+

with z =

(Zi,...

,

zn), such that the Taylor series of u at 0 is given by

a neighborhood of 0 in and the side of M defined by (9.2.2), and fl if u(z, s) extends up smoothly, i.e. there exists U E whose

restriction to M is u, then the Taylor series of U at 0 is also given by (9.5.1) and hence the coefficients ca.p in (9.5.1) are given by (f)" U(O). = If M' is another real-analytic hypersurface in given in normal coordinates by (9.2.5), and h is a CR mapping from M to M', with h (0) = 0, then by Proposition 2.3.3, we may write h = (f, g), withf = (ft,... , fe), where the components andg are CR functions on M. We denote by F(z, w) = (Fi(z, w),... , w)) and c(z, w) the corresponding holomorphic formal power series associated to f and g as in (9.5.1). Observe that if the CR mapping h is the restriction to M of a mapping H E with H = (F, G), and F = (F1,... , then the formal power series Fi(z, w),... ,Fn(Z, w), Q(z, w) are the Taylor series

atOofFi,... DEfiNITION 9.5.2. Let h = (f, g) be a CR mapping from M to M' as above. We shall say that h is not totally degenerate at 0 if

aZk

in the sense of formal power series in the independent variables Zi,...

,

The following proves the invariance of Definition 9.5.2.

PR0I'osmoN 9.5.3. If a CR mapping h is not totally degenerate with respect to normal coordinates (Z, w) for M and (z', w') for M', then h is not totally degenerate with respect to any set of normal coordinates for M and M'.

The proposition can be proved by applying Lemma 9.4.4 to changes of normal coordinates for M and for M' and observing that with the notation above, the normality of the coordinates implies that Q(z, 0) 0. Details are left to the reader. (The proof that Q(z,0) 0 is given in (9.9.13)—(9.9.14), which is independent of the intervening sections.) 0 PRooF.

IX. HOLOMORPHIC EXTENSION OF MAPPINGS OF HYPERSURFAcES

254

The notion of a mapping being not totally degenerate can also be expressed in terms of the CR vector fields j = 1,... , n, given by the basis (1.6.6).

PRoPosmoN 9.5.4. Let h = (f, g) be a CR mapping as above. Then h is not such that totally degenerate at 0 and only there exists a E

La(det(Ljfk)I<Jk
= where given by (1.6.6).

.

. .

and the

are a basis of CR vector fields on M near 0

We need the following lemma for the proof of Proposition 9.5.4 as well as for later use. Let j = I,... , n, be the basis of CR vector fields on M given by (1.6.6). Then for any multi-index a LEMMA 9.5.5.

(9.5.6)

aza

on M near0andj(z, w) the correspondIn addition, s) isa CR ingforinal series given by (9.5.1), then

Laj(0) =

(9.5.7) Hence,

J(z, 0)

04f and only

/

there exists a such that L(xj(0)

0.

all contain a PROOF. The nonconstant coefficients in the expansion of which vanishes at 0 since s) 0 by definition of factor of the form normal coordinates. The only constant coefficient in the expansion of L" is that This proves (9.5.6). To prove (9.5.7), we note that by (9.5.6) we have of

=

On the other hand,

where the left hand side is regarded as a Taylor series at 0. Now (9.5.7) follows since (0) = 0 for all using the normality of the coordinates. 0 PROOF OF PROPOSITION 9.5.4. By definition, h is not totally degenerate if and

only if there exists a multi-index a such that (0) ôZk

I

By expanding the determinant above and applying (9.5.7) to each term, we obtain the proposition. 0

§9.6. REFLECTION PRINCIPLE FOR ESSENTIALLY FINITE HYPERSURFACES

255

§9.6. Reflection Principle for EssentiaHy Finite Hypersurfaces In this section we shall state and begin the proof of a general reflection principle for hypersurfaces in C

HE

fl

be a real-analytic hypersurface through 0 and where is a neighborhood of 0 in Suppose

that H(0) = 0 and H(M) C M', where M' is an essentially finite real-analytic hypersuiface in If HIM is not totally degenerate at 0, then H extends to a holomorphic mapping in an open neighborhood of 0 in We choose normal coordinates (z. w) and (z'. w') at 0 for M and M' respectively. As in §9.2 we denote by (F, G) the normal components of H and by h = (f, g) their restriction to M. Using Proposition 4.2.12 we may write the defining equation for M' in the form ü/ z', w'), with Q(0, z', w')

Q(x',

0,

w')

w', so that the following identity holds on M in a neighborhood

of 0:

(9.6.2)

= Q(f, f, g).

We let L•,j = 1,... ,n, be the basis of CR vector fields forM near0 given by (1.6.6). We note that by applying U to (9.6.2) for any multi-index y E have, by the normality of the coordinates (z', w'),

we

(9.6.3) In this section we shall prove the following result, which is the first step in the proof of Theorem 9.6.1.

PRoPosmoN 9.6.4. Let M, M', and H satisfy the hypotheses of Theorem 9.6.1, and g) = (fit... , g) be as above. Then each satisfies a polynomial equation with coefficients which extend down smoothly. More precisely, there exists a finite set S of multi-indices in such that for each j, 1 j< there exists an integer N1 and functions v), with u = and holomorphic near u0 = v= and v0 = 0 such that, for

j=1

Nj -

(9.6.5)

I

+

a neighborhood of z =0, s =0. PROOF. By (9.6.2) we have

(9.6.6)

= Q(f, f, g) = Q(f, f, 0) + gP(f, f, g),

IX. HOLOMORPHIC EXTENSION OF MAPPINGS OF

256

As in the

where P is a holomorphic function in a neighborhood of 0 in notation of (9.4.13) we write

1,0) =

(9.6.7)

Qa(f)fa.

Since M' is essentially finite, by Proposition 9.4.12, for every Xo E small there exists a multi-index such that Qa0(XO) 0. Let

sufficiently

(Fi(z,w),... be

the formal holomorphic power series for the CR functions (ft,... , f,J as in

§9.5. We claim that for all Xo sufficiently small 0)

(9.6.8)

0,

where the left hand side of (9.6.8) is regarded as a formal power series in Indeed, by Proposition 5.3.5, since h is not totally degenerate, the components of F cannot

satisfy a nontrivial holomorphic relation, proving the claim. On the other hand, (9.6.8) is equivalent to the existence of a multi-index j9o such that

0))(0)

>

0.

By (9.5.7) of Lemma 9.5.5 this implies that

Xo. 0)(0) =

(9.6.9)

It follows from (9.6.9) that as

0.

varies over all multi-indices, the convergent power

series in x. X

Qa(X)LPfa(0),

(which are all finite sums over have no nontrivial common zeroes in any sufficiently small neighborhood of 0, and thus the ideal J generated by these functions has finite codimension in C{X) by Corollary 5.2.6. By the Noetherian property (ideals in C(X J are finitely generated) there exist finitely many multi-indices, , such that the convergent power series (9.6.10)

X

generate the ideal J in

j = 1,... ,r, (which is of finite codimension).

257

§9.7. POLYNOMIAL EQUATIONS FOR COMPONENTS OF A MAPPING

We now apply

to (9.6.6) to obtain

=

(9.6.11)

I g))

Rewriting (9.6.2) in the form g = Q(f, f,

and substituting for g in (9.6.11) we

have,forj=1,... ,r, fa + Q(f, f,

(9.6.12)

(P(f, f, Q(f, f, i))) —

=

()

considered as independent variables, for 1 = = UI,, and with 0 n, and y E 5, where S consists of all multi-indices y E p the latter being 0 by and = = lfi'I. We let = 0 for p = 1,... , n. We write u = (9.6.3). Also, note that Equation and and similarly uo = and v = = (9.6.12) can be written in the form

Let

R1(f,u,v)=0, j= I,... ,r,

(9.6.13)

where the R1 are holomorphic in all variables near (0, U0, vO). Furthermore, since

Q(z', 0,0) (9.6.14)

0, it follows from the definition of R1 and (9.6.12) that

R1(f, u0, v0) =

Hence the ideal generated by the functions U0, v0), j = 1,... , r, has finite codimension in J (see (9.6.10)). The conclusion of Proposition 9.6.4 now follows from an application of Theorem 5.3.9 to the equations (9.6.13). 0

§9.7. Polynomial Equations for Components of a Mapping We return to the proof of Theorem 9.6.1. The main step is the following. PROPOSITION 9.7.1. Assume M, M', and H satisfy the hypotheses of Theorem 9.6.1, with M' given by lii' = z', w') and(z, w), (z', w') normal coordinates at Ofor M and M' respectively. Let (f, g) be the components of H IM• Then there a holomorphic function in a neighborhood of 0 in exists V(z, w, such

that (9.7.2)

Q(x. 1' g)U. z, s) = V(z, s +

s), x).

IX. HOLOMORPHIC EXTENSION OP MAPPINGS OF HYPERSURFACES

258

for all sufficiently small z, x

(Y' and s

R.

We shall prove Proposition 9.7.1 in §9.8. Let us assume for the moment that Proposition 9.7.1 has been proved, and complete the proof of Theorem 9.6.1 under

this assumption. We first note that by the normality of the coordinates for M', Q(0, f, g) g, so that taking x = 0 in (9.7.2) yields the desired holomorphic extendability for g. For the rest of the proof of Theorem 9.6.1 we will need the following corollary of Proposition 9.7.1. COROLLARY 9.7.3. Under the assumptions of Theorem 9.6.1 the components off satisfy polynomial relations of the form N1

-I

j=1,...,n,

(9.7.4)

is a positive integerandthe coefficients bkJ (z. where each in a neighborhood of 0 in with bkj(O) = 0. PROOF. Expanding

s) are real-analytic

f, g) in powers of x yields

f' g) = where according to Proposition 9.7.1 we have

(9.7.5)

g)(z, z, s)

z, s), 0),

(z, s +

=

for all multi-indices a and (z, s) in a fixed neighborhood of (0,0). Observe that (9.7.5) implies in particular (as does (9.7.2)) that = (1/a V/dx'' vanishes at (0,0,0) for every a. Since M' is assumed essentially finite at 0, by the Noetherian property and Proposition 9.4.12 we may choose multi-indices a1,... , a' of positive length such that the ideal generated in C{z'} by the power series (z', 0),... , (z', 0) has finite codimension. Thus, by Theorem 5.3.9, there are Weierstrass polynomials , with x = A) near (0,0), such that in P1(x,

(9.7.6)

=

k

= 1,...

,

r,

implies (9.7.7)

A), z) = 0,

j = 1,... ,n,

=

259

§9.8. END OF PROOF OF THE REFLECTION PRINCIPLE

Recall also, as we remarked above, thatg(z, (9.7.5) implies that (9.7.8)

s)

=

s), 0).

V(z,

j

= 1,...

Thus,

,fl,

= s), 0), for a = 0, a1,... , &, = VXU(Z, s + j = 1,... ii. This proves Corollary (9.7.3), because the coefficients polynomials P1 z) are holomorphic in x and the functions a neighborhood of 0 in M and vanish at 0. where

,

s), of the

0

Now, the proof of Theorem 9.6.1 may be completed using Corollary 9.7.3 as follows. By Theorem 5.5.32 we conclude from (9.7.4) that each component is then real-analytic in a neighborhood of 0 in M. Since the are CR functions, it follows from Corollary 1.7.13 that each is a restriction to M of a holomorphic The proof of Theorem 9.6.1 function defined in a neighborhood of 0 in is done, modulo the proof of Proposition 9.7.1, which will be given in the next 0 section.

§9.8. End of Proof of the Reflection Principle To complete the proof of Theorem 9.6.1, it remains to prove Proposition 9.7.1. For the proof of this proposition, we first write the Taylor expansion of f, g) around x

(9.8.1)

=

f,g) =

(x

f,g).

note that the left hand side of (9.8.1), for fixed x sufficiently small, is a CR function on M near 0 which, by hypothesis, extends up smoothly. To prove

We

Proposition 9.7.1, it suffices, in light of Lemma 9.2.9, to show that each term in the series of the right hand side of (9.8.1) extends down (i.e. as a measurable function and for z fixed, holomorphic in the rectangle —RE as in §9.2) in such a way in that the resulting series is absolutely convergent in VE. Hence Proposition 9.7.1 is a consequence of the following lemma. 9.8.2. There exists a constant C > 0 and 6 > 0 such that for each multi-index a E and z E IzI <e, the function s ,-÷ f, g)(z, s) extends down holomorphically to the rectangle —RE, continuously in —RE, and such that the extension Q, (1, 1, g)(z, s + it) is a measurable function in and satisfies

(9.8.3)

1, g)(z,

s + it)I

CIaIa!,

V(z, s, t) E

IX. HOLOMORPHIC EXTENSION OF MAPPINGS OF HYPERSURFACES

260

Here RE and

are as in defined in § 9.2.

We first prove the following. LEMMA 9.8.4. There exists

0 such that for

and for every multi-index a,

exist functions aa and ba defined in 17E such < €, thefunctionss+it Izol are holomorphic in — RE, continuous in — RE with

there

and such

that

more,for z

the functions

outside a

s

(Zo,

set of measure 0

in

s) do {IzI

C",

i-÷

Isi

<E,

not vanish identically. Further-

<€} one can

s) =

ba(Z,

for each fixed Zo

= aa(zo,

ba(Zo,

(9.8.5)

that

take for al

>0

D :=

The functions a (zo, s) and ba (zo, s) depend on the point Zo. The lemma does not claim that these functions depend in any smooth way on Zo, although the proof will reveal that there is a dense open subset of {lzl < where aa and can be chosen to depend smoothly on Z. REMARK 9.8.6.

PROOF. For a = 0 the lemma is immediate; indeed

= Q(f' f,g),

(9.8.7)

and g extends up smoothly and hence, by Lemma 9.2.9, extends down smoothly. As in the statement of the lemma, let D(z, s) := det We first note that by Proposition 9.5.4 since H is not totally degenerate there exists E > 0 and a multi-index such that (9.8.8)

L'8'D(z,

s)

0, Vz E C", S

We claim that for any multi-index a, Ia I

Ra (independent of M, M' and

(9.8.9)

E

R, lzl, Isl <E.

> 0, there exists a univeTsal polynomial

H) such that g)(z,

s)

=

s).

To prove (9.8.9) for al = 1, we apply L1 to (9.8.7) to obtain

(9.8.10)

j=l,... ,n.

§9.8. END OF PROOF OF ThE REFLECTION PRINCIPLE

261

By (9.8.8), D(z, s) 0 on an open dense set for Izi, Isi <E, so we may apply Cramer's rule to solve for (f, f, g) in (9.8.10) at such points (z, s) and multiply through by the resulting denominator, D, to obtain (9.8.9) for = 1, which then holds for all (z, s) sufficiently small by continuity. Assume by induction that (9.8.9) holds for ko. We prove it for IcrI = k0 + 1 by dividing both sides of (9.8.9) (for the appropriate a) by applying the L1's to both sides of the resulting equation, and again using Cramer's rule. Here again we do this first for points (z, s) for which D(z, s) 0 and conclude that (9.8.9) holds at all points (z, s) sufficiently small by continuity. The details of this calculation are left to the reader. This establishes claim (9.8.9). Observe (by using Lemma 9.2.9) that for any multi-index the function s) extends down smoothly (in the sense of §9.2) and its extension s

+ it) is real-analytic in VE. Since D(z,

(9.8.8), we may choose (ZO, SO, consider the set

(9.8.11)

W

E

s

+ it)

such that D(zo,

0 in VE by

s0 + it0)

0 and

:

The real analyticity of D mentioned above implies that W is a proper real analytic subset of {IzI <E} and hence of measure 0. We also note that for IzI <€, z g w, the holomorphic function —RE s + it i-÷ D(z, s + it) does not vanish identically. Hence its trace D(z, s) vanishes at most on a set of measure 0 of the interval <€ by elementary property of boundary values of analytic functions (see e.g. Zygmund [1]). This shows that for z g W we may take ba = and for aa the right hand side of (9.8.9). This proves the last statement of the lemma. We now fix zo e with Izol <E. Denote by a multi-index such that (s i-÷

holds for

= fl0 but for no fi with IflI <

Such a multi-index (with 0 can be found in view of (9.8.8). A simple argument using Leibnitz' rule shows that for a positive integer v, (9.8.12)

s)) $ 0

(s i-÷

holds for = < but for no with If we apply to (9.8.9), set z = Zo, and use the fact that (21a1 — is the minimal length of a multi-index fi for which (9.8.12) (with v = (21a1 — 1)) holds, we deduce that (9.8.13)

(f, 1,

)(Zo,

=

s)

s),

262

LX. HOLOMORPHIC EXTENSION OF MAPPINGS OF HYPERSURFACES

This proves (9.8.5) with

=

s)

I

(9.8.14)

L(21a1_I)Po(D2(at_I)(ZO,

s),

s) =

1

s).

Note that for Zo g W (see (9.8.11)), fi0 = 0, and hence (9.8.14) is consistent with the last statement of the lemma. This completes the proof of Lemma 9.8.4. 0

Lemma 9.8.16 below will be used to give the extension down of f,g) and to prove the estimate (9.8.3). Denote by P1(z, s, X) the polynomials in X given by Proposition 9.6.4, i.e. N1

(9.8.15)

with

P1(z,

s)

-I

X) = XNI +

=

s)X', s). The coefficients dk' extend down

smoothly as was already noted in Proposition 9.6.4. We denote by p(z, s) for k = 1,... , N1, j = 1,... , n, and izI, Psi < €, the roots (counted with multiplicity) in X of the polynomial P1(z,

s,

X).

9.8.16. For 0 sufficiently small and any multi-index a, the coefficients of the polynomial in X

£i=I

where denotes —REforIzI <e.

(pr,...

,

(f,pe,

(x —

...

X) :=

(9.8.17)

Cju=I

£ = (t1,...

,

extend smoothly down to isa root of this polynomial for

IzHsI <E. PRoOF.

We expand the product in (9.8.17) and write N

-I

(9.8.18)

where

N = N1 +

Since for each j fixed the coefficients ek(Z, s) are .+ symmetric holomorphic functions of the roots p, I r N1, there are, in view of Theorem 5.3.4, convergent power series *k such that

(9.8.19)

. .

ek(z,

s) =

s),

§9.8. END OF PROOFOF THE REFLECTION PRINCIPLE

263

We deduce that the coefficients , denotes where (d')1 ) for (Izi < €) (in the sense of §9.2), if ek(z, s) extend smoothly down to E > 0 is chosen sufficiently small (independently of a). This proves the first part of the lemma. Since by Proposition 9.6.4, f, (z, s) is a root of the polynomial

P3 given by (9.8.15), it follows by the construction of the polynomial P given by (f, f, g) is a root of this polynomial. This completes the proof (9.8.17) that

ofLemma9.8.16.

U

We shall now prove Lemma 9.8.2, and with that we complete the proof of Proposition 9.7.1. PROOF OF LEMMA 9.8.2. We choose e > 0 satisfying the conclusions of both Lemma 9.8.4 and Lemma 9.8.16. Fix Zo E C" with Izol < E. Consider the + it) in — RE given by merormorphic function Ca (zo,

(9.8.20)

.

—

Ca(Z0,ZO,S+it)=

-

ba(ZO,

S + it)

S) 0 and ba are given by Lemma 9.8.4. We note that, since ba (ZO, s) extends smoothly as a holomorphic function in — RE by for Is I <E and ba (zo, Lemma 9.8.4. Hence by the elementary fact about analytic functions used above, there is a subset J of {s E R: Is I <eJ, whose complement has measure 0 such that

where

E

J.

Picks0

JandletUbeaneighborhoodofsoinCsuch Thenthemeromorphic E Uand—e
that s + it) defined by (9.8.20) is holomorphic in —RE fl U and function Ca(Z0, extends smoothly to J fl U. Moreover, if P (z, s, X) is the polynomial given by 0 for s flU by combining Lemmas (9.8.17), then P(zo, s)) s, Ca(ZO,

i

9.8.4and9.8.16. Thus, P(zo, 4,s+it,

Oin (—RE UJ)flU, 0. because it is a holomorphic function that equals 0 on an open interval in t

By analytic continuation, it follows that (9.8.21) for s

+ it in

+it, Ca(ZO, ZQ,S + it)) outside

the poles of CU(ZO,

S + it).

We shall show that (9.8.2 1) implies that there is Zo, such that (9.8.22)

ICa(ZO,

0

a constant

C,

independent of

s + it)I

for all s + it E — RE (outside the poles of the latter function); such an estimate will s + it) is holomorphic in —RE. In obtaining the in particular imply that Ca(ZO, estimate (9.8.22), we will assume that 0 was chosen a priori sufficiently small

IX. HOLOMORPHIC EXTENSION OF MAPPINGS OF HYPERSURFACES

264

(i.e. we may need to make 6 smaller than is needed to apply Lemmas 9.8.4 and 9.8.16). First, since Q(x. z, w) is holomorphic in some neighborhood of (0, 0, 0) there is a constant C and S > 0 such that IQx°(X, z,

for all IzI, Iwl < S and fal > 0. Moreover, since the polynomials (9.8.15) and vanish defining the roots are monic, with coefficients that are smooth on at 0, it follows from Lemma 5.5.1 that if e > 0 is chosen sufficiently small then

<8

(9.8.23) for

(z,s,t) E

and allj = 1,... ,n,k = 1,... ,N1. BychoosingE >0

even smaller if necessary, we may also assume that If(z, s + i:)I < S and 1Q(pC, f, < 8. It then follows from (9.8.17) that the coefficients ek of the polynomial P (see (9.8.18)) satisfy (9.8.24)

Iek(z,

s

+ it)I

C' is some other constant. (Recall that the polynomial P depends on the multi-index cr.) Again applying Lemma 5.5.1 we deduce, from Lemma 9.8.16, (9.8.2 1), and (9.8.24), that (9.8.22) holds with some third constant C = C" that does not depend on Zo nor on a. Since, as already mentioned above, it follows in particular from (9.8.22) that the meromorphic function ca (zo, s + it) is in fact holomorphic and bounded in —Re, it must have a distribution boundary value on {: = 0). (See Chapter VII for more general results on existence and smoothness of boundary values of holomorphic functions.) On the other hand, for s E J, t-+0

+1:) =

Since the function on the right hand side is smooth for Is I <6 and the complement off in the interval (IsI <6) is of measure 0, it follows that the distribution boundary value of + it) on: = 0 coincides with

Hence, we obtain that the function s i-* s) extends continf, g)(zo, uously as a holomorphic function in —Rf, e.g. by Theorem 7.2.6. (In fact the extension is also smooth up to the boundary by Theorem 7.5.1.) Estimate (9.8.3) is then a direct consequence of (9.8.22). To complete the proof of Lemma 9.8.2 (and hence that of Theorem 9.6.1), it remains only to prove that the function f, g)(z, s + it) is measurable in V. For this, as noted above recall that the function D(z, s + it), defined in

*9.9. REFLECTION PRINCIPLE FOR CR MAPPINGS

265

Lemma 9.8.4, is real-analytic in V( and does not vanish identically. Hence the set A defined by (8.9.25)

A := (z, s, t)

E

:

D(z,

s + it) = 0)

and is therefore of measure 0. It follows is a proper real analytic subvariety of from the argument above, that we have for (z, s, t) V \ A, —

(8.9.26)

—

.

f,g)(z,z,s +zt) =

aa(z,z,s+:t

+it)'

with aa given by (9.8.14) with fib = 0. Since the right hand side of (8.9.26) is \ A, where A has measure 0, it follows that the left hand side real analytic in which is the desired conclusion. The of (8.9.26) is a measurable function in proof of Lemma 9.8.2, and hence that of Theorem 9.6.1, is now complete. 0

§9.9. Reflection Principle for CR Mappings In this section, we shall use the general reflection principle given by Theorem 9.6.1 to give other holomorphic extendibility results for CR mappings. We first state a result for a CR diffeomorphism. THEOREM 9.9.1. Let M, M' C CN be real-analytic hypersurfaces with p E M and p' E M'. If h is a smooth CR dsffeomorphismfmm M to M', with h(p) = p', then h extends holomorphically toa neighborhood of p in M is essentially finite at p or M' is essentially finite at p'.

provided that either

Theorem 9.9.1 will be a consequence of a more general result (Theorem 9.9.26),

which we shall state and prove later in this section. For this, we need the following definition for CR mappings between abstract CR manifolds of hypersurface type (see §2.1).

DEfiNITION 9.9.2. Let h: M -÷ M' be a smooth CR mapping between abstract CR manifolds M, M' of hypersurface type. Then, h is CR-transversal at p E M if (9.9.3)

where p' = h(p) and V' denotes the CR bundle of M'. It will be useful to express CR-transversalhty in terms of regular coordinates for embedded hypersurfaces.

IX. HOLOMORPHIC EXTENSION OF MAPPINGS OF HYPERSURFACES

266

PROPOSITION 9.9.4. Let M, M' C C" be smooth hypersurfaces through p and p' respectively, wzdh: M —÷ M'asmooth CR mapping. LetZ = (z, w)be regular coordinates for M vanishing at p E M and Z' = (z', w') regular coordinates for M' vanishing at p' = h(p) E M'. Let (z, s) be the local coordinates on M near p given by the embedding (1.6.3). Then, the following are equivalent.

(1) h is CR-transversal at p.

(ii) If h(z, g(z,

(9.9.5)

s)

= (f(z,

s), g(z,

s))

with z' = f(z,

s)

and w' =

s), then 0.

as

(iii) If h = (f, g) as in (ii) and c(z, w) denotes the formal holomorphic power series at (0,0) associated to the CRfunction g given by Proposition 1.7.14, then

(9.9.6)

0.

9w

PROOF. As in the proof of Proposition 9.2.6, it is easy to see from (9.2.7) that the "transversal" derivatives

(9.9.7)

9w real and equal. This proves that (ii) and (iii) are equivalent. To prove that (1) s) and (z'. and (ii) are equivalent, we observe that in the coordinates s') on M and W respectively, we have are

(9.9.8)

j,

az1

,...

I.

Since h is a CR mapping, CR-transversality at p = 0 is equivalent to

(9.9.9)

øs

The equivalence of (i) and (ii) follows from (9.9.8), the reality of calculation

The proof of Proposition 9.9.4 is complete.

(0) and the

0

We shall say that a CR mapping h: M -÷ M', with h(p) = p', is finite at p is if the formal holomorphic series in Z given by + Z) — N(p), where the formal holomorphic series associated to the components of h by Proposition 1.7.14, is a finite formally holomorphic mapping in the sense of Definition 5.2.7.

267

§9.9. REFLECTION PRINCIPLE FOR CR MAPPINGS

be real-analytic hypersulfaces, and PRoPosrnoN 9.9.10. Let M, M' C M. If h M -+ M' is a smooth CR assume that M is essentially finite at p mapping which is CR-transversal at p. then the following hold.

(1) M' is essentially finite at p. (ii) The CR mapping h is finite at p. (iii) The CR mapping h is not totally degenerate at p. PROOF. We choose normal coordinates Z = (z, w) for M vanishing at p and w') for M' vanishing at h(p). Thus, we assume that M and M' are given = by (9.2.4) and (9.2.5), respectively. We write h = (f, g) as in §9.5 and denote w) the formal holomorphic power series corresponding to f by F(z, w) and and g. We first prove the following lemma.

LEMMA 9.9.11. There exists a formal power series a(z, x) in 2n variables (z, x) such that (9.9.12)

a(z,

with cx(O, 0) =

0), F(x, 0), 0),

0)

0).

PROOF. We begin with the identity corresponding to (9.2.7) using the formal power series (F(z, w), G(z, w)) associated to the mapping H at p = 0,

(99 13)

,s where = 4(z, 2. s). By setting s = 0 and 2 = 0 in (9.9.13), we deduce, from the normality of the coordinates, that

Q(z,0) -'-0.

(9.9.14)

By using (9.9.14), we obtain that there exists a formal power series

(9.9.15)

c(z, w) — Q(x. —w) 2i

Using the reality of

(0) (see proof of Proposition 9.9.4), we have

(9.9.16)

fi(O) = —iRe

aw

=

aw

x. w),

IX. HOLOMORPHIC EXTENSION OF MAPPINGS OF HYPERSURFACES

268

Similarly, using (9.9.14) and the fact formal series y(z, x' w), with y(O) = (9.9.17)

w),

—

F(x, —w),

= 0,

0,

we deduce that there exists a

such that

g(z,w)+c(x,—w) 2

+ y(z, x0 w)w. We now sets = 0 in (9.9.13). The lemma follows from (9.9.13), (9.9.15), (9.9.16), and (9.9.17) with (9.9.18)

cr(z,

x)

ifi(z,

—

iy(z,

X,O)).

0

This completes the proof of Lemma 9.9.11.

We return to the proof of Proposition 9.9.10. By Proposition 9.9.4, the CR transversality of h, and Lemma 9.9.11, we we have a(0, 0) we may divide both sides of (9.9.12) by a(z, x) to obtain

0 in (9.9.12). Thus,

(9.9.19)

By differentiating (9.9.19) repeatedly with respect to x and setting x = 0, we conclude, as in the proof of Proposition 9.4.7, that the ideal in generated by the derivatives alals)

(9.9.20)

___L(z, 0, 0),

aE

is contained in the ideal in generated by F1 (z, 0),... , 0). The ideal generated by (9.9.20) in C(z} has finite codimension by the essential finiteness of M; it follows from Corollary 5.1.26 that the ideal generated in by (9.9.20) also has finite codimension. Hence the ideal generated by Fi(z, 0),... , 0) in is also of finite codimension. On the other hand, by (9.9.14) and the CR-transversality of h (i.e. (9.9.5)), we conclude that the ideal generated by the components of fl(z, w) = (F(z. w), c(z, w)) in w]J is the same as the ideal generated by the set f F1 (z, 0),... , 0), w). It then follows easily from the above that this ideal is of finite codimension, and hence the mapping 71(z, w) is a finite formally holomorphic mapping. Thus, (ii) is proved. The statement (iii) follows by applying Theorem 5.1.37. That is, we have (9.9.21)

det

(z, 0)

0.

269

§9.9. REFLECTION PRINCIPLE FOR CR MAPPINGS

It remains to prove (1). By differentiating (9.9.19), as above, repeatedly with generated respect to x and setting x = 0, we conclude that the ideal in generated by the by the derivatives (9.9.20) is contained in the ideal in derivatives aIaI*

(9.9.22)

(F(z,0),0,0),

aE

Denote by This proves that the latter ideal has finite codimension in Fk (z, 0) the polynomial mapping, whose components have degrees less than or equal to k, obtained by truncating the series of F(z, 0) at level k. Clearly, we have alal

(9.9.23)

aza

(1k(z, 0), 0, 0)

alal,,,

aza

(F(z, 0), 0, 0)

mod k,

for each multi-index a. Thus, by choosing k sufficiently large, Proposition 5.1.8 generated by 0) is a finite mapping and the ideal in implies that

(9.9.24)

(Fk(z,0),O,0),

aE

equal to that generated by the series in (9.9.22). We will complete the proof of (i), and thus the proof of the proposition, by showing that the ideal generated by is

(z',O,O), has finite codimension in C{z'}. Reasoning by contradiction and using Corollary tending toO such that 5.2.6, assume that there is an infinite sequence z'1 E alal,,, (9.9.25)

0,0) = 0 aza (z",

Since the mapping z '-÷ Fk(Z, 0) is a finite holomorphic mapping and hence open (by Theorem 5.2.9), the inverse tending to 0, by Theorem 5.2.8. images (z'1) contain points in This contradicts the finite codimensionality of the ideal generated by (9.9.24) in Cfz}, again by Corollary 5.2.6. The proof of Proposition 9.9.10 is complete. 0 for all multi-indices a and j = 1, 2

A sharper version of Proposition 9.9.10 may be obtained if M' is assumed to be of finite type at h(p). We state without proof the following result.

270

IX. HOLOMORPHIC EXTENSION OF MAPPINGS OF HYPERSURFACES

be real-analytic hypersurfaces and let THEOREM 9.9.26. Let M, M' C h : M —÷ M' be a smooth CR mapping with h(p) = p'. If M is essentially finite at p and M' is essentially finite at p'. then the following are equivalent:

(i) h is not totally degenerate at p. (ii) h is finite at p, (iii) h is CR transversal at p. Theorem 9.9.26 will not be used in the sequel; we refer the reader to Baouendi and Rothschild [5] for the proof.

We are now ready to state and prove a general reflection principle for CR mappings of which Theorem 9.9.1 is a special case since a CR diffeomorphism is necessarily CR transversal at every point. THEOREM 9.9.27. Let h: M —÷ M' be a smooth CR mapping between real-

analytic hypersurfaces M, M' c if h is CR transversal at p E M and M is essentially finite at p. then h extends holomorphically to a full neighborhood of p in

PROOF. Since M is essentially finite at p. it follows from Proposition 9.4.16 that M is of finite type at p and hence minimal, by Proposition 1.5.13. Therefore we may make use of Theorems 8.1.1 and 7.5.1 to conclude that (after shrinking if necessary) h extends as a mapping H in c00(?F) By Proposition 9.9.10, h is not totally degenerate at p and M' is essentially finite at h(p). Hence, we may apply Theorem 9.6.1 to reach the desired conclusion. D

§9.10. Reflection Principle for Bounded Domains We give some applications of the local reflection principle (Theorem 9.9.27) to proper holomorphic mappings between bounded domains with real-analytic boundaries. A main ingredient here is that the real-analytic boundary of a bounded domain in CN is essentially finite at every point. This is proved in Chapter XI (Theorem 11.6.1), which is completely independent of the results in this chapter. We shall assume this fact in all proofs in this section. Recall that a continuous mapping F: X Y, where X and Y are topological spaces, is proper if F' (K) is compact for every compact K C X. It follows easily that if H is a proper holomorphic mapping between domains D, D' C C" that extends continuously to the boundary 8D, then H(8D) C RD'. THEOREM 9.10.1. Let 0, D' c C"' be two bounded domains with real-analytic

boundaries. Let H: 0 -÷ D' be a proper ho! omo rphic mapping which is smooth

§9.10. REFLECTION PRINCIPLE FOR BOUNDED DOMAINS

271

up to the boundary 3D. Then H extends holomorphically to a full neighborhood in CN of D, the closure of D.

