Analytic Functions of Several Complex Variables [1 ed.]

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PRENTICE-HALL SERIES IN MODERN ANALYSIS

R. CREIGHTON BUCK, editor

London Sydney CANADA, LTD., Toronto INDIA (PRIVATE) LTD., New Delhi JAPAN, INC., Tokyo

PRENTICE-HALL INTERNATIONAL, INC.,

PRENTICE-HALL OF AUSTRALIA PTY., LTD., PRENTICE-HALL OF PRENTICE-HALL OF PRENTICE-HALL OF

ANALYTIC FUNCTIONS of

SEVERAL COMPLEX VARIABLES

ROBERT C. GUNNING

HUGO ROSSI

Department of Mathematics Princeton University

Department of Mathematics Brandeis University

PRENTICE-HALL, INC. Englewood Cliffs, N.J.

©-1965 by Prentice-Hall, Inc., Englewood Cliffs, N.J, Copyright under International and Pan-American Copyright Conventions

All rights reserved. No part of this book may be reproduced in qny form, by mimeograph or any other means, without permission in writing from the publisher.

Library of Congress Catalog C3:rd Number: 65-11674

Current printing (last digit) :

12

11

10 9 8 7

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PRINTED IN THE UNITED STATES OF AMERICA

03366-C

To Our Teachers:

SalomJJn· Bochner

Isadore M. Singer

PREFACE

The general theory of analytic functions of several complex variables was formulated considerably later than the more familiar theory of analytic functions of a single complex variable. Some of the principal functiontheoretic problems were attacked and a basic foundation for the subject was laid late in the nineteenth· century by Weierstrass, and around the turn of the century by Cousin, Hartogs and Poincare. Certain central problems, either trivial in one variable or peculiar to several variables, were left open. Significant work in many directions was achieved by Bergman, Behnke, Bochner and others, in papers appearing from about the mid-1920's until the present time. The peculiarities of several complex variables were well exposed and the central difficulties clearly stated by the time of the appearance of the book of Behnke and Thullen, but the main problems were still there. Then K. Oka brought into the subject a brilliant collection of new ideas based primarily in the earlier work of H. Cartan and, in a series of papers written between 1936 and 1953, systematically eliminated these problems. But Oka's work had a far wider scope, and it was H. Cartan who realized this and developed the algebraic basis in the theory. This was essentially put into its present form'in the seminars of Cartan in Paris (1951-52 and 1953-54) and the vastly useful tools provided by sheaf theory were first systematically employed there. The deep and extensive work of Grauert and Remmert on complex analytic spaces was built upon this foundation, and the same is of course true for the impressive works of many others during the last decade and at the present time. The intention of the present volume is to provide an extensive introduction to the Oka-Cartan theory and some of its applications, and to the general theory-of analytic spaces. We have neither attempted to write an encyclopedia of the subject of analytic functions of several complex variables, nor even tried to cover everything that is known today in the two areas of principal emphasis. Many fascinating aspects of this broad and active field that might have been encompassed by a book of the same title have been omitted almost entirely; the reader must look elsewhere for the differential-geometric and algebraic-geometric sides of the subject, for the theory of automorphic functions and complex symmetric spaces, for the Bergman kernel function, and for applications to mathematical physics. An attempt has been made vii

viii

Preface

to append a rather complete bibliography of books and papers in the two areas on which this introduction concentrates, so that the reader can pursue these topics further at will. This book has been written with the prospective student of several complex variables in mind. In fact, the main reason for writing this book has been the untenable lack of an adequate introduction to one of the most active mathematical fields of the day. Further, there have been many recent results which cast a new light on much of the introductory material, and these results should properly be exposed early in the development of the subject. We have tried both to arrange this book so that the fundamental techniques will be exposed as soon as possible, and at the same time to give a firm foundation for their use. Of course, as a very active field, several complex variables is still in a state of flux. There are many different approaches that an introduction such as this could take, and one's choice of the ideal organization of the material varies from year to year. Indeed, were we to rewrite this book from scratch starting today it would probably turn out to be a quite different book. The prerequisites for reading this book are, essentially, a good undergraduate training in analysis (principally the classical theory of functions of one complex variable), algebra, and topology; references have been provided for any important bits of mathematical lore which we did not consider standard minimum equipment for beginning graduate students in mathematics or their equivalent. The book is divided into nine chapters numbered with Roman numerals; each chapter is subdivided into sections indicated by capital letters. The definitions, lemmas, theorems, jokes, etc., are numbered in one sequence within each Section; the principal formulas are numbered similarly in a separate sequence. An expression such as "Theorem Ill C2" indicates a reference to the second numbered entity (in this case, a theorem) in Section C of Chapter III; for references within the same Chapter the Roman numeral will be dropped, and for references within the same Section the letter will often be dropped as well. References given in other forms will be left to the reader to decipher, with our best wishes for his success. In somewhat more detail, the outline of the contents is as follows. Chapter I is in itself an introductory course in the subject (perhaps one semester in length). It presents, in outline, the essentials of the problems and some approaches to their solutions, and, in some special cases, includes the complete solutions. The discussion also shows the necessity for developing further techniques for tackling the problems, and thus motivates the remainder of the work. Except for Sections G and H, which are optional, the contents of this Chapter are prerequisite for what follows. However, to get into the subject most rapidly, bypassing the motivational portions, the reader may pass directly to Chapter 11 after reading Sections A, B, and C of Chapter I; Sections D through G are not needed until Chapter VI, and reading them can be

ix

Preface

postponed until specific references are given to them. Chapter II contains the local theory of analytic functions and varieties, and is the natural sequel to Chapter I. Beyond this point, there are several paths which the reader may follow, one of which of course is the straightforward plodding through the chapters as they occur. The reader mostly interested in the sheaf-theoretic aspects, especially in Cartan's famous Theorems A and B, may proceed next to Chapters IV, VI, and VIII. The reader interested rather in complex analytic spaces and their properties may proceed directly to Chapters III and V. The sheaf-theoretic notation and terminology introduced in Chapter IV are used in Chapter V for convenience, but none of the deeper properties are really required; however the discussion in Chapter VII does require some of the results of Chapter VI. The final chapter consists of an exposition, from the point of view of the preceding material, of pseudoconvexity. This book developed from joint courses in several complex variables given by the authors at Princeton University during the academic years 1960-61 and 1962-63. It is a deep pleasure to both ofus to be able to record here the debts of gratitude we owe to those who helped make this book possible. Lutz Bungart and Robin Hartshorne wrote and organized the lecture notes for our first course; these notes formed the kernel of the present work, and their reception encouraged us to proceed with the task. Thomas Bloom, William Fulton, Michael Gilmartin, and David Prill at Princeton aided in the revision of these notes and contributed many corrections and improvements. We are deeply indebted to Errett Bishop and Kenneth Hoffman, who read our tentative drafts with many helpful and inspiring comments. Finally, our thanks go to the typists of these various drafts; Caroline Browne, Eleanor Clark, Patricia Clark, and Elizabeth Epstein; and to the staff of Prentice-Hall, Inc. Princeton, New Jersey Waltham, Massachusetts

R. C.

GUNNING

H.

ROSSI

CONTENTS

Chapter 1-Holomorphic Functions A B C D E F G H

The Elementary Properties of Holomorphic Functions Holomorphic Mappings and Complex Manifolds Removable Singularities . . . . . . The Calculus of Differential Forms . The Cousin Theorem. . . . . . Polynomial Approximations. . . . . Envelopes of Holomorphy . . . . . Some Applications to Uniform Algebras Notes . . . . . . . . . . . . . . .

1 13 19 22 31 36 43 55 63

Chapter II-Local Rings of Holomorphic Functions A B C D E

The Elementary Properties of the Local Rings . The Weierstrass Theorems . . . . . . . . Modules Over the Local Rings . . . . . . The Extended Weierstrass Division Theorem Germs of Varieties . Notes . . . . . . . . . . . . . . . . .

65

65 67 73 79 85 92

Chapter III-Varieties

93

A The Nullstellensatz for Prime Ideals, and Local Parametrization 93 B Analytic Covers . 101 C Dimension 110 Notes . . . . . 117

Chapter IV-Analytic Sheaves A B C D

118

The Elementary Properties of Sheaves Sheaves of Modules . . . . . . . . Analytic Sheaves on Subdomains of llfll, (f - c)-1 EA, and we can choose g 0 = (f - c)-1 • As c ---+ oo, (f - c)-1 ---+ 0 uniformly, so lim A(c) = 0. Thus A(c) is an c-oo

entire function which tends to zero at infinity, so by Liouville's theorem, J.(c) 0. But by Corollary 5, .A is closed, so by the Hahn-Banach theorem. since g 0 ff .A, there is LE d* such that L(g 0) =fa 0 and L annihilates .A. Thus the corresponding A is nonzero at the origin, a contradiction. Thus there does exist a c E C such that f - c E .A .. Such a c is uniquely determined by f For if also f- c' E .A, then c - c' E .A, so c - c' is not invertible. Thus c = c'. Defining h(f) = c, clearly h is a complex homomorphism, h(I) = l, and ker h = .A.

