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 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 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) 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 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 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 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 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 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 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 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 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,
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
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
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 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 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 =
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 and a convex open cone F C R" such that if
= 0 and
where
(7.2.4) W :=
+iv,
w): w = s
IzI
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 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
0); here, Et is the open unit disc as in Chapter VI. Observe that for Isol 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 and a smooth mapping v A,, E A,,0(M, U), defined for v E Rd with lvl 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), 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 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 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 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 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 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 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
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 0} and {Z E = {Z E Q: p(Z, Z) 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 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 > 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. (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:
