Andreotti-Grauert theory by integral formulas (Progress in Mathematics)

Progress in Mathematics Volume 74

Series Editors J. Oesterle A. Weinstein

Gennadi M. Henkin Jurgen Leiterer

Andreotti-Grauert Theory by Integral Formulas

Authors

Prof. Dr. Gennadi M. Henkin Zentrales Okonomisch-Mathematisches Institut der Akademie der Wissenschaften der UdSSR. Moskau

Prof. Dr. Jurgen Leiterer Karl-Weierstrass-Institut fiir Mathcmatik der Akademie derWissenschaften der DDR, Berlin

Library of Congress Cataloging in Publication Data Henkin, Gennadi, 1942 -

Andreotti-Grauert theory by integral formulas / Gennadi M. Henkin, Jiirgen Leiterer. p. cm. -- (Progress in mathematics ; v. 74) Bibliography: p. Includes index. ISBN 3-7643-3413-4

1. Functions of several complex variables. 2. Complex manifolds. 3. Integral representations. 4. Caucliv-Riemann equations. 1. Leitcrcr, Jiirgen, 1945- . Ill. Series: Progress in II. Title. mathematics (Boston. Mass.) ; v. 74. 0A331.144524 515.9'4-dcl9

1988

CIP-Titelaufnahme der Deutscben Bibliothek Chenkin, Gennadij M.:

Andreotti Grauert theory by integral formulas / Gennadi M. Henkin ; Jiirgen Leiterer. - Boston ; Basel : Berlin Birkhauscr, 1988 (Progress in mathematics : Vol. 74) ISBN 3-7643-3413-4 (Basel ...) Pb. ISBN O-8176-3413-4 (Boston) Pb. NE: Leiterer, Jurgen:: GT

All rights reserved.

No part of this publication may he reproduced, stored in a retrieval system. or transmitted, in any form or by any means. electronic, mechanical, photocopying. recording or otherwise. without prior permission of the copyright owner. 1988 Akademic Vcrlag Berlin

Licensed edition for the distribution in all nonsocialist countries by Birkhauscr Boston 1988 Printed in GDR ISBN 11-8176-3413-4

ISBN 3-7643-3413-4

PREFACE

The contemporary analysis on complex manifolds has been developed by means of the theory of coherent analytic sheaves (see, for instance, [Gunning/Rossi 1965 and Grauert/Remmert 1971,1977,1984]) and/or harmonic analysis (see, for instance, (Chern 1956, Kohn 1964, Hormander 1966, Morrey 1966, Wells 1973]). However, the first fundamental contributions to the function theory of several complex variables, obtained in the period 1936-51 by K.Oka, were based on the classical constructive method of integral representations (see (Oka 1984]).

In the seventies this constructive approach has had a come-back in order to obtain in a strengthened form (with uniform estimates) the main results of the theory of functions on complex manifolds. A systematic development of the function theory in Cn which uses integral formulas as the principal tool was given in the books (Henkin/Leiterer 1984 and Range 1986]. The constructive method of integral representations is working with success also in several other fields of complex analysis: Cauchy-Riemann cohomology of complex manifolds, holomorphic vector bundles on complex manifolds, analysis on Cauohy-Riemann manifolds, Radon-Penrose transform, inverse scattering problem.

The authors intend to write a book where these fields will be presented from the viewpoint of integral formulas. The present monograph is a tentative version of the first part of that book. Here we develop in detail the basic facts on the Cauchy-Riemann cohomtlogy of complex manifolds, where the emphasis is on finiteness, vanishing, and separation theorems for a class of complex manifolds which lies between the Stein, and the compact manifolds. Theorems A and B of Oka-Cartan for Stein manifolds as well as the finiteness theorems of Kodaira for compact, and Grauert for pseudoconvex manifolds appear as special cases of more general theorems. The theory developed in the present monograph was mainly obtained in the articles [Andreotti/Grauert 1962, Andreotti/Vesentini 1965, Andreotti/Norguet 1966, Kohn/Rossi 1965, and Hbrmander 19651 (it is astonishing