=

Thus the spectrum of dis in one-one correspondence with the collection of maximal ideals of d. For this reason, S(d) is sometimes referred to as the maximal ideal space of d. 10. Corollary. Let d be a uniform algebra with spectrum S. Jff1 , .•• , fn Ed have no common zeros in S, then there exist g1 , .•• , gn E d such that L f1g1 = I. Proof" Let .f be the ideal generated by f 1 , ••• .fn• If .f is a proper ideal, then by Zorn's lemma .f is contained in a maximal ideal .A. If h is the homomorphism whose kernel is .A, then_(; E .A, sof;(h) = 0, 1 :,:;:: j:,;:; n. But the J; have no common zero by hypothesis, so .f is not contained in any maximal ideal. Thus .f = d, so l E .f. Thus there are g 1 , ••• , gn Ed such that 1 = L gJ;•

We are now in a position to generalize proposition 2. 11. Definition. Let d be a uniform algebra with spectrum S. Let f1 , d. The joint spectrum of f1 , ••. , fn is the set

We shall denote its polynomially convex hull by u(f1 ,

••• ,

••• ,

fn E

fn)-

Notice that for [i. . .. ,fn Ed, if p is a polynomial in n variables, then ••• ,fn) Ed, since dis an algebra. Thus by Oka's theorem (proposition 2), if Fis holomorphic in a neighborhood of u(f1 , ••• ,fn), then Fis a uniform limit of polynomials on a(Ji, .. . .fn), so F(f1 , • •• ,fn) Ed also.

p(f1 ,

60

Holomorphic Functions

Chap. I

Finally, this is true for functions holomorphic in a neighborhood of consisting of the first p - l rows of F must belong to the module ..ii; therefore, by the induction step, we can write q

F'

= Ih1G1 i=l

q

F"

=

F - Ih.1G;

=

(0, ... , 0, f") E .Aii(o;,>;

i=l

thus f"

E

.A~(O;r>• so again by the induction step, f" =

I• h;'g;' i=l

M = sup II &;;II 3.(o;,> i,:j

and

L= sup IIG;ll&• ;

it follows that q

F = Ih;G; i=l

q

s

q

+II h7a;;G; = Lh;G; i=l i=l

i=l

where h; E n(!),i(o:r> and llh;llii(o;r> ~ K IIF\13. ::,:;;

j-1

K IIFnlla· Thus the functions h,.n(z) are uniformly bounded on the compact set .1.(0; r); and, by Cauchy's integral formula, all partial derivatives are uniformly bounded on the compact subset .l(0; tr). We therefore have a normal family; hence, after perhaps taking a refinement of the sequence, we can assume that the functions h1n(z) converge uniform!)' on .l(0; tr) to a function h1(z), where of course h1(z) E (Oaco;r>· Then, whenever z E .l(0; ½r), a

F(z)

= limFn(z) = lim Ih;n(z)G,.(z) n ➔ oo

n ➔ oo

q

=

i=I

Ihlz)G;(z); i=l

and hence F E .H, as desired. E. Germs of Varieties

In the study of functions of several complex variables, as in the study of functions of a single variable, the set of zeros of a function (or, in higher dimensions, the set of common zeros of many functions) plays a very important role. In one dimension, the geometric situation is particularly simple, since such zero sets must be discrete. In several variables, the situation is far more complicated. Submanifolds are defined as (locally) zero sets of certain functions, but not all zero sets are submanifolds. Certainly the set {z1 z2 = 0} is not a manifold, since the origin separates it into two components. A more serious example is the zero set Z of the function w2 - z1 z2 • The Jacobian matrix for this function is (2w, -z2 , -z1), so at every point except the origin, Z is a submanifold. At the origin the set Z is not even topologically a manifold (that is, no neighborhood is homeomorphic to a four-dimensional cell). For the complement of a point in a four-dimensional cell is simply connected; however, no neighborhood of the origin on Z is simply connected. For there is a map of C 2 - {0, 0} onto Z - (0, 0, 0) which is a two-sheeted covering map: (ti, t2) ----+ Ct1t2, t~, t:); thus the image cannot be simply connected.

86

Local Rings of Holomorphic Functions

Chap. II

We shall make a detailed local study of the zero sets of holomorphic functions in this section and in the next chapter; but first we shall gather together some simple global properties (some of which we have already seen). 1. Definition. -Let U be a domain in en. A subset V of U is a subvariety if for every z in U there are a neighborhood Uz and functions f1 , . . • , ft holomorphic in u., such that V n Uz = {x E Uz; f1(x) = 0, ... , ft(x) = 0} = V(f1 , . . . , ft)2. Theorem. A subvariety V of U is a closed, nowhere dense subset of U. U is connected, then U - V is also connected.

If

Proof: Let V be a subvariety of U. Let Xn E V, and suppose xn converges to x E U. In a suitable neighborhood U., of x there are defined holomorphic functions f 1 , ••• ,ft such that V n U., = {f1 = 0, ... ,f1 = 0}. Thusf;(xn) = 0, for large enough n and allj. By continuity, then,f;(x) = 0, 1 ~j ~ t. Thus x E V. Finally, since Vis a thin set (as in Chapter I, Section C), Vis nowhere dense, and does not disconnect any connected open set (corollary I, C4).

3. Theorem. Let F be a collection offunctions, holomorphic on an open set U. Then V(F)-= {x EU; f(x) = 0 for all f E F} is a subvariety of U. Proof: Let z EU. Let f, be the ideal in n(!}, generated by {f,;f E F}. Since n(!} • is Noetherian, there are gi, ... , gk E f • which generate f •· Since the gt are in the ideal generated by F, there are /; E F and h11 E n(!} • such that gt :"" L ht;f;,z• Thus we may assume that the generators ~ have representatives g; E F. By th~orem D2, we can find a small enough polydisc ~(z; r) such that, if f E (!} A(z;r) and r. E J., then f = L higi in ~(z; r). Thus V(F) n ~(z; r) = V(g1 , ••• , gk) n ~(z; r). For if f E F, then f, E f,; so f = L h;g; in ~(z; r). Hence V(f):::, V(g1 , •.• , gJ. This holds for all f E F, so V(F):::, V(g1, .•• , gk). On the other hand, since the g; are in F, V(g1, •.• ,gk) n ~(z; r) = V(F) n ~(z; r). Thus the theorem is proven.

The notion of a variety has a local version, which we now introduce. 4. Definition. Let X, Y be subsets of en. The sets X and Y are said to be equivalent at 0 if there is a neighborhood U of 0 such that X n U = Y n U. An equivalence class of sets is called the germ of a set. The equivalence class ofX is to be denoted by X.

Let f be the germ of a function at 0. By V(f) we mean the equivalence class of the sets {x E U;f(x) = 0}, where f represents the germ f in the

Sec. E

Germs of Varieties

87

domain U. It is clear that if f 1 , f 2 represent f in the domains U1 , U2 respectively, the sets {x E U 1 ;fi(x) = 0}, {x E U 2 ;fix) = 0} are equivalent. As in the case of germs of a function, the collection of germs of a set inherits algebraic structure from the global objects from which it is derived. Thus, if X1, X2 are germs of a set, X1 U X2 , X1 X2 , X1 - X2 are welldefined germs of a set, and X 1 c X 2 is a well-defined relation on germs. For example, X 1 U X 2 is the equivalence class of sets X 1 U X 2 as X 1 and X 2 run through the equivalence classes X 1 and X 2 • (Notice that for an infinite collection {X"'} of germs of a set, U X« and X« are not necessarily well-defined.)

n

n

5. Definition. Let f on X if X c V(f).

E

/!J,

and let X be the germ of a set. We say f vanishes

6. Definition. A germ X is the germ of a variety ft E n@ such that X = V(f1) n · · · n V(ft),

if there are elements f 1 ,

.•. ,

We shall write V(f1) n · · · n V(ft) as V(f1 , . • . , ft), or as {f1 = 0, ... , ft= 0}. We shall refer to the collection of germs at Oat To recapitulate, X is the germ of a variety at 0 if there are a neighborhood U of 0 and functions Ji, ... ,ft holomorphic in U, such that {x E U;J;(x) = 0, 1 :S: i ::;; t} is a representative for X.

nm.

7. Proposition. Let V, WE

Proof· Let V

=

V(f1 ,

nm.

••• ,

Then V

n w,

=

W(g1,

ft), W

vn w=

V(f1 ,

••• ,

V

u WE nm.

.•. ,

gk). Then

f;, gi, ... , gJ

and

nm.

8. Definition. Let VE The ideal of V is defined to be the set id V = J(V) = {f E n(i); f vanishes on V}. Let d c n(i)· The locus of d is defined to be loc d = V(f), where the intersection is taken over all f E d.

n

9. Proposition (i) id V is an ideal of n@, (ii) loc dis a well-defined germ of a variety.

Proof: (i) If f, g vanish on V, so do f + g and fh, for any h E J!J. (ii) Let J be the ideal generated by d. Since n(i) is Noetherian by theorem B9, J is finitely generated, say by g1, .•• , gt. Then loc d = V(g 1 , . . • , gt). For every element of d vanishes on loc d, thus every element of J vanishes on loc d. In particular, since gi, ... , gt E J, V(g 1 , . . . , gt) ::::, loc d. On the other hand, every element of the ideal generated by g1 , . . • , g1

Local Rings of Holomorphic Functions

88

vanishes on V(gi, ... , gt). But this ideal is just J, and J V(g1, ... , gt).

Chap. II :::>

d, thusloc .sit:::>

10. Definition. If J is an ideal in a ring f!ll, we define the radical of J to be the ideal Rad J = {x E f!/l; there is a positive integer n such that xn E J}. 11. Theorem. The following relations hold between a variety and its ideal, an ideal and its locus: (i) V1 :::> V 2 implies id V1 c id V2 • (ii) V1 =I= V 2 implies id V1 =I= id V2. (iii) id V = Rad id V. (iv) .f1 :::> .f2 implies loc J 1 c loc J 2. (v) loc id V = V. (vi) loc J = loc Rad J. (vii) id loc J :::> Rad J.

Proof: (i) If f vanishes on V1, since V 1 :::> V 2 , f vanishes on V2. Thus id V1 C id V2, (ii) Since V1 =I= V2 , for any representatives V1 of V1, and V2ofV 2, Vi =I= V2 • Let V(/1, ... .ft), V(g1, ... , gk) be representatives of V 1 , V 2 , respectively, in a neighborhood U. Then for every neighborhood W c U of 0,

Thus there is a zw E (Vi - V2) U (V2 - Vi); so there is anfi such that ft V. Now, since V is the germ of variety, we can write V = V(f1 , . . . , f 1). Then f; vanishes on V, so f; Eid V. Thus f; vanishes on loc id V, and V(f;) :::> loc id V. This holds for all i, so that in fact V = V(f1 ) n · · · n V(ft) :::> loc id V, proving (v). (vi) Since J c Rad J, we have loc J :::> loc Rad J. But, if f E Rad J, some rn E J, so rn vanishes on loc J. Thus £vanishes on lac J, so loc Rad J :::> loc.f.

a

Sec. E

Germs of Varieties

89

(vii) Certainly, for any ideal .f, id loc .f => .f. Thus, using (vi),

= id loc Rad .f => Rad .f. VE nm is said to be irreducible

id loc .f

12. Definition. A germ for V1 , V 2 E nm implies that either V

=

V1 or V

=

if V = V1 U V2,

V2 .