PROOF. We shall give the proof under the additional condition

(9.10.2)

HIaD is CR-transversal at every point of 3D.

Under the additional condition (9.10.2) the proof follows from Theorem 9.9.27 since 3D is essentially finite at every point as mentioned above. In fact1 condition (9.10.2) is always satisfied under the rest of the assumptions of Theorem 9.10.1. We will not give the proof of this fact here and we refer the reader to the paper of

0

HuangandPan[1].

Although the following result for mappings between pseudoconvex domains

(see §2.2 for a definition) is outside the scope of this book, we shall state it (without proof) and use it for the proof of Theorem 9.10.4 given below. THEOREM 9.10.3 (BELL-CATLIN [1J, DIEDERICH-FORNAESS 13]). Let

D and

D' be bounded pseudoconvex domains in with real-analytic boundaries. Let H: D -+ D' be a proper holomorphic mapping. Then H extends smoothly to the boundary of D. We now state a reflection principle for proper mapping for pseudoconvex domains.

THEOREM 9.10.4. Let D, D' c CN be bounded pseudoconvex domains with real-analytic boundaries. Let H: D -+ D' be a proper holomorphic mapping. Then, H extends holomorphically to a full neighborhood of D, the closure of D inCN. PROOF.

By Theorem 9.10.3, we may assume that H is smooth up to the bound-

ary 3D and, since H is proper, maps 3D into 3D'. Theorem 9.10.4 will be a consequence of Theorem 9.10.1 if we prove that H is CR-transversal at every point of 3D. Pick a point E 3D and let c CN be a sufficiently small neighborhood of P0 such that \ 3D has two components, = D fl and = D. We also denote by M = 3D fl Similarly, we let = H(po) and we choose to be a sufficiently small neighborhood of c such that M', where as above M' = 3D' fl has two components, = D' 1t and = c' \ D'. Then, by shrinking further if necessary, H: —÷ is a holomorphic mapping which extends smoothly to M and sends M into M'. Moreover, in the terminology of §2.2 (see Remark 2.2.18 (i)), and are pseudoconvex sides of M and M', respectively; recall that this means that

—

log 8M and — log öM,

IX. HOLOMORPHIC EXTENSION OF MAPPINGS OF HYPERSURFACES

272

are plurisubharmonic where 8M(Z, Z) is the distance to M and similarly for (see §2.2) in and respectively. In view of the remarks above, the proof of Theorem 9.10.4 will be completed by the following proposition. PROPOSITION 9.10.5. L.et M, M' c Ci" be pseudoconvex smooth hypersutfaces E M'. Let be open one-sided neighborhoods of with pa E M, as above, and assume that are pseudoconvex sides of M, M', respectively. If H: -+ is a holomorphic mapping that extends smoothly to M and maps then H is CR-transversal at P0. M into M', with H(po) =

For the proof of Proposition 9.10.5, we shall need to use a particular defining function for M' at The existence of such a defining function is the content of the following lemma.

Let M C C" be a smooth pseudoconvex hypersuiface with < 1, there exist an open neighborhood C C"' of po and a smooth defining function p(Z, Z) of M in with the following properties: c \ M has two components, p(Z, 2) > 0} and {Z E = {Z E Q: p(Z, Z) <0}, where is apseudoconvex side of M, and is strictly plurisubharmonic in LEMMA 9.10.6.

P0 E M. Then for any 0 <

PROOF. In view of Theorem 2.2.17 (see also Remark 2.2.18 (ii)), there is an open neighborhood of P0 and a smooth defining function g (Z, 2) of M in c such

that—Ioga

= tZ

E

a(Z, Z) >

Furthermore, in the proof of Proposition 2.2.28, we obtain the following estimate (see (2.2.30)) for this particular defining function,

(9.10.7)

2)aJãk

—

VaECN, where as usual e.g.

d2a

= For each Z

aZJazk

we write

=

{a E C:

2)a1 = 0j

§9.10. REFLECTION PRINCIPLE FOR BOUNDED DOMAINS

273

for the orthogonal complement of V2 with respect to the usual Hermitian and inner product in CN. Then, (9.10.7) implies that (9.10.8)

2)a1ãk

Va E V2,

0,

VZ E

j.k=1

the vector a E CN as a = a' + a" with a' E

Decompose, for given Z

and a" E

It follows from (9.10.8) that

(9.10.9)

2)afãk

2Re

j.k=I

j.k=I

+ I

I

Since a is smooth in (9.10.9) implies that there is a constant C > of Z, such that for all a CN and Z (9.10. 10)

2)ajäk

0,

independent

lal.

'.Fc=l

Now, we also have, for each a" E (9.10.11)

Ia"I

and Z E

= az(Z, Z)I

Since we may assume oz (Z, Z)

0 in Q, it follows from (9.10.10) and (9.10.11) that there is a constant C > 0 such that, for all a E CN and Z

(9.10.12)

Z)aJäk

2)a4.

C jaI

j=I

j.k=I

with 0 < < 1. For simplicity of notation, we shall assume that Po = 0. We will show that, for K > 0 sufficiently large (depending on the defining Choose

function

p(Z, Z)

a(Z,

satisfies the conclusion of the lemma. A straightforward calculation shows that

(9.10.13)

= I1Q.l7_2e_J7KIZI

IX. HOLOMORPHIC EXTENSION OF MAPPINGS OF HYPERSURFACES

274

with (9. 10. 14)

((1

NJk =

+

—

+ Zkaz1) —

+ a2K(Sj& —

where denotes the Kronecker delta symbol. We claim that if we choose K > 0 sufficiently large and then make sufficiently small, then for any a C's' and we have Z

> !a21a12

(9. 10.15) j.k=I

which will complete the proof of Lemma 9.10.6. To prove the claim, we note that, by (9.10.12), (9. 10.14), and the Cauchy—Schwarz inequality, we have

(9.10.16)

(1

j=I

j.k=I

+ C) + Ka21a12(1 —

— aIaI

where C is the constant in the inequality (9.10.12). Now, let us choose

(9.10.17) and

K

i+

(C+ 1)2

so small that 1,

We

1,

YZ

c.

then get, for Z

(9.10.18)

j=I

j.k=I

j=I

K22 lal.

Completing the square on the right in (9.10.18), we find

(s/i

(9.10.19)

jk

I

j

I

fK

(C+1)2\

2

2

275

§9.10. REFLECTION PRINCIPLE FOR BOUNDED DOMAINS

l'his proves the claim (9.10.15) in view of the choice of K in (9.10.17).

0

We now turn to the proof of Proposition 9.10.5. PROOF OF PRoPosmoN 9.10.5. In view of Lemma 9.10.6, we may assume, by is an open is a component of \ M', where if necessary, that shrinking neighborhood of in which there exists a defining function p'(Z', Z') for M' with and —p2/3 is strictly plurisubharmonic in the property that p' is positive on into Consider the function accordingly so that H maps We shrink —

*(Z, Z) := _(p'(H(Z), H(Z)))

(9.10.20)

2/3 ,

zE

It is easy to see that the composition of a plurisubharmonic function with a holomorphic mapping is plurisubharmonic and, hence, is plurisubharmonic in Moreover, is negative in c1, extends continuously to M, and is identically 0 on -+ CN be an analytic disc (see Chapter VI) with A(0) = P0 and M. Let A: 0 for E such that the image of A is an embedded smooth complex curve C. We also require that the intersection between C and M is transversal, i.e. that E

with

EM. (WecantakeCtobee.g. asmailpieceofa

complex line that meets M transversally at p0.) By shrinking if necessary, we may assume that there is a smooth real curve y C C through 0 that divides into two components, and A2, and that A maps into and the curve y into M. We write = + flj. By a rotation in C, we may assume that y is tangent to the n-axis at = 0 and that contains a connected segment of the negative i-axis with 0 as a boundary point. which extends The function h = o A is a negative subharmonic function in continuously, with boundary values 0, to y. The following result is a consequence of the classical Hopf Lemma. For the reader's convenience, we also indicate how to give a direct proof using the Poisson kernel. LEMMA 9.10.21. Let C Candy C C be as above. Let h (or nonpositive but not identically 0) subharmonic function in

be a negative which extends

continuously to y in such a way that h(0, 0) = 0. Then, there exist C > 0 and t5 > 0 such that (9.10.22) PROOF.

LetE c Cbeasmalldiscsuchthat0E dEandE\{0J C

Then,

h extends continuously to E. By a translation and a simple scaling argument, we

IX. HOLOMORPHIC EXTENSION OF MAPPINGS OF HYPERSURFACES

276

may assume that E is the unit disc and that the conclusion we need is < —C(1

(9.10.23)

0<

—

<

1,

where C > 0 is some constant. To prove this inequality we observe that (9.10.24) h

the boundary of the unit disc E and Pk is the

Poisson transform of k (see (6.1.4)). Since/i 0 and/i 0 on a E, it is easy to see, using (6.1.4), that (9.10.23) holds for h replaced by Pk. The desired inequality (9.10.23) is then a consequence of (9.10.24). 0 We

return to the proof of Proposition 9.10.5. We apply Lemma 9.10.21 to —((p'

(9.10.25)

o

H o A)(C,

(p'

o

to deduce that

Ho

—8

<0.

H extends smoothly to M, and A is holomorphic in Since p' is smooth in the function on the left hand side of (9.10.25) is smooth up to y. By the inequality (9.10.25), we therefore conclude, since p' o H o A vanishes at 0, that (9.10.26)

a(p' oH 0 A)(00)

<0.

This implies, since p' is a defining function of M', that (9. 10.27)

(Ho

g

as is easily verified using the chain rule. On the other hand, since A maps y into M, we have (9.10.28)

X,,0 = A.

—

T,,0M.

\dq 0/

We claim that E \ T.M'. Indeed, since the holomorphic mapping H extends smoothly to M, we have (cf. Proposition 2.3.7), (9. 10.29)

(J 0 H.)(X,,0) = (H. 0 J)(Xpc)

and, since A is holomorphic,

= (A. 0 1)

=

where, by a slight abuse of notation, we denote by J the complex structure on the

appropriate complex space. The claim follows from (9.10.27). Hence, H is CR transversal at p0 and the proof of Proposition 9.10.5 is complete. 0 As mentioned above, this also completes the proof of Theorem 9.10.4.

0

§9.11. FURTHER RESULTS ON THE REFLECTION PRINCIPLE

277

§9.11. Further Results on the Reflection Principle In the main result of this chapter, Theorem 9.6.1, it is required that the target is essentially finite at 0 and that the mapping H defined hypersurface M' C on one side of M, taking M into M', and preserving the origin, is not totally degenerate at 0. As is shown by examples, some combination of assumptions and the mapping H is necessary in order on the hypersurfaces M, M' C for the mapping to extend holomorphically across the origin, but the conditions of Theorem 9.6.1 are by no means necessary. In this section, we state, without proofs, a few results where other combinations of assumptions are used, as well as a result for higher codimensional generic submanifolds. We use the same notation as in previous sections. We begin with M and M' being real-analytic hypersurfaces in C2, where the situation is better understood. Recall that in C2 the notions of essential finiteness, finite type, and D-finite type all coincide (Proposition 9.4.16). The following result states that if both hypersurfaces M, M' C C2 are assumed to be of finite type (or, equivalently, essentially finite) at 0, then one needs to assume nothing beyond continuity up to M of the mapping H. THEOREM 9.11.1 (HUANG [2J). Let M, M' C C2 be real-analytic hypersur. faces with 0 E M and 0 E M'. Assume that M and M' are both offinite type at 0. Then, any holomorphic mapping H E suchthatH(M) c M' and 11(0) =0, extends as a holomorphic mapping in a neighborhood of 0 in C2.

If instead the condition that the mapping H is smooth and not totally degenerate at 0 is kept, then it is possible (still in C2) to relax the conditions on the hypersurfaces, as is illustrated by the following result. We say that a real-analytic hypersurface M C is Levi flat at 0 if the Levi form is identically 0 (as a Hermitian form) at every point p E M near 0.

M, M' c

be realanalytic hypersurfaces with 0 E M and 0 E M'. Assume that M is not Levi flat w 0. Then any holomorphic mapping H E H is not totally degenerate at 0, extends as a C THEOREM 9.11.2 (BA0UET'IDI-RomscInLn 171). Let

C2

holomorphic mapping in a neighborhood of 0.

Notice, in particular, that M' need not be of finite type (essentially finite) at 0 for Theorem 9.11.2 to apply. Indeed, if M is e.g. of infinite type at 0, but not Levi flat, then the theorem applies and, under the assumptions of the theorem, M' is necessarily of infinite type at 0. We now return to the case of real-analytic hypersurfaces in for general n 1. Recall that when M is of finite type (minimal) at 0, all CR functions (and

IX. HOLOMORPHIC EXTENSION OF MAPPINGS OF HYPERSURFACES

278

hence all CR mappings) extend holomorphically to one side of M. However, when M is no longer assumed to be of finite type at 0, the class of CR mappings sending

M into M' is potentially larger. If M and M' are assumed to be algebraic (see Chapter XIII for a precise definition), then one can prove the following extension result for CR mappings (which are not assumed a priori to extend to either side). THEOREM 9.11.3 (BAOUENDI—HUANG-ROTHSCHILD 11]). Let

M, M' C

be connected real-algebraic hypersuifaces. Assume that there exists p E M at which M is essentially finite. If H: M -÷ M' is a smooth (C°°) CR mapping whichisadiffeomorphismatsomepointpi E M, then H extends asaholomorphic n

1,

mapping in an open neighborhood of M in Let us remark that the conclusion of Theorem 9.11.3 fails to hold if the condition that M and M' are algebraic is removed, as is illustrated by the following example.

EXAMPLE 9.11.4. Consider the following two real-analytic hypersurfaces in C2,

M = ((z,w):

(9115) where t

Im w =

=

w)

s) is the unique solution of the equation

s, t) = (0,0, 0). Both M and M' are of infinite type at the origin and neither is Levi flat at that point (and hence there exist essentially finite points arbitrarily close to 0). Consider the mapping, defined for w) M, H(z,w) = (z,g(z,w)), in a neighborhood of the point

if Re w > 0 and g(z, w) = 0 if Re w 0. The reader can check (or consult Ebenfelt [1]) that H is a CR mapping from M into M' with Jac H 0. Clearly, H does not extend as a holomorphic mapping in any neighborhood of the origin. Observe that M' is real-algebraic, but M is merely real-analytic. Thus, the only hypothesis of Theorem 9.11.3 that is violated is the required real-algebraicity of M. One can construct a similar example where M is real-algebraic but M' is not. where g(z, w) =

There are other results concerning the reflection principle for real-analytic hypersurfaces at nonessentially finite points, which assume various special forms of the defining functions. We mention a few articles in the notes below. Also in this chapter we have not addressed the question of holomorphic extension of

NOTES FOR CHAPTER IX

279

CR mappings between real-analytic generic submanifolds of higher codimension. However, many of the techniques used for hypersurfaces in this chapter can be extended to higher codimension. We conclude this chapter by giving, without proof, such a result for CR diffeomorphisms. The notion of essential finiteness defined in §9.4 for hypersurfaces may be extended to higher codimension; we refer the reader to Chapter XI for the precise definition.

[1]). Let M, M' c C" be 9.11.6 generic real-analytic submanzfolds and p E M. Assume that M is essentially that extends holomorphically finite at p. If h : M -+ M' is a CR to an wedge of edge M centered at p, then h extends as a biholomorphism in an open neighborhood of p in CN.

Combining Theorem 9.11.6 with the holomorphic extendability of CR functions

to a wedge for minimal generic submanifolds (Theorem 8.1.1), we obtain the following corollary. COROLLARY 9.11.7. Let M, M' c C"' be generic real-analytic and p E M. Assume that M is essentially finite and of finite type at p. If then h extends as a biholomorphism in an h : M -+ M' is a CR

open neighborhood of p in C"'. Notes

for Chapter IX

The classical Schwarz reflection principle essentially dates back to Schwarz [1] in 1869. Fefferman's theorem (Feffennan [1]), as stated in Remark 9.3.9, was published in 1974. The first results concerning local holomorphic extension of sufficiently smooth holomorphic mappings across real-analytic hypersurfaces in several complex variables are due to Pinchuk [1] and Lewy [3], who considered the case of Levi nondegenerate hypersurfaces (see Theorem 9.3.4). Later, Diederich and Webster [1] obtained some results about holomorphic extendability of biholomorphisms across certain Levi degenerate points on a real-analytic hypersurface. Theorem 9.6.1, in C2, was proved by Baouendi, Bell, and Rothschild [1], and later, in CN, by Baouendi and Rothschild [3], [5], and independently, in a slightly different formulation, by Diederich and Fornaess [4]. Theorem 9.10.3 was proved independently by Bell and Catlin [1], and Diederich and Fornaess [I]. Theorem 9.10.4, in C2, appeared in Baouendi, Bell, and Rothschild [1], and later, in independently in Baouendi and Rothschild [3], and Diederich and Fornaess [4]. Diederich and Pinchuk [1] generalized Theorem 9.10.4 in the case of C2 by dropping the condition of pseudoconvexity. The axguments in the proof of Proposition 9.10.5 are due to Diederich and Fornaess [1] (Lemma 9.10.6), and Fornaess [1]. (These methods were also used in Bell [1].)

280

IX. HOLOMORPHIC EXTENSION OF MAPPINGS OF HYPERSURFACES

As mentioned in the text, Theorems 9.11.1, 9.11.2, 9.11.3, and 9.11.6 are due, respectively, to Huang [2], Baouendi and Rothschild [7], Baouendi, Huang, and Rothschild [1], and Baouendi, Jacobowitz, and Treves [1]. Example 9.11.4 was given in Ebenfelt [1]. We mention here also the papers by Meylan [1], Maire and Meylan [1], Merker and Meylan [1], and Mir [1], where holomorphic extension results at not necessarily essentially finite points can be found.

CHAPTER X

SEGRE SETS

Summary In this chapter, we introduce the Segre sets and Segre manifolds of a generic realanalytic submanifold M C C" at a given point p. The Segre sets are a sequence of subsets of CN of increasing generic dimension. One of the main results in this chapter is that the intersection between M and the largest dimensional Segre set

at a point p E M coincides, generically, with the local CR orbit of p in M. As a consequence, M is minimal at p E M if and only if the maximal dimension of any Segre set at p equals N. The Segre sets and manifolds are essential tools in the study, in Chapter XII, of holomorphic mappings which take one generic real-analytic submanifold into another. They are also crucial in Chapter XIII for the study of mappings between real-algebraic subvarieties. We begin by defining the complexification of a generic real-analytic submanifold of C". The Segre manifold of order k of M at p is defined, in § 10.2, as a complex submanifold of CkN. The Segre set of order k is then defined as a projection into C" of the corresponding Segre manifold. After developing some basic properties of the Segre sets and manifolds in § 10.2 and § 10.4, we state the main theorem in § 10.5 and give some examples in detail. The proof of the main theorem is then given in § 10.6—s 10.7.

§10.1. Complexification of a Genenc Real-analytic Submanifold

Let M be a generic real-analytic submanifold in Ce'. We shall define the complexification M C C" x CN of M. We embed CN in C2N = x as the maximal totally real subspace (10.1.1)

:=

The natural anti-holomorphic involution

(10.1.2)

= 2}.

E

(Z,

:= 281

in Ci" defined by Z)

282

X. SEGRE SETS

leaves the totally real space the set in defined by

invariant. Given a set S in

'S :=

(10.1.3)

we denote by 'S

E 5).

We use the same notation for the corresponding transformation taking sets in

to sets in domain C

Note that if X is a complex analytic set defined near Z° in some by h,(Z) = ... = = 0, then 'X is the complex analytic = Z0 = ... = = 0. Here,givena

setin'c2 C

holomorphic function h(Z) we use the notation h(Z) = h(Z). The transformation * also preserves algebraicity of sets. Let U c CN be an open subset intersecting M such that

MflU = fZ €U: p(Z,Z)=0),

(10.1.4)

where p =

,

Pd) and the Pj are real-valued, real-analytic functions such

that azpi A. . AdZPd Oon MflU. We shall assume thatp(Z, in U x'U. We define the local complexification M = Mu C .

(10.1.5)

M : Mu := ((Z,

EU

x 'U: p(Z,

is holomorphic

of Mn U by

= 0).

We shall also assume that (10.1.6) a complex submanifold of codimension d in U x 'U. Note that M can be identified with a real-analytic submanifold in the totally real space such that M is its usual complexification. We observe that M is invariant under the involution defined in (10.1.2). Indeed, all the defining functions p(Z, Z) for M are real-valued, which implies that the holomorphic extensions p(Z, satisfy

(10.1.7)

Thus given (Z,

15(Z, E C2N we

have

= p(C. Z). 2) =

=

Z) = p(Z,

so

(10.1.8)

(Z,

M

(Z,

E

M.

We shall use the notation introduced in this section throughout this chapter.

283

§ 10.2. DEFINITION OF THE SEGRE MANIFOLDS AND SEGRE SETS

§10.2. Definition of the Segre Manifolds and Segre Sets is as in § 10.1. Weshaildefine a sequence of Pickapoint p U, where U c complex manifolds 6*(p) = (Sk(p, U) C CkN,k l,as follows. Fork = 2j+l with j 0, we define

(10.2.1)

(Z,

62j+1(P, U) (Z1,

),

. . .

'Si(p, U) = (Z

(10.2.2)

For k = 2j with j

(Z,

i'), (Z',

.

. .

,

(Z'', i'),

E

= 0 should be read as

where the above for j

(10.2.3)

,

U: (Z,

E M}.

we define

62j(P, U)

i), (Z',

),

. . .

,

(Z',

. . .

,

(ZJ',

(p.

.A'l}.

U) may be empty, but for p U sufficiently close Note that = to M it is not. Indeed, since p(q, = 0 for all q M fl U and (10.1.6) holds, we can make use of the implicit function theorem to prove the existence of Z U = 0, for p sufficienfly close to M. The reader such that (Z, j3) M, i.e. p(Z,

can easily check that (10.1.6) implies that if (p, U) is not empty, then it is a complex submanifold of U of codimension d. In what follows, we shall use the phrase "for p U sufficiently close to M" to mean sufficiently close so that 6i(p, U) is not empty. We claim that, for k 1, U) is nonempty if and only if U) is nonempty and, if so, = ek(p, U) is acomplex submanifold of codimension kd in CkN. Indeed, if U) is empty, there is no point Z U such that (Z, M, or E tU such that (p, E M. Clearly, it follows from (10.2.1) and (10.2.3) that 6k(p. U) is empty for all k 1. Conversely, if (5i(p, U) is not empty, then there exists Z U with (Z, j3) M. It follows that the point (Z, Z, p,... , Z) is in (p. U) and similarly for e21(p, U). The fact that if ek(P, U) is nonempty, then it is a complex submanifold of codimension kd in C&N follows from (10.1.6), (10.1.7), and the definitions (10.2.1) and (10.2.3). This proves the claim. We shall call the complex manifold U) defined by (lO.2.1)—(10.2.3) the Segre of p of order k. We denote by

x...xC" ÷CN

X. SEGRE SETS

284

the projection onto the first factor Cs'. We are now ready to define the Segre sets

Sk(p)= Sk(p,U)c U,k >_0,ofMatp. DEfINITION 10.2.4. Fork = 0, let So(p) = (p}. Fork

1, let

Sk(p) := Sk(p, U) := pr (ek(p, U)),

(10.2.5)

where (Sk(p,U) is defined by (10.2.1)—(10.2.3). The above definition is significant only if p U is sufficiently close to M since U) may be empty. Whenever U is considered fixed, we shall otherwise suppress the dependence of M, and Sk(p) on U. In general, the sets Sk(p, U), fork > 2, even if nonempty, are not submanifolds of U. However, for p sufficiently close to M, the set

U) =

(10.2.6)

U)

is nonempty and is a submanifold of U as mentioned above. It is called the Segre variety of M at p, first introduced by Segre [1] in the case of a hypersurface. (See also the notes at the end of this chapter). Another useful characterization of the Segre sets Sk (p) is given by the following inductive procedure. PRoposrrloN 10.2.7. The following holds for k

E*Sk_%(p,U),

Sk(p,U) = (Z EU: (10.2.8)

=

PROOF.

?:

eM)

Si(q,U).

U

q€SA_L(p.U)

We shall prove the equality

E *S2j(p, U), (Z,

S21+i(p, U) = (Z E U:

(10.2.9)

for j

I

E

1 and leave the remaining cases to the reader. Suppose that Z E Then, by Definition 10.2.4, there exist Z',... ç', Z') E x ... x *U x U such that U).

xU

MJ,

(10.2.10)

*

(Z, c'), (Z', i'), .

. .

,

(Z',

.

. .

,

E .A't.

We claim that E S21(p, U), which will imply that Z is in the right hand side of (10.2.9). Note that, by (10.1.8), we also have from (10.2.10)

(10.2.11)

Z'),

. . .

,

(c',

21),... (c', Z''), (p, Z') E 14, *

which proves the claim. The opposite inclusion is proved similarly.

0

* 10.2. DEFINITION OF THE SEGRE MANIFOLDS AND SEGRE SETS

PRoposmoN 10.2.12. For p (10.2.13)

M fl U, the following inclusions hold:

U) C S1(p, U) C

(p) =

U) C Sk÷I(p, U) C

C

In addition, for p (10.2. 14)

(p} =

285

U sufficiently close to M, the following inclusions hold: So(p,

U) C S2(p, U) C C S21(p, U) C S21÷2(p, U) C

and also

U) c

(10.2.15)

U) c ... c

U) C S21+3(p, U) c

To prove (10.2.13) we observe that if p E M, then p Si(p). The inclusions (10.2.13) follow from Proposition 10.2.7 by induction on k. To prove (10.2. 14), it suffices, by making use of Proposition 10.2.7 and an inductive argument similar to the one above, to prove that for p E U, sufficiently close to M, we *U such that have p S2(p). Thus, we must show that there is PROOF.

p(p,

= 0,

i.e. that (3i(p, U) = S1(p, U) is nonempty. As remarked above, this holds for p U sufficiently close to M. The proof of(10.2.15) follows the same lines. 0

Let U' c

c CN

§10.1, and let p

be open subsets satisfying the conditions imposed on U in U'. It is immediate that, for all k 1,

(10.2.16)

6k(P, U') C 6k(p. U),

U

S*(p, U') C Sk(p, U).

U') is nonempty, e.g. if p U' is sufficiently close toM, then U') is an open submanifold of ek(P, U). The following proposition is an easy consequence of the definition of the Segre manifolds given by (10.2. 1—10.2.3) and is left to the reader.

PROP0SmON 10.2.17. For p E U sufficiently close to M, there are natural U) -÷ 6k+2(P, U)givenasfollows. Fork = 2j,

holomorphic embeddings j ? 1, the map e2J(P, U) —* (10.2.18)

(Z, C'.... , Z'', C')

U) is given by

(Z,

. . .

,

Z'',

Z'',

X. SEGRE SETS

286

and similarly fork = 2j + 1, j

0, the map 62j+1(P, U)

U) is

given by

(10.2.18')

(Z,e,...

Furthe rmore, for p M there are natural holomorphic entheddings 3k(p, U) U)given by p) e21÷1(p, U), j 1, U) £1 i-+ (102 19) U) i-+ (52j+2(P, U), j 0. The Segre sets of M at p, with p E M, play a crucial role in the rest of this chapter and we restrict ourselves to this case for now. We observe that if p U) contains a distinguished point, namely each of the submanifolds

10220

M,

)

(depending on whether k is odd or even). Thus, p E Sk(p. U) for all p. k, and U. We define U) to be the connected component of U) containing the distinguished point given by (10.2.20). We observe that the generic rank of the projection pr restricted to any connected component of U) is constant. For every p M and every k 1, we let dk(p) be the integer defined by (10.2.21)

dk(p) :=

max

Ci" is the projection onto the first factor as given before Definition 10.2.4.) The reader should note that, in view of (10.2.16) and the fact that the projection is a holomorphic mapping (see the observation above), the number dk(p) is independent of the choice of the open set U containing p M. Although of less importance, the definition of the numbers dk(p), given by (10.2.21), can be generalized to points p U near M but not necessarily on M. where is given by (10.1.5). Let U be an open set as above and (p.9) 9, U) to be the connected component of the Segre manifold We define U) which contains the distinguished point (Here pr

(10.2.22) (p,0, p.... ,9, p)

U),

(p,0,... p,0) e21(p, U), ,

(depending on whether k is odd or even). As in (10.2.21) we define the number dk(p, 9) by (10.2.23)

dk(p, 0)

0E6(p.&U)

rank(prl

we have dk(p, Observe that for p E M and 9 = = dk(p). Also, as before, the number dk(p, 0) is independent of the choice of the open set U (as long as (p, 0) E Me). We have the following.

§10.2. DEFINITION OF THE SEGRE MANIFOLDS AND SEGRE SETS

287

PRoPosmoN 10.2.24. Let U be as in §10.1 and (p,0) E Mu (defined by (10.1.5)). Then the following inequalities hold: N, N —d =dt(p,0) (10.2.25) ... N. for p E M, the following hold:

...

N — d = di(p)

(10.2.26)

N.

U) given by observe that the embeddings (Sk(p, U) -+ (p.0, U) into the connected Proposition 10.2.17 map the connected component 0, U) and commutes with the projection pr (on the first factor component CN). The inequalities in (10.2.25) easily follow. Similarly, for p E M the U) into embeddings '3k(P. U) -÷ ek+I(P, U) given by (10.2. 19) maps (p. U) and commutes with the mapping pr, from which (10.2.26) follows. 0 This proves the proposition. PROOF. We

Another useful inductive description of the Segre manifolds is given by the following proposition, which is an immediate consequence of the definitions (10.2.1)— (10.2.3), and is left to the reader. PROPOSITION 10.2.27. Let U be as in § 10.1. Then the following holds for all

2andallp EU:

k

6k(p. U) =

EM).

U),

Z E U, £1 E

The following shows that one can choose arbitrarily small open sets U for which

the Segre manifolds ek(p, U) are connected.

PRoPosmoN 10.2.28. Let po E M, Uo an open neighborhood of po in and Ic0 an integer 1. Then there exist open neighborhoods U, U' of Po in C's' (satisfying the conditions imposed in § 10.1) with U' C U C U0 such that for all In U) is p E U' and all k, 1 k icc,, the Segre manifold there is a neighborhood basis B,,0 of Po in C" such thaifor all U E ek(p0, U) =

U), Vk,

Ic

1

k0.

PROOF. Let po E M and let (z, w) be normal coordinates for M vanishing at th) in a neighborhood of po = 0 as in

po such that M is given by w = Q(z,

Proposition 4.2.12. For r > 0, we shall use the notation

:=

(10.2.29) For

z = (Zi,...

,

(z E

:

Izi
we have used the notation

:= (w E C" : Iwl
=

sup 1z11 and similarly for

x x and that U0 contains (r0) x (ro). We shall fix these coordinates for the remainder of the proof, which will be based on the following lemma. Iwl. We may assume that Q is holomorphic in

X. SEGRE SETS

288

there exists 10.2.30. Forany integerk and any E with 0 < such that for all r, 0 < r < r1(k, E), and all r1(k, e) with 0 < ri(k, e) <
(10.2.31)

x

Sk(p,

x

C

PROOF. We prove the lemma by induction on k, and we begin by the case k = It is convenient to write

F(z,

(10.2.32)

Q(z, x,

r)

for

1.

r).

=

<E > 0,thereisS(€)suchthatlF(z, We claim that the conclusion of the lemma for (z, < I(z, (fork = I) holds with r1(l, E) = min(6(E), ro). Indeed, for any r with 0 < r <
BycontinuityofFatO E

(10.2.33)

U) = Si(p, U)

= {(z, w) E with

U=

:

choice of rj(1, E), it follows that (F(z, x from (10.2.33) that Sj(p, U) C

fork =



given by (10.2.32). However, by the

and F(z,

X

IzI

< E for (zi < r. Hence it follows which is the desired conclusion

1.

We assume by induction that the lemma holds for 1,... , k — I and we shall prove it for k, assuming k 2. We claim that the conclusion of the lemma holds r < rj(k,€). with Proposition 10.2.27, we have for p E B2(r) x (10.2.34) {(z,

il) E

U) = :

x with U = which is contained in

Izi


(WI
x hypothesis. Since Sk(p, U) = pr

E ek_L(P, U),

W

= F(z,

as in (10.2.34) we have pr(fl)

Sk_I(p, U),

by the choice of r1 (k, e) and the inductive

U)) and r

8(€), the claim follows

from the definition of (1(E). The proof of Lemma 10.2.30 is complete.

0

We may now complete the proof of Proposition 10.2.28. We proceed again by x (r0) C U0. In fact we shall induction on k0. As before, we assume prove the following stronger statement (Pk0) by induction on k0, from which the proposition follows.