0

that these remarkable results did not as yet enter into books). The novelty added here consists in new proofs based on integral formulas. As in the case of the theory of functions in C°, this makes it possible to prove all basic facts in a strengthened form: uniform estimates for solutions of the Cauchy-Riemann equation for differential forms on strictly q-convex and strictly q-concave domains, uniform approximation and uniform interpolation for the a-cohomology classes on strictly q-convex domains, solution of the E.Levi problem for the a-cohomology with uniform estimates, the Andreotti-Vesentini separation theorem with uniform estimates etc. A part of these results with uniform estimates was obtained already in the seventies [Fischer/Lieb 1974, Ovrelid 1976, Henkin 1977, Lieb 1979]. Some of these reusults are new. These results with uniform estimates admit important applications in the theory of holomorphic vector bundles and the theory of the tangent Cauchy-Riemann equation. Such applications will be the subject of the following parts of the pending book mentioned above - elements of this are contained in the articles [Ajrapetjan/Henkin 1984 and Henkin/Leiterer 1966]. Moreover, in the following parts of that book, we intend to present some developments of the Andreotti-Grauert theory in connection with the Radon-Penrose transform - elements of this can be found in [Henkin 1983, Henkin/Poljakov 1986 and Leiterer 1986]. Note. Further on the book [Henkin/Leiterer 1984] will be refered to as

[H/L]. The present monograph may be considered as a continuation of [H/L]. However, without proof, we use only results from the elementary Chapter 1 of [H/L]. Moreover, all basic results of Chapter 2 of [H/L], which is devoted to the theory of functions on completely pseudoconvex manifolds (= Stein manifolds, after solution of the E.Levi problem), are obtained anew, as the special case q=n-1 of Chapter 3 of the present work.

Acknowledgments. We thank C. Schmalz for reading portions of the

manuscript, catching many errors. We wish to thank also the Akademie-Verlag Berlin, and particulary Dr.R.HSppner, for support and cooperation.

6

CONTENTS

CHAPTER I.

INTEGRAL FORMULAS AND FIRST APPLICATIONS ................

9

Summary ............................................................ Generalities about differential forms and currents ............. 0. 1.

9 9

The Martinelli-Boehner-Koppelman formula and the Kodaira finiteness theorem ...................................................

21

2.

Cauohy=Fantappie formulas, the Poincare 8-lemma, the Dolbeault isomorphism and smoothing of the 3-cohomology ..................

34

3.

Piecewise Cauchy-Fantappie formulas ............................

46

CHAPTER II. q-CONVEX AND q-CONCAVE MANIFOLDS .......................

59

Summary .................................. I......... ................

59

4.

q-convex functions .............................................

59

5.

q-convex manifolds .............................................

65

8.

q-concave functions and q-concave manifolds ....................

73

CHAPTER III. THE CAUCHY-RIEMANN EQUATION ON q-CONVEX MANIFOLDS .....

77

Summary ............................................................

77

7.

Local solution of au=fo r on strictly q-convex domains with r>n-q ..........................................................

78

8.

Local approximation of 3-closed (0,n-q-1)-forms on strictly q-convex domains ...............................................

83

Uniform estimates for the local solutions of the a-equation constructed in Sect. 7 ......................................... 10. Local uniform approximation of a-closed (0,n-q-l)-forms on

9.

86

strictly q-convex domains ........................... ............ 11. Finiteness of the Dolbeault oohomology of order r with uniform estimates on strictly q-convex domains with r>n-q ..............

93 96

12. Global uniform approximation of a-closed (0,n-q-1)-forms and invariance of the Dolbeault cohomology of order >n-q with respect to q-convex extensions, and the Andreotti-Grauert finiteness theorem for the Dolbeault cohomology pf order >n-q on q-convex manifolds ..................................... ..

100

7

CHAPTER IV. THE CAUCHY-RIEMANN EQUATION ON q-CONCAVE MANIFOLDS ..... 117 Summary ............................................................ 117 13. Local solution of & =fO r on strictly q-concave domains with 10 and C Hr(X,l7E

(2.20)

such that, for all 0n+l, then,

(1<sn+l, Hr(X,oE) is isomorphic to

Hr-n(X,(CD n)E).

By (2.23), this yields (2.18).

For the remainder of this proof we may assume that 1 (CO,s-1) -> (ZO's )

-> 0

is exact. Let

Sour,."Hq(X,(Zcus)E) - Hq+1(X,(ZOug-1)E

(q>0)

(2.27)

be the ooboundary homomorphisms from the corresponding long exact cohomology sequence. Then it follows from (2.23') that (2.27) is an isomorphism if q>1, and the maps Seur,s induce isomorphisms

O,s(X E) ^

cur"H cur

1 our E H (X'(ZO,s-1)

(1 0 on a h K and x = 0 on X\W. Since V is relatively compact in UaD, we can find f>0 such that p +F x and p -E x are (q+l)-convex in V (of. Observation 4.15). Then the domains

D':= Du{zeV: 'p(z) < EZ/ (z)) and

D":= (D\V) u{zeV: e(z) < - Ex (z)) 69

have the required properties.