13. Theorem. V is irreducible if and only if id V is prime.

Proof: Suppose V = V1 U V2 with V =I=- V1 , V =I=- V 2 • Then id V =I=- id V1, id V =I=- id V2 , by theorem 11. But V => Vi implies id V c id Vi, so there is an element f 1 Eid Vi, f1 ff. id V. Thus f 1f 2 Eid V1 n id V2 = id V. But since neither f 1 nor f2 is in id V, it follows that id V is not prime. Conversely, if id V is not prime, we have f, g E J!J such that fg E id V but f ff. id V, g ff. id V. Then neither f nor g vanishes on V, so V n V(f) =I=- V, V n V(g) =I=- V. But V = [V n V(f)] u [V n V(g)]. For [V n V(f)] u [V n V(g)] = V n [V(f) U V(g)] = V n V(fg) = V, since fg EV. Thus V is not irreducible. 14. Corollary. V(f) is irreducible if and only if!= pn, where p is irreducible.

We now show that any variety has a unique decomposition into a union of irreducible subvarieties, called its irreducible branches. 15. Theorem. Let VE nm. We can write V = V1 u ... u vk, where the V; are irreducible and V; ¢ Vi for i =I=- j. V1, ... , Vk are uniquely determined byV. Proof· Note that germs of varieties satisfy a descending chain condition; that is, ifV 1 E nm and V1 => V2 => V3 . .• , then from some integer k necessarily Vk = V k+i = V k+2 = · · · . For if there were an infinite, strictly descending sequence of germs of varieties Vi, then from theorem 11 (i) and (ii), the ideals id Vi c ,/!J would form an infinite, strictly ascending sequence of ideals in n@, whic\1 cannot happen since J!J is Noetherian by theorem B9. Now suppose there were some germs of varieties which could not be written as the union of finitely many irreducible varieties, and let V be a minimal such variety. Since Vis not irreducible, it can be written in the form V = V1 U V2 for some proper subvarieties V1 , V2 ; but since V was minimal, both V1 and V 2 can be written as finite unions of irreducible varieties, hence so can V, a contradiction. We now verify the uniqueness. Suppose that also V = v~ u · · · u v~, with v; ¢ v; for i =I=- j. Then v; c v 1 u · · · u v., so v; = (V1 n v;) u · · · u (V. n V;). Since v; is irreducible, it follows that v; = V1 n v; for some j = j(i) so v; c V;w· By reversing the roles of the two representations, we obtain for allj, V; c V~c;c;»; but by assumption this implies that k(j(i)) = i. Similarly j(k(i)) = i, thus the correspondence i-+ j(i) is one-one onto. Finally v; C V;(i) C v~(;(i)) = v;, so v; = V;w, and the uniqueness of the representations is verified.

Local Rings of Holomorphic Functions

90

Chap. II

16. Corollary. Let V = V 1 U · • • U Vk be the decomposition of V into its irreducible branches. Then for all i, Vi ¢ U Vi. j s"i

Proof: The proof is in fact contained in the proof of theorem 15.

Among the formal relations between ideals and varieties derived in theorem 11, assertion (vii), that id loc J => Rad J, remains in a clearly unsatisfactory state; we should either show by example that this can be a strict containment, or prove th11-t it is an equality. It is actually true that id loc J = Rad J; this is the celebrated Nullstellensatz or zero-theorem, a result which lies considerably deeper than the other results in this direction which we have discussed thus far. The proof of this theorem requires a more careful analysis of the geometric and analytic properties of varieties and germs of varieties; we shall turn to these matters in the next chapter. At the present, to motivate the later discussion, we shall examine some simpler aspects of the Nullstellensatz. For the case of principal ideals, the theorem can actually be demonstrated without difficulty as follows. For any element f E /!J, let (f) = /!J · f be the ideal generated by fin /!J. 17. Lemma. Let f, g E i!J be relatively prime. We can choose coordinates in a neighborhood of O such that for some A, µ E nm, we have Af + µg = p, where p E n-i(O and p =f=. 0. z 1 , ••• , Zn

Proof: By the Weierstrass preparation theorem we can choose coordinates in a neighborhood of O so that we can write

f=uP,

g=vQ

where u, v are units, and P, Q are Weierstrass polynomials in n-i(O[znl• Then P, Q must be relatively prime (by lemma BS). Let ffe be the quotient field of n- 1(0. Then P, Q E ffe[znl, and are relatively prime. By the Euclidean algorithm (c.f. [I, K]) there are elements h, k E ffe[znl such that hP

+ kQ =

1.

We can write h = '.A'/a, k = µ'/b, where A', µ' zero elements of ,._ 1 (0. Now '.A'P

+ µ'Q =

E

n-i(O[znl and a, bare non-

ab;

and putting A= '.A'u-1 , µ = µ'v-1, p = ab, we obtain the desired result. 18. Theorem (Nullstellensatz for Principal Ideals). Let g E n(O be irreducible. Then id V(g) = (g).

Sec. E

Germs of Varieties

91

Proof· We may assume g =I=- 0. By the Weierstrass preparation theorem we can choose coordinates z1 , • . . , zn so that g = uP, where u is a unit in /!J and Pis a Weierstrass polynomial in n-i(!}[znl• Since u is a unit, V(g) = V(P), and (g) = (P); so we can replace g by P. Now suppose that f Eid V(P). If P is not a factor off, P and fare relatively prime (since P is irreducible). Thus by lemma 17 (we may need to find new coordinates, but we can still assume that P is a Weierstrass polynomial), there are elements A, µ E n@ such that (1)

11.f+ µP=p

is a nonzero element of n- 10. Now choose a polycylindrical neighborhood Ll(0, r) in which (1) is an equation involving functions rather than germs. Since P is a Weierstrass polynomial in zn with coefficients holomorphic in Ll' = Ll(0, (r1 , ••• , rn-i)) c cn-1 , there is, by a proper choice of .!l(0, r) (see lemma I, B3), at least one z~ for every z 0 E .!l' such that (z 0, z~) E .!l

and

P(z 0 , z~)

= 0.

Then also f (z z~) = 0, and, by (1), = 0. But z 0 is any point in .!l', so p == 0 in .!l', contradicting p =I=- 0. Thus P must factor f, so f E (P), and the desired result follows. p(z0 )

0,

To demonstrate the usefulness of even this special case of the Nullstellensatz, we observe the following result. s

19. Theorem. Let f En(!), and let f =

IT pfi be its factorization into irreducible i=l

factors. Then V(f) = V(p1 ) U · · · U V(p8 ) is the decomposition of V into its irreducible branches. Proof: Since Pi is irreducible, the ideal (Pi) generated by Pi is a prime ideal, and loc (pi) = V(pi) is thus irreducible. Now

V(f) = V(Il pf')

=

u V(p;') = u V(P;)

is a decomposition of V(f) into irreducible varieties. If V(p;) c V(p;), then id V(p;) => id V(p;), By theorem 18, then, (P;} => (P;), so P; divides P;, contradicting the assumption that P; is irreducible. Thus for i =I=- j, V(p;) ¢ V(p1), and the V(P;) are the irreducible branches of V(f). _The proof of the Nullstellensatz for any prime ideal is essentially a generalization of the proof of theorem 18. However, the special variable zn is not so easily found; it requires multiple use of the preparation theorem, and the use of the algebraic theorem of the primitive element. Once this is done, the Weierstrass polynomial Pis easily found. We have to replace the Euclidean algorithm by a more subtle argument, but the final analysis is the

92

Local Rings of Holomorphic Functions

Chap. II

same: if a function vanishes on the variety and depends only on some of the variables (the right ones), then it is identically zero. Details of the argument appear in the next chapter. We conclude the present discussion by showing that, actually, it suffices later to prove the Nullstellensatz only for prime ideals. 20. Theorem. If the Nullstellensatz holds for prime ideals, it holds for all ideals; that is, id loc J = Rad J for any ideal J, if id loc fl' = Rad fl' for any prime ideal fl'.

Proof: We can write the irredundant representation for any ideal J of J = .P2 1 n · · · n .P2 8 , where the ideals .PL; are primary, and the ideals Rad .PL;-= fl'; are prime. The prime ideals f/'1 , ••• , fl', are uniquely determined by J. Since loc .PL;= loc fl';, then n(!):

loc J

= loc f/'1 U · · · U loc fl',,

and id loc J = id loc f/' 1

n · · · n id loc

f/' 8 •

By the Nullstellensatz for prime ideals, id loc J

= f/'1 n · · · n fl' = Rad J, 8

thus theorem 20 is proved. Let us now observe that, in view of the Nullstellensatz, the mapping V--+ id V maps nm onto the set of radical ideals inn(!)· For if J = Rad J, then id (loc J) = Rad_J = J. We may summarize this by-saying that the mapping V--+ id V is an antihomomorphism of the lattice nm onto ~he lattice of radical ideals in n(!)· We remark that this is not an anti-isomorphism of nm into the lattice of ideals of n(!) since, in general, id (V1 n V2) =I=- id V1 + id V2• For an example, take V1 = V(x2 - y 3), V 2 = V(x 3 - y 2). Then V 1 n V 2 = {O}, but x ff= id V1 + id V2 • Notes

The Weierstrass preparation and division theorems can be proved by arguing directly with power series, and thus can be extended to formal power series rings over quite general fields; this version of the theorems is of considerable use in algebraic geometry. The advantage of the function-theoretic proof given here is mainly that it simplifies the derivation of the extended form of the division theorem in Section D. Oka's lemma (theorem C3), which is fundamental to the modem development of the subject, occurs implicitly in [186, VII]; see also [64] and [66].

CHAPTER III

VARIETIES

A. The Nullstellensatz for Prime Ideals, and Local Parameterization

In this section we shall finish the proof of the Nullstellensatz. The proof is as important as the theorem itself, for it furnishes us with a very effective way of studying irreducible varieties. Let f!J' be a prime ideal in J!J with f!J'. cf= (0), f!J' cf= J!J (in these special cases, id Joe f!J' = f!J' trivially). Since f!J' is prime, J!J / f!J' is an integral domain. We shall find coordinates z1 , . . . , zn and an integer k such that ,l!J n f!J'·= (0). Then, for the natural homomorphism 7T: n(!)-+ n(!)/ f!J', 1T is an isomorphism of k(!) onto 1T(k{!}). Let nffe" be the quotient field of n(!)/ f!J', and ,? be the quotient field of k(!)· By the choice of the coordinates z1 , .• : , zn, we shall see that [nffe": ,?] < oo, and that n(!)/f!J' is integral over k(!).t 1. Definition. Let f!J' be a prime ideal in n(!)· A coordinate choice Zi, ••• , zn is said to be a regular system of coordinates for the ideal f!J' if the following

are satisfied: (i) f!J' n k(!) = (0) for some k; (ii) n(!)/f!J' is integral over k(!); (iii) nffe" is generated over ~ by the single element

'l'/k+1

=

7T(Zk+ 1).

The integer k is called the dimension of f!J' relative to the coordinate system z,,.. We shall eventually see that this is independent of the coordinate choice, and is the dimension (in a topological sense) of the variety loc !:!'. z1 ,

•.• ,

2. Theorem. Any prime ideal f!J' C n(!) has a regular system of coordinates.

Proof· We first prove by induction on n that, by a linear change of coordinates, we can attain (i) and (ii) above and in addition (iv)

n(!)/ f!J'

is generated over

k(!)

by the elements 11! =

7TZJ,

k

+ 1 :,;;: j

t We use the notation [,.jo°: k.FJ to denote the degree of the extension k.F c 93

,..F.

:,;;: n.

94

Varieties

Chap. III

If PJ' = (0) take k = n; if f!lJ = n0 take k = 0. Thus we may assume throughout that PJ' is a proper prime ideal. In the case n = 0, n0 = k, where (aii) is nonsingular, then z1 , • . . , zk; z~+I• ... , i>k z~ are semiregular coordinates for f!I'. (b) Let z1 , . . . , zk; zk+1, ... , Zn be semiregular coordinates for f!I'. There exists a linear transformation z; = 2 aiizi, for j > k, such that Zi, ... , zk; l>k z~+ 1, ... , z~ is a regular coordinate system for f!I'.

Varieties

112

Chap. III

Proof" (a) If 1r: n(!)--+ n(!)f PJ is the natural homomorphism discussed in theorem A2, then ri; = ~ aurJ; (rJ; = 77(z;)) are integral over /!J. Thus there exi~ a polynomial g; E k(!)[X] such that g;(ri~) = O; and so g;(z~) E PJJ. Notice that, as in lemma A3, the g; are Weierstrass polynomials. (b) We need only verify that conditions (i), (ii), (iv) of theorem A2 are satisfied-by the coordinates z1 , . . • , zk; zk+I• ... , zm and then the existence of the desired linear transformation follows from the primitive element theorem, as in theorem A2. However, (i), (ii), (iv) are obvious. 6. Lemma. Let z1 , n@. Let 77j(z1, ••• ,

loc (PJ

n

i(!))

••• , Zn Zn)

be regular coordinates for the prime ideal PJ in ~ k, 7Ti maps lac PJJ onto

= (z 1, •.• , zi)- Then, for j

properly.

Proof: Let il, ilk, q;, T;, D be an admissible representation for PJ, and let V represent loc PJ in il. Then by theorem AlO, 1rk maps V properly onto ilk. If j > k, let il1 = 1r1(il). Then PJ n /!J is a prime ideal in ;@, and il;, ilk, q;, Tk, (i, k -::;,j), D is an admissible representation of PJJ n ;@. Let V; represent loc (PJ n ;(!)) in il;. Then, for 1r;iz1 , • •• , z,) = (z1 , ••. , zk), 7T;k maps V; properly onto ilk. Since 1rk = 7T;k 77;, it follows that 77; is proper. Now, {f 0 77;;/E PJJ n ;(!)} c PJ; so (shrinking the neighborhoods· if necessary), 77·;1(V;) => V. Thus 77; maps V into V;- Next let (z~, . .. , z']) E V; - V(D). If j = k, let ~+i be a root of qk+l(z~, .. . , z~; X) = 0. In all other cases, define zJ = TtC~+1 )/ D(z~, ... , ~) for i > j. Then 77;(~, ... , z~) = (~, ... , zJ), and (z~, . .. , z~) E V - V(D). Thus 1r (V) ::i V; - V(D). But since 77; is proper on V, 77;(V) is closed. Thus 1r1 maps V onto V;. O

Let us analyze the above lemma in the particular case j = k + l. Then Vk+l = {z E ilk+l; qk+1(z1 , ••• , zk; zk+1) = 0}; 77k+l maps V properly onto Vk+ 1 , and 77k+1: V - V(D)--+ Vk+ 1 - V(D) is a homeomorphism. Thus the other coordinates serve only to separate points over V(D) which may be identified in Vk+ 1 •

n {z1 = 0, ... , is part of a semiregular coordinate system , Zt; Zt+I• •.. , Zn is a semiregular coordinate

7. Theorem. Let PJ be a prime ideal inn(!), and suppose loc PJ Zt

= 0} = {0}. Then

Zt+1, ... , Zn

for PJ. If dim PJJ = t, then z1, system for PJ.

.••

Proof: We prove the theorem by induction on n. The case n = I is trivial. If t = n, there is nothing to prove, so assume t < n. Let V be a representative of loc PJ in a neighborhood of 0. Since V n {z1 = 0, ... , zt = 0} = {0}, there is a function f E PJJ such that f(O, . .. , 0, an):#- 0 for some an:#- 0. Then f is regular in zn, so f = u · P, where Pis a Weierstrass polynomial _with coefficients in n-I(!), and u is a unit. By lemma 6 we also

Dimension

Sec. C

113

have loc (f!lJ n n- 1 0) n {z1 = 0, ... , z 1 = 0} = {O}. We can thus apply the jnduction assumption to find new coordinates zi, ... , z; such that Zi, ... , z;, z1+1, .•. , zn-l are semiregular coordinates for f!1i n n- 1 0. Let k = dim 0 V. Of course, if t = k, we will not have to change the z;'s for i :,;; t. Now, zn-v . .. , zt+l are integral over k(P (mod g'J), by condition (ii) of definition 4; ~ zn is integral over n-i0 (mod g'J); so there is also an element gn E k0[znl n f!lJ. Thus the theorem is proved. Notice that the g/s can be taken to be Weierstrass polynomials. The following corollaries are trivial. 8. Corollary. Let V be an irreducible variety at O in en, ·and let H be a hyperplane of dimension t through Osuch that V n H = {O}. Then dim 0 V :,;: n - t. 9. Corollary. Suppose V is an irreducible variety at O of dimension k, and H is a hyperplane through 0 of dimension n - k, and V n H = {0}. Then

any coordinate system z1, ••• , semiregular coordinate system.

Zn

such that H = {z1 = 0, ... , zk = 0} is a

IffE /!J, it is clear that dim 0 V(f) < n unless f = O; in fact, we shall see that if f is not invertible, nor identically zero, then dim 0 V(f) = n - l. Indeed it is true on any irreducible variety that the null· set of a nonzero function is a subvariety of codimension one in the variety. In order to prove this, we must introduce the concept of dimension of a reducible variety. 10. Definition. Let V be the germ of a variety at O in en. Let V = V1 u ... U Vt be the decomposition of V into irreducible branches. The dimension of V is de.fined to be dim V = max dim V;1 :Si:St

We shall say that Vis pure dimensional if all the branches ofV are of the same dimension. 11. Theorem. (a) Let f E n0, and suppose that f(0) = 0 but f -:f::. 0. Then V(f) is of pure dimension n - 1. (b) If V is a germ of a variety of pure dimension n - I, then f(V) is a principal ideal.

Proof (a) Write f = u · II Pt', where the p, are irreducible and u is a unit. Then V(f) = U V(p;), and the V(p;) are the irreducible branches of V(f). We saw in the proof of the Nullstel/ensatz for principal ideals (theorem II, EIS), that coordinates can be chosen so that (p,) n n-i0 = (0). Thus, dim V(p;) = n - l.

Chap. III

Varieties

114

(b) Let V be a variety of pure dimension n - I, and let V = V 1 u · · · u V, be its decomposition into irreducible branches. Now, for each i, we can find coordinates z1 , •.• , zn_ 1 , zn which are regular for V;. If q~ is the Weierstrass polynomial found for zn then q~ is irreducible, so id V(q~) = (q~)Now, iff vanishes on V, then f vanishes on V;; so f vanishes on V(qn). Thus q~ divides f. This holds for all i, so II q~ divides f; hence J(V) = (II q~). i

i

Theorem l l(a) continues to hold on arbitrary analytic varieties. 12. Lemma. Let V be a variety at O in en of dimension k. There exist coordinates z1, ••. , Zn such that V n {z1 = 0, ... , zk = O} = {O}. Proof: The point is that we can treat simultaneously all the branches of V. Let f Eid V, f =fa 0. Let Zi, • .• , zn_ 1 ; zn be coordinates such that f is regular in zn. Then V(f) n {z1 = 0, ... , zn-l = 0} = {O}, so certainly tqis is true for V. Let V' = Joe (id V n n-i{!)). As in lemma 6, ?Tn_ 1 : V---+ V' is light and dim V' = k. By induction, we may find new coordinates z1 , • . . , zn·-l so that V' n {z1 = 0, ... , zk ~ 0} = (1Tn_1)-1 (0) is discrete, so (as a germ at 0) is just {O}.

13. Lemma. Let F = {f1 , . . . , ft} be holomorphic in a neighborhood U of the origin in en. Let Lx(F) = {z EU; F(z) = F(x)}for all x EU; and suppose dim Lo(F) = k. Then, in some neighborhood W of 0, dimx Lx(F) :::;: k for all xEW. Proof- By lemma 12, we can find coordinates z1 , small enough e > 0,

.•• ,

zn such that, for

n {!1 = 0, ... ,ft = 0, z 1 = 0, ... , zk = 0} = {0}. In particular, this set is disjoint from {z E en; lz,I = e, i > k}. Thus, by' continuity, there is an 'Y/ > 0 such that ~(0; e)

~(0; e)

n {lz;I = n

ti

IJ;I

e, i

>

k}

+ 11z;I < (t + k)rJ} =

Let W be a neighborhood of 0, W c U, so that 1 :::;: i :::;; k, in W. Then, for x E W, the variety L.,(F)

n {z1 = x1, ••• , zk =

0.

I.Iii
6(V). Then V - A is connected and dense in V.

Proof' Choose representatives V, A of V and A, defined in a polycylinder L\, such that L\, L\k are admissible for V. Let C be a component of V - A. Then (C n L\, 1r, L\k) is an analytic cover. Thus, by theorem Bl8, C n L\

Varieties

116

Chap. III

is a subvariety of Vin ~- But C n ~ contains C, an open subset of V; so that C n ~ is not a proper subvariety of V, and hence C n ~ = V. ·

17. Theorem. Let V be a variety at O in en. Let S be a connected germ of an open set in 9?(V). Then {f E n(!;) 0 : f vanishes on S} is a prime ideal in n(!;}· Proof· This is trivial.

18. Corollary. Let V be a variety at O in

en.

If Vis regular, it is irreducible.

19. Theorem. Let V be a variety at O in en. Let 9?(V) = U S; be the decomposition of 9?(V) into connected components. Then V = U 81 is the decomposition ofV into irreducible branches. Proof" Let V1 , . . . , V, be the irreducible branches of V. Observe that 9?(V) = U (V, - A,), where A,= 6(Vi) u (UV, ri V;)- A, is contained i*i

i

in a proper subvariety of V, (since 6(V,) is proper, and Vi ¢ UV;)- Thus, Ni

by theorem 16, V; - A, is connected and dense in V;. If S;; = (V; - A;) u (V1 - A1) is connected, then by theorem 17, {f E n(!;) 0 ; f vanishes on S;;} is prime. But this ideal is just id V; ri id V1. Thus the V; - A; are just the components of 9?(V). We remark, for future use, that in the above theorem we can replace 6(V) by any nowhere dense subvariety A such that V' => 6(V). The proof is the same:

20. Theorem. Let V be a variety at O in

en.

Let A be a nowhere dense sub-

variety ofV such that A::> 6(V). Let V - A=

.t

U W; be the decomposition i=l

of V - A into connected components. Then V = ofV into irreducible branches.

t -

U W; is the decomposition i=l

Now let V be a subvariety of a domain U. By the preceding local analysis we see that 9?(V) is dense in V; and if Wis a component of 9?(V) that W is a subvariety of V. The closures of the components of 9?( V) are defined to be the (global) irreducible branches of V. In particular, V is irreducible if and only if 9?(V) is connected. This generalizes the local concept. We have the following decomposition by dimension:

21. Theorem. Let V be a subvariety ofU. Let Wk= {z EV; Vx has a branch of dimension k at x}. Then Wk is a subvariety of U of pure dimension k.

Notes

117

Proof· Wk is the closure of the points in 9l(V) of dimension k, and thus is the closure of a union of components of 9{( V). 22. Corollary.

wk= {x EV; dimx V 2 k} is a subvariety of u. Notes

The proof of the Nullstellensatz developed here is due to Rilckert [207]. The resulting local parametrization is also due to Rilckert, although the proof of theorem AIO is developed from ideas of Bishop [35]. The concept of an analytic cover was introduced by Behnke and Stein [22] in order to generalize to higher dimensions the idea of a Riemann surface as an "analytic configuration." As noted in the text, Grauert and Remmert [111] proved that analytic covers are analytic varieties. The systematic use of the concept of analytic covers, and, in particular, the use of arguments involving symmetric polynomials, in the study of global problems on analytic varieties, was exploited with notable success by Bishop [33, 35]. Some of these uses have appeared in this chapter; others will appear in Chapter V. We wish to draw special attention to theorem Bl 9, which will be used to simplify substantially many of the arguments dealing with varieties. Theorem B22 is a generalization of a theorem due to Oka [186, VIII] (Cartan [69]), which is of special relevance in dealing with the normalization of a variety. We have still another use for it (theorei:n V, BS). The normalization of a variety V can be defined as follows: it is a pair (W, p) where Wis a variety, pis a holomorphic, light, proper mapping of W onto V, such that p: W - p-1 (6( V))---+ V 6(V) is biholomorphic, and, finally, w/1Jw is the integral closure of vm.,. The

z

p(w)=x

first proof of the existence of the normalization was given by Oka [186, VIII]. There are now several easier proofs [l, 153].

CHAPTER IV

ANALYTIC SHEAVES

A. The Elementary Properties of Sheavest

In the last two chapters our interest has centered principally on local questions, on germs of functions and germs of varieties at a point. In pursuing our investigations further, we are led to the problem of amalgamating these local results into global statements; and questions arise about such points as the relationships between the ring (!Ju of functions holomorphic in an open set U c 0. Let D be a branch of L.,(B V {g}) of dimension k. Since L.,(B) is of dimension k at x, and D c L.,(B), D must be a branch of L.,(B). Then, there is a point y ED near x such that L.,(B) 11 = D 11 = L.,(B V {g})11 • Since dim11 D > 0, either dim D n Vi> 0 or dim D n V2 > 0 or there is a j such that dim D n W1 > 0 in a neighborhood of y. For otherwise D = (D n V1) v (D n V2) VU (D n WJ is a countable set. But since j

L.,(B) = L.,(B V {g}) = D in a neighborhood of y, g fails to cut down the dimension of a positive-dimensional level set of B on Vi or V2 or some W1, whichever it is which intersects D in a positive-dimensional set. Thus g is not in one of t;g1 , t;g 2, or t;gi for some j. Thus W is of second category. 4. Theorem. Let (X, xl!J) be an analytic space of pure dimension d. Let be a Frechet algebra of holomorphic functions on X such that the injection i: ff (!)x is continuous. Suppose X is ff-complete. The set W of g E §d such that Xis g-complete is of second category in §d.

§

Proof" Let f = (/1, ••• ,fd) E § 0. We may suppose that f(x 0 ) = 0, and since f,, - f,,(xk) also converges uniformly to f, that f,,(xk) = 0. Now, if dim., 0 L.,.(f) = 0, there is (in a suitable choice of local coordinates) a polydisc Li such that f never vanishes on oLi. Then, for k large enough, f,, never vanishes on oLi, so V(fk) n Li is a compact variety and thus of dimension zero. This contradicts Thus/E§n• Thus t:g is dense, and its complement is a countable union of closed sets, so t:g is of second category. This theorem is fundamental in the study of the global theory of analytic spaces, and is the first step in an attempt to embed analytic spaces in en. A different application is the following extension theorem on analytic subvarieties: 5. Theorem (Remmert-Stein). Let V be a subvariety of the polydisc D in cm. Suppose W is an irreducible subvariety of D - V, of dimension n. If n > dim V, then the closure W of W in D is an irreducible subvariety of dimension n.

Proof· Let x E W. We have to show that there is a neighborhood Li of x such that W n Li is a subvariety of Li. If x ff- V, there is nothing to prove. If x E V, let W~ be the germ of a variety of pure dimension n such that W~ ::::i V., (such a one is easily found). Choose a representative W' in a neighborhood U of x such that W'::::, V n U. We now restrict attention to U. Consider the disjoint union w 0 of W' and Was an analytic space of pure dimension n. The algebra § = 0u/J, where J is the ideal of functions holomorphic on U which vanish on WU W', is a Frechet algebra of functions holomorphic on w 0 in a topology stronger than that induced by 0W". Clearly w 0 is §-complete. Then there is F =(Ji, ... ,fn) E 0"& such that L(F) n w 0 is 0-dimensional, for all p E w0 • We may assume F(x) = 0. Now, we return to W = WU W' in U. There is a polydisc Li(x; e) such that Fis not zero on oLi(x; e) n W. For if not, L.,(F) n w0 would be uncountable, and hence not a zero-dimensional variety. Let 'Y/ < inf {IF(p)I; p E oLi(x; e)}. Let Li= Li(x; e) n F- 1(Li(0; 'Y/)). Then

F: W n Li--+ Li(0; 'YJ) is proper. In particular, F: V n Li---+ Li(0; 'Y/) is proper, so F(V) is a subvariety of Li(0; 'YJ). Since dim V < n, dim F(V) < n also, and p- 1F(V) is a subvariety of w0 of dimension less than n; so W1 = W - P-1F(V) is

170

Analytic Spaces

Chap. V

dense in W n 6., and F: W1 - - 6.1 = 6.(0; 'Y/) - F(V) is a proper map. Thus F(6(W1)) is a subvariety of W1 of dimension less than n, and as in the above, W2 = W1 - P-1 F(6(W1 )) is dense in W n 6., and F: W2 - - 6. 2 = 6. 1 - F(6(W1)) is proper. Let 6(F) = {x E W2 ; rank ,,F < n}; again F(6(F)) is a subvariety of 6. 2, and W3 = W2 - P-1F(6(F)) is dense in W n 6.. Thus A = F(6(F)) u F(6(W1)) u F(V) is negligible in 6.(0; 'Y/), and F: W3 --(6.(0; 'Y/) - A) is proper and nonsingular, and thus a covering map. Since Wis connected, all the hypotheses of theorem III, Bl9, are verified, so W n 6. is an analytic subvariety of 6. and the theorem is proved. 6. Corollary. Let X be an analytic space, and suppose that Y is an analytic subvariety of X, and W is an analytic subvariety of X - Y. If W has finitely many branches, and the dimension of every branch of W is greater than dim Y, then W is an analytic subvariety ofX.

Proof: Since the problem is a local one, we may consider X embedded in a polydisc D in en. By the theorem, the closure of every branch W, of W is a variety. Since W = U W, (the union is finite), W is a variety in D, and thus a subvariety in X.

Finally, we use the above theorem to prove that every subvariety of projective space is a projective variety. 7. Theorem (Chow). Let z 0 , ••• , zn be homogeneous coordinates for pn. Let V be a subvariety of f!Jn. Then there are homogenf!ous polynomials fi, ... , ft such that V = V(f1 , ••• , ft). Proof- The obvious mapping 77; en+i - {0}-- pn has rank n everywhere, so 7T-1 (V) is a variety in en+1 - {0} of dimension at least 1 everywhere. Since Vis compact, it has finitely many branches, thus also 7T-1 (V) has finitely many branches. Thus, by corollary 6, V' = 1T-1(V) u {0} is a 00

subvariety of

en. Suppose g E n+l(90 vanishes on v~. Expand g = Ign n=l

into a series of homogeneous polynomials. Then for any z, and t Ee, 00

g(tz) =

I

giz)tn. If z

E

V', then g(tz) vanishes identically in t. But then

n=l

all the coefficients are zero, so gn(z) = 0. Thus every gn vanishes on V~. Let V~ = V(g1, ... , g"), and expand each gi in a series of homogeneous 00

polynomials, gi =

I

g~. Then V~

n=l

= V(g~), and since

n+1(9o

is Noetherian,

there are finitely many homogeneous polynomials Ji, ... ,ft (selected from the g!) such that V~ = V(fi, ... , ft). Then, clearly V = V(f;_, ... ,ft).

Notes

171

Notes The definition of ringed space in Section A should be compared with the more general notion due to Grothendieck (see [113]). It would not have suited our purposes in this chapter to introduce this more general concept ~f function. For a more detailed exposition of complex manifolds, and of differential geometric methods on these, the reader is referred to A. Weil [243]. The discussion on tangent spaces to analytic spaces is also not as full as it could be, and serves only as an introduction. The reader is urged to consult the paper of Whitney [250] which discusses more geometric candidates for the tangential structure and has many interesting examples. Theorem BS is due to Grauert and Remmert [111]; the proper mapping theorem is due to Remmert [189, 191], but the proofs in Sections Band C differ from the original pro.ofs. Theorem D4 is due to Grauert [99], but the present proof is an extraction from the proof of a more general theorem of Bishop [33]. The original proof of the Remmert-Stein theorem (theorem DS) can be found in [193]; there one finds a somewhat different local parametrization of analytic spaces. Our proof of theorem DS is due to Bishop. The original proof of theorem D7 (Chow's theorem) is in [77]; the enormous simplification presented here is due to Remmert and Stein [193].

CHAPTER VI

COHOMOLOGY THEORY

Suppose given over a topological space D an exact sequence of sheaves )---+ HQ(D, (!)i>-1)-+ HH 1(D, 0). Since Hq(D, 0) = HH1(D, 0) = 0, then Hq(D, (!Ji>)= HQ(D, (!)•H), and hence clearly Hq(D, (!Ji>)= 0 whenever q 2 I, p 2 l. Now, assuming that the theorem holds for all terminating chains of syzygies of lengths ::=; r - I, the exact sequence (3) can be reduced to the two shorter exact sequences µr

µ.

µ1

O---+ {!)Pr---+ {!)Pr-1---+ • • • ---+ (!)1>1---+ PJt---+ 0, (4) Q -+

PJl ---+ {!)Po

---':.+ !f' ---+ 0'

where Pit is by definition the kernel of µ. By the induction hypothesis, Hq(D, Pit)= 0 for all q 2 I. Since Hq(D, (!)i> 0 ) = 0 for all q 2 I, by the preceding paragraph, the exact cohomology sequence corresponding to the second exact sequence of sheaves in (4) shows that Hq(D, !7) ~ HH1(D, Pit) for all q 2 I; and hence Hq(D, !7) = 0, as desired. 5. Corollary. If D c en is a simply connected polydomain, not necessarily having compact closure, then to an exact sequence of sheaves of the form (3) there corresponds an exact sequence of groups of sections of the form

(5)

µr

µ,

0---+ r(D, {!)Pr)---+· · · ---+ r(D, (!JP,)---+ r(D, {!)Po) ~ r(D, !7) ---+ 0.

Proof" The proof is again by induction on the length r of the chain of syzygies (3). When r = 0, the result holds trivially; so assume that the theorem holds for all terminating chains of syzygies of lengths ::=; r - I, and decompose the exact sequence (3) into the two shorter exact sequences (4). By the induction hypothesis applied to the first exact sequence in (4), there is an exact sequence of sections (6)

and by the exact cohomology sequence corresponding to the second exact sheaf sequence in (4), the following sequence is also exact: µ

(7) 0---+ r(D, Pit)---+ r(D, {!)Po)---+ r(D, !7)---+ H 1(D, Pit)---+ · · · . From theorem 4, however, H 1(D, Pit)= O; and then, combining (6) and (7), it follows that (5) is exact, as desired. D. Leray's Theorem on Cohomology

The technique of Dolbeault, using a resolution by differential forms, provided one method for calculating the cohomology groups of a manifold with coefficients in the sheaf of germs of holomorphic functions. We shall

Sec. D

Leray's Theorem on Cohomology

187

now discuss a result of Leray which will furnish a different method of calculating these cohomology groups. This method may be considered as an introduction to Cech cohomology theory as well, although we shall not pursue that matter further. Let D be a paracompact Hausdorff topological space, and U = {U} be a covering of D by open subsets. To this covering of D there is associated a simplicial complex called the nerve of the covering U, N(U), which is defined as follows. A p-simplex a E N(U) is an ordered (p + !)-tuple of open sets of the covering U, which have nonvacuous intersection. If a = ( U0 , ••• , U.,) E N(U) is such a simplex, its support is the nonempty open set Jal = U0 n U1 n · · · n U., c D. Let!/ be a sheaf of abelian groups over the space D. A p-cochain of N(U) with coefficients in the sheaf!/ is a function f which associates to each p-simplex a E N(U) a section f(a) E r(Jal, !/). Since the set of sections is an abelian group, the set of all p-cochains form an abelian group cv(N(U), !/). There is then an operatorb.,: CP(N(U), ! / ) CP+l(N(U), !/), defined for all dimensions p :2'. 0 and called the coboundary operator, which has the following form. If/ E CP(N(U), !/) and a= (U0, ••• , U.,+1) E N(U), then b.,f E CP+1(N(U), !/) has the value 'lJ+l

(bJ)(a)

(1)

= 2,(-l)1r f(U 0, 11

••• ,

U;-1, U;+ 1, ... , U.,+1),

i=O

where r 11 denotes the restriction of sections to the open set Jal. 1. Lemma.

bP+tbP

= 0,for all indices p ::2:: 0.

Proof· If f E CP(N(U), !/) and if a= (U0 , clearly

(b.,+ib.,f)(a) v+2 = 2,(-l)1ra(b.,f)(U0 , i=O

••• ,

••• ,

U.,+ 2) E N(U), then

U 1_ 1 , U;+ 1, ... , U.,+2) -

P+2

=

2. (-1);+1cr

11

f(Uo, ... , U1c-1, U7c+1, ... , U;-1, U;+1,. • •, U.,+2)

i,k=O

le~ Yi U' -o (13)

;_,,. i t

ii

.l.o j y

y

o-c01 U")1>m-•• ·-c01 U")1>-YI u"-o Vm

Vi

P

Am are isomorphisms defined over the subset U = U' n U", such that (13) becomes a commutative diagram when restricted to U. Each homomorphism A; will be represented by a nonsingular, holomorphic, matrix-valued function F;(z) defined over U. By Cartan's lemma, there are holomorphic, nonsingular, matrix-valued functions F;(z) and F;(z) defined in K' and K" respectively, such that F;(z) = P;'(zr1F;(zr1 for all Z EK. These matrices in turn define isomorphisms A;: (0 K')1>• (0 K')1>• and A;: (0 K")1>• (0 K")1>•, such that A;= (A;)-1 (A;)-1 when restricted to K. Altogether, we

I

I

I

I

Sec. F

Amalgamation of Syzygies

205

can write out the diagram (14) of exact sequences. The mappings J. 0 ,

I

µ,',,.

µ,~

I

••• ,

Am

I

µ,'

0 - - ({!) K')'Pm - - · · · - - ({!) K')'P - - .'7 K' - - 0

~!

I

0 - - ({!) K')'Pm (14)

~

~!

'!

•••~({!)I K')'P-':.+ .'7 J K' - - 0

Am,),

Aot

i

~

o--cm I K")'Pm~ ···~({!)I K")'P ~ .,al of(p, q) such thatf(p') c/= f(q') for all (p', q') E N,i>,a>· Cover V - N with finitely many such neighborhoods and let f r+l• .•• , f. be the associated functions. Then, for any p, q E 0, pc/= q, there is an /j such that f;(p) c/= f;(q). Thus the functions f 1 , ••• ,.f. embed U as a subvariety of e•. We now wish to add more functions so as to make the mapping proper. First, since O is compact, we may

Chap. VII

Stein Spaces, Geometric Theory

212

multiply each fj by a constant so that llf;II u ::;; I, and none of the desired properties is destroyed. For p Eau, since p if= K, there is an /E 0x such that 1/(p)I > 11/IIK• By modifying/ by a constant we may assume 1/(p)I > I > 11/IIK- By continuity there is a neighborhood U'P of p such that 1/(q)I > 1 for all q E U'P. Cover aU by finitely many such neighborhoods, and let f ,+1, ... ,ft be the associated functions. Then W = {q EU; 1/;(q)I < I, I :::;;J ::;; t} is the required neighborhood, and cp = (/1 , . . • ,ft) is the required mapping. Surely cp is biholomorphic on W; we need only verify that cp: W---+ .l(0, l) is proper. Let Sc .l(0, l) be a compact set. Then qi-1(S) n W = cp-1 (S) n U = qi-1(S) n V; for if p Eau, there is an/; such that IJ;(p)I > l, so qi(p) ff= .l(0, 1). But qi-1(S) is closed, and Vis compact, so qi-1(S) n Vis compact. 4. Corollary. If (X, x(!}) is a Stein space, there are Oka-Weil domains W n 00

defined by global functions such that Wn

U Wn,

Wn+i, and X =

c

n=l

Proof Since X has a countable topology and is locally compact, we can 00

U

write X =

Kn, with Kn compact and Kn

c

Kn+l• We shall now choose

n=l

by induction a sequence Wn satisfying the requirements. Let W1 be an OkaWeil domain containing K1 . Suppose W1 , .•• , Wn-l are chosen so that K1_ 1 U W 1_ 1 c W 1 for j ::;; n - I. Since Wn-l 0 Kn-l is compact, its holomorphically convex hull is also compact, and so has a neighborhood Wn which is an Oka-Weil domain. Then Wn :::, Wn-l u Kn-i• Thus the sequence 00

{Wn} can be chosen. quence.

00

U Wn:::, U Kn-i =

n=l

X, so {Wn} is the desired se-

n=l

5. Lemma. Let V be a closed subvariety of the polydisc .l(0, r). Let r' < r, and let f be a holomorphic function on V in Li(O, r'). There is a function F holomorphic on Li(O, r') such that F = f on V. Proof Let J be the sheaf of ideals of V. Then v0 n(!}/J I V. Since f E H 0(Li(O, r') n V, v0), f defines a section a of n(!}/J on ii(O, r'), by extending it to be zero off of V. Now, the foilowing sequence is exact:

~

q

o-J-n(!}-n(!)f.f---+0,

so the corresponding cohomology sequence is also exact: 0---+ H 0(.l(0, r'), .f)---+ H 0 (.l(0, r'), n(!})

..!!._. H 0(.l(0, r'), n(!}f_f)

---+ H1( .l(0, r'),

J) ---+ · · · .

By theorem VI, F6, H1(.l(0, r'), J) = 0, so the mapping q is surjective. Let FE n(!} f. Thus, if x EK - L, lg(x)I > llgllL, so x ¢L. Thus L =Lis holomorphically convex.

9. Corollary. Let (X, x(!)) be a Stein space and Y an open subset of X. The following statements are equivalent:

I

(1) (Y, x(!) Y) is a Stein space, and 0x is dense in 0y on Y; (2) if K is a compact subset oJY, K(0y, Y) = K(0x, X); (3) Y is 0x-convex.

Proof' Assume (I) holds. Let K be a compact subset of Y. First, we prove that K(0x, Y) = K(0y, Y). Since 0y => 0x Y, K(0x, Y) => K(0y, Y). Suppose p E Y, and there is an/E 0y such that 1/(p)I > 11/llx, Since 0x is dense in 0y, there is a g E 0x close enough to f on Ku {p} so that also lg(p)I > llgllx, Thus K(0y, Y) = K(0x, Y). Since Y is a Stein space, it follows that K(0x, Y) is compact, and thus an open and closed subset of K(0x, X). Then by the above corollary, K(0x, Y) is 0x-convex in X, and thus K(0x, Y) = K(0x, X). Thus (I) implies (2). Suppose (2) holds. Let Kbe com_pact in Y. Then K(0x, X) is a compact subset of Y and thus K(0x, X) = K(0x, Y). Thus for all compact sets K, also Y is 0 x-convex. Finally (3) implies (1). Suppose (3) holds. If K is a compact subset of Y then K((!)x, Y) is compact. But K(0x, Y) => K(0y, Y), so K(0y, Y) is compact. Thus Y is holomorphically convex, and since it is a subdomain of a Stein space, Y is also a Stein space. Now, for K compact, K(0x, Y) is a compact subset of Y, and so is an open arid closed subset of K(0x, X). Thus again by the previous corollary, K(0x, Y) = K(0x, X) => K. Thus if f is holomorphic on Y, by theorem 6',Jis approximable in ll·llx by functions in @x, Thus @xis dense in (!)y.

I

Sec. B

Special Analytic Polyhedra

215

10. Theorem. Suppose (X, x0) is an analytic space, and X = U X; with X 1 c:: X;+1• Suppose (X;, x0 IX;) is a Stein space, and is 0x1+1-convex. Then Xis a Stein space. Proof First, we prove for every i, that 0x IX; is dense in @x,· Let K 0 be a compact subset of X;, Choose compact sets K1 c X;+ 1 such that Ki+ 1 ::i K; ::i K 0 for all j > 0 and X = U K 1• Let lo E 0 x . By induction, we now choose a sequence {f;} such that J; E 0x;+; and II.ft - J;_1 IIK;-i < 2-ie. Choose 11 so that Ji. E 0x1+ 1 and 11/i - lllKo < e/2. If Ji., ... ,fk are already chosen, since 0x.i+k+l I X;+k is dense in 0 x i+k , there is an J'1~ 0, such that 11/;lln :::::; I - 'f/, I :::::; i :::::; k. Let D 0 = oD 0 = B. Now, Sn Vis a compact subset of D n V, so 'fJ > 2e > 0, such that 11/;llsr,v < I - 2e, l :::::; i ::;;; k. {/1 c/= 0} is $'-complete, and k - l n, the set of (k - !)tuples in § k - l which give light maps on U n {/1 c/= 0} is dense in §'k-l by theorem V, D4. Thus we can find (h 2, ••• , hk) E § 1,-1 with llh;llv < e, such

Proof: Let U sµch that there is an 'f/, D - V. Then there is an e, Since U n

av= B u

z

that (h 2 + fd1- 1 , ••• , hk + fd1- 1 ) is light on U n {/1 c/= 0}. Define g 1 = f 1 , g; = (1 - e)-1 [h;/1 + f;] for i > I, and let Q = D 0 u {p E V, !g;(p)I < I, l :::::; i ::;;; k}. Then Q has the desired properties. First, Q c D. For suppose p ff= D. Then, if p ff= V, we have p ff= D 0 , so also p ¢ Q. Thus we may assume p EV - D. Then there is an i such that lf;(p)I z I. If i = l, it follows that p ¢ Q. For i > 2 we compute !g;(p)I: !g;(p)I = (1 - e)-1 1/;(p) + h;(p)fi(p)I ;:::: (I - e)-1 (1 - lh;(p)l lfi(p)I) z (I - e)-1 (1 - e) = I. Thus p ¢ D implies p ¢ Q, so Q c D.

217

Special Analytic Polyhedra

Sec. B

Now S c Q. In fact, if p ff D and 1/;(p)I < l - 2e for all i, then p For then lg;(p)I = (I - e)-1 lf;(p) h;(p)fi(p)I

E

Q.

+

+ e(l

::;;; (1 - e)-1 ((1 - 2e)

- 2e)) ::S:: I.

Also, Q is a polyhedral domain with frame (V, g 1 , ••• , g1c)- For oQ c V. If p EB, 1/;(p)I < l - 'YJ < l - 2e for all i, sop E Q. On the other hand, oQ c lJ, so oQ n B' = 0. Thus oQ is a compact subset of V. Finally g;g1\ l ::S:: i ::S:: k, gives a light map on U n {g1 =I- O}, since

g;g11 = (1 - e)-l(h;

+ fJ-;1).

2. Lemma. Let (X, x0) be a pure n-dimensional analytic space and D a prepared polyhedral domain with frame (U, f1, . : • , fk). Let V be a relatively compact neighborhood of oD contained in U, so that IIJ; II av l"'ID < r-1, 1 ::S:: i ::S:: k, for some r > 1. Let QN be the union of the components of

V; l(rf;(p))N - (rfi(p))NI < 1, 2 ::S:: i ::S:: k} which intersect D - V. Then RN= (D - V) u QN is a polyhedral domain {p

E

with frame (D n V, (rf2)N - (rf1)N, ... , (rfk)N - (rf1)N) contained in D, if N is sufficiently large. Proof" oRN c (oV n D) u (lJ n V) in any case. Since llr/;llavl"'ln < 1, if N is large enough, ll(r/;)N - (rJ;Yllav/"'\D < 1, so oRN c lJ n V. Thus, we need only verify that RN n oD = 0 for N large enough. Suppose this were false. Let c be such that 1 > c > r- 1, and let V0 = V - {p E V; lf;(p)I < c, 1 ::S:: i::;;; k}. Let B = oV0 n D. By assumption, there are connected components TN of QN n lJ n V0 such that TN n oD =I- 0 and TN n B =I- 0 for arbitrarily large N. TN is compact. If x E TN, there is an i such that lf;(x)I > c. Then lrNf1(x)I ~ (cr)N - 1 ~ 2rrN for sufficiently large N. In particular f 1(x)

l(J;/f1)N -

=I- 0 on TN,

and

11 = l(J;r)-NI

lrNf f - rNJ{I ::;;; ((rc).v - 1)-1 :::::; (27TNr1

for x E TN . Thus for N suffici_ently large, there are disjoint circular neighborhoods Df of the Nth roots of unity such that

LJ

f;(x)E Df fi(x) i=l for x E TN. Since TNis connected, there is for each Nan index j(N, i) such that (J;/f1 )(TN) c Df(N). Let 'YJcN,i) be the corresponding root of unity. Choose a sequence Nm so that for each i the sequence 'YJ converges, say to 'YJ;• By further refining the sequence {Nm} we may also assume that the

218

Stein Spaces, Geometric Theory

Chap. VII

sets TNm converge to a compact set T (in the compact metrizable space of compact subsets of P). Tis again connected, and T n oD -=I=- 0, T n B -=l=0. Since B n oD = 0, T must have uncountably many points. But f;/f1 is constantly equal to 'YJ; on T, so, since D is prepared, the set T n (!1 -=I=- 0) is finite. But for x E T, some lf;(x)I > c > 0, so also fi(x) -=I=- 0. Thus T n {Ji -=I=- 0} = T is countable. Thus we have a contradiction, so the lemma must be valid. · 3. Theorem. Let (X, xl!J) be a pure n-dimensional analytic space. Let D be a polyhedral domain with frame (V, f1 , ••• , fk) such that f1 , ••• , fk belong to a Frechet algebra ffe of holomorphic functions on V such that the mapping ffe mu is continuous. Suppose that V is ffe-complete. Then, for any closed subset S of D such that S n oD = 0, there are g1, ••• , gn E ffe such that (V n D, g1 , ••• , gn) is a frame for a polyhedral domain Q such that s C QC D.

Proof" The proof is by induction on k - n. If k - n = 0 we are through. Suppose now k > n. By lemma I, there is a prepared polyhedral domain Q1 with frame in ffe such that S c Q1 c D. By lemma 2 there is a polyhedral domain Q 2 defined by k - l functions in ffe such that Sc Q 2 c Q1 • By the induction hypothesis there is a Q of the desired type such that SC QC Q 2 • 4. Corollary. Let (X, xl!J) be a pure dimensional Stein space and let P be an

analytic polyhedron in X. Let S be a compact subset of P. There is a special analytic polyhedron Q such that S c Q c P. If P is defined by global functions, so also is Q. Proof" Let P = {lf;I < I, I :::;; i :::;; k}, where the f; are holomorphic in an open set Uc X, U ::> P. The mapping (f1 , ••• ,fk) of Pinto L\(0; I) c (Ck is proper, so (!1 , ••• ,fk) has compact level sets. Since Xis Stein, these level sets are zero-dimensional. Let ffe be the Frechet subalgebra of l!Ju generated by f 1 , ••• ,fk. Then ffe has zero-dimensional level sets. Thus, since P is a polyhedral domain, there are g 1, ••• , gn (n = dim X) which are in ffe, and which are a frame for a polyhedral domain Q such that S c Q c P. In particular, Q is compact, and since the g; are holomorphic on all of P, by the maximum principle, Q = {p EP; \g;(p)I < I, I :::;; i:::;; n}. Thus Q is a special analytic polyhedron. 5. Corollary. If(X, xl!J) is a pure dimensional Stein space, and K is a compact subset of X, there is a special analytic polyhedron P such that K c P c X.

If (X, xl!J) is a pure dimensional Stein space, there is a sequence Pn of special analytic polyhedra defined by global functions such that Pn c Pn+l and X = U Pn· 6. Corollary.

Sec. C

The Imbedding Theorem

219

Proof" Let X = u wn, wn C wn+l• where the wn are Oka-Weil domains. Let Pn be an analytic polyhedron such that wn C pn C / \ C wn+l• Then pn C wn+l C pn+l and X = u Pn.