289

§ 10.3 EXAMPLES OF SEGRE SETS AND SEGRE MANIFOLDS

For any r, 0 < r U) is connected for 1

STATEMENT Segre manifold

U=

(k, ro)),

k

U',

the

lco, with

U' =

x

and p

x

where r1 (k, E) is the function given by Lemma 10.2.30.

We begin by proving (Pk0) for Ico = 1. It follows from (10.2.33) and the conclusion of Lemma 10.2.30 with k = I that we have U)

(10.2.35)

{(z, w)

IzI


F(z,

r1(1, r0), and p U' = Bz(r) x x 0 < r with U = (p, U) is a graph over the ball Izi
6k(P, U)

w=

:IzI
< r} x which shows that U) is a graph over U). Since is connected, by the inductive hypothesis, it follows that the manifold (p. U) 6k—1 2 6k(P' U) is also connected. The proof of Proposition 10.2.28 is complete. REMARK 10.2.37. As a consequence of Proposition 10.2.28, it follows that for the number dk(p) can be defined by replacing U) by Sk(p, U) in (10.2.21) for any U 8,,, where B,, is the neighborhood basis given by Proposition 10.2.28. Similarly, for (p.9) M, the number dk(p, 9) can be defined by replacing 9, U) by U), provided U, U' are as in the proposition and

pM (p,O)

Mu'.

§103 Examples of Segre Sets and Segre Manifolds In this section we give some examples of Segre sets and Segre manifolds. Other examples will be given at the end of § 10.6. EXAMPLE 10.3.1. Let M C (C3 be the generic manifold of codimension 2 given by

(10.3.2)

Im w1

=

Im w2

=

X. SEGRE SETS

290

weuse the notation Z = (z, WI, w2) and = (x1 ti, r2). We take U =C3. The defining equations of M = Mu are then Here,

(10.3.3)

Wi

=

W2 = T2+1Z2X2.

We choose p = (Zp, Wp) = (Zp, WIpi W2p) and, by applying Proposition 10.2.7, we find that (10.3.4) Using

S1(p)=

(p), we obtain, by making use of Proposition 10.2.7,

this to calculate

(10.3.5)

W2=W2p+i(ZZP)2J.

((z,w)EC3: WI

S2(p) = {(z, w)

E

C3:

E

C,

WI=WIpIX'Zp+IZX1, W2=W2pi(X'Zp)2+i(ZX')2} and by repeating this procedure we find

(10.3.6)

S3(p) = {(z, w)

E

C3

z1

:

E C,

=Wip+Z1Zp1Z1X1+IZX1, 11)2 = 11)21) + — i(z'x52 + i(zx')2). WI

U) are graphs over Hence (Sk(p, U) is connected, k =

The reader can also check that the Segre manifolds

connected sets for all p E C3 since U = C3. 8) = 3, 1,2, 3. One may then check that di(p, 8) = 1, d2(p, 8) = 2, and fork 3, for all (p. 8) E M. (We refer to Example 10.5.7 for more details in the

case p = 0.) 10.3.7. Let M C C2 be the hypersurface (10.3.8)

M = ((z, w): 1mw

= Re wIzI2}.

We can take here the defining function of M as

(10.3.9)

p(Z,

= i(t

—

w) + (w +

where Z =

(z, W) and = (x1 r). We shall describe the Segre manifolds of points p E M sufficiently close to the origin.

LetU CC2 betheopenset (10.3.10)

U = {(z,

W): IZI


§ 10.3 EXAMPLES OF SEGRE SETS AND SEGRE MANIFOLDS

where r <

1

is a positive number to be determined later. Since U = *fJ Mu is

defined by

Let

p=

(Zp,

1(z, w,

1

E C2

—

with Iz,1
U)

(10.3.11)

x U: w = l+izX

I

=

w) EU: w =

= S1(p, U) =

j

and similarly,

(10.3.12)

G2(P, U)

w=wp

l+ZXZp

I

1+1ZXJ

Y""• 1ZX I+SXZp

and U) are graphs over shall choose r so that (S1(p, U) and (r) x (r) respectively. For this, let r with 0 < r < be chosen so that We

1

(1±2)2 Since the function t i-÷ is increasing for 0
U) = j(z, w) E C2:

IzI


1

(

and similarly (10.3.14)

U) =

w,

r)

E C4:

IzI
T=wp1_ixzp, w=w Hence with this choice of r, U, and p. the Segre manifolds (p, U) and U) are connected. If, in addition, p is in M, we may make use of(10.2.21) with U) replaced by U), fork = 1,2. Then we havedi(p) = 1. Alsod2(p) = 1 =0,

andd2(p)

X. SEGRE SETS

292

C C2 be the hypersurface of Example 10.3.7 and let defining function of M given by (10.3.9). We take p = 0. We shall show that one can choose a connected open neighborhood U of 0 in C2 satisfying 0 for (Z, U x *(J and such the condition in § 10.1, i.e. such that that (52(0, U) is not connected. Let C C be the open set defined by EXAMPLE 10.3.15. Let M

p(Z,

be the

(10.3.16) and U C C2 the open set defined by

Iw—izl<0.05}.

(10.3.17) Note that Also

(10.3.18)

and U

are starshaped

*U

(with respect to the origin) and hence connected.

r)

=

With the notation Z = (z, w),

C2: X E c,

IT

+


r) we have for (Z,

=

E C2

{w+r=0 and i+zx =0).

(10.3.19)

We leave as an exercise to the reader to check that it follows from (10.3.16)— 0 for (Z, (10.3.19) that pz(Z, E U x *U. With the choice of U given by (10.3.17), the reader can also easily check that U) = {(z, w)

(10.3.20)

<0.05, w

E C2:

and

(10.3.21)

(52(0,U)=

Ix'I

<0.05,

r' =0,

ZEcZ,

Iw—izl <0.05, is clear that (0, U) is a connected complex manifold of dimension 1. However, the two dimensional complex manifold (52(0, U) is disconnected. The connected component U) of (52(0, U) containing the origin is given by It

((z,w,x',r')

Ix'I <0.05,

r' =0 IzI

<0.05,

w=0J.

Furthermore, it is easy to check that e2(o.u)) (Q)

=

2,

1.

Here d1 (0) = d2(0) = 1, which shows that U) cannot be replaced by (p. U) in (10.2.21). Note also that with this choice of U, the Segre set S2(0, U) contains a nonempty open subset of C2, while this does not happen if U is chosen

as in Example 10.3.7 since in that case the corresponding Segre set is contained in the hyperplane w = 0.

BASIC PROPERTIES OF THE SEGRE SETS

293

§ 10.4. Basic Properties of the Segre Sets We shall need the following characterization of the Segre sets in normal coordinates as introduced in §4.2. We keep the notation of the previous sections of x this chapter as well as that of §4.2. Let p0 E M and let Z = (z, w) E x be normal coordinates for M vanishing at Po. We choose U of the form to be an open neighborhood of Po = (0, 0) contained in the coordinate neighborhood of Po• In particular, M fl U is given by w = Q(z. th), where the function x *(J It will also be convenient to assume that U is Q(z, x. r) is defined in x *U. sufficiently small so that the d x d matrix QT(z, (Recall that QT(0, 0,0) = Idxd.) PROPOSITION

10.4.1. Let M C CN, p0 E M, and U =

(a) lip = (zr,

w)

(10.4.2)

U

x

as above.

in Si(p, U)

and only if

w = Q(z, i,,, ü,,,).

(b) For evely j ? 1 and p = (Zp.

E U, there exists an open set

C

*UZxUzx...X*UzxUzx*UzCC(2)_I)nSuchthat(Z,w)E UisinS2J(p,U) 4f and only if there exists (x'. z',... (10.4.3)

w

= Q(z,

x'.

,

x''.

x')

E V21 with

z', Q(z',

(2(x'. (c) For every j

1 and p = (Zp,

wp)))...))).

U, there exists an open set V2J+j C

Uzx*Uzx...xUzx*UzCC2JPZ such that (z,w)€Uis in S2j+i(p,U)if and only if there exist (x (10.4.4)

z'

... x'.

E V2J+I with

w=Q(z,x',Q(x',z',Q(z',x2,... Q(z3',

Q(z3,

iii,,)))

.. .

The fact that S1(p, U) is given by (10.4.2) follows immediately from (10.2.2). The rest of the proof is by induction using Proposition 10.2.7. Assume that the proposition has been proved for Sk(p, U) with k 2j. By Proposition 10.2.7, U) consists of those (z, w) U for which there exists (x. v) E PROOF.

*S(p U) with w = Q(z, x' r).

(10.4.5)

The desired representation of (p. U) follows by expressing U) using (10.4.2) and substituting this result in (10.4.5). The open set consists of those

(X',z',...,X',zOforwhich(X',z' Q(z1,x2,

x',

z',

i,,,

E

X. SEGRE SETS

294

Since a similar argument holds if the proposition has been proved for all k, k

0

2j + 1, the proof of Proposition 10.4.1 is complete.

In Proposition 10.4.1 the Segre manifolds 6k(p, U) need not be connected. In the following proposition the Segre manifolds are connected, as in Proposition 10.2.28, and in this case the Segre sets may be described more explicitly. PROPOSITION 10.4.6. Let p0 E M, and (z, w) normal coordinates vanishing at

po as in Proposition 10.4.1. Then for every ro > 0 sufficiently small, and every x positive integer k0, there exists r, 0 < r < r0, such that if U = x one has E and p = (Zp,

(10.4.7) andfor any

positive integer j

S21(p,

(10.4.8)

w) E CN:

=

U)

with

2j + 1


Q(z,

k0 the following hold.

U) =

{ (z, w)

IzI

< r, 2(x', z',

w = Q(z.

(10.4.9)

IzI

. . .

,

Q(z',

z1,

S2j+I(P' U) = f (z, w)

CN

:

IzI < r,

w = Q(z,

z',

.

z') E

. .

z', Q(z',

Q(z'',

.

z', Q(z',

time)))

.

The proof of Proposition 10.4.6 is essentially contained in that of Proposition 10.2.28, and the details are left to the reader. 0 PROOF.

We now prove the following symmetry property of the Segre sets for the generic

submanifold M c C". PRoPosmoN 10.4.10. Let M, and U be as in Proposition 10.4.1. The following holds for all k ? 0 and all p, q U sufficiently close to M: (10.4.11)

q

Sk(p, U)

p

Sk(q, U).

295

§10.4. BASIC PROPERTIES OF THE SEGRE SETS

More precisely, we have, fork = 2j,

:= (p.

Z',

. .

.

, Z'',

,

U)

E

(10.4.12)

Z'',

£1 := (q,

. . .

Z',

,

e21(p, U),

and, fork = 2j + 1,

Z',...

:= (p.

Z')

U)

E

(10.4.13)

(q, 2', c'... Z', and

U).

E

,

if U is sufficiently small, p. q E U are sufficiently close to M, and satisfy (10.4.12) or (10.4.13), then

=

(10.4.14)

Clearly, (10.4.11) follows from (10.4.12) and (10.4.13). We shall prove Proposition 10.4.10 in the case k = 2j + 1, and leave the even case to the reader. Recall Z',... , Z') E 62j+,(q, U) if and only if = (p, PROOF.

(p, c'), (Z', c'),

. . .

,

(Z',

(Z',

. . .

c'), (Z1,

,

.A'lu.

By (10.1.8), this is equivalent to

(c'',

(i', 2'),

. . .

,

a", 2'),

2'),

.

. .

ii...

,

a", , 2',

(q,

which in turn is equivalent to = (q, 2', U). This E proves (10.4.13). It remains to prove (10.4.14). Since we assumed that U is contained in a normal coordinate neighborhood of some p0 E M as in the setup of Proposition 10.4.1, we denote the normal coordinates by Z = (z, w) and, accordingly, p = (Zp, we), and Zk = (4, By Proposition 10.4.1 (or, more q = (zq, wq), = precisely, by an inspection of its proof), it follows that

(104 15)

= thq)))...))).

Xo"

We introduce the map T:

C", defined in

a neighborhood

of the

origin,

follows: (10.4.16)

T(x',z',...

,X',z'):=Q(zp,X',Q(X',z',Q(z',X2,..., x',

z',

Q(z1,

thq)))

.

.

as

X. SEGRE SETS

296

and similarly we write X0 = For brevity, we write X = (x',z',... the fixed point Xo. It follows that 4,... . 4) for

= n +rankT(Xo).

(10.4.17)

Also note, by the first parts of the proposition, that (10.4.15) is equivalent to E e2J+l(P, U). The derivative of the right hand side of (10.4.15) with respect thq), since to tiJq is invertible at (X0, x 'U. Thus, by the implicit function theorem, we may solve (10.4.15) for Wq and we obtain' (10.4.18)

=

liJq

X, z,,,

for some smooth function F. The implicit function theorem implies that the rank equals the rank ofT at X0. Hence, (10.4.18) implies that of X0, z,,,

= n + rank T(Xo).

rank

0

This completes the proof of the proposition, in view of (10.4.17).

PRoPosmoN 10.4.19. For every positive integer k, every p0 E M, and every open neighborhood U c CN of p0 satisfying (10.1.6), there exists E 62k(po. U) such that

=

(10.4.20)

rank

P0.

PROOF. The proof is slightly different depending on whether k is odd or even. We give the proof here only for the case k = 2j and leave the odd case to the reader. It suffices to prove the proposition for U replaced by an arbitrary smaller neighborhood of po, in view of (10.2.16). For fixed k, by Proposition 10.2.28, we x is chosen so that e1(pO, U) is may assume that p0 = 0 and that U = 2k = 4j. (Here (z, w) are normal coordinates vanishing at connected for £ p0 = 0 as in Proposition 10.2.28.) The proof is based on the following observation, which is easily checked from the definition of the Segre manifolds. We have (10.4.2 1)

U) =

(Z,

Z',

. . .

(Z',

,

Z11,

U),

E ,

z2i',

E

U)}.

'Using the first part of the proposition and Proposition 10.4.1, one can find an explicit formula for X. Zp. Wp) similar to (10.4.15).

297

* 10.4. BASIC PROPERTIES OF THE SEGRE SETS

U) and the definition (10.2.18) of d21(po), there

By connectedness of exist points

:= arbitrarily close to the point (p0' Poe...

,

such that

Po,

= d21 (p0).

rank

(10.4.22)

U),

E

,

Now, put

jI

1

73l ,.J

71

'so

with

=

(10.4.23)

By Proposition 10.4.10,

(10.4.24)

...

=

=

,

=

U) and, in addition,

E

rank (prle2J (Z u)X'410)

= d21(po),

= rank

where the last equality is (10.4.22) above. Observe that

((Z,

Z',

. . .

,

Z'',

. .

(Z,

c'):

. ,

Z',

. . .

E

,

U))

is a submanifold of e4J(PO, U). The desired conclusion (10.4.20) then follows from (10.4.24) by taking 0 = The following proposition is not essential for the results on mappings that will be presented later in the book but is included for completeness.

PRoPosmoN 10.4.25. Let po E M, and assume there are positive integers k, £ and points (p.8) E M arbitrarily close to (po, such that 8) £. Then there are points M arbitrarily close to po such that d*(pi) t.

We shall make use of Proposition 10.2.28 and assume that P0 = 0 and that there exist neighborhoods U and U' of 0 in CN such that U) is connected for all p E U'. Hence, by assumption and the definition of the number dk(p, 8) given by (10.2.23), there are points p E U' arbitrarily close to po such PROOF.

that (10.4.26)

aEet(p.u)

rank(prIe,(P U))

? £.

X. SEGRE SETS

298

As in the proof of Propositions 10.2.28 we may assume that U and U' are contained

in a fixed normal coordinate neighborhood of 0. In view of (10.2.25), we may assumet > n = N —d. Wealsoassumek = 2j + 1 andleavetheevencase x CN ...+ Cd, to the reader. We introduce the holomorphic mapping T: defined in a neighborhood of the origin, as follows: (1AA')1\/ •

T(

'Af '' I''p' —

Q(zp,

p

z', Q(z', x'' Q(z''.x',

.

...))).

brevity, we write X = (x'. z',... x'1 z') and write T(z, X, for the expression given by (10.4.27). The argument used in the proof of Proposition For

10.4.7 shows that (10.4.26) is equivalent to (10.4.28)

rank Tx(z, X, fr)

£ — n

This means that there for some fixed (z. X) arbitrarily close to 0 in of size at least (t — n) x (t — n) of the Jacobian matrix is a minor m(z, X, X, which does not vanish identically as a function of (z, X) near the origin for p fixed arbitrarily close to the origin. Consider the convergent power series expansion of the function m,

m(z, X,

=

By assumption, there exist a0, such that the antiholomorphic function p i-+ does not vanish identically in a neighborhood of 0 in We claim that maopo(fr)

the restriction of this function to M does not vanish identically on any neighborhood of P0 in M. For, if it did, then M, near the origin, would be contained in a proper complex analytic subvariety of namely the zero locus of the function This is impossible since M is generic and no proper holomorphic Z i+ subvariety can contain a generic submanifold. (The latter statement is an immediate consequence of Proposition 1.8.11.) It follows that there exist points p' E M arbitrarily close to p0 such that (10.4.28) holds for p replaced by pi. This proves (10.4.26) with p replaced by p'. The proof of Proposition 10.4.25 is done. 0 We conclude this section by discussing mapping properties of the Segre sets. Let M C Ci" and M' c CN be two generic, real-analytic submanifolds. Let

H(Z) be a holomorphic mapping from U to U', where U C C's' and U' c CN' are open subsets satisfying the conditions imposed in § 10.1 and suppose that H

§ 10.4. BASIC PROPERTIES OF THE SEGRE SETS

maps

M into M'. For each k = 2j + 1, with j

299

0, we define the holomorphic

mappings

as follows:

(10.4.29)

112J+I(Z,

Z',

. . .

,

(H(Z), Similarly, fork = 2j, with j>

1,

H(Z'),

. . .

,

we define

(10.4.30)

H(Z'),

(H(Z),

. . .

,

We With this notation, fl1 (Z) = H(Z), and we shall write fl(Z, = fl2(Z, shall denote the complexification of M', the Segre sets and Segre manifolds of M' at p', etc. by adding "primes" to the symbols of the corresponding objects for M at p.

PROPOSITION 10.4.31. Under the assumptions above, we have the following 1: properties for every k

C M' U)) C

U')

H(Sk(p, U)) C

U').

(10.4.32)

PROOF. The second and third inclusions in (10.4.32) follow easily from the first. To prove the first inclusion, let p' be a defining function for M' and consider the function valued in C1, (10.4.33)

(Z,

i-÷ p'(H(Z),

By assumption, this function vanishes for (Z, such that Z E M and = Z. This set of points is a maximally real submanifold of the complex manifold M C Hence (10.4.33) vanishes identically for (Z, E M, which is the same as saying

thatflmapsMintoM'.

0

REMARK 10.4.34. The second inclusion in (10.4.32) also holds with ek(P, U)

and ek(H(p), U') replaced by e U sufficiently close to M as the reader can easily check.

X. SEGRE SETS

300

§ 10.5. Segre Sets, CR Orbits, and Minimality In this section, we consider the Segre sets of a real-analytic generic submanifold M at a point po E M. Recall, from Proposition 10.2.24, that we have

n=di(po)
(10.5.1)

wheredisthecodimensionofM inC" andN = n+d. Wedefinejo

ito

be the smallest integer such that d10(po) is the maximal among all

It is clear from (10.5.1) that the integer Jo is well-defined, and we shall call Jo the Segre number of M at Recall that if U is a sufficiently small neighborhood of p0. then the Segre sets Sk(po, U) are defined in § 10.2 for all nonnegative integers k. (with 2 < m1 < ... < As in §4.3, we denote by m1,... , the Hörmander numbers of M at p0 and , 1,, their multiplicities. We shall write r = and denote by the Hörmander num... bers counted with multiplicity, i.e. = m1 for 1 < j < = m2 for £2, etc. As in §4.1, we denote by the local CR orbit of po +1 in M. In view of Nagano's Theorem (Theorem 3.1.4), Orb1,0 is a germ at P0 of a 2n + r-dimensional real-analytic CR submanifold of M. The main results in this section are the following three theorems.

j

THEOREM 10.5.2. Let M be a real-analytic generic submançfold in

of CR

dimension n, po E M, and Jo the Segre number of M at P0. Assume that there are r (finite) Hörmander numbers of M at P0. counted with multiplicity Then the following hold.

(a) There is a holomorphic submanifold X c

CN

of (complex) dimension n + r through Po such that for every sufficiently small open neighborhood U of Po in Ce", S30(po, U) is contained in X, and U) contains a relatively open subset of X. In = n + r.

(b) The germ of the holomorphic X at p0 coincides with the intrinsic cation of the local CR orbit of po in M, and the germ at p° of M fl X is Orb,,0. REMARK 10.5.3. It will follow from the proof of Theorem 10.5.2 that the in-

equalities in (10.5.1) are in fact strict inequalities up to d10(po) and stabilize after

that, i.e. d1(Po) = for J 2: Jo. This fact implies, in particular, that d + 1. However, the Segre sets themselves do not stabilize. Indeed, the Jo following is a consequence of Proposition 10.4.19 and Theorem 10.5.2. THEOREM 10.5.4. Let M be a real-analytic generic submanifold in CN, p0 E M, and Jo the Segre number of M at P0. Then for every sufficiently small

§10.5. SEGRE SETS, CR ORBITS, AND MINIMALITY

301

open neighborhood U of P0 in CN the Segre set S2JO (poe U) contains a complex submanifold of C" through po of complex dimension dj0(po), and the germ of S210(po, U) fl M at P0 coincides with the local CR orbit Orb1,, of Po in M. PROOF OF THEOREM 10.5.4. An inspection of the proof of Theorem 10.5.2 will

0 and every sufficiently small open neighborhood U of show that for every j Ce", S1(po, U) C X, where X is given by Theorem 10.5.2. It follows from P0 in Proposition 10.4.19 and Theorem 10.5.2 that S2)0 (po. U) contains a relatively open The theorem now follows from Theorem 10.5.2. D subset of X containing Theorems 10.5.2 and 10.5.4 can be combined to give the following new criterion for M to be of finite type (or minimal) at p0.

THEOREM 10.5.5. Let M be a real-analytic generic submanifold in Ce", Pa M, and Jo the Segre number of M at Pa. Then the following are equivalent.

(1) M is minimal at p°. (ii) = N. (iii) For every sufficiently small open neighborhood U of Po in CN, the Segre

set Sjo(Po, U) contains an open set in (iv) For every sufficiently small open neighborhood U of Po in Ce" the Segre set S210(po, U) contains an open neighborhood of in PROOF OF THEOREM 10.5.5. Clearly, if

= N, then, by Theorem 10.5.2,

X is an open subset of CN and, hence, X fl M contains a relatively open neighborhood of P0 in M. Thus, by Theorem 4.1.3, M is minimal at Po• Conversely, assume that M is minimal at po. Then, again by Theorem 4.1.3 and Theorem 10.5.2, X must contain a relatively open neighborhood of P0 in M. Since M is generic, no open piece of it can be contained in a proper holomorphic submanifold of C"'. (See Proposition 1.8.11.) Thus, the dimension of X, which coincides with is N. This proves the equivalence of(i) and (ii). It follows from the definition of (pa) that (ii) and (iii) are equivalent. Finally, by Theorem 10.5.4, (ii) and (iv) are equivalent. This completes the proof of Theorem 10.5.5. D For the case when M is a hypersurface, the proof of Theorem 10.5.2 is simple. For the reader's convenience, before proceeding to the general case, we shall restate and prove Theorem 10.5.2 in the hypersurface case. PROPOSITION 10.5.6. Let M be a real-analytic in CN, P0 E M, and Jo the Segre number of M at pa. Then Jo equals either one or two. = N) if and only is minimal at Po. Jo = 2 (in which case

X. SEGRE SETS

302

PROOF. We choose normal coordinates (z, w) E C" x C vanishing at po so that

M is given by w = Q(z, th) as in Proposition 4.2.12. Let U be a neighborhood x using the notation in (10.2.29). If we of 0 in of the form U =
U) = ((z, w) E

Izi


Izi
w=Q(z,x,0)).

0) 0, in which case S2(po, U) = U) and dk(po) = n 1, i.e. Jo = 1, or (b) Q(z, x' 0) $ 0. If (a) holds, then the complex hypersurface (w = Oils contained in M and, hence, M is not minimal at 0 and, thus, If (b) holds, then, by normality of the coordinates, (z, x, 0) d2(po) = n + 1 = N, i.e. Jo = 2. Also, if (b) holds, then it is easy to check (e.g.

Either (a) Q(z,

for all k

0

by Theorem 1.6.7) that M is minimal at Po•

The proof in the case of a generic manifold M of higher codimension is much more involved, partly since minimality of M cannot be easily characterized by the form of a set of defining equations. Before entering into the proof of Theorem 10.5.2 in the general case, we shall give a few examples. The first was considered in Example 10.3.1.

10.5.7. Let M C C3 be the generic submanifold defined by Im WI

=

1mw2 =

Then M is of finite type at 0 with HOrmander numbers 2,4, each with multiplicity 1. For j = 1,2,3, we denote by the Segre set Sj(O, C3). By (lO.3.4)—(1O.3.6) we have S1

1058

2

= {(Z,wI,w2) = ((z, w1,

=

E C3: w1

W2) E

C3:

w1, W2) E C3: w1

=0,

W2

=O},

E C, w1

=

izX1 . W2

= i(zx')2}

Z1 E C,

=ix'(z—z'),

W2—i(X')2(Z2(Z')2)).

52 is not an algebraic subset of C3, the closure of S2 in C3 is algebraic and is given by

Although

(10.5.9)

=

{(z, WI, w2) E C3: w2 =

Similarly, is not an algebraic subset of C3, but we note that 53 contains a dense open subset of C3. Indeed, given any (z, WI, W2) E C3 with w2 — 0 and

§10.5. SEGRE SETS, CR ORBITS, AND MINIMALITY WI

0, we may solve explicitly for

303

z' E C in the last equations in (10.5.8) to

show that(z,wl,w2) ES3. 10.5.10. Consider M C C3 defined by

Imw1=1z12,

hnw2=0.

Here, 2 is the only (finite) Hörmander number at the origin (with multiplicity one).

Again, S1 is given by the first equation in (10.5.8), and

It is easy to see that the subsequent Segre sets Sj

d1(0) =

= {(z, wi, w2) e

=

C3: W2

j ? 3.

= 0},

2. Here Jo =

and the intersection of the closure of S2 with M equals the CR orbit of 0 given by M fl (to2 = 0). The holomorphic submanifold X of Theorem 10.5.2 is the hyperplane (w2 = 0). Thus,

1,

and

2

for j

2

We now establish some notation that will be used in the proof of Theorem 10.5.2, which we shall give in the next sections. As mentioned before, we shall work in normal coordinates (z, w), vanishing at and assume that M is given by w = Q(z, th), where Q is as in Proposition 4.2.12. It will be convenient to write (10.5.11)

Q(z,

r) = T +q(z,

r),

where

(10.5.12)

q(z,0,r) =—q(O,x,t)

In C2", we choose coordinates (Z, with Z = (z, w) and = r), where C" and w, r E z, Thus, in view of the reality condition (10.1.7), the complex manifold M is defined by either of the equations (10.5.13)

w = Q(z,

r) or v =

z, w).

By Proposition 10.4.6, given a positive integer k0, if U = x with r and r0 sufficiently small as in that proposition, then the Segre set 5k (p0, U), 1 k k0, is the image of the holomorphic map (10.5. 14)

Bz(r)k 3 A

i-÷ (z,

vk(A))

E

CN,

X. SEGRE SETS

304

where vk is defined as follows. For k

= 2j

(10.5.15)

A=(z,X1,z',...

where the

are defined recursively by

re = wt +

(10.5.16)

+ 1

zt, wt), re_I

withw = £

fort = 1,2,... ,j; forj =0, we have v1

0,

+ q(zC,

tt_1), £ ? 2

£=1

0. Similarily, fork = 2j, we define

v21(A) as follows. We take

A=

(10.5.17) where

z',

(z,

. . .

x'). v2'(A) =

,

t'

x'' r'),

is defined recursively by

= We+4(X I,zC,Wo) with

(10.5.18)

=

fort=1,... ,j—landr'=O. We conclude this section with an example showing how the construction of the Segre sets can be used to determine if a given generic submanifold is minimal.

Let M C C3

10.5.19.

be

the generic submanifold of codimension

two and CR dimension one given by (10.5.20)

=

Im

Im 7.02

=

where (z,w) = (Z,WI,w2) E C3 ands = (SI,52) = Re w, andk a positive integer. The complexification M C C6 is then given by W1

= T1 +IZX,

where (z, w1, w2, x'

T2) are the

setsofMatp=OwithU Si =

((z,0,0)

:

= T2+1(2T1

coordinates

+izx)kzx

in C6. We shall

compute the Segre

=C3. Wehave

z E

S2 = {(z, WI, w2)

W2

C),

E C3:

3xI E C,

WI

=

= iZX',

).

We observe that the closure

of S2 in C3 is the submanifold given A straightforward calculation shows that

S3 = {(z, w1,

W2)

C3: 3 X1, Z1 E C, W2 =

W1

= ix'(z

(_ixl)

—

by w2

=

z'), — (2z' — z)kz)}.

§10.6. HOMOGENEOUS SUBMANIFOLDS OF CR DIMENSION ONE

Observe that, fork =

1,

53 =

52

=

{(z, Wi, W2) E

305

C3: w2 = wi). Hence,

d2(0) = d3(0), so by Remark 10.5.3 (or by direct calculation), (0) = d2(0), for j 2, i.e. Jo = 2. By Theorem 10.5.2, M fl S2 coincides with the local CR orbit of 0 in M, and M is not minimal at 0, for k = 1. Fork > 1, a straightforward calculation shows that the generic rank of the mapping

ix'(z—z'),

C3

—(2z' _z)kz)) EC3

equals three, and hence d3(0) = 3 and Jo = 3. By Theorem 10.5.5, it follows that M is minimal at 0 for k> 1. The reader can check that for k> 1, the Hönnander numbers at 0 are m1 = 2, m2 = 2(k + 1). § 10.6.

Homogeneous Submanifolds of CR Dimension One

Before we prove Theorem 10.5.2 (in § 10.7) we first discuss the homogeneous case because the proof of the theorem will essentially reduce to this case. We consider in this section submanifolds M in CN, N = n + d, of the form WI

w,

,th1...1)

M:

(10.6.1)

Wr

Wr+I = Wr+!

Wd = Wd,

r

d is an integer (r = 0 corresponds to the canonically flat submanifold), and each qj. for j = 1,... , r, is a weighted homogeneous polynomial of degree m1. The weight of each z1 is I and the weight of Wk, fork = 1,... , r, ism*, where 0

as in §4.4. We take the open set U, of §10.1 to be CN. Because of the form of the = (0, CN) defining equations of M, given by (10.6.1), the Segre manifolds are complex algebraic connected submanifolds of The closures of the Segre sets = S1 (0, CN) are therefore complex algebraic varieties in of dimenwhere d1(p) is defined by (10.2.21). (This is a consequence of sion d1 = the Tarski-Seidenberg Theorem, since a closed semi-algebraic subset of complex space is algebraic; see e.g. Dimca [1], but this fact will not be used here.)

X. SEGRE SETS

306

We let

for j = 2,... , d + 1, be the projection

defined

by

(10.6.2)

Wj,

. . .

,

:= (z, wi, .. .

,

(and hence M = By the form to be define M' c is the generic manifold of (10.6.1) of M, it follows that each M' C We

codimension j — 1 defined by the j — first equations of (10.6.1). Throughout this section, we work under the assumption that M satisfies the following, which in view of Theorem 4.4.1 is equivalent to M having r + 1 (finite) Härmander numbers, counted with multiplicities, at (0, 0). 1

for j = 2,... , r + 1, is offinite

CONDITION 10.6.3. The generic man jfold type at 0.

For clarity, we consider first the case where the CR dimension, n, is one, i.e. C. The purpose of the following proposition is to relate the Segre number z Jo = jo(0), defined in the beginning of § 10.5, to the integer r in (10.6.1) and to give, by induction on j, parametrizations of a particular form of open pieces of

S1'... Let M be of the form (10.6.1) with CR dimension n = 1 be the Segre sets and assume that M satisfies Condition 10.6.3. Let S1,... , of M at0, and letd1,... , be their dimensions. Then Jo = r + 1 andd1 = j, r + 1. Furthermore, for each j = 1,... ,r + 1, there isa proper for! complex algebraic subvariety V1 C C' (possibly empty) such that 5, satisfies PROPOSiTION 10.6.4.

j

(10.6.5)

n ((C' \

x

(z, Wi, ...

,

W,j) E ((C) \

= fj*(z, w1, ...

,

k = j, ... , dj,

where each fork = j,... , r, is a (multi-valued) algebraic function (with b1k disjoint holomorphic branches) outside and each 0 fork = r +1,... , d.

Clearly, the first statement of the proposition follows from the second. Thus, it suffices to prove that, for each j = 1,... ,r +1, there is a proper algebraic subvariety V1 such that (10.6.5) holds. The proof of this is by induction on j. Since = {(z, w): w = 0},(10.6.5)holdsforj = 1 with V1 = 0. Weassume such that (10.6.5) holds for that 2 e r + 1, and that there are V1... , 1,... — 1. By Proposition 10.2.7, we have , j= PROOF.

(10.6.6)

= {(z, w) E

r)

E

(z, W, X'

M}.

307

QNE

ASSERTION 10.6.7. The set of points (z, WI,...

C' such that there

,

with the and t) E *(S fl (V1_1 x exists (we,... , Wd) E property that (z, w, r) E M is contained in a proper algebraic subvariety

PROOF OFASSERTION 10.6.7. Let Ebethesetofpoints(z, Wi,... described in the assertion. Then (z. w1,... , w1_1) E C' is in E if

=

(10.6.8)

Ct

j = 1,... ,t —1,

+4J(x,z,Wi,...

forsome(x, ti,... ,

E

,

E (n1(S1_1)fl(V1_1 xC)). (Recall the two equivalent

sets of defining equations, (10.5.13), for M. The operation * here is taken in C', i.e. mapping sets in riiY) We claim that the set E wi-i) to is contained in a proper algebraic subvariety At C C'. To see this, note first implies that that (10.6.5) (which, by the induction hypothesis, holds for is contained in a proper irreducible algebraic subvariety in C'. Let P1

Ti,...

be a (nontrivial) polynomial that vanishes on *

,

C

be a (nontrivial) irreducible polynomial that vanishes T11... , C such that on*n.i(st_i). Thus, if (z, w1,... , w1_1) E Ethen there

and let

(10.6.9i)

Pi(x, z, w1, . WI

P1

(10.6.9ii)

+ 4i

z),

P2(X,z, w1,... P2(X, w1 +

i.e. A(z, WI,...

,

:=

. . ,

. . .

,

+

z, WI,

. . .

,

= 0,

,

wt_2)) = 0,

:=

,

z),

•

. .

+ 4e—i(x,

w1, ..

.

= 0 if we denote by k the resultant of P1 and P2 as

polynomials in x. The proof will be complete (with A1 = R'(O)) if we can show that R is not identically 0, i.e. P1 and P2 have no common factors (it is easy to see

that neither P1 nor P2 is identically 0). Note that, for arbitrary ti,...

,

we

have (cf. (10.5.13)) (10.6.10)

P2(X,z,ri +qi(z,x)

,T1_2))

= P2(X, T1, .

It follows from this that P2

. .

,

is irreducible (since p2 is irreducible). Thus, and P2 cannot have any common factors because P2 itself is the only nontrivial factor of P2 and, by the form (10.6.5) of SII, P2 is not independent of This completes the proof of Assertion 10.6.7. 0

X. SEGRE SETS

308

We proceed with the proof of Proposition 10.6.4. Let us denote by C the proper algebraic subvariety with the property that (z, Wi,... , w(_2) E \ implies that the polynomial P1(X, z, defined by (10.6.9), , considered as a polynomial in X, has the maximal number of distinct roots. Let C1 C Ct be defined by

xC),

(10.6.11)

is the zero locus of the polynomial P1(z, w1,... , (Note that by the choice of P1, we have fixed, let For (z, w1,... , C C C be the domain obtained by removing from C the roots (z, w1,... , in X of the polynomial equation where G1 C

P1(X, z, WI,... , w1_2) = 0. In view of Assertion 10.6.7 and the inductive hypothesis that (10.6.5) holds for

St_i, it follows from (10.6.6) that (10.6. 12) f (z,

S1 fl ((Ct \

W1,...

,

E

C1) X Cd_€+I) =

((Ct

x Cd_t+I):

\ C1)

wk = gek(x, Z,

E

WI,...

cC,

,

k = £ — 1,...

,

,

where

(10.6.13)

gtt(x, z, wi,... WI +

:=

,

z), .

.

qk(z, x, wi +

+ qc—2(x, Z, WI,

. ,

z),

...

. . .

, Wi_I + qk—I(x, z,

,

WI,...

,

fork = t—1,... ,d. Note thateach gek, fork = £—1,... ,r,isa(multi-valued) algebraic function such that all branches are holomorphic in a neighborhood of every point z, w) considered in (10.6.12), and gtk 0 fork = r + Now, suppose that actually depends on z, WI,... , (10.6. 14) Then, for each

z, wi,...

ax

z°, wy,...

,

d.

i.e.

0.

,

such that one branch g of gt.t—I is

,

holomorphic near (x°. z°, w?,... (10.6.15)

1,...

,

with

z°, to?,...

,

0

309

§10.6. HOMOGENEOUS SUBMANIFOLDS OF CR DIMENSION ONE

and 0 , z , w1, .

0

(lO.o.lu)

W1_.1 —

. . ,

we may apply the (algebraic) implicit function theorem (see Theorem 5.4.6) and

deduce that there is a holomorphic branch 9(z, w1,... such that function near (z°, w?,... Z, WI,... ,

wI_I — g(8(z, WI,... ,

(10.6.17)

of an algebraic 0.

Since gg.c—i is an algebraic function, which in particular means that any two can be choices of branches g at (possibly different) points (x°. z0, W?,... , space avoiding the singularities of connected via a path in z, w1,... , it follows that any solution 9 and also avoiding the zeros of can be analytically continued to any of (10.6.17) near a point (z°, w?,... , other solution near a (possibly different) point. Thus, all solutions 9 are branches As of the same algebraic function, and we denote that algebraic function by a consequence, there is an irreducible polynomial R1(X, z, w1,... , WI_I) such that X = WI,... ,WI_I) is its root. Let D1 C Ct be the zero locus of the discriminant of R1 as a polynomial in X. Outside (Cg U D1) x c in the equation we can, by solving for x = WI,... ,

(10.6.18)

WI_I = ge.e—i(x, Z, WI,

. . .

,

describe S1 as the (multi-sheeted) graph

(10.6.19)

Wk

= flk(Z, w1, .

.

. ,

:= ga(Oc(z, W1, .

fork =

t,... ,d.

Clearly,

wehaveflk

. .

,

WI_I), WI,

. . .

,

Ofork = r+ 1,... ,d. Bytaking

V1 to be the union of C1 U D1 and the proper algebraic subvariety consisting of points where any two distinct branches of coincide (for some k = £,... , d), we have completed the proof of the inductive step for j = £ under the assumption that Z, WI,... , w(_2) actually depends on x. Now, we complete the proof of the proposition by showing that Condition 10.6.3 forces (10.6.15) to hold as long as £ — 1 r. Assume, in order to reach a contradiction, that does not depend on x. It is easy to Z, WI,... , verify from the form (10.6.1)ofM that the sets Jrk(SJ), for j = 1,... , k, are the Segre sets of Mk at 0. Let us denote these sets by (Mk) (so that with this notation we have Now, note that if we pick (z°, w?,... , = E Mt

then

(10.6.20)

W'?

+

z°),...

,

+

z°, wy,...

,

X. SEGRE SETS

310

Since the generic real submanifold Me C

cannot be contained in a proper

algebraic subvariety (Proposition 1.8.11), Ce fl Me is a proper real algebraic subset of Me, and hence we may pick the point(z°, w?,... E Me such that it is not on the algebraic subvariety Ct. Then by construction of Ce given by (10.6.11), the point $

(10.6.21)

,Wt_2)

=

w?

z°),...

+

+

,

z°,

wy,...

,

is not in (Ve_1), since as observed following (10.6.11), we have Vt_i C Ge_i. = S1_!(Me) consists ofa be_i.e_i-sheeted By the induction hypothesis,

graph (each sheet, disjoint from the other, corresponds to a branch of fci.e.-..i) above a neighborhood of the point (z°, w?,... , In view of (10.6.21) and the definition of P1 given by (10.6.9 i), we have E Since w?, . the defining gee—i is assumed independent of x equation . .

(10.6.22)

Wt_I = ge.e—i(x, Z,

WC_2)

for near the point (z°, w?,... , From the definition (10.6.13) of gee—i and (10.6.21) it follows that Se(Mt) also consists of a b-sheeted graph, with

be_i.e_i, (each sheet corresponds to a choice of a branch of the function th?, ... wg_2)) above a neighborhood of (z°, wy,... , fe_i.e_i at b

$

C Sg(MC), we must have b = Since branch there is possibly another branch (z, Wi,... we_2) the following holds

and, moreover, for each such that for every point

$

(10.6.23)

ft_i._i(z, W1, .. .

(z, WI + qe—i (z,

$ WC_2)

(Z, z),...

Z, W1 +

= + qe_2(Z, Z, W1,... , WC..3))+

,

z), .

.

.

$ We_2 + qe_2(z, Z, Wi,

.

.

.

,

we_3)).

Since all the sheets of the graphs are disjoint, the mapping k k' is a permutation. We average over k and k', restrict to points (z, wi,... , we_a) Me_i, and obtain, by (10.6.23) and the equations for Mt_i (similar to (10.6.21)),

(10.6.24)

wy,...

> 11)1,

. .

,

,

We_2)

+

= thi

WC_2).

§ 10.6. HOMOGENEOUS SUBMANIFOLDS OF CR DIMENSION ONE

defined by

Let us denote by f the holomorphic function near (z°. w?,... , 1

(10.6.25)

f(z, WI,

. .

w,, .

:

. ,

311

.

.

,

k=l

and by K C Ct the CR manifold of CR dimension 1 defined near

f(z°, w?,...

(z°. w?,...

(10.6.26)

,

by

K

f(z, Wi,... ,we_i): (z, Wi,...

E

,

= f(z, Wi,... Observe that dimR K = dimR Me—I
,

W(_2)}.

Equation (10.6.24) immediately

implies that K C M1 and hence Mt is not minimal at the point given by (10.6.26).

On the other hand, by Condition 10.6.3, M' is of finite type at 0 and hence of finite type and minimal in a neighborhood of 0 in M (see § 1.5). Note that, by the form (10.6.1) of M, the condition that Mt is of finite type at a point (i.e. not on Wt_I) is only a condition on (z, WI,... , (z, WI,... , sufficiently close wi_i). Thus, by picking the point (z°, w?,... , E

fl Mt is a proper real to 0 (which is possible since, as we mentioned above, algebraic subset of M'), we contradict Condition 10.6.3 with j = t. (Recall that 2 < £ < r + 1.) Hence (10.6.15) holds, and the proof of Proposition 10.6.4 is 0 complete. For the case of arbitrary CR dimension n, the following is the analogue of Proposition 10.6.4.

PRoPosmoN 10.6.27. L.a M be of theform (10.6.1) and assume that M satisfies Condition 10.6.3. Let Si,... , be the Segre sets of M at 0. Then, for each j= 1,... ,jothereisapartitionofthese:{1,2,... ,r}intolj ={i1,i2,... ,iaj} and = {k1, k2,... , such that (10.6.28)

and there is a proper algebraic subvariety (10.6.29)

n

\ V1) x

= {(z. w1,...

,

Wd) E

C

such

that

x

\

x

c"—')

satisfies

X. SEGRE SETS

312

and (Wk1,... , Here (z, w,1,... , w0) E E = each is a (multi-valued) algebraic function outside V, and, moreovei I,... , b1, for all > is independent of

The proof of Proposition 10.6.27 is similar to that of Proposition 10.6.4 with some additional technical difficulties. We omit this proof, and refer the reader to the authors' paper [1]. § 10.7.

Proof of Theorem 10.5.2

We start by proving (a). Since the Segre sets of M at po are invariantly defined, we may choose any holomorphic coordinates near p0. Let /Ll •.. S jig. be repeated according to their multiplicities. the Hörmander numbers of M at By using Theorems 4.3.2 and 4.5.1, we can find normal canonical coordinates x vanishing at P0. After solving in w we obtain that M near p0 (z. w) e is given by

=

+qj(z,

+

,

d

Wk=Wk+

hkt(Z,z,W)We,

j = 1,...

ü,),

,

r,

k=r+1,... ,d,

t=r+l

for j = 1,...

is weighted homogeneous of r, q1(z, th1,... , degree ü) is a real-analytic function whose Taylor expansion at the origin consists of terms of weights at least + 1, and the hk( are real-analytic functions that vanish at the origin. Here, z is assigned the weight 1, w1 the weight where,

,

forj = r+1,...

forj = 1,...

,d. Moreover, the

homogeneous manifold M0 C CN defined by

j = 1,... ,r,

I w1 =

(10.7.2)

1

-

k=r+1,...

WkWk,

satisfies Condition 10.6.3 in view of Theorem 4.4.1. For E > defined by scaled coordinates ü) E

(10.7.3)

0,

,d,

we introduce the

I

(W1 = w1(W; E) = E 1W1,

j = 1,... ,d,

are the Hörmander numbers, counted with multiplicity for j = 1,... , r, and = j.t, + 1 for k = r + 1,... , d. We write hkt for the function

where the

(10.7.4)

ii'; E) := !hk€(Z(Z;

th(th;

313

§10.1. PROOF OF THEOREM 103.2

and similarly,

(10.7.5)

th; E)

:=

E), ,ï,(ü,; E)).

E),

th; E) are real-analytic functions of Note that both th; E) and CN x R. In the scaled coordinates, ü'; E) in a neighborhood of (0,0; 0) in

M is represented by the equations

j=l,...,r k=r+1,... ,d.

wk=wk+E Let U = Sj(O, U), for

x 1

j

by (10.5.14). For 0 < corresponding mappings (10.7.6)

with r and r0 sufficiently small so that the Segre sets

2(d + 1), is the image of the holomorpic map given th), we consider the 1, in the dilated variables E)

:

A i-÷

-+ C', and i3'(A;

E

Here A = EA, where A isas in (10.5.15) and (10.5.17). We observe that

E))

and El < 1. is defined and holomorphic for A E The generic dimension of the Segre set S1 (0, U) is the generic rank of the U) is connected) 0 (since the Segre manifold mapping (10.7.6) with e for and is independent of E. We shall denote this rank by d1(M), and write By the real-analytic the generic rank of the mapping given by (10.7.6) for E = 0. dependence on and the upper semi-continuity of rank, we have

j=l,2 of the homogeneous For 0 the mappings (10.7.6) parametrize the Segre sets manifold M0 defined by (10.7.2). By Proposition 10.6.27, applied to the Segre sets of M0 at 0, we deduce that the generic dimension of the maximal Segre set is the generic dimension of the of M0 at 0 is n + r. Thus, d10 n + r, where maximal Segre set of M at po. On the other hand, if we go back to the unscaled coordinates (z. w), we note from the construction of the Segre sets that each S1 is = ... = = 0). Thus contained in the complex manifold X = {(z. w): n + r, so that we obtain the desired equality d10 = n + r. This proves part (a) of the theorem. It follows from (10.7.1) that the CR vector fields of M are all tangent to MflX = ((z. w) E M: to1 = 0, j = r + 1,... , d). Thus, the local CR orbit of po is

314

X. SEGRE SETS

contained in M X. Also, since there arc r Hörmander numbers, counted with multiplicity, the CR orbit of po has dimension 2n + r. Since the dimension of M fl X is 2n + r as well, it follows that the local CR orbit of po is M fl X. This D proves part (b) of the theorem.

Notes for Chapter X The "Segre varieties" (i.e. the Segre sets or manifolds of order one) of a realanalytic hypersurface were introduced by Segre [1] in 1931. Much later, Segre varieties were used in the study of the reflection principle (see Chapter IX and its notes) and the study of mappings (see Chapters XII, XIII, and their notes). Webster [1], [2] used the Segre varieties to study the algebraic mapping problem as well as the reflection principle. Further use of the Segre varieties in the study of the reflection principle occurs in the work of Diedcrich and Webster El] and [1]. (The Segre later in the work of Diederich and Fornaess [4] and varieties are also implicitly used in the definition of essential finiteness in the work of Baouendi, Jacobowitz, and Treves [1].) For more recent use of the Segre varieties in the reflection principle, see also Diederich and Pinchuk [I] and Huang [2J. Furthermore, the Segre varieties have been used by Chern and Ji [11, [2] to study the Riemann mapping problem. The definition and construction of the Segre sets (of degree greater than one) for generic real-analytic submanifolds, as given here, appeared in the work by

Baouendi, Ebenfelt, and Rothschild [1], where most of the main results, e.g. Theorem 10.5.2, were proved. (However, some results, such as Theorem 10.5.4, appear here for the first time.) This seems to be the first generalization and use of Segre-type methods for submanifolds of codimension greater than one. Further use of the Segre sets in connection with holomorphic mappings appears in Baouendi, Ebenfelt, and Rothschild [2], [3] (see Chapters XII and XIII). The presentation of the Segre sets and manifolds given here has been influenced by the later work of Zaitsev [1]. Finally, we mention here the striking similarity between the construction of the Segre sets and their properties and the notions of curvature of mappings in studied in the work of Christ, Nagel, Stein and Wainger [I].

CHAPTER XI

NONDEGENERACY CONDITIONS FOR MANIFOLDS

Summary In this chapter we introduce three notions of nondegeneracy at a point of a CR manifold and study the relations between them. These nondegeneracy conditions

are used in Chapters XII and XIII for the study of mappings between generic submanifolds. In § 11.1, we define the concept of finite nondegeneracy for a smooth abstract CR manifold, and in § 11.2 we give an interpretation of this condition for an embedded generic submanifold of Ch' in terms of its defining function. For a realanalytic generic submanifold of CN, holomorphic nondegeneracy is introduced

in § 11.3 and essential finiteness in § 11.4. One of the results proved in § 11.3 is that holomorphic nondegeneracy at a point p E M propagates to all points on the connected component of p in M. The relationships between these three notions, for real-analytic generic submanifolds, are explored and some examples are given in § 11.5. It is shown that finite nondegeneracy implies essential finiteness which, in turn, implies holomorphic nondegeneracy at a given point. For hypersurfaces, Levi nondegeneracy at a point implies finite nondegeneracy at that point, and essential finiteness implies finite type. In § 11.6 we show that a compact real-analytic generic

submanifold is essentially finite at all points. Holomorphic nondegeneracy and essential finiteness for smooth generic submanifolds are defined in § 11.7—11.8, and their basic properties are proved.

§11.1. Finite Nondegeneracy of Abstract CR Manifolds In this section, we shall define a nondegeneracy notion, extending that of Levi nondegeneracy (see §2.2), for a C°° CR manifold. Let (M, V) be a smooth (abstract) CR manifold with CR dimension n and CR codimension d with V being its CR bundle (see §2.1). We denote by T°M C T'M the characteristic bundle of the CR vector fields, i.e. for each p M, (11.1.1)

and by T'M C CT*M the bundle defined by (11.1.2)

:=

J..

E

315

XI. NONDEGENERACY CONDITIONS FOR MANIFOLDS

316

Observe that T°M can be identified with a subbundle of T'M and that the real

fiber dimension of T°M is d, whereas the complex fiber dimension of T'M is n + d. Given a CR vector field L, we define the operator TL, mapping the space of smooth sections of T'M, f'(M, T'M), into itself, as follows (11.1.3)

TLW

:=

WE

r(M, T'M),

where j denotes the usual contraction of a differential form by a vector field, i.e. If X, Y are smooth vector fields and cr is a 2-form then

(Xja, Y) =

(11.1.4)

(a, X A Y).

To see that 71 maps r(M, T'M) into itself, we let L' be any CR vector field. We then have, by using the definition of 71 and a well known identity from differential geometry (see e.g. Helgason [1]),

(11.1.5)

(71W, L') = (dcv, LA L')

=L((co,L'))—L'((cv,L))—(w,[L,L'])=O, where the last equality follows from the fact that cv is orthogonal to CR vector fields and that V is involutive. If f is any smooth function on M, cv is a 1-form valued in T'M, and L is a CR vector field, then the following are consequences of the definition (11.1.3): (11.1.6)

TfLcv =

fljw,

TL(fw) = (Lf)cv+ fTLW.

Indeed, the first identity is immediate. To prove the second, we observe that

71(1 cv) = Ljd(fco) =

Lj(df A cv + fdw)

=(Lf)w—(w,L)df+fTLcv = (Lf)w + f2jw, where the last equality follows from the fact that L is orthogonal to cv. IfS = (L', L2,... , Lk) is a string of CR vector fields, we shall denote by

ISI=k (11.1.7)

and

TS:=TL*O...OTLI.

Hence, for any such string 5, T5 maps F(M, T'M) into itself. We use the convention that if ISI = 0, then Ts is the identity.

§ 11.1. FINITE NONDEGENERACY OF ABSTRACT CR MANIFOLDS

317

The CR manifold (M, V) is finitely nondegenerate at p e if there exists a finite integer k such that M DEFINTION 11.1.8.

span {TSO(p):

(11.1.9)

VS,

k, Y0 E F(M, T°M)) =

151

The CR manifold M is ko-nondegenerate at p M jfk0 is the smallest k for which

(11.1.9) holds. If no such k exists, then M is said to be finitely degenerate at p.

The following is an easy consequence of Definition 11.1.8 and (11.1.6) and gives a characterization of k-nondegeracy in terms of local bases for the CR vector fields and the characteristic forms. We leave the details to the reader.

PROPOSITION 11.1.10. Forp EM, letL1,... bealocalbasisfortheCR vector fields near p, and 01,... Gd a local basis for sections of T°M near p. Then, M is k-nondegenerare at p jf and only if (11.1.11) span

YR

(L1,...

,

0<

k, Yj

(1,...

We have the following relationship between Levi nondegeneracy (see §2.2) and finite nondegeneracy.

PRoiosmoN 11.1.12. A smooth CR manifold M of hypersurface type is Levi nondegenerate a: p M if and only if M is 1-nondegenerate at p. PROOF. Recall that being of hypersurface type means that the CR codimension

is one. LetO

Obeacharacteristicformnearp €M, andletL1,...

be a local basis for the CR vector fields near p. By Proposition 11.1.10, M is 1nondegenerateatp Mifandonlyifthecovectors(0(p), = l,...n}, span By the definition of T,M, this is equivalent to the covectors TL1O(p), for j = 1, . . n, forming a basis for the dual of Hence, by linear algebra, 1-nondegeneracy at p is equivalent to nondegeneracy of the Hermitian form .

(11.1.13)

(L1, L*)

(TUG, Lk),

at p. Moreover, by a calculation similar to (11.1.5), we have

(11.1.14)

A L*}

= L1

((0, Lk))

—

Lk ((0, L1))

—

(o,

[L1, Lk]) = (0, [Li, Lk]).

This completes the proof, in view of Definition 2.2.3 (cf. (2.2.23)).

0

REMARK 11.1.15. For a CR manifold M of higher codimension, a similar argument as in the proof of Proposition 11.1.12 shows that 1 -nondegeneracy at a

XI. NONDEGENERACY CONDITIONS FOR MANIFOLDS

318

point p E M is equivalent to the Levi map of M at p being nondegenerate (see §2.2). The latter can also be formulated as follows. If for some X,, E VP,

= (0,

(11.1.16)

for all 0 E

then

and Y,, E extending the vectors Xe,, 1'p

[X, 21) = 0,

=

0.

Here, X, Y are CR vector fields

The property of being k-nondegenerate at p

M is a CR invariant condition of M at p. We can also identify finer invariants for any CR manifold M at p E M as follows. Denote by E0 = C® and, for v> 1, let C (11.1.17)

E(v) = span {TSO(p): VS.

v, VO E r(M, T°M)} c

Then (11.1.18)

E(v) We define the positive integer 01 to be the smallest v for which E0 (provided such a number exists). We also define the integer A1 (1,... , n } by

A1 :=dimE(o1)—d.

(11.1.19)

and A1 Assuming that we have defined 1 01 < ... A,_i. we define inductively to be the smallest integer v for which >

A,
E(v) (provided such exists), and (11.1.20)

A1

In view of (11.1.18), there exists an integer (which could be 0) such that this process stops at j = Lo. Arguing as in Proposition 11.1.10, one can compute the numbers (11.1.21)

1

<...
01

andA1,A2,... ,A10,with to

(11.1.22) j=1

using any basis L1,... , for the CR vector fields and any basis Od for the characteristic forms. Observe that M is k-nondegenerate at p if and only if to

(11.1.23)

EA1=n. j=1

In that case, k =

1.2. FINITE NONDEGENERACY OF GENERIC SUBMANIFOLDS OF C"

319

§11.2. Finite Nondegeneracy of Generic Submanifolds of C" In this section, we shall present finite nondegeneracy in terms of local defining of codimension d and CR dimension equations for a generic submanifold M C n, i.e. N = n + d. Let e: M -÷ C"' be the embedding of the manifold M CT*CN -+ CT*M the corresponding pullback. If in CN, and denote by Z = (Z1,... , ZN) are coordinates in C", then the covectors L*dZj(p), for J = at eachp E M. Thisfollows from theobservation 1,... , N, formabasis for that e*dZj (p) J.. V,, for every p E M, by the definition of and the fact that L*dZJ(p), for j = 1,... , N, are linearly independent, as is easy to verify e.g. using suitable local coordinates, since M is generic. Also, if for some open set U c CN intersecting M (11.2.1)

where p =

Pd) and ap1 A

. .

L*(iapi)(p),

(11.2.2)

0 in U, then the covectors

.

.

,

for every p E Mn U. The reality of the covector (i ap1 )(p) form a basis for follows from the fact that L*(dpj) 0, d = a + a, and Pj is real-valued. Using The linear independence of the covectors this, we find that L*(iOpj)(p) E (11.2.2) is easily checked e.g. using suitable local coordinates. Thus, we can take

as a basis for local sections of T°M the forms

= L*(iapi),

(11.2.3)

Let L1,... , a = (a1,...

,

j = 1,...

,d.

be a local basis for the CR vector fields in Mfl U. For a multi-index we use the notation 1a1

with the convention that the vector valued function

is the identity. We also use the notation api/az for 8p1

(8Pj

PROPOSITION 11.2.4. LetMbeasmoothgenericsubmanifoldinCN, p E MflU and L11... , L,, a basis of CR vector fields near p as above. Then M is knondegenerate at p if and only if

(11.2.5)

span

p5): j

E

(1,... ,d},

IaI

k}

C".

XI. NONDEGENERACY CONDITIONS FOR MANIFOLDS

320

PROOF. We first observe that (11.2.5) is equivalent to

j E(i,...

(11.2.6)

,d},

SE (L1,... ,

0

e


since commutators of CR vector fields remain CR vector fields. Let L be a CR vector field on M fl U, extended smoothly to U, i.e.

L=

(11.2.7)

Then, ifO1 is as in(l1.2.3), (11.2.8)

=

TLOJ = iLjL*dapj =

We shall compute L.jd8p1,

(11.2.9)

=

A dZns)

aL k.m

azkazm

dZm

=

L

dZrn.

Thus, we have for p E M (11.2.10)

TLOJ =

L

Similarly, we obtain for S E (L1, ...

(11.2.11)

Ts81 =

S

The conclusion of the proposition follows from the fact that z*dZ1, j

= 1,... , N,

0

form a basis for the sections of T'M. REMARK 11.2.12.

A similar argument as in the proof of Proposition 11.2.4,

shows that one can compute the invariants 0J' A1, Lo at p. introduced at the end of the previous section, using the subspaces

(11.2.13)

span {La

(p,

j E (I,... ,d},

jaj < v}.

We leave the details to the reader.

The following is useful for verifying finite nondegeneracy in normal coordinates.

§ 11.2. FINITE NONDEGENERACY OF GENERIC SUBMANIFOLDS OF CN

321

COROLLARY 11.2.14. Let M C be a real-analytic generic submanifold of x Cd codimension d and CR dimension n. Let p0 E M and Z = (z, w) be normal coordinates vanishing at p0 SO that M is given by th = z, w) as in Pmposirion 4.2.12. For p = (Zp, E M sufficiently close to p if and only if

Iia\aaQJ

(11.2.15)

-

M is k-nondegenerate at p a

(11.2.16)

and only

-

a

jE{1,...,d), Also, M is k-nondegenerate at P0 if and only

(11.2.17)

j (1,... ,d), 1

span

IaI

k) =

To prove the first and the last part of Corollary 11.2.14, we apply Proposition 11.2.4 with p(Z, 2) = — z, w) and PROOF.

--

a

(11.2.18)

a

aw

j=1,...,n.

The details are left to the reader. To prove the second part, we take p(Z, 2) w — Q(z, th). Then,

=

2) =

(11.2.19)

where Cd is the vector (0,... ,0, 1,0,... ,0) with 1 in the jth position. From this it is easy to see that M is k-nondegenerate at p if and only if

(11.2.20)

span (La

(Zp,

jEf1,...,d},

322

XI. NONDEGENERACY CONDITIONS FOR MANIFOLDS

Now, since

=

z,,, w,,), (11.2.14) follows from the identity

w))) =

(11.2.21)

r). The identity (11.2.2 1) follows from for every holomorphic function F(z, D the form (1 1.2.18) of L3 and the chain rule. 11.2.22. Consider the real-analytic hypersurface M C C3 given by (11.2.23)

1mw = 1Z212

+

+

We take as a basis for the CR vector fields (11.2.24)

L1 =

dw dw dz2 The reader can use Proposition 11.2.4 and the remark following its proof to check that for every p M, oj(p) = 1, (11.2.25)

£o(p) =

2, (Zp)2

= 0,

A1(p)

=

1,

= 0,

and for p E M with (zp)2 = 0, a2(p) = 2 and A2(p) = 1. Hence, at every point A1(p) = 2,so that M is finitely nondegenerate at all pM. More p E M, precisely, M is 2-nondegenerate at those points p E M for which (zp)2 = 0, and I -nondegenerate (i.e. Levi nondegenerate) at all other points.

§11.3. Holomorphic Nondegeneracy We shall introduce another notion of nondegeneracy that will be shown to be related to finite nondegeneracy introduced the previous sections. We let M be a real-analytic submanifold of C's'. By a germ at p E of a holomorphic vector field, we shall mean a germ of holomorphic section of TI.OCN, i.e. a (1, 0)-vector field X of the form (11.3.1) with

a1 germs at p of holomorphic functions.

DEfiNITION 11.3.2. A real-analytic submanifold M C

is holomorphically

at p E M if there is no germ at p of a holomorphic vector field X tangent to M (i.e. with a representative tangent to M in a neighborhood of p) nondegenerate

such that X IM $ 0. If M is not holomorphically nondegenerate at p. we say that it

is holomorphically degenerate at p. We shall also say that M is holomorphically nondegenerate if it is so at every point. For connected generic submanifolds, we have the following result.

323

§11.3. HOLOMORPHIC NONDEGENERACY

THEOREM 11.3.3. Let M be a real-analytic generic connected submanjfold of and only C" and let P2 E M. Then M is holomorphically degenerate a: p' if it is holomorphically degenerate at p2. PROOF. We start with an arbitrary point p0 E M and we choose normal coordix C', where n is the CR dimension and d the codimension w) nates Z =

z, w), as in

of M, vanishing at Po• We assume that M is given by Proposition 4.2.12, for (z, w) near P0 = (0,0). We write

QCx,z,w)

(11.3.4)

< 8. We shall assume that 8 is chosen sufficiently small so that is a the right hand side of (11.3.4) is absolutely convergent. Here = In what follows, we holomorphic function defined for IzI, IwI <8 valued in denote the polydisk {IzI, IwI <8) by U.

for I z I, Ix I,

1w I

LEMMA1I.3.5. Let (z1,w5 E

MflU. IfXisagermat(z',w')ofaholo-

morphic vector field in dv, then X is tangent to M (land only if (11.3.6)

X=

w)—

and

W)4a.zj(Z, w) E 0,

with a1 holomorphic in a neighborhood of (z', &), and with the identity in (11.3.6)

valid for all multi-indices a and all (z, w) in a neighborhood of (z', w1) in CN. given by (11.3.4). are the derivatives with respect to Zj of the Here the PROOF. Suppose (11.3.7)

X=

a holomorphic vector tangent to M with coefficients holomorphic in a neigh-

borhood of(z', w') in C/v. Then, (11.3.8)

z, w) z, w)),

for some d x d-matrix valued function c. Setting

= 0 and

in (11.3.8), we obtain (11.3.9)

£.= 1... ,d,

=

z, w)

=

w

XI. NONDEGENERACY CONDITIONS FOR MANiFOLDS

324

since z, w) = 0, and this in (11.3.8) and setting ü) =

z, w) is thed x d identity matrix. Substituting w), we find

(11.3.10)

w)

0.

The fact that X satisfies (11.3.6) now follows from (11.3.4) by expanding (11.3.10) Conversely, if X is a holomorphic vector field that satisfies (11.3.6), then it in is easy to check that it is tangent to M. The proof of Lemma 11.3.5 is done. 0

Mfl U LEMMA 11.3.11. If there exists (z', holomorphic C'1 -valued function a(z, w) such that a(z, w)

w)

anda germ at(z', w') of a

0,

for all a, j = 1,... ,d, and all (z, w) in a neighborhood of(z', &),

then there

exists a holomorphic C" valued function a'(z, w) in U such that

(11.3.12)

a'(z, w)

for all a, j = 1,... ,d, and all

w)

0,

w) E U.

PROOF. The existence of a (z, w) near (z', w') implies that for any n x n matrix-

valued function r(z, w) with columns obtained from the infinite set of vectors w): a E j — 1,... d}, the determinant of r, detr(z, w), vanishes identically near (z', w'). Since r(z, w) is holomorphic in all of U, detr(z, w) vanishes identically in U. Let 0(U) denote the ring of holomorphic functions in U, and K = M(U) the field of fractions of 0(U). The fact that all determinants detr(z, w) 0 in U, where r(z, w) is as above, implies that the linear span over K of the vectors w): a E j = 1,... , d) is a proper subspace of K". Hence, there is ã(z, w) K" which is orthogonal to the subspace generated w) is 0 as an element of K for all a by these vectors, i.e. à(z, w) . and j = 1,... , d. Thus, by clearing the denominators of à(z, w), we obtain a'(z, w) E 0(U)" satisfying (11.3.12), which completes the proof of Lemma 11.3.11.

0

The conclusion of Theorem 11.3.3 now follows by the existence of normal coordinates at every point and the connectedness of M. 0

In view of Theorem 11.3.3, if M is a connected real-analytic generic submanifold in CN, we shall say that M is holomorphically nondegenerate if it is holomorphically nondegenerate at some point, and hence at every point, of M.

§11.4. ESSENTIAL FiNITENESS FOR REAL-ANALYTIC SUBMANIFOLDS

325

§11.4. Essential Finiteness for Real-analytic Submknifolds

We shall give a geometric definition of essential finiteness for any generic real-analytic submanifold M. Recall that a definition of essential finiteness for hypersurfaces was given in Chapter IX. The geometric definition given here coincides, in the case of a hypersurface, with the one mentioned above. We assume that U c C's' is an open set, intersecting M, satisfying the conditions in § 10.1—2. We shall use the notation of those sections for the Segre manifolds 6k(p. U) and the Segre sets Sk(p. U). For p E M fl U, the set (11.4.1)

A(p,U) := {z EU:

EM,

E

is a holomorphic subvariety in U, in view of Theorem 5.2.2. It coincides with the following intersection of holomorphic subvarieties

A(p,U)=

(11.4.2)

fl

61(q,U).

qE6i(p.U)

The subvariety A(p, U) can also be described as

(11.4.3)

A(p, U) = {Z EU: (Z,*61(p, U)) C (52(p, U)}.

Observe that p belongs to the subvariety A(p, U), since q E 6i(p, U) if and only if p E U) by Proposition 10.4.10. We claim that

A(p, U) C M.

(11.4.4)

To prove this, we use the easily verified fact that, for any point q E U, q E U) if and only ifq E Mfl U. Suppose that p' E A(p, U). Since p E M, it follows that p e1(p). By applying (11.4.2) with q = p. we deduce that the point p' E 6i(p, U). By applying (11.4.2) again, this time with q = pi, we obtain E '5i(pi, U). Thus, E M, proving the claim. We also observe that (11.4.3) implies that (11.4.5)

A(p, U) C S2(p, U).

Clearly, if U' c U and p E M fl U', then

p E A(p, U) flU' C A(p, U').

(11.4.6)

We will show that, in fact, for U sufficiently small the inclusion in (11.4.6) becomes an equality. This is a consequence of the following proposition. Letp E M and let Z = (z. w) be normal coordinates, vanishing at p. defined in an open neighborhood

U of p so that M is defined by (11.4.7)

w=

or

w=

in Proposition 4.2.12, in U. We expand Q(x. z, w) in as in (11.3.4), near (z, w, x) = (0,0,0). In what follows, we take U in product form, i.e. U = x where and are polydiscs centered at the origin in and respectively. as

XI. NONDEGENERACY CONDITIONS FOR MANIFOLDS

326

PRoPosmoN 11.4.8. With the notation above, if U is sufficiently small, then for every pi = (z',w') E Uwe have (11.4.9)

A(pi, U) = {(z, w1)

w') = 4a(z', w'),

U:

Vcr}.

In

A(p, U) = {(z,0) EU:

(11.4.10)

=0, Va}.

PROOF. We choose U so small that the series in (11.3.4) converges uniformly x x Then, it follows from(11.4.1)that (11.4.11)

A(pi,U)=

{(z,w)EU:w—Q(z,x,r)=0,V(x,r)EUwithr=Q(x,z',w')j. Since (x. r) = (0, w') E U, we may take x = 0 and r =

w1

in (11.4.11), and

we conclude that

A(pi, U) = f(z, w') EU: w1 = Q(z, x,

(11.4.12)

Q(x,z',

w1)),

w1))

In view of the identity (which follows from the definition of Q(z, x,

E

U}.

r) given in

(4.2.13))

which holds for all written (11.4.13)

z,

w

Q(z, x'

w) E

x

z,

x

w)), the

equation in (11.4.12) can be

Q(z,x,Q(x,z,w'))=Q(z,x,Q(x,z1,w')).

If we choose U sufficiently small, then

0) is invertible and hence, by the

inverse function theorem, (11.4.13) is equivalent to (11.4.14)

Q(x,z, w') = Q(x,z', w').

Since in the definition of A(pi, U) (11.4.14) must hold for all x near 0, the first statement of Proposition 11.4.8 follows from the expansion (11.3.4). The last statement follows from the first by taking (z', w') = (0,0) and using the normality of the coordinates.

0

11.4.15. For the conclusion (11.4.10) to hold, it is easy to check that no assumption on the form of the open set U is necessary, but merely that U is a sufficiently small open set containing p.

§11.4. ESSENTIAL FINITENESS FOR REAL-ANALYTIC SUBMANIFOLDS

327

It immediately follows from Proposition 11.4.8 and Remark 11.4.15 (as well as (11.4.6)) that, for p E U' c U with U sufficiently small,

A(p, U) flU' = A(p, U').

(11.4.16)

Thus, the germ at p of the holomorphic subvariety A(p, U) (see §5.2),

A(p) := A(p,

(11.4.17)

U sufficiently small, justifying the notation A(p). We

refer to A(p) as the essential variety of p. DEfINITION 11.4.18. A real-analytic generic submanifold M is essentially fi-

nite at p E M if A(p) = (p}, where (p} here means the germ at p of the holomorphic subvariety consisting solely of the point p. It is easily checked, in view of Proposition 11.4.8, that Definition 11.4.18 is a generalization of the definition of essential finiteness for hypersurfaces given in §6.

As in the hypersurface case, a generic real-analytic submanifold is said to be of D-finite type at p0 E M if there is no germ at po of a nontrivial holomorphic subvariety contained in M. It follows immediately from (11.4.4) and the definitions that if M is of D-fInite type at po. then it is also essentially finite at P0. We shall now present a description of A(p), similar to that given in Proposition 11.4.8, in a fixed coordinate system allowing the point p to vary. This is the description that will be used in the proof of Theorem 11.5.1 below. Let U be a sufficiently small open set such that M fl U is given by p(Z, 2) 0. Let

L/=>bkJ(Z,Z)_a_,

(11.4.19)

j=l,...n

be a basis for the CR vector fields in Mn U, extended to all of U. We also assume that the coefficients are holomorphic in U x *(J p M fl U, we denote by

j=1,...n.

(11.4.20) k=I

Note that the vector fields X,,.1, j

= 1,...

,

n, span the tangent space of *6i (p. U).

We define a collection of functions in U x U x *U by (11.4.21) for £ =

1,

.

. .

d,

a

Cta(Z, p. C) :=

C),

=

.. . Xv,,.

and with

XI. NONDEGENERACY CONDITIONS FOR MANIFOLDS

328

PROPOSITION 11.4.22. With notation as above, for U sufficiently small, and

p e M fl U let (11.4.23) C is a holomorphic subvariety of U containing p such that its germ at p is the essential variety at p, i.e.

A(p) =

(11.4.24) (Here C,, denotes the germ of C

C,,.

at p.)

11.4.25. It follows from Definition 11.4.18 and Proposition 11.4.22 that M is essentially finite at p if and only if C,, = (p). By the Nullstellensatz (see Theorem 5.2.5), this is equivalent to the fact that the ideal generated by the cta(Z, p, has finite codimension in C(Z — p}. Thus, the definition of essentially finite given here agrees with that given for hypersurfaces in Definition 9.4.3. PROOF OF PRoPosmoN 11.4.22. The fact that C is a holomorphic subvariety of U follows from Theorem 5.2.2. A point Z E U is in C if and only if

(11.4.26)

0,

span foreverymulti-indexaande = 1,... ,d. E 6,(p, (p, U) at (11.4.26) holds if and only if the restriction the tangent space of to the connected component of *(51(p U) containing of the function is identically 0. Since e1(p, U') = e,(p, U) fl U' for U' C U, there is a

(possibly smaller) open neighborhood U' of p such that if Z E U' and satisfies (11.4.26), the holomorphic subvariety A(p, U') also contains Z, in view of the definition (11.4.1). Conversely, if Z E A(p, U), then the restriction of pe(Z,.)

(p. U) is identically 0 and hence Z E C. This concludes the proof of

to

0

Proposition 11.4.22. We now list some basic properties of the subvarieties A (p. U). PROPOSITION 11.4.27. Let p

E M and U C C" be as in Proposition 11.4.8.

Then, the following hold.

(i)

lip', p2 E U with p2 E A(p,, U), then A(p,, U) = A(p2, U).

(11.4.28)

(ii) If pi, P2 E U, then (11.4.29)

p2EA(p,,U)

p, EA(p2,U).

(iii) If P1' P2 E U, then (11.4.30)

P2

A(p,, U)

U) = IS,(p2, U).

§ 11.5. COMPARISON OF NONDEGENERACY CONDITIONS

329

PROOF. To prove (i), we use the normal coordinates used in Proposition 11.4.8 and write = (z', w'), = (z2, w2). By Proposition 11.4.8, we have

(11431)

A(pi,U) = {(z,w') EU: A(p2, U) = {(z. w2) E U:

Va} w2)

=

w2), Va).

Since p2 E A(pi, U), the conclusion of (i) follows. The conclusion of (ii) follows from (I) and the fact that P' E A(pi, U) (see (11.4.6)).

It remains to prove (iii). We first claim that P2 E A(pi, U) implies that we U) U), then P2 E U). Indeed, ifq E have U) C by (11.4.2). Now, the claim follows from the symmetry property (10.4.11). The reverse inclusion follows by using (ii) and repeating the argument. Thus, we have U) = (51(p2, U). The opposite proved that E A(pi, U) implies that 0 implication follows from (11.4.2). This completes the proof.

We conclude this section with the following description of the set of points where M is not essentially finite. THEOREM 11.4.32. Let M C CN be a real-analytic generic

Then

the set of points at which M is not essentially finite is a (possibly empty) realanalytic subvariety of M. PROOF. Since being real-analytic is a purely local property of a set, it suffices to prove the statement of the theorem in M fl U, where U C CN is a sufficiently small open set (so that Proposition 11.4.22 holds). For an integer r 0, denote by E, c M fl U the set of points p E M fl U at which the codimension of the ideal (in C{Z — p}) generated by the convergent Taylor series in Z — p of the functions is greater than r. By Remark j = I,... ,d and a E Z i-+ cja(Z, p. 11.4.25, the set of points p at which M is not essentially finite is the intersection of all the Er, r 0. Since the intersection of a sequence of real analytic subvarieties is again a real-analytic subvariety (see Remark 5.2.4), it suffices to show that each is a real-analytic subvariety of Mn U. For this, we observe that the coefficients of the convergent Taylor series in Z — p of the functions Cja(Z, p, are realanalytic in p. The real-analyticity of Er then follows immediately from Theorem 5.1.15. This completes the proof of Theorem 11.4.32. 0

§11.5. Comparison of Nondegeneracy Conditions We shall connect the notions of finite nondegeneracy, holomorphic nondegeneracy, and essential finiteness. We have the following result.

XI. NONDEGENERACY CONDITIONS FOR MANIFOLDS

330

THEOREM 11.5.1. Let M C be a connected real analytic generic manifold of codimension d and CR dimension n. Then the following conditions are

equivalent.

(1) M is holonwrphically nondegenerate. (ii) There exists E M and k > 0 such that M is k-nondegenerate at

(iii) There exists V. a proper real analytic subset of M and an integer £ =

t(M),

t(M)

n, such that M is e-nondegenerate at every p E M\V. (iv) There exists p1 E M such that M is essentially finite at (v) There exists V. a proper real-analytic subset of M, such that M is essentially finite at all points in M \ V. 1

We shall call the number £(M) given in (iii) above the Levi number of M. PROOF. We shall first prove the equivalence of (i),(ii), and (iii). It is clear that

(iii) implies (ii). We shall now prove that (ii) implies (I). Assume that M is knondegenerate at pi. We take normal coordinates Z = (z, w) vanishing at so that M is given by w = Q(z, ü) (or, equivalently, ii' = z, w)), where Q(z, ü) is as in Proposition 4.2.12, near P1 = (0,0). We can take for a basis of CR vector fields a

(11.5.2)

d

k=I

a

ôwk

j=l,...,n.

The hypothesis (ii) implies, in view of Corollary 11.2.14, that the vectors

j1,... ,d,

(11.5.3)

CN. By the normality of the coordinates, this implies that the 0), JaJ
.

Hence M is not holomorphically degenerate at 0, in view of Lemma 11.3.5, which proves (i). To show that (i) implies (iii), we shall need the following two lemmas.

11.5.4. Let , fd(X) bed holomorphic functions defined in an open connected set in Cl', valued in CN, and linearly independent at some Jftheaafj(x), j = 1,... ,d, a E point in atapointin then the aafJ(x), j = 1,... , d, IaI N — d, span for X E outsidea (possibly empty) proper holomorphic subvariety of c. PROOF. First, observe that, since , fa(x) are holomorphic and linearly independent at a point, and is assumed to be connected, fi (x).... fd(x)

§115. COMPARISON OF NONDEGENERACY CONDITIONS

331

are linearly independent at every point x

riety of

outside a proper holomorphic subvaSimilarly, it follows from the hypotheses in the lemma that the

j = I,... ,d,a

E

We shall

prove the following property

(Ps) for every integer t, 0

e

('P1)

E

j


such thatdimspan

I

d,0

N

—

d.

d-i-e.

above observation, the lemma follows from (Pt) by taking e = N — d. We now prove (Ps) by induction on t. By assumption, (Ps) holds for £ = 0. We assume that (Pr) holds fort to
(11.5.5)

span

I

j

d, Ial

S

=span(dafj(x):

I

and the dimension of the vector space (11.5.5) is d + to. Choosing U sufficiently

small, we may assume that there exist C"-valued functions, here denoted by d, IaI Lo) such that their gi(x),... , gi+0(x), among the fd"fj(x) : I values at x span the vector space (11.5.5), for every x U. It follows from the equality (11.5.5) and Cramer's rule that for every flwith = + I and every in U such j = 1,... d,thereexistholomorphicfunctions4'(x),... ,

j

,

that

=

(11.5.6)

d+t0

for every x U. By differentiating (11.5.6) and using a simple induction on the length of we obtain a decomposition of analogous to (11.5.6) for all multi-indices This implies that (11.5.7)

dim span {

a

d+

for all x E U, contradicting the hypotheses of the lemma, since d + to is assumed to be strictly less than N. This proves (P10÷1) and, hence, the proof of the lemma is complete. 0 Let (z, w) E be normal coordinates for M as above, and z, w) a holomorphic function in 2n + d variables defined in a connected neighborhood in of z = z', w = w1, X = with (z', w') M. If 0, for (z, w) M near(z', w1), then h z, w) 0. LEMMA 11.5.8.

PROOF. We can

think

of

EM

z,

w) as defined in a neighborhood of the

point

Since

Xl. NONDEGENERACY CONDITIONS FOR MANIFOLDS

332

z, w) = 0 M ismaximallyrealinM z, w) = OonM, for (z, w, x' r) E M (see Proposition 1.3.11). However, since M is defined by = Q(x, z, w), the variables (x. z, w) are independent on M. The lemma

o

follows.

We now return to the proof of Theorem 11.5.1. To prove (i) implies (iii), we

again take (z, w) to be normal coordinates around some point P0 E M. By the assumption (1) and (11.3.6), it follows that the w), j = 1,... , d, all a, span for (z, w) outside a (possibly empty) proper analytic variety. Equivalently, by z, w) span CN for the normality of the coordinates, we obtain that the (z, w) outside the same analytic variety. Now choose (z°, w0) E M so that the span of the z (0, z0, w°) equals CM. (Such points exist since M, a generic submanifold, cannot be contained in a proper analytic variety by Proposition 1.8.11). We apply Lemma 11.5.4 with z°, w0), j = 1, . .. , d, to conclude = £, span CN for x that there exists £ N —d such that the z°, w°), IczI outside some (possibly empty) proper analytic variety. We claim that this implies that the £, span Ce', for (z, w) E M outside a (possibly z, w), IaI empty) proper real-analytic variety V C M. Indeed, the set of points (z, w) E M t, does not span C/V is a real-analytic variety where the z, w), V C M. We must prove that V is proper. If it were not, then the z, w), w), in view of Lemma 11.5.8, £, would not span CN at any point IcrI contradicting the conclusion reached above. This proves the claim, and also the (iii), since the statement of the claim is precisely (iii). This implication (1) ; completes the proof of the equivalence of (i), (ii), and (iii). It remains to show that (i), (ii), and (iii) are equivalent to (iv) and (v). Observe, by Theorem 11.4.32, that (iv) and (v) are equivalent. We shall use the notation of § 11.4. We show first that (iii) implies (iv). Let p' E M be any £-nondegenerate point, i.e. the span of

j
(11.5.9) is

IaI

CN. On the other hand, it follows from (11.4.21) that

(11.5.10)

Cja.Z(Pi, p"

=

Hence by the inverse mapping theorem the only common zero, near

of the functions Z i'-÷ Cja(Z, pi' is p" which proves that M is essentially finite at in view of Proposition 11.4.22, hence (iv). Next, assume that (v) holds. We shall need the following lemma.

§11.5. COMPARISON OF NONDEGENERACY CONDITIONS

11.5.11. Let 1,2,... , a sequence in

333

be an open neighborhood ofO E C" and F1(Z), j = = Ofor all j, and Assume that

dimspan{Fj.z(Z):j=1,... ,}=k.
Then, there exists a complex manifold W of dimension N — k in

containing 0 such that

1w

0, for evely j.

By renumbering the F1 and shrinking if necessary, we may assume By renumbering the variables, if = k for Z E that rank(F1(Z),... , x Ck, and applying the implicit function necessary, writing Z = (x, y) E theorem, we can solve for y = y(x), with y(0) = 0, in the k equations F1(x, y) = y(x)) 0 for all j. This follows from = Fk(x, y) = 0. We claim that the fact that dim span (Fz(Z): j = 1,... , } = k for all Z E and by implicit 0 differentiation of F1(x, y(x)). PROOF.

If the rank of the (LGpJ.Z)(Z, 2) were less than N generically on M, then of maximal rank near 0 in M, in view of (11.5.10) and Lemma at any point 11.5.11, there would exist a complex manifold W of positive dimension through = 0 for all Z E W and all j, a. Hence M would not Pi such that cja(Z, p" by Proposition 11.4.22, contradicting (v), since P' can be essentially finite at be chosen outside a proper real-analytic variety. Thus, we have proved that (v) implies (iii). This completes the proof of Theorem 11.5.1. 0 Although it follows from Theorem 11.4.32 that the set of points where M is essentially finite is open (possibly empty), we shall conclude this section by giving a direct proof of this fact (not based on the algebraic theorem 5.1.15). For this we need the following lemma.

LEMMA 11.5.12. Let (fj}JEJ be holomorphic in a neighborhood of 0 in Suppose that Z = 0 is an isolated zem ofthefiinctions — f,(O), j J. Then there exists 8 > 0 such that for IZ0I <8, Z = 0 is an isolated zero of the functions +4) — j E J. PROOF. For j E J, let FJ(Z, which is holomorphic near = fj(Z) — Oin C2N. Let V be the variety of zeros of the F). We claim that there exists E > 0

andö is

:

Zf

discrete. Indeed, by assumption there exists e > 0 such that V fl fIZI =

= 0) = 0. Therefore by compactness, there exists 8, 0 < 8 < E, such that <8) = 0. Henceforany <8, the set Vn(IZI <E, = Vfl(IZI = e, is discrete. Hence the zero Z = of F(Z, is isolated, which completes the proof of the lemma.

0

XI. NONDEGENERACY CONDITIONS FOR MANIFOLDS

334

essentially finite points of M is open. We that M is essentially finite at some point (otherwise there is nothing to prove). Choose normal coordinates Z = (z, w) around a point E M at which M is essentially finite and observe that if p = (Zp, Wp) is in this local chart, we

We may now prove that the set of

assume

have

(11.5.13)

=

we conclude from (11.5.13) that M is essentially finite for By Lemma 11.5.12, any p0 in a neighborhood of P1. Hence, the set of essentially finite points is open.

Rtkrtl 11.5.14. (i) Let M be as in Theorem 11.5.1, and let p E M. An inspection of the proof above shows that the following implications hold: M isjini:ely nondegenerate at p M is essentially finite at p M is holomorphically

at p. For completeness, we recall here the implication, mentioned in § 11.4, M is of D-flnite type at p M is essentially nondegenerare

finite at p. We give below simple examples which show that the converse implications fail to hold in general, even in the case of a hypersurface. (ii) For hypersurfaces in N > 1, we also have the implication M is

essentially finite at p M is of finite type at p (see Proposition ; 9.4.16). However, for higher codimensional generic manifolds there is no obvious relation between finite type and the nondegeneracy conditions considered in this chapter. EXAMPLE 11.5.15. Let M C C2 be the hypersurface given by

(11.5.16)

Imw=1z14.

The reader can easily verify that M is essentially finite at every point but is not finitely nondegenerate at points on M where z = 0. EXAMPLE 11.5.17. Let M C C2 be the hypersurface given by

(11.5.18)

1mw =

Re

w

1z12.

case, M is holomorphically nondegenerate at every point but is essentially finite only at points where w 0. In

this

EXAMPLE 11.5.19. Let M C C3

(11.5.20)

be the

1mw = IziI2

hypersurface given by — 1Z212.

In this case, M is essentially finite at every point, but is not of D-finite type, say, at 0. Indeed, the nontrivial complex manifold (ZI = Z2, w = 0) is contained in M.

§11.6. COMPACT REAL-ANALYTIC GENERIC SUBMANIFOLDS

335

§ 11.6. Compact Real-analytic Generic Submanifolds The purpose of this section is to prove the following result. THEOREM 11.6.1. Let M

c C" be a compact generic real-analytic submani-

fold. Then M is essentially finite at every point.

PROOF. Let W C M denote the set of points at which M is not essentially finite. Assume, in order to reach a contradiction, that W is not empty. Then by Theorem 11.5.1(v), the set W is a closed subset of M and, hence, W is compact. Thus, there is a point p W at which the Euclidean distance 8(.) from the origin, restricted to W, is maximized. We let U be an open neighborhood of p as in Proposition 11.4.8, chosen so small that A(p) = A(p, (see (11.4.17)). Let V be a connected component, containing p, of the holomorphic subvariety A(p, U) with V of positive dimension. We claim that V C W. Indeed, by Proposition 11.4.27 (1), we have, for p' (11.6.2)

V.

V C A(p, U) = A(p1, U).

Hence, the germ A(pi) has positive dimension, by (11.4.6) and (11.6.2). This proves the claim. We have now found a positive dimensional irreducible holomorphic subvariety V on which the restriction of the Euclidean distance 8(.) from the —* V be a nonorigin reaches a maximum at an interior point p E V. Let A: constant analytic disc with A(0) = p as given by Theorem 5.2.10. The function 8 o A is a subharmonic function in which has a maximum at the interior point 0. Thus, by the maximum principle for subhannonic function, 8 o A is constant. It follows that the nonconstant analytic disc is contained in the sphere of radius 8(p) centered at 0. This is a contradiction in view of the following lemma. —÷ CN be an analytic disc and assume that LEMMA 11.6.3. Let A: contained in the unit sphere. Then, A is constant.

PROOF. Let A = (A1,...

,

is

AN) and consider the function in N

(11.6.4)

— 1.

By assumption, we have

0. Taking the Laplacian off, we deduce

(11.6.5)

j=l which completes the proof of the lemma.

0

XI. NONDEGENERACY CONDITIONS FOR MANIFOLDS

336

0

The proof of Theorem 11.6.1 is now complete.

As a consequence of Theorem 11.6.1, the real-analytic (smooth) boundary of a bounded domain in CN is essentially finite at all points. We conclude this section with an example of a compact real-analytic generic submanifold of codimension two.

LetMcC4bethesetdefinedby IZ1(2+1Z212= I,

(11.6.7)

1Z312+1Z412=

1.

The reader can check that M is a generic compact real-analytic submanifold. Hence, by Theorem 11.6.1, it is essentially finite at every point. § 11.7.

Nondegeneracy for Smooth Generic Submanifolds

In this section we shall extend the notion of holomorphic nondegeneracy, previously defined for real-analytic submanifolds, to smooth submanifolds and relate it to other nondegeneracy conditions. Let M c C" be a smooth generic submanifold of codimensiond and po E M. Assume thatp(Z, Z) = (p1(Z, Z),... pd(Z, 2))

a set of smooth defining functions for M in a neighborhood of Po• A formal holomorphic vector field at po is a vector field of the form is

(11.7.1)

X=

with a1(Z) series in Z

where Poll denotes the ring of formal power — P0 with complex coefficients. We say that the formal vector field X is tangent to M if there exists a d x d matrix c(Z, 2) with matrix elements that are formal power series in Z — po and 2 — flo such that

(11.7.2)

Xp(Z, Z)

c(Z, Z)p(Z, 2),

means equality of formal power series in Z — P0 and Z — The reader can easily check that the existence of a nontrivial formal holomorphic vector field (i.e. not all coefficients are 0) at Po tangent to M is independent of the choice of holomorpbic coordinates and defining equations near Po• where

DEfINITION 11.7.3. A smooth generic submanifold M C CN is holomorphically nondegenerate (in the formal sense) at p0 E M if there is no nontrivial formal holomorphic vector field at Po tangent to M.

§11.7. NONDEGENERACY FOR SMOOTH GENERIC SUBMANIFOLDS

337

and PR0P0sm0N 11.7.4. If M C CN is a real-analytic generic sense) at P0 if p0 E M, then M is holomorphically nondegenerate (in the formal and only jf it is holomorphically nondegenerate at p0 (see Definition 11.3.2). One implication in the proposition is trivial. The other will follow from Theorem 11.7.5 below, which relates holomorphic nondegeneracy (in the formal sense) to finite nondegeneracy as defined in § 11.1—2, and Theorem 11.5.1. We leave it to the reader to deduce Proposition 11.7.4 from Theorem 11.7.5. In what follows, we shall simply refer to holomorphic nondegeneracy (in the formal sense) as holomorphic nondegeneracy.

be a smooth generic submanjfold. Then, the

THEOREM 11.7.5. Let M C following hold.

M, then M is holomorphically nondegenerate at p. (ii) If M is holomorphically nondegenerate at p E M, then there is a sequence of points p1 E M, tending to p. such that M is finitely nondegenerate at

(i) If M is finitely nondegenerate at p

Pj•

(iii) flu C M is a nonemply open subset with M finitely degenerate at every p E U, then M is holomorphically nondegenerate at every p E U, the closure of U in M. (iv) If M is holomorphically nondegenerate at p. then there exists an open neighborhood U C M of p such that M is holomorphically nondegeneraze

ateveryq EU. We shall need the following characterization of holomorphic nondegeneracy, which is related to Lemmas 11.3.5 and 11.3.11 in the real-analytic case. Let Z = (z, w) E C" x Cd be holomorphic coordinates vanishing at p E M such that M is defined near p = (0,0) by (11.7.6)

Q(z,

w= w, ii,) is a smooth C"-valued function in

satisfying (4.2.18)

(such coordinates exist by Theorem 4.2.16 and Proposition 4.2.17 with e.g. K = 1). Since w, th) is flat on M, the formal Taylor series of the smooth function Q(z, w, ti) has no w terms. We shall denote this formal series in (z, th) by z, w) the formal Taylor series at 0 of the Q(z, ti). Similarly, we denote by conjugated function z, th, w).

LEMMA 11.7.7. The smooth generic submanifold M c C's', defined near (0,0)

XI. NONDEGENERACY CONDITIONS FOR MANIFOLDS

338

by (11.7.6), is holomorphically nondegenerate at (0,0) (land only (11.7.8)

span

where F denotes the quotient field of

wJ].

PROOF. Let X be a formal holomorphic vector field of the form

X=

(11.7.9)

+

The vector field X is tangent to M if and only if

w)

(11.7.10)

c(z, w, where

b = (b1,...

,

bd) and c(z, w,

th)

Hence, taking (11.7.12)

—

Q(L z, w)),

is a d x d matrix with formal power z, w), we obtain

series coefficients. Substituting ü, = (11.7.11)

th)(th

b(z,w) "-'

= 0, we obtain 0, Q(0, z, w)).

b(z, w)

Substituting back into (11.7.11) we obtain (11.7.13)

w) (QZk(z,

z, w)) — QZL(Z,0, Q(O, z, w)))

0.

(z, w) satisfying (11.7.13), then by taking b(z, w) given by (11.7.12) we obtain a nontrivial formal holomorphic vector field (11.7.9) tangent to M. To complete the proof, we must show that (11.7.8) is equivalent to the fact that the only formal series a1(z, w) satisfying (11.7.13) are the trivial ones. To prove this, we expand Conversely, if there exist nontrivial power series

QZL(z,

z, w)) —

0, Q(0, z, w))

§11.7. NONDEGENERACY FOR SMOOTH GENERIC SUBMANIFOLDS

339

a Taylor series in and apply a standard linear algebra argument as in the proof 0 of Lemma 11.3.11. The proof of Lemma 11.7.7 is complete as

begin by proving (i). Thus, we assume that M is k0-nondegenerate at p, for some nonnegative integer k0. We choose approximate x of order K >> k0 (see Proposition 4.2.17) normal coordinates (z, w) E for M vanishing at p. Hence, M is given near p = (0, 0) by (11.7.6) satisfying also the conditions of (4.2.19). As in the proof of (11.2.17) of Corollary 11.2.14, one can show that ko-nondegeneracy of M at (0,0) implies (since K >> ko) PROOF OF THEOREM 11.7.5. We

(11.7.14)

j = l,...d,

span

1

IaI

On the other hand, the constant term of the power series (11.7. 15)

for any IaI

z.

k0, is precisely (sihce K >> /

a

(-) az

(11.7.16)

k0)

Q1.2(O,O,O,O).

By (11.7.14), this implies that M is holomorphically nondegenerate, in view of

Lemma 11.7.7. This completes the proof of (i). Since (iii) implies (ii), as can be easily checked, we shall prove only (iii). Thus, assume that M is finitely degenerate in a nonempty open subset U C M, and let p E U. We take approximate normal coordinates (z, w) of order, say, one for M vanishing at p, i.e. M is given near p = (0, 0) by (11.7.6) where Q(z, w, th) also satisfies the conditions of Proposition 4.2.17. Since w, th) is flat on M, one can check, as in the proof of Corollary 11.2.14, that finite degeneracy in U is equivalent to (11.7.17)

span {La

(zr,

the),

E U, where L = (L1,... , is a local basis for the CR all p = (Zp, vector fields on M near p. We shall need the following lemma. for

LEMMA 11.7.18. Let F(z,

w, th) be a smooth function in defined in a is fiat at 0 (i.e. vanishes to

neighborhood of 0. Then the restriction of F to M

XI. NONDEGENERACY CONDITIONS FOR MANIFOLDS

340

infinite order at 0) if and only F(z, the Taylor series of F at 0. PROOF.

w,

z, w))

We parametrize M near (0,0)

x Rd

(11.7.19)

0,

where F(z,

w, ü) is

by

EM.

'-÷

s), s — i4'(z, s)) is flat Hence, the function f(z, s) = F(z, s + x Rd if and only if its Taylor series in (z, s) at (0, 0,0) at (0, 0,0) E is identically 0. Using the fact that ñ, = z, w) is the unique power series

solution of the equation (11.7.20)

can check that the formal power series .F(z, w, z, w)) is obtained by taking the Taylor series of f(z, s) at (0, 0,0) and then substituting s = z, w))/2. This completes the proof ofthe lemma. 0 (w + one

Let D(z, w, ti') be a determinant formed by n vectors chosen from the left hand side of (11.7.17). By (11.7.17), DIu 0 and, since p = (0,0) E U, DIM is flat at p. By Lemma 11.7.18, the formal power series V(z, w, z, w)) 0, at (0, 0). Thus, we where V(z, w, th) is the Taylor series of D(z, w, deduce that (11.7.21)

span

z, w)),

(z,

wi], £ = (4C1,...

where fr denotes the quotient field of

=

(11.7.22)

2- +

z, 1=1

Now, for any formal power series F(z,

4), and

awt

w), we have

w)) =

(11.7.23)

,

w))),

which is similar to the identity (11.2.21) in the holomorphic case. Thus, we obtain

(11.7.24)

(a)U span {

(z,

z, w)),

§ 11.7. NONDEGENERACY FOR SMO(YFH GENERIC SUBMANIFOLDS

where

341

is as above. Consequently, we have

(11.7.25)

span

(z,

z, w))I,

where F is as in the statement of Lemma 11.7.7. This proves that M is holomor-

phically degenerate at p = (0,0), which completes the proof of (iii). To prove (iv), we take coordinates (z, w), vanishing at p. such that M is given near p = (0, 0) by (11.7.6). By Lemma 11.7.7, there is a determinant D(z, w) w). formed from the vectors in (11.7.8) which is not Oas a formal power series in If we choose q = (Zq, Wq) E M sufficiently close top, then the coordinates (z', w') defined by (z. w) = (z' + Zq. W' + Wq) vanish at q and M is defined by

(11.7.26)

W'

= Q'(z',

w', ill),

where

(11.7.27)

Q'(z', Z', W', W') = Wq + Q(z' + Zq. Z' +

W' + Wq. th' + thq).

Lemma 11.7.7 applies in the coordinates (z'. w'). Thus, M is holomorphically

nondegenerate at q M if and only if (11.7.8) holds with Q replaced by Q', where Q' is the Taylor series of Q' at (z', w') = (0, 0). The coefficients of this series are small perturbations of the coeffiecients of Q if q is sufficiently close to p = 0, in view of (11.7.27). Hence, if q is chosen in a sufficiently small open neighborhood U c M of p, then the determinant D'(z', u?), corresponding to D(z, w) above, is not 0 as a formal power series. Thus, M is holomorphically nondegenerate at q, in view of Lemma 11.7.7. This completes the proof of (iv) and hence that of Theorem 11.7.5. 0 The following is an immediate corollary of Theorem 11.7.5.

CokouARY 11.7.28. Let M c C's' be a smooth generic submanifold, U c M an open subset, and U its closure in M. Then, the following are equivalent: (1) M is finitely degenerate at every p U; (ii) M is holomorphically degenerate at everj p E U; (üi) M is finitely degenerate at every p U; (iv) M is holomorphically degenerate at every pU. We conclude this section by giving an example showing that holomorphic nondegeneracy of a smooth generic submanifold need not propagate as it does in the real-analytic case, i.e. the analogue of Theorem 11.3.3 does not hold.

XL NONDEGENERACY CONDITIONS FOR MANIFOLDS

342

EXAMPLE 11.7.29. Let M C C2 be the hypersurface defined by

Imw=exp(_

(11.7.30)

1z12+(Rew)2)

Then M is holomorphically degenerate at (0, 0) since the holomorphic vector field

a/az

is

tangent to M (in the formal sense) at (0, 0). On the other hand, the hy-

persurface = M \ {(0, 0)} is connected, real-analytic, and Levi nondegenerate, say, at (0, 1 + By Theorem 11.5.1, M is holomorphically nondegenerate at every point.

11.8. Essential Finiteness of Smooth Generic Submanifolds We shall extend the notion of essential finiteness, defined for real-analytic generic submanifolds in § 11.4, to smooth generic submanifolds. Let M c CN be a smooth generic submanifold of codimension d with defining equation p (Z, Z) = 0 near E M,wherep =(pi = N,bea Pd). Let L1... smooth local basis for the CR vector fields on M near Po• We write

j= 1,...n.

(11.8.1) For p E M near

we denote by

j=

(11.8.2) ack

where

is the formal power series obtained from the Taylor Z Z = i.e. if b(Z, Z) =

E

series of

.

Z

Z) and

(11.8.3)

b(Z, Z)

denotes its Taylor series at Z

(11.8.4)

>bap(Z — =

b(p,

—

p, then —

'.-' p

We define a collection of formal power series in (11.8.5)

p,

p,

—

by

C).

The following proposition, whose proof is straightforward and left to the reader, will show that the definitions of essential finiteness and essential type given below are invariant.

118. ESSENTIAL FINITENESS OF SMOOTH GENERIC StJBMANIFOLDS

343

PROPOsmON 11.8.6. The ideal Ik(Cta) C p1], where Ik(CC.a) denotes d and the ideal generated by the formal power series cja(Z, p.
P0.

DEfiNITION 11.8.7. A smooth generic submanifold M c

is essentially

finite at p M if (11.8.8)

00,

denotes the ideal generated by the formal power £ If (11.8.8) holds, then the for 1 d and all a E series Cta(Z, p. integer on the left hand side of (11.8.8) is called the essential type of M at p. written For convenience, we write — oo if M is not essentially finite at p. where I (Ct.a) C

pJJ

It follows from Remark 11.4.25 and Corollary 5.1.26 that Definition 11.8.7 agrees with Definition 11.4.18 for real-analytic generic submanifolds. Moreover, for a real-analytic generic submanifold, the essential type, defined by the left hand side of (11.8.8), can also be computed by replacing p}] by C(Z — pJ. Hence, Definition 11.8.7 is consistent with Definition 9.4.3 for a hypersurface.

The reader should note that Proposition 11.8.6 implies that if M and M' are smooth generic submanifolds of CN and H is a germ at p E M of a biholomorphism sending M into M', with H(p) = p' M', then M is essentially finite at p if and only if M' is essentially finite at p'. In fact, the same conclusion holds even if H is merely assumed to be a germ at p of a smooth CR diffeomorphism from M to M'. Indeed, the proof of this is essentially the same as in the biholomorphic case in view of Proposition 1.7.14.

We shall make use of Theorem 5.1.15 to prove that essential finiteness is an open condition. THEOREM 11.8.9. Let M c function

M p '-+ is upper semicontinuous. In finite is open in M.

be a smooth generic submanifold. Then, the E Z÷ U too)

p E M where M is essentially

PROOF. Let r be a positive integer (or co). We must show that the set of p E M at which
(11.8.10)

p,

cta(Z + p,

XI. NONDEGENERACY CONDITIONS FOR MANIFOLDS

344

Observe that (11.8.11)

p,

(p,

where the

ideal in

are smooth

generated by

of I,,. Let J(Z, p,

j

the functions of p near po. Denote by Then, is the codimension p, i = 1,... ,k, be chosen so that the p.

k, generate Denote by K1,... ,KN1 the polynomials variables given by Theorem 5.1.15 where n, k, p in that theorem are taken to be N, k, r0, respectively. By Theorem 5.1.15, there is with I N1, such that 0, where a(f) is as defined in flo))) I

in k(r +

(5.1.18). By continuity, there exists an open neighborhood U c M of po such that p. p))) 0, for each p U. By Theorem 5.1.15, the ideal generated by f1(., p. i),... , has codimension ro < r. The desired upper p, semicontinuity follows by observing that the latter ideal is contained in 0 REMARK 11.8.12. If M is a real-analytic generic submanifold, then, for every integer r 0, the set of p E M at which (M) > r is a real-analytic subvariety in M. Indeed, this is precisely the content of the proof of Theorem 11.4.32, since the set of points where (M) > r equals the set Er introduced in that proof. We shall relate essential finiteness to the notions of holomorphic nondegeneracy (defined in §11.7) and finite nondegeneracy (defined in §11.1—2).

THEOREM 11.8.13. LetM C beasmoorhgeneric subman (fold and p E M. Consider the following properties:

(i) M is finitely nondegenerate at p; (ii) M is essentially finite at p; (iii) M is holomorphically nondegenerate at p. Then the following implications hold: (i) (ii) PROOF. We first prove that (i)

(iii).

(ii). Assume that M is ko-nondegenerate

at p. Choose approximate normal coordinates Z = (z, w) E C" x with n + d = N, of order K >> k0, vanishing at p (see §4.2). Thus, we assume that M is given near p = (0,0) by (4.2.19). By using (11.7.14), we deduce that the ideal in

generated by the formal power series

(11.8.14)

I

I

Icri

coincides with the maximal ideal mz, which has codimension one in taking (11.8.15)

L1 =

aw1

j = 1,... ,n,

By

11.8. ESSENTIAL FINITENESS OF SMOOTH GENERIC SUBMANIFOLDS

we obtain, with p = (0,0) and

=

(11.8.16)

r),

= a

345

d

+

0,

a

Xj

We find, with p(Z,

—

Q(z,

r), mod K,

ctO((Z, w), p. j5)

(11.8.17)

K for IaI by Wt, 1

w

w), p, —

£

I

--

mod K — IaI — 1,

0, 0),

and £ = 1,... , d. Observe that the ideal in C([z, WI] generated d, and the series in (11.8.14) coincides with the maximal ideal

wJ] since M was assumed to be ko-nondegenerate at p. If we choose c K >> k0 sufficiently large, then we can apply Proposition 5.1.8 to conclude that k0, in the ideal generated by I £ d and 0 Icri w), p, It follows that this ideal has w]] coincides with the maximal ideal finite codimension and, hence, M is essentially finite at p. This completes the (ii). proof of(i) We shall now prove that (ii) (iii) by contradiction. Thus, we assume that (ii) holds, but M is holomorphically degenerate at p. Let X be a nontrivial formal holomorphic vector field at p tangent to M. Denote by K1 the order of vanishing of X at p. i.e. K1 is the smallest integer for which there exists at least one coefficient of X (in any set of coordinates near p) that vanishes precisely of order K1 at p. Observe that K1 is defined independently of the choice of holomorphic coordinates near p. If (z, w) are approximate normal coordinates of order K for M, vanishing at p, then we can write

(11.8.18) where a1, bk E WI] and at least one of these vanishes of order exactly K1 at (0,0). We make use of Proposition 4.2.28. Thus, we assume that M is given by

(11.8.19)

w_QK(z,2,w,th)=0

or

where the Taylor series Q"(z, ü) of Q" at (0,0) satisfies (4.2.29) with the formal series Q(z, w) given by Proposition 4.2.28. The fact that (z, w) are approximate normal coordinates of order K implies, as in the proof of Lemma 11.3.5, that

(

b(z, w)

0,

mod K,

z, w)

0,

mod K — Ian,

n .

.

j=1

XI. NONDEGENERACY CONDITIONS FOR MANIFOLDS

346

K. We shall choose K > K1 + 1, so that we may assume that at for every lal least one of the a1 (z, w) vanishes exactly of order K1 at (0, 0). In view of (4.2.29), the last relation in (11.8.20) can be written (11.8.21)

w)

= Ir(Cta) C

Denote by

I=

0,

p]] the ideal generated by the power series

= I,... ,dandlal $ r,andby

p1] the ideal generated by all the

C

and Definition 11.8.7. By using (11.8.5) and p(z, w, we obtain (11.8.22)

for t = 1,...

mod K — al.

as in Propos ton 11.8.6 ü) iii — z, w),

mod K —lal— 1,

cCa((Z,W),0,0)

Note that each individual power series cia depends on K, although the ideal Ir, for r sufficiently large, and I do not, in view of Proposition 11.8.6. Using (4.2.29), we obtain (11.8.23)

,

d and

Ia I

K

— 1.

cea((z, w), 0,0) '-'

z, w),

mod K — laI

— 1,

fort = 1,... ,d and lal

K — 1. Chooser0 so large that = land such that d and Ial < r0, equals generated by Qe!a(0,z, w), fort = I z, w). Proposition 5.1.8 and (11.8.23) the ideal I generated by all the

the ideal

imply that, for K sufficiently large (depending only on I), for I has finite codimension in w]]. Since I= I z, w) -'£ = I,... , d, the ideal I has finite codimension if and only if the ideal generated 1, has finite codimension. By applying Theorem in by (0, z, 0), Ial 1, we deduce that there exist 5.1.37 to the ideal generated by z, 0), lal €,, a', i = 1, .. . n, such that (11.8.24)

det

(0, z, w))

I$i.j$n

0.

By definition of the number r0 above, we have Ia1 I

r0. We define SO to be the smallest integer for which the homogeneous part of degree So of the determinant in (11.8.24) is not identically 0. We write (11.8.21), fort = £, and a = a' as follows

(11.8.25)

w)

S1(z, w),

i = I,

. . .

NOTES FOR CHAPTER Xl

347

where S,(z, w) is a formal power series that is O(K —r0 —1). By solving(l 1.8.25) for a1 (z, w) using Cramer's rule, we deduce that K1

K — ro

—

1

SO

or, equivalently,

K S K1s0 + To +

(11.8.26)

1.

Since the right hand side of(1 1.8.26) is independent of K, and the integer K can be taken arbitrarily large, we have reached the desired contradiction. Thus, the proof of Theorem 11.8.13 is complete. 0 The more precise relationship between finite nondegeneracy and esential finiteness is given in the following proposition, whose proof is left to the reader.

PRoPosrrloN 11.8.27. Let M C

be a smooth generic p E M. The following two conditions are equivalent.

(i) M

and

is finitely nondegenerate at p.

(ii) M is essentially finite at p and

=

1.

Notes for Chapter XI Holomorphic nondegeneracy was introduced by Stanton [1], [2] in the study of infinitesimal CR automorphisms of real-analytic hypersurfaces (see Chapter XII and its notes). Theorem 11.3.3 for the case of hypersurfaces was proved in Baouendi and Rothschild [Ill, and also in Stanton [2]. The higher codimensional case of this theorem was treated in Baouendi, Ebenfelt, and Rothschild [11. Finite nondegeneracy for embedded hypersurfaces was defined in the work of Baouendi, Huang, and Rothschild [I] on the reflection principle for algebraic hypersurfaces. The definition for submanifolds of higher codimension can be found in the paper by Baouendi, Ebenfelt, and Rothschild [I]. (The abstract definition given here is influenced by that given in Ebenfelt [2] for hypersurfaces.) The notion of essential finiteness, as stated here, was given in Baouendi, Jacobowitz, and Treves [1] for real-analytic generic submanifolds. (The importance of this concept for hypersurfaces is already implicit in Diederich and Webster [1].) Theorem 11.5.1 was proved for hypersurfaces in Baouendi, Huang, and Rothschild [1] and in the general case in Baouendi, Ebenfelt, Rothschild [1]. Theorem 11.6.1 follows from the more general result of Diederich and Fornaess [2] (see also Diederich and Webster [11).

348

XL NONDEC}ENERACY CONDITIONS FOR MANIFOLDS

The definition of holomorphic nondegeneracy for smooth generic submanifolds given in §11.7 is new. Essential finiteness in the smooth case, as defined in §11.8, follows earlier definitions for hypersurfaces by D'Angelo [2] (see also the book of D'Angelo [31) and by Baouendi and Rothschild [5]. The results presented in § 11.7 and § 11.8 have not previously appeared.

CHAFFER XII

HOLOMORPHIC MAPPINGS OF SUBMANIFOLDS

In this chapter we study germs of holomorphic mappings that take one realanalytic generic submanifold M C CN into another M' c CN'. Under favorable conditions, genus of such mappings are parametrized by their jets of finite order. We begin with a discussion on jets of holomorphic mappings and jet manifolds. In § 12.2, we establish a basic identity for germs of holomorphic mappings which map M into M', with M' assumed to be finitely nondegenerate. This reflectiontype identity is crucial for all the results proved in Chapters XII and Xffl. As an application of the basic identity and the Segre sets (Chapter X), we prove in § 12.3 that the set of germs of biholomorphic maps taking M into M' is parametrized by the set of jets of a fixed finite order, provided that M is of finite type and finitely nondegenerate. In §12.4—12.5, we study the infinitesimal CR automorphisms of a real-analytic

generic submanifold M. One of the main results is that the vector space of infinitesimal CR automorphisms is finite dimensional at every point if M is holomorphically nondegenerate and of finite type at some point. It is also proved that this space is infinite dimensional if M is holomorphically degenerate. Chapter XII concludes with a discussion of the analytic dependence on jets of a fixed finite order for genus of mappings taking M into M'. In particular, it is shown that the stability group of M at p E M, i.e. the group of biholomorphisms preserving M and fixing the point p. is a Lie group provided M is finitely nondegenerate and of finite type at p.

§12.1. Jet Spaces and Jets of Holomorphlc Mappings Let X, X' be complex manifolds and x E X. We denote by E(X, X')1 the set of germs at x of holomorphic mappings, H: U —+ X', where U C X is an open neighborhood of x in X which may depend on H. We write E(X, X') for the set of germs of holomorphic mappings from (open subsets of) X to X', i.e. the disjoint 349

XII. HOLOMORPHIC MAPPiNGS OF SUBMANIFOLDS

350

union

E(X, X') :=

(12.1.1)

U

E(X,

XEX

For x' E X', we also denote by E(X,

the subset of E(X, X')1 consisting

of those germs H for which H(x) = x'. For X = CN, p E CN, Xl = CN'I and k a nonnegative integer, we shalt as follows. Consider two germs of holomorphic E(CN, mappings, H', H2 Wesaythat H' and H2 areequivalentmodulo H2, if H' (Z)— H2(Z) is i.e. theirTaylorseries at k, written H' p agree up to order k. Observe that this notion is independent of the holomorphic local coordinates used near p and p' := H'(p) = H2(p), respectively. The jet space Jk(CN, is the space of these equivalence classes, called jets of we denote by j(H) its jet of order k at order k at p. For H E E(CN, define the jet space

the subset of p. For p' E CN' we also denote by consisting of those jets with H(p) = p'. Since the notion of the jet of a germ at p of a holomorphic mapping H with H(p) = p' is independent of the coordinates used near p CN and p' CN', respectively, we can define the jet space X')(*.e) for any complex manifolds X and X' with x E X and x' E X'. In what follows, we write N = dim X and X')(X.X') has a natural complex manifold N' = dim X'. The jet space structure. In given coordinates Z and Z' near x E X and x' E X', respectively, the jet space jk (X, X') (x.x') can be identified with the space of CN' valued polynomials

in Z

—

p,

with p = Z(x), of degree at most k, mapping p to p' = Z'(x').

Coordinates on X')(X.X') can then be taken to be the coefficients A = (Au) of the polynomials corresponding to the jets. The dimension of

is independent of x and x'. We shalt denote it by K(k). Thus, as a complex X')(xx') is isomorphic to manifold Given H E E(X, we abuse notation slightly and write Z' = H(Z) for the mapping Z' o H o Z'. The jet (H) is then identified with the Taylor series of H (Z) at x, truncated to order k.

If X = X' and x = x', then the composition of germs in E(X, X)(XX) induces a composition of jets in J'(X, of Thus,

We write for the open subset consisting of those jets that are invertible under composition. has a natural group structure, which is in fact a complex Lie group

structure. We refer to this group as the jet group of order k of X at x. The reader can also note that the mapping is a group homomorphism from to Gk where is the group, under composition, consisting of those germs in E(X, X)(X.X) which are invertible.

§12.1. JET SPACES AND JETS OF HOLOMORPHIC MAPPINGS

351

The jet manifold of order k, J'(X, X'), is a complex manifold of dimension for any x E X, defined by N + N' + K(k), or equivalently N + dim Jk(X, the projections Jk(XX1) ) XxX' ,t.

x = Jk(X,

where

x' E X'. (Further details about the structure of

X and X') can be found e.g. in

Malgrange [1], Golubitsky and Guillemin [11.) If k and £ are nonnegative integers, then there is a natural projection

Jk+t(X X')

(12.1.2)

Je(X X').

Since J°(X, X') can be identified with X x X', we have ,r = JTkO and jr1 = where Jrx denotes the natural projection X x X' —÷ X. If U and U' are nx x U') is a natural = coordinate charts on X and X', then J'(X, An element of coordinate chart on J'(X, X'), isomorphic to U x U' x with x E X and H E E(X, Jk(X X') will be regarded as a pair (x, For the rest of this section, we shall consider only the case X = CN and

= Cs'.

For future reference, we record here the formula

j(H) = (Aa)O
(12.1.3)

in given coordinates Z, Z' near p. H(p), respectively. Let (5, (5' be germs of complex submanifolds of the same dimension at p E CN', respectively. We shall distinguish a complex submanifold and p' /,fk((5, (5') of the set of jets as follows. Define (12.1.4)

Nk(cS6l)

fj(H): HE H((5) C (5's HI6 biholomorphismj.

We claim that ftfk((5, (5') is a complex submanifold of Jk(CN,

To

prove this, we choose coordinates Z = (z, w) and Z' = (z', w'), vanishing at p and p' respectively, such that (5 = (w = 0) and (5' = (w' = 0). For H E E(CN, we write H = (f, g), i.e. z' = f(z, w) and w' = g(z, w). This givescoordinatesA
where 1

aIal+IPlf

a!fl! azaawp

(0,0),

1

aIaI+IflI

XII. HOLOMORPHIC MAPPINGS OF SUBMANIFOLDS

352

In these coordinates, we have

=0, al

=

(12.1.6)

which proves the claim above. in § 10.1. In particular, observe Recall the operation * defined for subsets of C") We shall extend this operation to subsets of = that

E(CN, C""),, and a representative of H extends to an open follows. If H = neighborhood U of p. then we define the mapping H from *(J to *CN' by

We define (p. j(H)) E Jk(*CN, *CN) as follows

(p, j(H)) =

(12.1.7)

In any coordinates Z, Z' near p and p', if A = j(H), then we obtain A = by taking the complex conjugate of the right hand side of (12.1.3). Thus, we extend the operation * to an operation taking subsets S C

c

*CN') by

(p,j(H)) Es).

=

(12.1.8)

§12.2.

Basic Identity for Holomorphic Mappings

Before stating the basic identity, we need some further notation. Let M C Ci" and M' C CN' be real-analytic generic submanifolds of the same CR dimension n. We write N = n + d and N' = n + d'. Choose p0 E M and M'. Recall (p0, U) denotes the first Segre manifold of M at P0 in U. We shall denote that its germ at po simply by 6; we use similar notation for (pa, U') and write t3' for its germ at Observe that both dim6 = dim 6' = n. For each nonnegative integer k, we use the notation of (12.1.4) (12.2.1)

gk

,) c Jk(CN,CN').

6') c Jk(CN CN')(

For nonnegative integers k and £, by a slight abuse of the notation in (12.1.2), we

denote by

the natural projection Jrk.c:

C" x Jk+t(*CN *CN')

x

*CN')

induced by the natural projection (12.1.2). Recall the notation M for the cornplexification of M, as in § 10.1.

§12.2. BASIC IDENTITY FOR HOLOMORPHIC MAPPINGS

353

THEoREM 12.2.2 (BAsic IDENTITY). Let M, M', p'o, and fiP be as above. Then there exist an open neighborhood Assume that M' is k0-nondegenerate at

SC") of {po} x *NkO such that, for every integer £ 2: 0, x V in defined in there exists a holomorphic Jt(C", C") valued function such that H(M) C M' satisfying the following. For every H E E(CN, = TAM', there exists an open neighborhood U,, of p0 in CN such and E V and for every £ 0 that (f(Z, E UH x lJjj then (Z,

=

(12.2.3) (Z,

(z.

Y(Z,

Mfl(UH x

REMARKS 12.2.4.

(i) Theremaybeno germs H E E(CN,

satisfying the assumptions in the theorem. Even in this case, the functions 'I's are constructed in the proof of the theorem.

(ii) Since the CR dimensions of M and M' are equal, the assumption that -+ TAM' = T;0M' is equivalent to the linear map being an isomorphism.

(iii) The assumption H(M) C M' implies, in view of Proposition 10.4.31, for any k. Therefore, for Z that H(6) c 6' and hence E stays in V. This is the condition that sufficiently close to (Z, essentially defines U,, in the theorem; of course, U,, is a priori assumed so small that H extends to it. It suffices to prove Theorem 12.2.2 in any set of coordinates near p0 and Therefore, Theorem 12.2.2 is a consequence of the following reformulation in x Cd be normal coordinates for M at Po normal coordinates. Let (z, w) E defined in U and (z', w') E x C" normal coordinates for W at p'o defined in U', vanishing at P0 and respectively. As in §12.1, there is an associated U x U' x coordinate system on We denote these coordinates by (z, w, z', w',

(12.2.5) where

(z, w, z', w') E U x U'.

If p E U, p' E U', and H E E(CN, then we write H = (f, g), w) and W' w). In the coordinates introduced above, we where z' = f(z, = g(z, have, for any nonnegative integer t, (12.2.6)

(p,

= (zr, wi,, z,', w,.,

XII. HOLOMORPHIC MAPPINGS OF SUBMANIFOLDS

354

where p = (Zp, we), p' = (z,., w,.), and

:=

1

aIaI+Iflhf

dZadwfl

:=

(zr, w,,),

1

aIrI+IoIg

(zr, we).

y!S!

In these coordinates, the projection Jr*.c in (12.1.2) becomes

Jrkt (z, w, z', w',

= (z, w, z', w', Also, we have

=

lzzowr):

= 0,

#0)

£.

1

by (12.1.6). Similarly, there is an associated coordinate system on X

X

We

denote these coordinates by

(x. r.

x •U' We leave it to the reader to check, using the where r, coordinates introduced above, that Theorem 12.2.2 follows from Theorem 12.2.7 below. THEOREM 12.2.7 (BAsic IDENTiTY IN NORMAL COORDINATES). Let M

and M'

be real-analytic generic submanifolds through the origin in C" and CN' respectively, both with CR dimension n. Let Z = (z, w) E C" x and Z' = (z', w') C" x C" be normal coordinates at the origin for M and M' respectively Assume that M' is ko-nondegenerate at 0. Then, there exists an open neighborhood V of (12.2.8)

((0,0,0)} x 1(A

e

= 0,

1

IcrI

oj.

k0,

x *CN x x *CK(ko) such that, for every multi-index v in a holomorphic CN' valued function defined in V x the following. For every H = (f, g)

satisfying

such that H(M) C M'

0, there exists an open neighborhood Ujj of 0 in C"' such

and that (12.2.9)

E(CN,

there is

(z,

V.

V(Z,

U11 X

355

§ 12.2. BASIC IDENTITY FOR HOLOMORPHIC MAPPINGS

and, for every v E

=

(12.2.10)

(z, XUH).

PROOF'. We write the defining equations of M as z, w) and the = z', w') (see Proposition 4.2.12). If equations of M' as defining =

H(M) C M', where H = (f, g)

E(CN, CN')(O.O), then

holds for points (z, w) E M near 0. By complexifying, we obtain

r) =

(12.2.11)

r), f(z, w), g(z, w)),

for all (z, w, x' r) E M near 0. We define the holomorphic vector fields 4 in C2N tangent toM (and resulting from the complexification of the CR vector fields

of M)by (12.2. 12)

z, w)-,

4 = i—. +

j = 1,... , n.

We shall also need the following vector fields tangent to M and given by

j1,...,n, (12.2.13)

j = 1,...,d,

7; = I

V1

k

k=1

=4 —Qk.Z,(z,x,r)Tk,

j=

After applying the 4 to (12.2.11) Ia I times, and applying Cramer's rule after each application, we obtain for (z, w, r) M near the origin, any multi-index a,

andt=1,... ,d', (12.2.

r), f(z, w), g(z, w))

14)

=

t))Pa8

t))i
reader should note that the constructions given in this proof can be carried out regardless of whether or not a mapping H as in the theorem exists.

XII. HOLOMORPHIC MAPPINGS OP SUBMANIFOLDS

356

where = ó(z, w, r) = det[L11k(X, r)], and are universal polynomials, i.e. independent of M, M', and H. Note that for any germ at 0 of a holomorphic function r) and any multi-index fi we have Thus, A(0) = 0) 0. Also, = 0 by the normality of 0 by the assumption that det the coordinates. Hence the right hand side of (12.2.14) vanishes at the origin. It also follows that (0,0, is in the submanifold (12.2.8). By the assumption that M' is ko-nondegenerate at 0 and Corollary 11.2.14, there

existnmulti-indicesa',...

(1,...

S lal

d'} such that det

E

0. Hence, by the implicit function

theorem, there exists a unique C" valued holomorphic function S(X', t', r) in an xC", sothatS(0, 0,0) =0 open neighborhood U1 xU2xU3 of(0, 0,0) C" r', r) solves the system of equations and X = (x's X, Q'(X,

(12.2.15)

r')) = r1,

j = 1,...

,

with r = (r1,... , r,,). For any positive integer k, we shall introduce the temporary vector valued variables corresponding (vp)IpI
(12.2.16)

Wedefine

We write (U7')i
((uy)i
21a1—I

)1sk.msn)

Observe that is a universal rational function that vanishes when = 0, and has poles only on the hypersurface det(u7')I
from the above that there is an open neighborhood U (depending on H) of 0 in x *CN such that for (z, w, r) E M flU we have the identity (12.2.17)

f(z, w) = 0 ((O'f(x,

T))I$ISko),

where

(12.2.18)

O((uy)Iyg
:= s

vo,

Observe that 0 is holomorphic when uO E U1, V0 E U2, and the point

§ 12.2. BASIC IDENTITY FOR HOLOMORPHIC MAPPINGS

357

is in an open neighborhood (14 of the relatively open subset of a plane defined by

= 0,

(12.2.19)

15

det((u7')I
ko,

0.

Here U4 is detenrnned by the condition that

(12.2.20)

((uy)i
(13,

0.

The neighborhoods U1, U2, U3, and U4 are independent of the mapping H. Now since f(z, w) is a function of (z, w) only, in view of (12.2.13) we have for any multi-index v = (v', v"),

f(z, w) =

(12.2.21)

(z, w).

By applying VU'TU" to the identity (12.2.17), we obtain alvi

(12.2.22)

f

,, (z, w) azv awv ,

f(x,

=

where we have used the notation A = (A', A") and (VIL'

88 ((VA'

f E U1,

i,

= (jil, .i"). The functions are holomorphic when

E U2 and

(12.2.23)

U4.

The identity (12.2.22) holds when (z. w, v) E M flU. Observe that the vector fields V3. 2, and £j have coefficients that are holomorphic functions of (z, w, r) in a fixed neighborhood (depending only on M) U5 x * u5 of the origin in C2N. By substituting (12.2.13) in (12.2.22), we obtain for(z, w, x' r) E Mfl(UH x *UH) where is an open neighborhood of (0,0) depending on H,

alvif (12.2.24)

azv'awv"

W—

U,, Here the function (z, W, X'

aaix,

XII. HOLOMORPHIC MAPPINGS OF SUBMANIFOLDS

358

where V1 is

is holomorphic in a neighborhood of the form V1 x a neighborhood of

((0,0,0)) X

(12.2.25)

E

0,

in

CN

J.LXOrO

x

*CN x *CN' x

1 <

(We have used the notation

= x'

and

= r'.)

we need to find the components correTo complete the construction of sponding tog and its derivatives. For this we rewrite (12.2.11) in the form

g(z, w) = Q'(f(z, w), f(x,

(12.2.26)

r)),

substitute (12.2.24) with v = 0 in (12.2.26), and apply the vector fields V1 and to the identity thus obtained as above. We obtain a neighborhood V C V1 of the submanifold (12.2.25) in x *CN x *CN' x *CK(ko) and C" valued holomorphic functions (z, W, X' T, in V x K(ko), such that, for (z. w, x *UH), where UH C 0H is an open neighborhood of (0, 0) depending on H, (12.2.27)

w)

=

w,

(rf(x,

omit the details of this construction since it is similar to the one above for the component f. Note, by inspecting the construction above, that the function is defined only in terms of the defining equations of M and M' and does not depend on the existence or choice of a mapping H. The proof of Theorem 12.2.7 We

0

is complete.

§ 12.3.

Determination of Holomorphic Mappings by Finite Jets

In this section we shall combine the basic identity of § 12.2 with the Segre sets of Chapter X to obtain uniqueness results for holomorphIc mappings which map one real-analytic generic submanifold, M C C", into another, M' C of the same dimension. We fix points po E M and E M'. The reader should recall the definitions of finite nondegeneracy and holomorphic nondegeneracy in Chapter XI. We have the following result.

§ 12.3. DETERMINATION OF HOLOMORPHIC MAPPINGS BY FINITE JETS

359

THEOREM 12.3.1. Let M C CN bea connected, real-analytic, holomorphically nondegenerate, generic submanifold of codimension d and assume that M is minimal at some point. Then for all p E M outside a proper real-analytic subvariety with of M the following holds. If M' C CN is a real-analytic generic dimE M' = dima M, and H I, H2 are germs at p of biholonwrphisms that map M into M' such that

daIH2

(12.3.2) then W

aza

(au)

<(d+

Va:

=

1)(N—d),

H2.

The following is an immediate corollary of Theorem 12.3.1. COROLLARY 12.3.3. Let M be a connected real-analytic, holomorphically nondegenerate, generic of codimension d which is minimal at some point.

Then for all p E M outside a proper real-analytic subvariety of M the following holds. If H is a germ at p of a local biholomorphism of CN mapping M into itself fixing p. and satisfying (12.3.4)

= 0,

= 8jk' 1

j, k

N,

(d + 1)(N — d),

2

then H is the identity map. Theorem 12.3.1 will follow from more general results given below. We shall show, under the assumption that M' is finitely nondegenerate at that a biholomorphic mapping with H(M) C M', H(po) = is determined along the Segre sets of M by a finite number (depending only on M and M') of derivatives at po. In fact, we shall prove the following more general theorem.

THEOREM 12.3.5. Let M C C's', M' C CN be real-analytic generic submanof the same CR dimension and let P0 M, M'. Assume that M' is ko-nondegenerate at

Let 111, H2 be germs at P0 of holomorphic mappings

satisfying HJ(M) C M', HJ(po) =

and

=

M'for j = 1,2.

Then there exists an open neighborhood U of P0 in CN such that for all pairs = jrko+eH2 then of nonnegative integers r, £ the following holds. If for all Z in the Segre set S,(po, U).

An immediate consequence of the basic identity, Theorem 12.2.2, is the following. If H is a germ at P0 M of a holomorphic mapping satisfying the hypotheses of Theorem 12.2.2, UH the neighborhood of po provided by that PROOF.

XII. HOLOMORPHIC MAPPINGS OF SUBMANIFOLDS

360

theorem, and if (Z,

E

M fl (U,, x S U,,), then for any £

0,

determines uniquely

the corresponding Let H' and H2 be as in Theorem 12.3.5 and UHI and be an open neighborhood open sets given by Theorem 12.2.2. Let U C UHI fl of p0 in C" satisfying (10.1.6). We shall prove the theorem under the assumption that r is odd. We write r = 2j + 1. (The even case is analogous and left to the Z',... , reader.) If Z S,(po, U), then by Definition such that

(Z,

(12.3.6)

Z',...

Z1) E 62j+I(PO, U).

,

By (10.2.1) this is equivalent to the inclusion (12.3.7)

{(ZJ,

(c',

(Z'',

(Z'2, c.'')

. ..

2'—'), ,

(c', Z'), (Z, c')) C M fl(U flU).