If aD is of class Ck, then, by Corollary 5.11, we can assume that is Ck and 4(z)#0 for all ztUaD. Since V is relatively compact in UaD, ) then also d(Q + i # 0 on V if E is sufficiently small.

x

5.13. Theorem. Let D C X be a domain with Ck boundary (20 and all with w=(wq+2I...1wn)ECn-q-1

D,,CM+1 9 0 there exist a neighborhood W C Cn of DnCW+1 and a holomorphic function hw on Cn which is independent of zq+2,...,zn such that Ih-hwIO.

In the proof of this theorem we use the following 9.2. Lemma. Let (U,g,(p,D] be a q-convex configuration in Cn, D2:=(zEU: and h(z,x):= Im<w(z,x),x-z> for z,xaU (Im:= imaginary

p(z)n

94

for some wcDiv(Q) and some constant c (of. Sect. 3.5 and the proof of Proposition 0.9). In view of (7.4) and (10.4), we can find open sets Wk (j=0,...,N(k); k=1,2,... ) such that

D u({zcU: if(x) # 0

if x.dD n supp bk and zeWi.

(10.10)

Therefore, the right hand side of (10.9) defines a Coo form gk on Wk with

91 = Lv(bkf)

on D.

Since w(z,x) depends holomorphically on zl,...,zq+1 and since r>n-q-1, it follows that

gk 6z0 r(Wk) (observe that

(10.12)

gk=0 if r>n-q). Moreover, by (10.10) and the definition

of Tv we obtain continuous (0,r-1)-forms hk on Wk which are C°O in Wk\(D n supp bk) such that

hk = Tv(bkf)

on D.

(10.13)

Since hk is Coo in Wk \ (DA supp bk) and by (10.8), one obtains that

(10.14)

By (10.8), (10.11) and (10.13), we have the relation fk=gk+ahk on D. Therefore, taking into account (10.12), (10.14), and (10.5), we can find neighborhoods Vk of D and numbers ck>0 (j=1,...,N(k); k=1,2,... ) such such that, by setting

Fk(z) = (gk+8hk)(z + akvk),

zeVk, forms Fk E ZO r(Vk) with

95

1

(10.15)

k

are defined.

By (10.1) and (10.2), it follows from estimate (9.2) in Theorem 9.1 that

lim

k---) oo

IIT'"(5b0 f)IIo

D

= 0.

(10.16)

Since f is continuous at 0, without loss of generality we can assume that, moreover, f(0)=0. Then it is clear that also lim

= 0 for

DE

k -* oo. Together with (10.16) this implies that

lim

k - moo Setting fk =

iIfk1IO,D6= 0.

(10.17)

N(O) Fk, now we obtain a sequence f of 3-closed

(0,r)-forms in some neighborhoods Vk of D.

In view of (10.7), it follows from (10.15) and (10.17) that lim Afk-fIIO DE = 0 for

k -p oo. Without loss of generality we may assume that & is so small that [U,9, (p-&,D I is also a q-convex configuration. If r=n-q-1, then the proof can be completed by applying Theorem 8.1 to (U,yo, (p-&,L) and the forms fk.

If r>n-q, then we proceed as follows: For all k, we

take a Coo function Xk on en with supp xk == Vk and ,/k = 1 on

.

By

Theorem 7.8, after shrinking Vk, we may assume that fk=&tk for some '"

,r-1(Vk)' Therefore, fk:=a(xkuk) is a continuous 3-closed (0,r)-form on Cn with fk=fk on 5

11. Finiteness of the

. O

Dolbeault cohomology of order r with uniform

estimates on strictly q-convex domains with r>n-q

11.1. Definition. Let D X be a domain in an n-dimensional complex manifold X. Suppose E is a holomorphic vector bundle over X or, more generally, E is a C°CC7 vector bundle over b (for the notion of a C°'12

vector bundle, see Sect. 0.12), where 0O

O,r

(5,E) into

C0 r(D,E) such that 3T - K is the identity operator on ED r (D,E). IV.

If 3 is oc-regular at Zg r(D,E) and, moreover,

dim[Z00 r(D,E)/ED r

(D,E)] < oo,

then 3 is also J3-regular at Z0 r(D,E) for all 0n-q with respect to q-convex extensions, and the Andreotti-Grauert finiteness theorem for the Dolbeault cohomology of order >n-q on q-convex manifolds

12.1. Definition. Let X be an n-dimensional complex manifold, and O