It is clear that this section is independent of Chapter VI and the preceding section. We wish to point out that the rest of this chapter is also independent of Chapter VI. In particular, the imbedding theorem can be proved on the basis of the geometric theory and the analytic results of Chapter I. In order to recognize this, we must note only that the following result has this property. 7. Proposition. Let X be a closed subvariety of pure dimension k of a domain D in en. If D is polynomially convex, then X is also polynomially convex. Proof· Let K be a compact subset of X; we show that the polynomially convex hull K of K is contained in X. Since D is polynomially convex, it is Stein, so we can find an analytic polyhedron P such that P :::> K. Now, by theorem 3, there is a special analytic polyhedron Q = {x EX; /g;(x)/ < I, I :::;; i:::;; k} such that g; E (!JD and (P n X) :::> Q :::> K. Let D' = {z E P; /g;(z)/ < I, I :::;; i :::;; k}, and g: D' 0, there is an f E 0x such that [If - l [IK < e, [[flle-K < e. Proof: Let C = {x EX; /g;(x)/ :::;; !, I :::;; i:::;; t} with g; E 0x. Add functions g tw ... , gk E (!} x so that, for some neighborhood U of C, ( U, (g1, ••• , gk)) is an Oka-Weil domain. Let g = (g1 , ••• , gk). Then g(U) is a closed subvariety of Li(O, I), so is polynomially convex. Thus g(C) is a polynomially conv~x compact set and g(C) = g(K) u g(C - K), and this union is disjoint. Now the function which is I on g(K), and O on g(C - K), is holomorphic in a neighborhood of g( C), so by theorem l, F8, given e > 0 there is a polynomial p such that lip - l lla < e, [[p[[ 0 < e. Then f = p(g1, ... , gk) is the desired function.

C. The lmbedding Theorem

In this section we shall prove that a Stein manifold can be realized as a closed submanifold of Euclidean space. Such an imbedding F: X en will allow us to express X as an increasing union. of Oka-Weil domains, all defined by the same n-tuple of functions. First we shall show that we can exhaust X by special analytic polyhedra defined by the same n-tuple of functions. We shall build, upon this foundation, functions which separate points everywhere and give local coordinates everywhere. By means of category arguments we shall be able to keep the number of functions finite.

Stein Spaces, Geometric Theory

220

Chap. VII

1. Definition. Let X, Y be topological spaces. A mapping f: X ---+ Y is almost proper if each component of r 1 (K) is compact in X whenever K is compact in Y. _2. Theorem. Let (X, m) be a Stein space of pure dimension n. The set ':§ of f E mi which gives an almost proper map of X into en is dense in m~.

Proof: Let h

f

EG

such that

mi, e >

E

0, and K a compact set in X. We have to find 00

llf - hllK
0. Let L be a component off-1(.:'.l(0; oc)). Choose i so that P; n L =l=0, and let m > max (ix, i). We claim that L n 8P m = 0. If this is the case, we are through, for then L c Pm (since L is connected and intersects Pm). If x E 8Pm, there is a j such that lgt,.(x)I = I. Then 00

IUx)I = I 2 ocigl(x))t• + h;(x)I k=O 00

:::::: ocm

lg{,.(x)I -

:::::: ocm -

Thus f(x)

1n•-1(x)I

117-1(x)I -

-

2

ock lgi(xW•

k=m+l

rm+i :::::: m

+1-

2-m+l

>

oc.

ff: Ll(0; oc), so x ff: L.

3. Lemma. Let (X, m) be a Stein space of pure dimension n, and suppose f: X---+ en is an almost proper holomorphic map. Let K be a compact subset of X. There are only a finite number of components H of f- 1 (.15.(0; r)) which meet K. Furthermore, max {rankx f; x E 9iX} = n, and f(H) = L\(0; r). Proof: Let L be a component off-1 (.15.(0; r)). Since {x E 8L;f(x) = 0} = 0 , V(f) n L is a compact subvariety of int L, and thus a finite set of points.

Sec. C

The lmbedding Theorem

221

.Thus V(f) is a discrete set of points in X. Let H be the union of the components ofJ-1(~(0; r)) which meet K. Choose p > r so that/(K) c L\(O; p). Then H c J-1(1'.\(0; p)); in fact His contained in the union U of the components of J-1(L\(O; p)) which contain K. Since K is compact, only finitely many components are involved, and so U is relatively compact (since f is almost proper). Thus His compact, and/: intH---+ L\(O; r) is proper. Thus, as in theorem V, C5(a), or theorem III, B20, /(int H) = L\(O; r), and max {rank.,/; x EX, n (int H)} = n. In particular H could be a single component of 1(~(0; r)), so V(f) intersects every such component. Since V(f) n His finite, H has only finitely many components.

.r-

4. Theorem. Let X be a Stein space of pure dimension n, and suppose f: X---+ en is an almost proper map. The set of fn+l E 0x such that (f, fn+l): X---+ cn+1 is a proper map is dense in 0x.

Proof· Let h E 0 x, K a compact subset of X, and e > 0 be given. Let 00

X =

U X;, where the

X; are Oka-Weil domains and

X;

c

int

X;+i·

Let H1c

i-0

be the union of the connected components ofJ-1(3.(0, k)) which intersect X1c. 00

We may assume Kc H 1. Clearly X

= U H1c. H1c is holomorphically con1c-1

vex. For certainly H1c is compact, and its hull b1c must be contained in J-1(5.(0, k)); and thus H1c is an open and closed subset of bk. Thus, by 1(3.(0, k)) n H1c+1) - H1c, Gk is also corollary BS, H7c = bk. Let Gk= compact since G1c u Hk = J-1(~(0, k)) n H1c+1 and H1c is open and closed in G1c U H1c. Observe that G1c u H1c is also holomorphically convex. We now

u-

00

construct fn+i = _! 0. By assumption, there are points Xa E ~(15, oc) such that x 6 tends to the boundary of D as 15---+ 150 ; thus d(x6 )---+ 0. But x 6 EK, since x 6 is an interior point of a disc whose boundary is contained in /(_; and this contradicts the hypothesis that Dis pseudoconvex, i.e., d(xa) z d(K) > 0. Thus we must have ~(15, oc) c D for all 15 < 1; consequently

{tp(w)

+ J.e-h · a; iJ.I

< 1, lwl

:S::: 1} c

D,

i.e., da(tp(w)) 2 e-h, or -log da(tp(w)) :S:: h(w) for all w, lwl :S::: 1. Now, as in Chapter I, the difficulty in proving that a pseudoconvex Riemann domain is Stein occurs in the case that D is not finitely sheeted. For if Dis finitely sheeted, then the sets {x ED; -log d(x) :s;; oc} are compact for oc < 0, so D is p-convex and we can apply theorem C14. Our problem thus is to construct from -log d a psh. function cp such that the sets {x ED; cp(x) :s;; oc} are compact. 2. Lemma. Let (D, p) be a connected Riemann domain. Let x0 ED, and r < d(x 0 ) < oo. There is a C 2 psh.function cp z Ode.fined on o = {y ED; d(y) > r} such that {y E o; cp(y) < oc} is relatively compact in D.

Proof· Let e, p be real numbers, with O < e < p/2 < r/2. We now define an increasing sequence {Vk} of open sets such that Vk c D

, flk is compact,

Oka's Pseudoconvexity Theorem

Sec.D

281

and fl

= U Vk. Let VJ= Li(x0, e) n D

. Since e < p, certainly VJ is compact, and vg c D

. Suppose VJ, . •• , v~- 1 are chosen. Define

Certainly V~ c D

; we now show that fl~ is compact. Let {xn} be a sequence of points in Each Xn is in some Li(yn, e) with Yn E v~- 1• Since ft- 1 is compact, we may assume that Yn-+ y 0 E Vt- 1 • In particular, Yo E jj

; so d(y0 ) 2 p > e. Thus &(y0, e') is compact for some e', e < e' < p. Since d(ym xn) < e, eventually all the xn are in &(y0, e'), so the xn have a limit point in D. Qlearly, if fl(P) is connected, narticular, it is possible to prove theorem B6 directly for locally free sheaves, without any of the machinery evolved here [142]. The methods of J. J. Kohn [142] are very sensitive to boundary behavior, and open up a field of study which is apparently inaccessible in the sheaf-theoretic approach. It should also be mentioned that strict pseudoconvexity is only an extreme case of the properties of being pseudoconvex or pseudoconcave. (Cf. the papers of Rothstein [205], Ehrenpreis [3, 85], Kohn [142], Andr~otti-Grauert [5], Andreotti [6], KohnRossi [143].) Except for proposition A4 (due to Kohn) the results in Chapter IX, up to lemma B5, are due to E. E. Levi [162]. The results in the rest of Section B are due to Grauert [104]; (an earlier argument, using other techniques is due to Ehrenpreis [3]). The extension to spaces is due to Narasimhan [180]. The argument in proposition C5 is a trivial case of an important theorem of Grauert [105], which goes as follows. Suppose ,r: X--+ Y is a proper mapping of analytic spaces, and .9' is a coherent sheaf on X. If U is open in Y, define r u = H 0(,r-1( U), .9'). The collection {r u} defines a presheaf on Y, and the associated sheaf ,r0(.9') is called the direct image sheaf. Grauert's theorem is that ,r0(.9') is a coherent sheaf on X. In particular, in proposition C5, what is used is that ,r0(0E) is coherent. Remmert's proper mappings theorem (theorem V, C8) is easily deduced from this theorem. There are many ways of handling the rest of Sections C, D: the reader is refered to the papers of Bremermann [50--52], Docquier-Grauert [82], Behnke [16]. The theorem we refer to as Oka's theorem in Section D was first proved in C 2 by Oka [186, VJ], and then in ([:n by Bremermann [49], Norguet [185], Oka [186, IX]. The theorems in Section E are to be found in Grauert [106] (originally in Kodaira [138]). Grauert actually proved theorem E3 in the case that A has a weakly negative vector bundle of any rank. The proof is exactly the same as_ in the case of a line bundle; however, at the end A gets mapped into a Grassmannian rather than projective space.

APPENDIX A

PARTITIONS OF UNITY

For the sake of completeness we gather here the proofs of a few simple standard results, centered about the existence of partitions of unity, which have not yet generally found their way into the undergraduate curriculum. We shall prove these results only for the very special cases in which they have been needed in the preceding discussion. 1. Lemma. Suppose that K is a compact subset of an open set V c