The theorem follows by iterating, r = 2j + 1 times, the consequence of Theorem 12.2.2 stated at the beginning of the proof. 0 H', and H2 be as in Theorem 12.3.1. COROLLARY 12.3.8. Let M, M', po, Denote by d the codimension of M. If for any integer £ ? 0, J(d+I)ko+tHI = then

local CR orbit

=

all Z in some representative of the

in M of po•

PROOF. Take r = Jo in Theorem 12.3.5, where Jo is the Segre number of M at 10.5. Recall from Remark 10.5.3 that Jo d+ 1. P0 as defined in the beginning The conclusion of the corollary follows from Theorems 12.3.5 and 10.5.2. 0 REMARK 12.3.9. It will follow from the proof that the number d +1 in Corollary

12.3.8 can be replaced by Jo, the Segre number of M at

If X denotes the

complex submanifold of CN defined in Theorem 10.5.2, then under the hypotheses and H2 coincide for Z E X. of the corollary, PROOF OF THEoREM 12.3.1. Since M is holomorphically nondegenerate, there exists a proper real-analytic subvariety of M such that M is £(M)-nondegenerate, where £(M) < N — d is the Levi number of M as defined in § 11.5 at every M outside this subvariety (see Theorem 11.5.1). Since M is minimal (or p equivalently, since M is real-analytic, of finite type) at some point, it follows from Theorem 1.5.10 that M is of finite type (and hence minimal) at every point outside a proper real-analytic subvariety. Hence, at every point of M outside a proper realanalytic subvariety, M is both minimal and £(M)-nondegenerate. If H' and H2

§12.4. INFINITESIMAL CR AUTOMORPHISMS

361

are germs of biholomorphisms at such a point p, then M' is t(M)-nondegenerate at p' = H' (p) = H2(p). Moreover, the hypotheses of Theorem 12.3.5 are satisfied with P0 = p and k0 = t(M). We may assume that H, and H2 are holomorphic mappings defined in a connected open neighborhood U of p in CN. We apply Theorem 12.3.5 with r = jo, where Jo is the Segre number of M at P0 as defined 0. We conclude, after shrinking U if necessary, that in § 10.5 and £

H'(Z) = H2(Z),

(12.3.10)

VZ E

U).

U) By Theorem 10.5.5, d30(p) = N since M is minimal at p. Hence, contains an open subset of CN. The proof is complete, in view of (12.3.10) and

unique continuation of holomorphic functions from open subsets.

D

We conclude this section with an application to Aut(M, po). the stability group is a real submanifold and po M, of M at po M defined as follows. If M C we define Aut(M, P0) as the set of H (see § 12.1 for the notation) such

that H(M) C M. Clearly, Aut(M, Po) is a group under composition. We have the following result.

THEOREM 12.3.11. Let M C Ce" be a real-analytic generic submanifold and let P0 E M. Assume that M is k0-nondegenerate and minimal at P0. Then

Aut(M, P0)

(12.3.12)

infective group homomorphism from the stability group of M at Po to the jet group of order (d + 1)k0 of C" at Po• is an

REMARK 12.3.13. As in Remark 12.3.9, the numberd + 1 can be replaced by here = N. the Segre number of M at PROOF OF THEOREM 12.3.11. Theorem 12.3.11 follows from Theorem 12.3.5 by an argument similar to that in the proof of Theorem 12.3.1 above.

§ 12.4.

Infinitesimal CR Automorphisms

Here we shall prove some basic results on infinitesimal automorphisms of a CR manifold. Let (M, V) be a smooth CR manifold, with V c CTM being its CR bundle (see Chapter II).

DEnNrrION 12.4.1. A smooth real vector field X defined in an open subset U of M is called an infinitesimal CR automorphism of M if, for every p E U, there exists e > 0 and an open neighborhood U' C U of p such that the diffeoexptX . q U are CR mappings for each fri E. morphisms U' q The following characterizes the infinitesimal CR automorphisms of M.

XII. HOLOMORPHIC MAPPINGS OF SUBMANIFOLDS

362

THEOREM 12.4.2. Let X be a real vectorfield defined in an open subset U C M. Then X is an infinitesimal CR automorphism of M if and only if for every smooth CR vector field L in U, the commutator [X, LI is again a CR vector field.

In the proof we shall use the notation 4(p, t) = 4),(p) for the flow exp t X . p of the vector field X. We use the following formula for commutators PROOF.

of vector fields with X (see e.g. Nomizu Li, Chapter I, Sect. 2]), (12.4.3)

[X, L](p) = lim!(L(p) — (4),).L(4)_,(p))).

If X is an infinitesimal CR automorphism, then q i-+

4),(q) is a local CR diffeomorphism near p. for sufficiently small t, and, hence, E VP if L is a CR vector field. It follows that the right hand side of (12.4.3) is a CR vector at p, since the vector space is closed. This proves the "only if' part of the theorem. To prove the converse, we choose a local basis L1,... , of the CR vector fields in an open neighborhood U' C U of some given point p E U. (Since being an infinitesimal CR automorphism is a local property, it suffices to prove that the flows 4), of X are CR near p.) Assuming that [X, L] is a CR vector field if L is, we have, for every q E U',

[X,

(12.4.4)

=

where aJk are smooth functions in U'. The proof is complete if we show that there exist functions CJk(q, t) in U' x [—E, El such that

= > CJk(q, t)Lk(q),

(12.4.5)

for q E U', provided that we choose U' sufficiently small. To prove (12.4.5), we must show, for every q U', o 4),)}(cD_,(q)) =

(12.4.6) We fix q

(12.4.7)

U' and

E

C°°(U'). We denote by u1(t) the function

u1(t) := (L1(* o 4),))(4)_,(q)),

Using (12.4.3), we obtain

(12.4.8)

CJk(q,

=

—([X,

j = 1,

. . .

, n.

§ 12.4. INFINITESiMAL CR AUTOMORPHISMS

363

We shall compute 8u1 /8t at t and show that the u3 satisfy a system of linear = 4), o

ordinary differential equations. Since

u1(t + t') = {L3(i/,

(12.4.9)

we have

o

o4),, = (q). Differentiating (12.4.9) with respect to t', setting = 0, and using (12.4.8), we obtain

=

where

%k

= ([X,

(12.4.10)

Hence, by (12.4.4), we have L(t) =

(12.4.11) We write this as

t) = A(q, t)u(t),

(12.4.12)

where u = (u1,... , and A(q, r) is ann x n matrix with smooth coefficients Observe that in U' x [—e, E] that do not depend on u(0) =

(12.4.13)

If V' (q, t),...

,

V'1

,

(q, t) are linearly independent solutions of (12.4.12) such that

0) =

(12.4. 14)

where 8jk denotes the Kronecker symbol, then it follows from standard uniqueness

for ordinary differential equations that

u(t) =

(12.4.15)

that V',...

are independent of the function tfr. Thus, (12.4.6) holds with CJk(q, t) = V"(q :). This completes the proof of Theorem 12.4.2. 0 Observe

,

12.4.16. If(M, V) is a smooth abstract CR manifold and po M, then aut(M, P0) 15 the set of germs of infinitesimal CR automorphisms at po.

An immediate consequence of Theorem 12.4.2 is that aut(M, p0) is a Lie algebra with the usual notion of commutators of vector fields. For an embedded smooth generic submanifold M C CN, we have the following characterization of infinitesimal CR automorphisms.

XII. HOLOMORPHIC MAPPINGS OF SUBMANIFOLDS

364

be a smooth generic submamfold, and let COROLLARY 12.4.17. Let M C X be a smooth real vector field on M defined in an open subset U C M. Then X is an infinitesimal CR automorphism of M if and only if (12.4.18) where the a1,

j = 1,...

,

N, are CR functions in U.

has a PROOF. Observe that X, being a real smooth vector field on M C in U. We must show that representation (12.4.18) for some smooth functions X is an infinitesimal CR automorphism if and only if the a1, j = 1,... , N, are CR functions. For this, let (12.4.19)

be

a CR vector field in U. Then, we have

[X, LI =

(12.4.20)

((xbl

—

Laj)—t-

—

The right hand side of (12.4.20) is a CR vector field in U if and only if Laj

j = 1,...

0,

N, in U. Thus the conclusion of Corollary 12.4.17 follows from 0 Theorem 12.4.2. ,

DEfiNITION 12.4.2 1. If (M, V) is a real-analytic abstract CR manifold and

po E M, then hol(M, po) C aut(M, Po) is the set of germs of real-analytic

infinitesimal CR automorphisms at p0.

Clearly, hol(M, P0) is a Lie subalgebra of aut(M, po). We have the following characterization of real-analytic infinitesimal CR automorphisms of real-analytic generic submanifolds in CN. PROPOSITION 12.4.22. LetM C C's" bea real-analytic genericsubmanifold and X a germ at p0 of a real, real-analytic vector field on M. Then, X E hol(M, po) if and only if there exists a germ Z at Po of a holomorphic vector field (as in (11.3.1)) in such that Re Z is tangent to M and X = Re ZIM.

PROOF. The proposition is an immediate consequence of Corollary 12.4.17, since real-analytic CR functions on M near p0 extend as holomorphic functions in a neighborhood of p0 in CN by Corollary 1.7.13. 0

365

§ 12.4. INFINITESLMAL CR AUTOMORPHISMS

Proposition 12.4.22 is important because of the following basic fact. We identify CN with R2N in the usual way, i.e. Z = x + iy with x, y E RN. If

N1 (12.4.23)

a

X=

Yl

I

j=1 is a smooth real vector field in

a

+

CN, then its flow

c1(x,y,t)=4'(x,y,t)+i(Z)"(x,y,t)

(12.4.24)

Z") be the flow of X considered as a real vector field is defined by letting in R2N. We abuse notation in the usual way and write 4(Z, Z, t). In general 4(Z, Z, t) is not holomorphic. However, we have the following. PROPOSITION 12.4.25. Let Z be a holomorphic vector field in an open subset c CN and X = Re 2. Then, the holomorphic flow r) of 2 (with r E C) coincides,for t = t E ]R, with the real flow of X written in the form (12.4.24).

PROOF. The proof consists of taking the real and imaginary parts of the holo0 morphic flow for real time. The details are left to the reader.

We end this section by showing that the holomorphic flow of a holomorphic vector field tangent to a real manifold M gives rise to a one (complex) parameter group of automorphisms of M.

PRoPosmoN 12.4.26. LetM c CN bea real smooth subman (fold and po M. Let 2 be a germ at P0 ofa holomorphic vectorfield tangent to M and let 4'(Z, t) be its holomorphic flow. Then, for It I sufficiently small, the mappings Z i—* are local biholomorphisms in a neighborhood of po mapping M into itself

r)

PROOF. For A E C, with IAI = 1, and Z E M near po, define the smooth R'1-valued function for t in a neighborhood of 0 in R by

h(t) =

(12.4.27)

At), cD(Z, At)),

where M has codimension d and is defined near P0 by p(Z, 2) = 0. Observe that

h(0) =Oand (12.4.28)

=

At), (D(Z,

+

At)A

At), c1(Z,

At)A

= A(ZP)(4(Z, At), '1(Z, At)) + A(ZP)(4(Z. At), 4(Z, At)).

XII. HOLOMORPHIC MAPPINGS OF SUBMANIFOLDS

366

Since Z is tangent to M, we have Zp(Z, Z) = c(Z, Z)p(Z, 2) and Zp(Z, Z) = c(Z, Z)p(Z, 2), where c is a smooth d x d-matrix valued function. It follows from (12.4.28) that (12.4.29)

=

+

where = c(4(Z, At), ct(Z, At)). By uniqueness of solutions of ordinary differential equations, it follows that h(t) 0, for sufficiently small t so that h(z) is defined. Since A E C, XI = I, and Z E M were chosen arbitrarily, we conclude that 4'(Z, r) is in M, for Z E M near po and r C with In sufficiently small. This completes the proof of Proposition 12.4.26. 0

§123. Finite Dimensionality of Infinitesimal CR Automorphisms In this section we shall prove that aut(M, p), the space of infinitesimal CR automorphisms, is finite dimensional under the hypotheses of holomorphic nondegeneracy and minimality. We start by observing that a holomorphically degenerate generic submanifold has many infinitesimal CR automorphisms. More precisely, we have the following.

PRoPosmoN 12.5.1. Let M C C" be a real-analytic, connected, generic submanifold. If M is holomorphically degenerate, then for every p E M (12.5.2)

dimR hol(M, p) = cx.

PROOF. Let p E M and 3) be a nontrivial holomorphic vector field near p tangent to M. Observe that if Z1 is the first coordinate of Z, then the holomorphic vector fields for every j = 0, 1, ..., are all tangent to M. It is easy to

check that they are linearly independent. As a consequence, the vector fields are tangent to M, their restrictions X1 to M are in hol(M, P0) (in Re view of Proposition 12.4.22). The X are linearly independent over R since the restriction of the real part of a nonzero holomorphic vector field tangent to M cannot vanish identically on the generic manifold M (see Proposition 1.8.11). This proves (12.5.2).

0

Under the additional condition that M is minimal at some point, the converse of Proposition 12.5.1 holds. Indeed, we have the following.

§12.5. FINITE DIMENSIONALITY OF INFINITESIMAL CR AUTOMORPHISMS

367

be a real-analytic, connected, generic subTHEOREM 12.5.3. Let M C manifold. If M is holomorphically nondegenerate and minimal at some point, then

dimRaut(M, p)
(12.5.4)

for all p E M. As a corollary of Theorem 12.5.3, we obtain COROLLARY 12.5.5. Let M C C's' be a real-analytic, connected hypersurface. The following are equivalent.

(i) M is holomorphically nondegenerate;

(ii) dimRaut(M, p) < 00 for all p E M; (iii) dimR hol(M, p) < oo for all p E M. PROOF. The implication (ii)

(iii) is obvious in view of Definition 12.4.21.

(i) is the content of Proposition 12.5.1. Since M is a The implication (iii) hypersurface, Theorem 11.5.1 and Proposition 9.4.16 together with the assumption that M is holomorphically nondegenerate imply that M is also minimal (i.e. of finite type) at some point. Thus (i) (ii) by Theorem 12.5.3. 0 PROOF OF THEOREM 12.5.3. Choose P0 E M. Let X', .. . , E aut(M, P0) be linearly independent over R. Let x = (x1, . . . , x,.) be a local coordinate system on M vanishing at po. In this coordinate system, we may write

For

y=

y1X' +

= X'(x).

X' =

(12.5.6)

.. . .

.+

.

,

y,,,)

ax

E Rm, we denote by 4(t, x, y) the flow of the vector field i.e. the solution of

=

(12.5.7)

(1)(O,x,y)

=x.

By choosing 8 > 0 sufficiently small, there is c> 0 such that the flows c1(t, x, y) are smooth (C°°) in {(t, x, y) E ö}. This follows Itt <2, IxI c, from the identity (12.5.8)

c1(st,x,y) = c1(t,x,sy),

SE R,

XII. HOLOMORPHIC MAPPINGS OF SUBMANIFOLDS

368

which, in turn, follows from the fact that the solution of (12.5.7) is unique (the reader can verify that the left hand side of (12.5.8) solves the initial value problem (12.5.7) that defines the right hand side of (12.5.8)). Denote by F(x, y) the corresponding time-one maps, i.e.

F(x, y) := D(1, x, y).

(12.5.9)

To complete the proof of Theorem 12.5.3, we shall need the following. LEMMA 12.5.10. ly2I

8',

0 <6' <6, such tharforanyfixed y', y2 with F(x,y2)for lxi c, then necessarily y' = y2.

There existsS',

if F(x,y')

12.5.10. Note from (12.5.8) that, with the notation e, for ith

PROOF OF

unit vector in (12.5.11)

ôy,

=

ds

s=o

at

Thus, denoting by X(x) the r x rn-matrix with column vectors X'(x), we have

0) = X(x).

(12.5.12)

If J(x) = (Ji(x),... , Jr(x)) is a continuous mapping into R' for lxi

c,

then

we put (12.5.13)

llJ(x)li := sup

By Taylor expansion we obtain

(12.5.14)

1IF(x, y2) — F(x, y')lI

y') .

(y2

—

C1y2

—

—

where C

> 0 is some uniform constant for 1y1 I, ly2l <6. Now, by assumption, the vector fields X1,... , x" are linearly independent over R. This means precisely that there is a constant C' such that (12.5.15)

llk(x) yll ? C'iyl.

Using (12.5.12), (12.5.15), the smoothness of F, and a standard compactness argument, we deduce from (12.5.14) the conclusion of Lemma 12.5.10.

0

§12.5. FINITE DIMENSIONALITY OF INFINITESIMAL CR AUTOMORPHISMS

369

We continue the proof of Theorem 12.5.3. We shall give a complete proof only

of the weaker result

dimRhol(M, p)
(12.5. 16)

for all p E M, under the assumptions of Theorem 12.5.3. We shall then indicate how the proof of (12.5.16) can be modified to obtain the more general result (12.5.4). Thus, we let p0 E M be as above and assume that the vector fields X',... ,Xm are real-analytic, i.e. in hol (M, p0). We shall make use of Theorem 12.3.1. Denote

by U the open neighborhood of po on M given by lxi < c, where x and c are as in the beginning of the proof. In view of Propositions 12.4.22 and 12.4.25, it 8', there are biholomorphisms Z i-+ F(Z, y) for follows that for fixed y, Z in some open neighborhood c c CN of U C M such that the time-one maps x i-+ F(x, y) satisiS' F(x, y) = F(Z(x), y), where Z(x) is the parametrization of M near p0. As in the proof of Theorem 12.3.1, we can find a point p E U at which M is both minimal and £(M)-nondegenerate, where t(M) is the Levi number of M. By choosing the number 8' > 0 in Lemma 12.5.10 even smaller if necessary, we may assume that the maps Z i—÷ fr(Z, y), for < 8', are biholomorphisms Consider the real-analytic in such that F(U, y) c M. Let k = (d + defined by mapping from <8' into the jet manifold

(12.5.17) where

j

y l•-*

denotes

jICF(. y)

Jk(CN,CN)P,

the jet of order k in Z at p. This mapping is injective for

<8' in view of Theorem 12.3.1 and Lemma 12.5.10. Consequently, we have a real-analytic injective mapping from a neighborhood of the origin in Rm into Jk(CN, This implies thatm
a point pj. We claim that each CR diffeomorphism x i-+ F(x, y) extends as a biholomorphism near p1, for j = 1,... , v, in CN. In the hypersurface case, i.e. d = 1, this follows from Theorem 9.9.1, since M is essentially finite at pj

370

XII. HOLOMORPHIC MAPPINGS OF SUBMANIFOLDS

by Remark 11.5.14. For d > 1, the biholomorphic extension of F(x, y) is a consequence of Corollary 9.11.7. Applying Theorem 12.3.1 to the extensions FJ(Z, y) around each p1, we deduce that x i-± F(x, y) is uniquely determined in a neighborhood of Pi by the jet of its extension FJ(Z, y) of order k = (d + at p3. We claim that, in fact, x ,-+ F(x, y) is determined in the whole component by its k-jet at pj. Indeed, if two CR functions on a connected minimal CR manifold agree in an open subset, then they agree on the whole manifold (as is easily proved e.g. by combining the wedge extendability results in Chapter VIII with a connectedness argument). The conclusion of the proof follows as above by considering the mapping

(12.5.18)

y '—*

(J1.

E

Complete details of the latter part of the proof can be found in Baouendi, Ebenfelt,

0

and Rothschild [2].

REMARK 12.5.19. The proof of Theorem 12.5.3 provides upper bounds on the dimensions of hol (M, po) and aut(M, namely

dimhol(M,po) dimaut(M,po)

(12.5.20)

K((d+ l)t(M))+N, v(K((d+ l)€(M))+N)

where K(j) is as defined in § 12.1 (with N' = N) and v is the number of components of (M \ V) fl U that meet P0 as defined in the proof above. § 12.6.

Iterations of the Basic Identity

In the proof of the uniqueness for holomorphic mappings. Theorem 12.3.5, we iterated the basic identity (12.2.3) several times. For other applications, it will be useful to have the following, more precise and technical, description of this iteration. THEOREM 12.6.1. Let M C Ct', M' c CN' be real-analytic generic submanLet 110 be E M'. Assume that M' is ko-nondegenera:e at ifolds, po M, a germ at po of a holomorphic mapping with H°(M) C M', H0(po) = and M) = M'. Then for all nonnegative integers t. r, there exist an open neighborhood Vtr in

...

x

1262)

rodd,

mines

CN x

*CNx

x C's' x *CN m times

CN')PO,

r even

371

ITERATIONS OF THE BASIC IDENTITY

§ 12.6.

of rodd, rllmes

1263

reven r

times

anda holomorphic JC(CN,

satisfying the

F(,. defined in

following, if H is a germ at P0 of a holomorphic mapping, with H(M) C M', M) =

M',

r, and if for some nonnegative integers £ and

(Poe

(1264)

Po....

V(r, E V(.r,

(p0,

=

if r is even,

c CN, such that

then there exists an open neighborhood of

(Z,

if r is odd,

(z,

V(Z,

(1265)

r

if r is even.

PROOF. We shall only give the proof in the odd case, i.e. r = 2j + 1. Let U0 = UHO be the 12.2.2, and

provided by the basic identity, Theorem

neighborhood of

let

(Z,

(12.6.6)

Z',... ,

E

(Jo).

This is equivalent to the following inclusion of a set of 2j + I elements, namely

(1267)

{(Z3,

...

,

Z'), (Z,

c .M 11 (U0 fl 5U0).

By applying the basic identity to each of the pairs of points in (12.6.7), we obtain

(z,

= =

(12.6.8)

(z.',

= W2jk0+C

(c.',

(2',

XII. HOLOMORPHIC MAPPINGS OF SUBMANIFOLDS

372

Taking the complex conjugates of every second identity in (12.6.8), starting with the second one from the bottom, and then substituting the right hand side of every identity thus obtained into the right hand side of the identity above it, we obtain (12.6.9)

=

(Z,

(z,

J9i+I)ko+e(ko)))))

(c,... —

F

1Z ,',1.1, z'

'... '., ' zi

.(2J+I)ko+e(Ro

Observe that

(12.6.10)

(p0, (Poi

= (po,

E

for any nonnegative integer k in view of Remark 12.2.4 (iii). Thus, for every pair of nonnegative integers £ and r, there exists, by continuity, an open neighborhood in

(12.6.11)

c"

x *CN 2j+I

x ...

x

xJ(21 I)kO+C(*CN .CN')

times

of the point (12.6.12)

(po} X (Po} x ... x 2j+I times

such that Fe.,, defined by (12.6.9), is holomorphic in

It remains to show that if H is a germ at of a holomorphic mapping as in the statement of the theorem and, if for some 1, r, (12.6.4) is satisfied, then there exists

an open neighborhood U,' of P0 such that (12.6.5) holds. Let U,, be the open neighborhood of Po provided by the basic identity, Theorem 12.2.2. Observe that if (12.6.4) holds, then there exists an open neighborhood U,' C U,, of P0 such , that the function (z, 1, zi, is holomorphic for

*j4,r (Z, z',... , Zi) E . "uç x U,'. On the other hand, since we chose U,' C UH, all identities in (12.6.8) hold, with H° replaced Z',... , Z1) E by H for (Z, x Uj' x ... x . .

The conclusion (12.6.5) then follows from the construction (12.6.9) of F1,.. This completes the proof of Theorem 12.6.1 for the case of r odd; we leave the case of r even to the reader. 0

§ 12.7. ANALYTIC DEPENDENCE OF MAPPINGS ON JETS

373

§ 12.7. Analytic Dependence of Mappings on Jets In this section we show that under some general hypotheses holomorphic mappings taking one generic submanifold into another depend analytically on their finite jets at a given point. The precise statement is the following.

THEOREM 12.7.1. Let M C Ci", M' C CN' be real-analytic generic submanP0 E M, M'. Denote the codimension of M by d. Assume that M is minimal at P0 and assume that M' is ko-nondegenerare at Let H° be a

germ at P0 of a holomorphic mapping with H°(M) C M', H°(po) = pt,, and

=

Then, there exist an open neighborhood Uo of Po in CN, an

)k0 )k0 (CN, CM' open neighborhood V0 c of H°, and a holomorphic CN' valuedfunction F, defined in U0 x V0. satisfying the following. If H is a germ at P0 of a holomorphic mapping, with H(M) C M', H E V0. then

H(Z) = F(Z, j41)koH),

(12.7.2)

for Z in a neighborhood of

in CN.

from § 10.5 that if M C CN is a real-analytic generic submanifold, p0 E M, and M is minimal at po, then Jo, the Segre number of M at p0. is defined to be the smallest integer such that dj0 (p0) = N, where d1 (P0) is as defined in § 10.2 (cf. also § 10.5). Recall also, from Remark 10.5.3, that Jo d + 1. We shall prove a slightly stronger result, namely Theorem 12.7.1 with d + 1 replaced by Jo in the statement of the theorem. For this we need the following result, which will be used to obtain a lifting of a neighborhood of po in CN to the Segre manifold '52J0(P0, U). PROOF. Recall

Ctm x Ct be a domain, 6 C a complex connected and Ctm (1p4 Yo = (xo, Yo) E 6 with Denote by pr Cm. the natural projection cm x Ce -+ Assume that there exists a sequence of

LEMMA 12.7.3. Let c c

points pj E 6, J = 1, 2,..., tending to P0 such that pr(pj) = xo and the rank of prI6 at P1 is m. Then, there exists an open neighborhood c > (0, 1) of {xo) x (0, 1) and a real-analytic mapping v: co 6, with v (., t) holomorphic, (12.7.4)

prov(x,t)=x,

and

(12.7.5)

urn v(x0, t) = P0. r-+0

V(x,t)€w

XII. HOLOMORPHIC MAPPINGS OF SUBMANIFOLDS

374

PRoOF. Since the statement of the lemma is local near the point po, we may assume that 6 is defined near po by

G(x,y)=O, is a holomorphic where we use the coordinates (x, y) E cm x Ct and G: -+ Here q denotes the codimension of 6. A simple linear mapping with rank q in at the algebra argument, that we leave to the reader, shows that the projection point p1 = (x0, y1), j = 1,2, . ,has rank m if and only if . .

(12.7.6)

rank —(XO, y1)

ay

= q.

It follows, in particular, that q £ and that we can write y = (y', y") E k = 1,2,..., of {p1}, such that, fora subsequence = (xO, rank

(12.7.7) Define

the holomorphic subvariety 2t C (6 fl pC' (xo)) as follows

(12.7.8)

If

21=

E

6:


21, then the conclusion of the proof of the lemma is easy and left to the reader. Thus, we assume that (x0, Yo) e 21. We claim that at Least one irreducible component of (6 fl pr' (x0)) that contains po is not contained in 21. This follows easily from the existence of the subsequence Pjt converging to p0 and satisfying (12.7.7). Hence, 21 fl 'B is a proper holomorphic subvariety of the irreducible variety 'B. By the curve selection lemma (see e.g. Bierstone and Milman or Lojasiewicz [11), there exists a real-analytic curve a: (0, 1) —÷ 6 such that o((0, 1)) c 'B \2tand (x0, Yo)

(12.7.9)

lima(t) = r—+0

P0.

x Consider the function We write 0(1) = (xo, a'(t), a"(t)) E cm x g(x, r, y") = G(x, a'(t), y"). A standard application of the implicit function for any to E (0, 1), shows that theorem, near the point (x, C, y") = (x0, t0, there exists an open neighborhood w0 of x0 in cm and a unique function u0(x, t), defined in wo x (to — E, to + E) for some e > 0, holomorphic in x for each t E (to—e, to+e) and real-analytic in w0 x(to—e, to+E), such thatg(x, t, uo(x, t)) 0 and UO(X0, t) = o"(t). By uniqueness of u0(x, 1) near to, we obtain a function

DEPENDENCE OP MAPPINGS ON JETS

§ 12.7.

375

u(x, t), defined in a neighborhood cv C Ctm x (0, 1) of {xoJ x (0, 1). This completes 0 the proof of Lemma 12.7.3 with v(x, t) = (x, u(x, t)). We now return to the proof of our main result in this section, Theorem 12.7.1. We will use the iterated basic identity, Theorem 12.6.1. More precisely, we shall use Theorem 12.6.1 with £ = 0 and r = 2j0. By a slight abuse of notation, A) defined in the product we find a holomorphic CN' valued function S2j)(po, U) x V, where U is an open neighborhood of P0 in Ce" and V is an open joko in J2joko (Ci", such that the following holds. If neighborhood of V0 C V is a sufficiently small neighborhood of H°, then for all H as in the statement of the theorem,

=

(12.7.10)

j2iOkOH)

UH), where UH is a neighborhood of P0 in C1' depending on E H. By Proposition 10.4.19, we may apply Lemma 12.7.3 with = U) = N. We obtain an open connected neighborhood cv C CN x (0, 1) since of {po) x (0, 1)andareal-analytic lifting v: cv -+ U) with the properties for all

(12.7.11) pro v(Z, t) = Z,

t)) = N,

rank

V(Z, t) Ecu

and (12.7. 12)

lim v(po, t) = (p0, Poe...

,

p0, Po).

Pick E (52jo(PO, UHO), where UHO is a neighborhood of po in CN such that (12.7.10) holds for = v(po, to) for E UHO) and H = H°, with some to E (0, 1). We define the holomorphic CN' valued function (12.7.13)

F(Z, A)

to), A)

for (Z, A) E

x Vo by possibly shrinking the neighborhood V0 of as defined above, if necessary, and by taking U0 C U to be a sufficiently small open neighborhood of Po in Observe that since u(po, to) = E 62j0(po, U,,o) it follows that v(Z, t0) E UH0) for Z sufficiently close to Po• Consequently, the iterated basic identity (12.7.10) implies that (12.7. 14)

H°(Z) = F(Z,

for Z sufficiently close to P0. To complete the proof, we must show that if the neighborhood V0 of J2Joko H° in J210k0(CN, CN'),,0 is sufficiently small and if H is

as in the statement of the theorem, then (12.7.15)

H(Z) = F(Z,

376

XII. HOLOMORPHIC MAPPINGS OF SUBMANIFOLDS

for Z sufficiently close to p0. Observe that, for all (Z, t) E cv such that v(Z, t) 62jo(P0, UH), we have

t), j2iokoji)

(12.7. 16)

H(Z) = F02

in view of (12.7.10).

By (12.7.12), v(Z, t)

62j0(Po, UH) for (Z,:) in a

nonempty open subset of cv. Since 10(v(Z , t), tion in cv, it follows from (12.7.16) that

(12.7.17)

Fo.2J0(v(Z, :),

=

is a real-analytic func-

F02 10(v(Z, to),

forall (Z, t)

cv such that (Z, to) cv. In particular, there is an open neighborhood of P0 in C/v for which (12.7.15) holds. Hence the the proof of Theorem 12.7.1 is complete. 0

We conclude this section by giving an application of Theorem 12.7.1, related to Theorem 12.3.11. Recall that if M c CN is a submanifold and p E M, then Aut(M, p) C denotes the stability group of M, i.e. those germs at p of biholomorphisms mapping M into itself and fixing p. We equip with the natural inductive limit topology of uniform convergence on compact neighborhoods of p. i.e. a sequence H1 E converges to H E there is a compact neighborhood K of p in CN such that H1 and H are holomorphic

in a neighborhood of K and such that H1 converge uniformly to H on K. The stability group Aut(M, p) is equipped with the topology inherited as a subset of Also, recall the jet group have the following result.

C

CN)(p.p) from § 12.1. We

THEOREM 12.7.18. Let M C CTM be a real-analytic generic subman:fold and p M. Assume that M is ko-nondegenerate and minimal at p. Then,

(12.7.19) is a

Aut(M, p) —÷

continuous infective group homomorphism and its image is a real Lie subgroup of

(M. p))

REMARK 12.7.20. The reader should note that the first part of the conclusion of Theorem 12.7.18 is a weaker version of Theorem 12.3.11. The content of this theorem is that the image of (12.7.19) is a closed real-analytic submanifold of the jet group We do not know if the image of the map (12.3.12) in Theorem 12.3.11 is also a real-analytic submanifold in general. However, if M is a hypersurface, this is indeed the case (see Baouendi, Ebenfelt, and Rothschild [31), i.e. the exponent 2(d + 1 )k0 in Theorem 12.7.18 can be replaced by (d + 1 )ko.

NOTES FOR CHAPTER XII

377

The continuity of the mapping (12.7.19) is immediate since uniform convergence on a compact neighborhood of p of holomorphic mappings implies the convergence of their jets at p. To see that (12.7.19) is an injective group homomorphism, we observe that the following diagram is commutative PRooF.

Aut(M, p)

IP)

(12.7.21) G210k0

is the natural projection induced by (12.1.2). By Theorem where = The is injective and hence, by (12.7.21), so is 12.3.11, the mapping 12.1. fact that the jet mappings are homomorphisms was already pointed out in § (M, p)) is a Lie subgroup of G210k0 (Ca' To show that the image it suffices to show that it is locally closed (see e.g. Helgason [1]). We shall show that (Aut(M, p)) by realj2(d+I)ko(Aut(M, p)) is defined neareach point A E analytic equations and, hence, is locally closed. Pick a point A0 =

where H° E Aut(M, p). By Theorem 12.7.1, there exists a holomorphic CN valued function F(Z, A), defined in an open neighborhood U0 x V0 of (p, Ao), such that for all H E Aut(M, p), with j2(d+l)ke(H) V0. we have (12.7.2) for all p)) fl V0 Z in an open neighborhood of p. Thus the intersection A) satisfies F(Z, consists of those A for which the mapping FA(Z) (12.7.22)

FA(M) C M,

j2Jo&OFA

= A.

The first condition in (12.7.21) amounts to the following, (12.7.23)

p(FA(Z), FA(Z))

0,

Z E M.

By parametrizing M near p and Taylorexpanding (12.7.23), we obtain real-analytic equations on A that are equivalent to the first condition in (12.7.22). The second

condition is equivalent to a set of holomorphic equations on A. This completes 0 the proof in view of the remarks above.

Notes for Chapter XII The notions of jets and jet manifolds as used in this chapter are standard. The reader can consult e.g. the book of Golubitsky and Guillemin [1]. Theorem 12.2.2 is a reformulation and generalization of a result proved in Baouendi, Ebenfelt, and

378

XII. HOLOMORPHIC MAPPINGS OF SUBMANIFOLDS

Rothschild [1]. The results of § 12.3 and § 12.5 were given in Baouendi, Ebenfelt, (ii) in Corollary 12.5.5 was first and Rothschild [21. The equivalence (i) proved by Stanton [2]. The special case where the hypersurface M is also assumed to be Levi nondegenerate is due to Tanaka [1]. Theorems 12.7.1 and 12.7.18 are due to Zaitsev [1]. However, the proof given here is based on new results in Chapter X. Sharper versions (see Remark 12.7.20) of Theorems 12.7.1 and 12.7.18 in the case of a hypersurface were obtained earlier by Baouendi, Ebenfelt, and Rothschild [3). We should mention that all the main results of sections §12.3—12.7 in the case of Levi nondegenerate real-analytic hypersurfaces follow from the work of Chern and Moser [1) (see also Burns and Shnider [1)). Further work on parametrizations of local transformation groups of Levi nondegenerate hypersurfaces has been done by the Russian school, see e.g. the survey papers of Vitushkin [1] and Kruzhilin [1]. For higher codimensions, see related work by Tumanov and Henkin [1] on Siegel domains.

CHAFFER XIII

MAPPINGS OF REAL-ALGEBRAIC SUBVARIETIES

Summary In this chapter, we consider real-algebraic generic submanifolds and, more generally, subvarieties of C's". We address the following question. Let H be a germ

of a holomorphic mapping of C" taking one real-algebraic submanifold (or subvariety) M into another M' (of the same dimension). Under what conditions must H be an algebraic mapping? In § 13.1, it is proved that if H is a biholomorphism, then it must be algebraic provided that M is holomorphically nondegenerate and of finite type at some point. Another result proved in this section is that the CR orbit of a real-algebraic CR submanifold is real-algebraic. The necessity of holomorphic nondegeneracy for the algebraicity of all mappings as above is proved in § 13.2. For the special case of quadratic submanifolds, explicit necessary and sufficient conditions are derived. The conditions of being generic and of finite type are then reformulated in terms of holomorphic functions in § 13.3, and an example is discussed in detail.

§13.1. Mappings between Generic Real-algebraic Submanifolds

Let M, M' c be generic real-algebraic submanifolds. Recall from §5.4 that Mis real-algebraic if near each point p E M there are local defining functions p(Z, Z) of M which are real-algebraic functions of (Z, 2). In this section we shall find conditions on M and M' that in particular imply that all holomorphic mappings H, with H(M) C M' and Jac (H) 0, are algebraic. Recall, also from §5.4, that a holomorphic mapping is called algebraic if each component H1 (Z), j = 1,... , N, is an algebraic function. The sufficient conditions in the theorem below are motivated by the following simple examples. 13.1.1. Let M C C3 be the generic real-algebraic hypersurface defined by

(13.1.2)

1mw =

1z1z212.

379

XIII. MAPPINGS OF REAL-ALGEBRAIC SUBVARIETIES

380

Observe that M is minimal (i.e. of finite type) at every point. On the other hand, note that the hotomorphic vector field — z28/8z2 is tangent to M. Hence, M is holomorphically degenerate (see Definition 11.3.2). The holomorphic mapping (Zi, Z2, w)

(zjez2,

z2e2, w)

is biholomorphic near the origin, maps M into itself fixing the origin, but algebraic.

is not

EXAMPLE 13.1.3. Let M be the real line lit in the complex plane C. Then M is generic and holomorphically nondegenerate, but not minimal at any point. The mapping z i-+ zez is biholomorphic near the origin, maps M into itself fixing the origin, and is not algebraic.

By requiring M to be holomorphically nondegenerate and minimal at some point, we eliminate both Examples 13.1.1 and 13.1.3. We have the following result.

and M' c Ca" be connected, generic, realTHEOREM 13.1.4. Let M C algebraic submanjfolds of the same dimension. Assume that M is holomorphically nondegenerate and minimal at some point. If H is a germ at some point p E M of a biholomorphic mapping, with (-1(M) C M', then H is algebraic. Before proving Theorem 13.1.4, we state a corollary with conditions involving the Levi map of M (see §2.2). COROLLARY 13.1.5. Let M, M' c CN be connected, generic, real-algebraic submanifolds of the same dimension. Assume that there exists P2 E M such that the Levi map of M at PitS nondegenerate and the Levi map at P2 is surf ective.

If H is a germ at p M of a biholomorphic mapping, with H(M) C M', then H is algebraic. In particular if M and M' are connected real-algebraic hypersurfaces and M is Levi nondegenerate at some point, then the same conclusion holds. PROOF. The fact that the Levi map is nondegenerate at is equivalent to M being I -nondegenerate at p', in view of Remark 11.1.15. Thus, M is holomorphically nondegenerate, by Theorem 11.5.1. Also, the fact that the Levi map at P2 is surjective implies that M is minimal (i.e. of finite type) at P2, by Proposition 4.3.23. Hence Theorem 13.1.4 applies and the first part of the corollary is proved. To prove the second part, note that if a hypersurface is Levi nondegenerate at a point, then the Levi map at that point is both nondegenerate and surjective. 0 Theorem 13.1.4 is a consequence of the following more general statement.

§ 13.1. MAPPINGS BETWEEN GENERIC REAL-ALGEBRAIC SUBMANIFOLDS

381

THEOREM 13.1.6. Let M c C" and M' C CN' be connected real-algebraic CR submanifolds of the same dimension and CR dimension. Assume that M' is holomorphically nondegenerate and that M is generic and minimal at some point. if H is a germ at p M of a holomorphic mapping, with H(M) a relatively open subset of M', then H is algebraic.

PROOF. Let H be a holomorphic mapping, defined in an open neighborhood U

representing the germ given in the statement of the theorem. In order of p in to make use of the basic identity, Theorem 12.2.2, we need a point P0 E M fl U satisfying the following three conditions:

(i) M is minimal at Po (ii) M' is £(M')-nondegenerate at H(po); is an isomorphism. -÷ (iii) We observe that the set of P0 M fl U for which (i) holds is an open and dense subset of Mfl U (see Theorem 1.5.10 and Remark 1.5.11). Since H(M) is open in M' and HIM is real-analytic, the rank of H: M fl U —+ M' equals dim M' E M' at which M' on an open and dense subset of M fl U. Since the set of is £(M')-nondegenerate is open and dense in M', in view of Theorem 11.5.1, we deduce that the set of p0 M for which (ii) holds is also open and dense in M fl U. Moreover, since H: M 0 U —* M' is a CR mapping, dim M = dim M',

and CRdimM = CRdimM',(iii)holdsiftherankofH: MflU —+ M' at po equals dim M' (= dim M, so that H is a diffeomorphism at pa). Hence, there exists a dense and open subset of points P0 E M fl U for which (i), (ii), and (iii) above hold. We fix such a point P0 M 0 U for the remainder of the proof. of M at p0 satisfies N, By Theorem 10.5.5 the Segre number, where d1(po) is the maximal rank of the projection pr: (Jo) —+ (Jo) is and U0 is any sufficiently small neighborhood of P0 for which (3j(Po, connected. We shall apply the iterated basic identity, Theorem 12.6.1, to the mapping H atpo, with £ = 0, r = Jo. and H° H. We take Uo above so small that U0 C where the latter is given by Theorem 12.6A. At this point, it is important to note that the complexifications M and M', as well as all the Segre manifolds of M and M', are pieces of algebraic subvarieties. Indeed, their defining equations are obtained directly from the defining equations of M and M' which, by assumption, are algebraic in (Z, 2). As in § 12.1, we consider the jet CN') manifolds x C" x = with = N + N' + K(k). An inspection of the proof of the basic identity, Theorem 12.2.2, shows (by using the algebraic implicit function theorem, Theorem 5.4.6) that the functions 'l's, in (12.2.3), are algebraic functions in x Jko+t(CN C""). It follows (Proposition 5.4.3) that all functions in Theorem 12.6.1, being compositions of the 'I's, are algebraic.

XIII. MAPPINGS OF REAL-ALGEBRAIC SUB VARIETIES

382

Let

(p0, Uo) be a point at which the projection Pr:

(po, (Jo) -+

C"

has rank N. We claim that there is a holomorphic algebraic mapping v: U1 -+ Un), where U1 C Uo is an open neighborhood of Z = prq3, such that pr o v is the identity on U1. This follows from the general lemma given below.

c

x C' be an open connected neighborhood of (1,9) Ctm x C', and F: -+ C" a holomorphic mapping with F(.f, 9) = 0 be the complex defined by and rk(F) = p in Q. Let (5 C F(x, y) = 0). Denote by pr: (5 -÷ Ctm the restriction to (5 of the {(x, y) E LEMMA 13.1.7.

Let

Ctm

£ and Ctm. Assume that rk(pr)(1, 9) = m. Then p there exists an open neighborhood U1 C Ctm of I and a holomorphic mapping v: U1 -+ (5such that v() = (1,9) and pr o v is the identity map on U1. Moreover, F is holomorphic algebraic, then v can be chosen to be holomorphic projection Ctm x C' -÷

algebraic.

PROOF. A simple linear algebra argument, that we leave to the reader, shows that (13.1.8)

9)) = p

rk

since rk(F) = p in Q and rk(pr)(I, 9) = m. This implies, in particular, that 1. Now, the implicit function theorem implies that, after renumbering the p coordinates if necessary, we can write y = (y', y") E C1' x

and solve the

equation F(x, (y', y")) = 0 for y' = V(x, y") in a neighborhood of the point (1, 9). It suffices to take v(x) = (x, V(x, 9"), 9"), where 9 = (9', 9"). To prove that v can be chosen holomorphic algebraic, whenever F is, it suffices to use the algebraic implicit function theorem (Theorem 5.4.6) in the argument above. This completes the proof of the lemma. 0

We rewrite (12.6.5), with £ = 0 and r = Jo. in the form (13 1 9)

(pr'41,

= Fo.j0(q3,

(prqi,

=

if J0 Jioe(M')H)

is odd

if Jo is even.

By Theorem 12.6.1, (13.1.9) holds for every = (Sjo(po. (Jo). By taking v(Z), for Z E U1, we deduce, by Proposition 5.4.3, that H(Z) is holomorphic algebraic in U1. The proof of Theorem 13.1.6 is complete since a holomorphic function in a connected open set in C" which is algebraic in an open subset is algebraic. 0

We conclude this section by proving that the CR orbits of a real-algebraic submanifold are algebraic. The theorem of Nagano, Theorem 3.1.4, states that

§13.2. NECESSARY CONDITIONS FOR ALGEBRAICITY OF MAPPINGS

383

the integral manifolds of systems of vector fields with real-analytic coefficients are real-analytic. Thus, the CR orbits of a real-analytic CR manifold M are realanalytic submanifolds of M. However, in general the integral manifolds of systems of vector fields with real-algebraic coefficients are not algebraic manifolds, as can be readily seen by examples. Hence, one cannot use Nagano's Theorem to deduce that the orbits of an algebraic CR manifold are algebraic.

13.1.10. Let M be a real-algebraic generic subman (fold and P0 E M. Then the local CR orbit of Po in M is a real-algebraic subman (fold of M. PROOF. As mentioned in the proof of Theorem 13.1.6, the Segre manifolds for any sufficiently small open neighborhood U of p0. are (po, U) of M at complex algebraic manifolds. Let jo be the Segre number of M at P0 (as defined in Then by Proposition 10.4.19, for any sufficiently small open neighborhood U) such that pr(q30) = p0 and E U of po, there exists a point By the algebraic implicit function theorem (ci'. = the proof of Lemma 13.1.7), it follows that the Segre set S210(pO, U) is an algebraic On the other hand, by Theorem 10.5.4, the local CR orbit submanifold at § 10.5).

of P0 in M, Orb1,0 coincides with the germ at P0 of M fl

U). Thus

is defined by real-algebraic equations near po and, hence, is a real-algebraic o submanifold.

§

13.2. Some Necessary Conditions for Algebraicity of Mappings

In this section, we shall show that the conditions in Theorem 13.1.4 are essentially necessary for the conclusion of that theorem. We first show that holomorphic nondegeneracy is a necessary condition for the algebraicity of all biholomorphic self mappings between algebraic generic submanifolds of Ci". More precisely, we have the following.

THEOREM 13.2.1. Let M C CN be a generic, real-algebraic, connected subman jfold and assume that M is holomorphically degenerate. Thenfor every p E M there exists a germ H at p of a nonalgebraic biholomorphic mapping satisfying

H(M) C M and H(p) =

p.

Combining Theorem 13.2.1 with Theorem 13.1.4 and Remark 11.5.14, we obtain the following result for hypersurfaces. THEOREM 13.2.2. Let M C C", N> 1, be a real-algebraic connected hypersuiface. The following are equivalent. (i) M is holomorphically nondegenerate;

XIII. MAPPINGS OF REAL-ALGEBRAIC SUB VARIETIES

384

(ii) For every p E M and every germ H at p of a biholomorphic mapping with H(M) c M', where M' is another real-algebraic hypersurface, H is algebraic. PRoOF OF THEOREM 13.2.1. We first prove that under the assumptions of the theorem, given p M, there exists a germ at p of a nontrivial holomorphic vector field, tangent to M, with hotomorphic algebraic coefficients. We claim that this (z, w) in (11.3.6) are algebraic, in follows from Lemma 11.3.5. Indeed, the view of the algebraic implicit function theorem (Theorem 5.4.6). Hence, if M is holomorphically degenerate, then we can find nontrivial algebraic holomorphic solutions a3(z, w) of (11.3.6) by linear algebra over the field of fractions of holomorphic algebraic functions (and then clearing denominators). Thus Theorem 13.2.1 is a consequence of the following lemma. LEMMA 13.2.3. Let

M be a generic real-algebraic submanifold in Ce" and

X a nontrivial germ at po

M of a holomorphic vector field tangent to M

with algebraic holomorphic coefficients. There exists f, a germ of a holomorphic and arbitra rily small r C such that 1'(Z, r) is the holomorphic function at flow of the vectorfield Y = fX, then the mapping Z i-÷ 4(Z, r) is a nonalgebraic local biholomorphism mapping M into itself and fixing PROoF. If X = the group relation

ai(Z)& and cI(Z, i) its flow, then by differentiating

(13.2.4)

t(Z, r + a) =

a), r), we obtain

with respect to a, setting a = 0, and using the definition of (13.2.5)

r)

ak(cZ(Z, r)), k = 1,... , N.

After multiplying X by a nontrivial algebraic holomorphic function vanishing at if necessary, we may assume X (i'o) = 0. If for some arbitrarily small r the map Z i-* c1(Z, is not algebraic, there is nothing to prove since this mapping preserves M by Proposition 12.4.26. Otherwise, afterrenumbering the coordinates if necessary, we assume a1 0 and let f(Z) = eZI and Y = eZI X. We denote by W(Z, r) the holomorphic flow of Y. By (13.2.5), for the vector field Y and flow W(Z, v) instead of X and D(Z, r), and taking k = 1 we have (13.2.6)

r) =

r)).

§ 13.2. NECESSARY CONDITIONS FOR ALGEBRAICITY OF MAPPINGS

385

Hence we have

r) = a1('V(Z, r)).

(13.2.7)

To complete the proof of Lemma 13.2.3, we shall show that for some r arbitrarily close to 0, the map Z ,-* 'V(Z, r) is not algebraic. We reason by contradiction. If Z i-+ 'V(Z, r) were algebraic for some fixed r, then since all the coefficients ak are algebraic, it would follow from (13.2.7) that the function Z is also algebraic. Note that Z ,-+ Z1 — 'V1 (Z, r) is algebraic and not constant (since a1 0). However, if A (Z) is any nonconstant algebraic holomorphic cannot be algebraic. (This can be checked function, then the function Z i-+ by taking A(Z) as a holomorphic algebraic coordinate near a point Z0 where aA/aZ(Z0) 0.) Hence we reach a contradiction, which completes the proof of 0 the lemma. The following result shows that minimality of M in Theorem 13.1.4 is also necthrough Oofcodimension essary in model cases. A generic submanifold M C (see also Chapter N) if it is given near 0 homogeneous d is here called weighted by equations

p1(Z, Z) =... = pd(Z, Z) = 0,

(13.2.8) where

(13.2.9)

dp1

near 0 and pi,... ,

A ... A 8Pd

0

real-valued polynomials which are homogeneous with respect to some set of weights, i.e. for t > 0

p1(t"Z1, . ..

(13.2. 10)

for

j = 1,...

Pd are

,

d, with Vk,

r1

,

. . .

,

=

Z),

positive integers.

THEOREM 13.2.11. Let M C CN be a weighted homogeneous, generic subthrough 0. JIM is not minimal at 0, then there exists a germ at 0 of a man nonalgebraic biholomorphic mapping taking M into itself and fixing 0. We refer to Baouendi, Ebenfelt, and Rothschild [1] for the proof of this theorem, which is not difficult. We conclude this section with the following result on quadratic submanifolds.

XIII. MAPPINGS OF REAL-ALGEBRAIC SUB VARIETIES

386

THEOREM 13.2.12. Let

be a generic quadratic submanifold given

M c

by

Im w = p(z,

(13.2.13)

where z E C", w E C", and p = (pI,... , p") is an R'1-valued quadratic polynomial. Then every germ at P0 e M of a biholomorphic mapping H with H(M) C M', where M' is a real-algebraic submanifold of the same dimension, is Hermitian matrices, algebraic and only (13.2. 14)

are linearly independent (13.2.15)

£ = 1,...

At = over R

and

flkerAt =

(0).

PROOF. The sufficiency of the conditions on M follows from Corollary 13.1.5 and the following two facts: (a) (13.2.15) is equivalent to nondegeneracy of the Levi map of M at 0 and (b) linear independence of the matrices At is equivalent to surjectivity of the Levi map, by Propositions 4.3.23 and 4.3.25. We now prove the necessity of the conditions on M. If (13.2.15) does not hold,

then M is not I -nondegenerate at 0, by Corollary 11.2.14 (see also Proposition 2.2.10). In this case, since the defining equations of M are quadratic, it is easy to see that M is finitely degenerate at every point. Hence M is holomorphically degenerate at every point by Theorem 11.5.1. The necessity of (13.2.5) follows from Theorem 13.2.1. are linearly dependent over To complete the proof, assume that the matrices R. After a quadratic change of coordinates, we may assume that each polynomial p'(0, 0. Since the p'(z, has no harmonic terms, i.e. pt(z, 0) are assumed linearly dependent, we may also, by a real linear change of the w coordinates, assume that pj 0. The following map (13.2.16)

,Wd_1, Wd)I-+ (Z,wi,...

,

is a nonalgebraic biholomorphism near 0, fixing 0, and mapping M into itself. This completes the proof. 0

§13.3. MAPPINGS OF REAL-ALGEBRAIC SUBVARIETIES

387

§13.3. Mappings of Real.algebraic Subvarietles the set of regular points, Recall that for a real-algebraic subvariety A c isa real-algebraic submanifoid of CN (see §5.4) but need not be either dense in A or connected. The following is an application of Theorem 13.1.4.

THEoREM 13.3.1. Let A C CN be an irreducible real-algebraic subvariety. Suppose the following two conditions hold: (i)

is holomorphically nondegenerate.

(ii) If f is a germ of a holomorphic function at some p E A. and I

is

real-valued, then f is constant.

in A, and any germ H at P0 of a Z. the closure of Then for any po biholomorphic mapping such that H(A) C A', where A' C CN is a real-algebraic subvariey with dim A = dim A', necessarily the mapping H is algebraic. In order to prove Theorem 13.3.1 we shall need the following lemma. LEMMA 13.3.2. Let E C ing are equivalent:

bea real-analytic subman(fold. Then, the follow-

(a) E is generic and minimal in a dense open subset of E.

(b) 1ff is a germ of a holomorphicfiinction at some p E E, and real-valued, then

is

is constant.

PROOF. We first prove that (a) implies (b). Suppose p E E and f is a germ

at p of a holomorphic function whose restriction to E is real-valued. Since f is holomorphic, L(fIE) 0 for all CR vector fields L on E near p. The reality of L as above. As a consequence, any lIE implies that L(fIE) commutator C of any length of CR vector fields and their conjugates annihilates

fIE.

Since E is assumed to be minimal or, equivalently, of finite type on a dense

open subset of E, we conclude that f c for some constant c E R. It follows by Proposition 1.8.11 that f is constant in a neighborhood of p in CN since f — c is a holomorphic function that vanishes on the generic submanifold E. To prove the converse, observe that E is CR on a dense open subset E1 C E. Recall, from Corollary 1.8.10, that if E1 is not generic at a point p E E1, then the intrinsic complexification of E1 at p (see Definition 1.8.12) is a proper complex submanifoid X C C" through p containing an open neighborhood of p in E,. In this case, (b) cannot hold at p. since the local defining functions of X are not constant but vanish on E,. Thus, (b) implies that E1 is generic. To complete the proof, we must show that (b) also implies that E is minimal on a dense open subset. Let g be the Lie algebra spanned by the Sections of Note that the set of points p E E1 at which has locally constant C dimension is a dense open subset E2 C E1. We claim that E1 is minimal at each

XIII. MAPPINGS OF REAL-ALGEBRAIC SUBVARIETIES

388

p E2. Suppose, in order to obtain a contradiction, that there exists 1 the codimension of which E1 is not minimal. Denote by r

E2 at

By the local Frobenius Theorem (cf. Theorem 2.1.9), there exist real-valued real-analytic functions ft,... , fr defined in an open neighborhood E of P1 in E, vanishing at p' and with linearly independent differentials, such that every CR vector field L and its complex conjugate L are tangent for each q to the submanifold defined by q

E2 near

(13.3.3)

(p

= fi(q), ... fr(P) = fr(q)}.

E:

,

are CR functions on and since they are also realIt follows easily that , analytic they extend as holomorphic functions , f,. in an open neighborhood by Corollary 1.7.13. Each of these functions is not constant since they of pi in have linearly independent differentials. This contradicts (b) since fj and 0 f1 is real-valued for j = 1,... , r. This completes the proof of the lemma. PROOF OF THEOREM 13.3.1. Let M be a connected component of the dense (provided by Lemma 13.3.2) consisting of the points at which open subset of is generic and minimal. We choose the component M so that the point

as given by the theorem, is in the closure M of M. By assumption (i), M is holomorphically nondegenerate. Let H be a biholomorphic mapping defined in an open neighborhood U of p0 representing a germ of a biholomorphic mapping as in the statement of the theorem. Since po M, Mfl U isa nonempty open subset

of M. The image M' = H(M fl U) is a real-analytic submanifold of dimension

dim M = dim A since H is a biholomorphism in U. Moreover, M' C A' by assumption. Since dim A' = dim A = dim M', M' is a real-algebraic submanifold by Theorem 5.4.10. The conclusion of Theorem 13.3.1 follows from Theorem

0

13.1.4.

We conclude this section with an example. EXAMPLE 13.3.4 (CARTAN'S UMBRELLA). Consider A C C3 defined by

(13.3.5)

Re Zi((Re Z1)2 + (Re Z2)2) — (Re Z1)3 = 0.

This is an irreducible real-algebraic subvariety, which consists of the hypersurface

(13.3.6)

Re Z3

(Re Z1)3

= (Re Z1)2 + (Re Z2)2

and the 4-plane (13.3.7)

((Z1, Z2, Z3): Re Z1 = Re Z2 = 0).

NOTES FOR CHAPTER XIII

The regular part (13.3.8)

Note that

389

consist of the hypersurface (13.3.6) minus the 3-plane

{(Z1, Z2, Z3): Re Z1 = Re Z2 = Re Z3 =0).

is not dense in A. The reader can check that the hypotheses of

Theorem 13.3.1 are satisfied at any point p0 on the hypersurface (13.3.6), including the singular points on the 3-plane (13.3.8). Thus, in particular, any germ at such a point po of a biholomorphic self-mapping of A is algebraic. However, at any point p' on the 4-plane (13.3.7) not also on the 3-plane (13.3.8), there exists a germ at This follows p' of a nonalgebraic bihotomorphic self-mapping of A fixing from Theorem 13.2.1 since (13.3.7) is holomorphically degenerate at every point. (Explicit examples of such maps are also easy to find.)

Notes for Chapter XIII The main results in this chapter appeared in Baouendi, Ebenfelt, and Rothschild

[1). However, the study of the algebraicity of holomorphic mappings sending a real-algebraic submanifold into another has a long history. Early in this century Poincaré [1] proved that if a biholomorphism, defined in an open set in C2, maps an open piece of a sphere into another, it is necessarily a rational map. This result was extended by Tanaka [1] to spheres in higher dimensions. Theorem 13.1.4, in the case of Levi nondegenerate hypersurfaces, is due to Webster [I], who used this result to obtain a classification of ellipsoids in CN. Later, Webster's result was extended in some cases to mappings between Levi-nondegenerate hypersurfaces in complex spaces of different dimensions (see e.g. Webster [3], [1], Huang [1]). See also Bedford and Bell [1] for other results related to mappings between algebraic hypersurfaces. Theorem 13.1.4 in the case of hypersurfaces is proved in Baouendi and Rothschild [11]. In the higher codimensional case, related work on quadratic submanifolds was done by Tumanov [1]; Theorem 13.2.12 can essentially be found in this paper. See also more recent results of Sharipov and Sukhov [1] using Levi form criteria; some of these results are special cases of Theorems 13.1.4 and 13.2.12. It should perhaps be mentioned that the algebraicity results presented in this chapter are deduced from local analyticity in contrast with the general "G.A.G.A. principle" of Serre [1], which deals with the algebraicity of global analytic objects.

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INDEX

characteristic bundle, 315 HOlder space, 16L 126 codimension of a submanifold, 1.0 holomorphic, 12 of an ideal, 119 commutators (of vector fields), 17 complex algebraic subvariety, 141 bundle, 9 structure, 8 tangent space, 8 complexification (of a submanifold), 281 complexified tangent bundle, conjugate function, 159 conormal bundle, 219 convergent power series (ring of), 119 coordinates

abstract CR manifold, 36 algebraic function (analytic), 139. 380 implicit function theorem, 1.40 subvariety, 141 almost complex, 41) almost holomorphic extension, 99. 123

A(M, U), 128 analytic algebraic function, L39. 1.44. 380 analytic disc, 11L 155. 162 defect of, 221 attached (to a subset), 162. 181 analytic subvanety, 128 antiholomorphic tangent vector, 2 approximately normal coordinates, 98.

101.112 approximation theorem, 5g., 57

Aut(M, p), stability group, 326 aut(M. p), infinitesimal CR automorphisms, 363

approximately normal 28. IS1L 1-1.2 canonical, 112

holomorphic, 4. 12 normal, 112 regular, 10 ring, 141 transversal, 95 CR (Cauchy-Riemann) abstract manifold, 36 bundle, 36 codimension, 36

Baker-Campbell-Hausdorff Formula, Banach submanifold, 166 basic identity (for mappings), 354 basic solutions, 36 Bernstein's inequality, isi Bishop equation, 165 boundary value (of a function), 1.5. 186

bvf, boundary value of f, 186

dimension, 9.36 distribution, 36

canonical

fUflCtlon, 15. 36

coordinates, 26 102 flj

mapping, 49

form (of vector fields), 76 82 form, Cartan's umbrella, Cauchy transform, 1

otbit (lOCal), 94.

383

submanifold, 2 transversal mapping, 265 401

INDEX

402

CR vector field, 15, 36

CTM,

6

C(z}, convergent power series, 11.9 formal power series, 119

generic submanifold, 10 germ of a subvariety, 129 graded nilpotent Lie algebra, 86 Hubert transform, 159

D-finite type, 252, 327, 334 def A, defect of an analytic disc, 221 defect of an analytic disc, 221 unit disc derivative map, 224 dimension of a subvariety, 128, 141 VN, space of analytic discs, 178 direction (of a wedge). 184 distribution boundary value, edge (of a wedge). 184, 205 Edge-of-the-Wedge Theorem, 204 elementary symmetric functions, 133 essential 343 type, variety, 252, 327 essentially finite, 242. 327, 334, 343 evaluation map, 224 exp, 63—65

Exp, 77 extends up (down) continuously, 246 up (down) differentiably, 246 up (down) smoothly, 246 field of rational functions, 141 finite

codimension (of an ideal), L1.2 map, 130, 131, 266 type (submanifold), 17 type (vector fields), 74 finitely degenerate (manifold), 317 finitely nondegenerate (manifold), 317, 319 flat hypersurface, 16 flow of a vector field, 63—65, 384 formal CR structure, 36 holomorphic series, 29 holomorphic vector field, 336 power series (ring of), 119 formally integrable structure, 35

Frobenius theorem, 38,39

hol(M, p), 3M holomorphic algebraic function, 139 codimension, 12 coordinates, 12 extension (of CR functions), 26 flow (of a vector field), 384 function of slow growth, 184 hypersurface, 12 series, formal, 22 submanifold, 12 subvariety, 129 tangent vector, 7 vector field, 319, 322 vector field (formal), 336 holomorphically degenerate, 322 holomorphically nondegenerate, 322, 324, 334, 336 homogeneous submanifold, 385 Hopf Lemma, 275 hypersurface type, 36 Holder spaces, 158 HOrmander numbers, 74,92, 101

ideal of finite codimension, 112 implicit function theorem, 167 infinite type, 82 infinitesimal CR automorphism, 361 integrable CR structure, 36 integrable structure, 35 intrinsic complexification (of a CR submanifold), inverse mapping theorem, 167 involutive structure, 35 irreducible subvanety, 128, 130, 141

I (complex structure), 8 jet group, 350 manifold , of a function, 350, 35J. space, 350

INDEX

403

KruIl's Lemma, 120 Kuranishi-Akahori Theorem, 49

pseudogroup of local diffeomorphisms, 69 Puiseux expansion, 148

Levi flat hypersurface, 277 form, 41 map, 41, 107

quadratic submanifold, 386

nondegenerate, 41, 246, 317 number, 330 Lewy hypersurface, 4 local basis (CR vector fields), 16, 22 local defining function, 3, 6 locally embedded CR manifold, 36

manifold of attached discs, 178 maximally totally real submarnfold 10,56 minimal submanifold, 20 missing direction, 18 Mizohata structure, 38 multiplicity (of HOrmander numbers), 75 Nagano leaf (global), 63 leaf (local), 63 Theorem, 63 Nash function, 139 Newlander-Nirenberg Theorem, 39 nondegenerate Levi map, 41 normal coordinates, 96, 112 component, 245 Nullstellensatz, 130 Os,, germs of holomorphic functions, 129 orbit, 68, 73

plurisubhannonic function, 44 Poisson kernel, 157 transform, 157 polygonal path of integral curves, 68 principal value, 159 Privalov's Theorem, 159 proper mapping, 270 pseudoconvex, 41 pseudoconvex side, 44

radical (of an ideaL), 121 rank theorem, 173 real hypersurface, 3 structure, 37 submanifold, 6 tangent space, 8 real-algebraic submanifold, 379 subvariety, 141 real-analytic algebraic function, 139 formally integrable structure, 38 integrable structure, 38 submanifold, 4 subvariety, 129 reducible subvariety, 141 reflection principle, 242 Levi nondegenerate hypersurfaces, 247 general, 255 Schwarz 242 regular coordinates, 10 point (of a subvariety), 128, 141

unit circle Schwarz reflection principle, 242 Segre manifold, 283 number, 300 set, 284 set and minmality (theorem), 301 variety, 284 simple convergence (of formal series), 120 singular point (of a subvariety), 128, 141 singular variety, 128 slow growth (of a function), 185, 186 solutions (system of vector fields), 52 stability group, 376 strictly plurisubharmonic, 44 strictly pseudoconvex, 41

INDEX

404

structure formal CR, 36 formally integrable, integrable, 35 involutive, 35 real, 37 submanifold Banach, 166 complexification of, 281 CR, 9 D-finite type, 252, 334 essentially finite, 249, 327, 334, 343 generic, 10 holomorphically nondegenerate, 322,

334,336 submanifold homogeneous, 305 maximally totally real, 10, 56 minimal, 20 quadratic, 386 real, 6 real-algebraic, 379 real-analytic, 4 totally real, 10, 56 weighted homogeneous, 108, 385 subvariety algebraic, 141 analytic, 128 complex algebraic, 141 dimension of, 128 germ of, 129 irreducible, 128, 130, 141 real-algebraic, 141 reducible, 141 regular point of, 128, 141 singular, 128 singular point of, 128, 141

Sussman orbit global, 72, local, 68 Sussman Theorem, 68 symmetric power series, 133 tangent bundle, 6 tangent space complex, 8 complexified, 6 Banach manifold, 171 manifold of discs, 212 real, 6

TM, 8 TM,6 TAM. 6

totally degenerate mapping, 253 totally real submanifold 10, 56 transversal component, 245 transversal coordinate, 95 Thmanov's Theorem, 205 type (at a point), 18 Hilbert transform vanishing at 4, 164 7

wedge, 184, 205 Weierstrass polynomial, 135 Preparation Theorem, 132 weighted homogeneous coordinates, 74 differential operator, 74,91 polynomial, 74 submanifold, 108 vector field, 74, 91 submanifold, 108, 385

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