Basic Oka Theory in Several Complex Variables (Universitext) [2024 ed.] 9819720559, 9789819720552

Table of contents :
Preface
Contents
Conventions
Chapter 1 Holomorphic Functions
1.1 Holomorphic Functions of Several Variables
1.1.1 Open Balls and Polydisks of Cn
1.1.2 Definition of Holomorphic Functions
1.1.3 Sequences and Series of Functions
1.1.4 Power Series of Several Variables
1.1.5 Elementary Properties of Holomorphic Functions of Several Variables
1.2 Analytic Continuation and Hartogs’ Phenomenon
1.3 Runge Approximation on Convex Cylinder Domains
1.3.1 Cousin Integral
1.4 Implicit and Inverse Function Theorems
1.5 Analytic Subsets
Exercises
Chapter 2 Coherent Sheaves and Oka’s Joku-Iko Principle
2.1 Notion of Analytic Sheaves
2.1.1 Definitions of Rings and Modules
2.1.2 Analytic Sheaves
2.2 Coherent Sheaves
2.2.1 Locally Finite Sheaves
2.2.2 Coherent Sheaves
2.3 Oka’s First Coherence Theorem
2.3.1 Weierstrass’ Preparation Theorem
2.3.2 Oka’s First Coherence Theorem
2.3.3 Coherence of Ideal Sheaves of Complex Submanifolds
2.4 Cartan’s Merging Lemma
2.4.1 Matrices and Matrix-Valued Functions
2.4.2 Cartan’s Matrix Decomposition
2.4.3 Cartan’s Merging Lemma
2.5 Oka’s Joku-Iko Principle
2.5.1 Oka Syzygy
2.5.2 Oka Extension of the Joku-Iko Principle
Exercises
Chapter 3 Domains of Holomorphy
3.1 Definitions and Elementary Properties
3.1.1 Relatively Compact Hull
3.1.2 Domain of Holomorphy and Holomorphic Convexity
3.2 Cartan–Thullen Theorem
3.3 Analytic Polyhedron and Oka–Weil Approximation
3.3.1 Analytic Polyhedron
3.3.2 Oka–Weil Approximation Theorem
3.3.3 Runge Approximation Theorem (One Variable)
3.4 Cousin Problem
3.4.1 Cousin I Problem
3.4.2 Continuous Cousin Problem
3.4.3 Cousin I Problem—continued
3.4.4 Hartogs Extension over a Compact Subset
3.4.5 Mittag-Leffler Theorem (One Variable)
3.4.6 Cousin II Problem and Oka Principle
3.4.7 Weierstrass’ Theorem (One Variable)
3.4.8 ¯∂-Equation
3.5 Analytic Interpolation Problem
3.6 Unramified Domains over Cn
3.7 Stein Domains over Cn
3.8 Supplement: Ideal Boundary
Exercises
Chapter 4 Pseudoconvex Domains I — Problem and Reduction
4.1 Plurisubharmonic Functions
4.1.1 Subharmonic Functions (One Variable)
4.1.2 Plurisubharmonic Functions
4.1.3 Smoothing
4.2 Hartogs’ Separate Analyticity
4.2.1 Baire Category Theorem
4.2.2 Separate Analyticity
4.3 Pseudoconvexity
4.3.1 Pseudoconvexity Problem
4.3.2 Bochner’s Tube Theorem
4.3.3 Pseudoconvex Boundary
4.3.4 Levi Pseudoconvexity
4.3.5 Strongly Pseudoconvex Boundary Points and Stein Domains
Exercises
Chapter 5 Pseudoconvex Domains II — Solution
5.1 The Oka Extension with Estimate
5.1.1 Preparation from Topological Vector Spaces
5.1.2 The Oka Extension with Estimate
5.2 Strongly Pseudoconvex Domains
5.2.1 Oka’s Method
5.2.2 Grauert’s Method
5.3 Oka’s Pseudoconvexity Theorem
Exercises
Afterword— Historical Comments
References
Index
Symbols

Citation preview

Universitext

Junjiro Noguchi

Basic Oka Theory in Several Complex Variables

Universitext Series Editors Nathanaël Berestycki, Universität Wien, Vienna, Austria Carles Casacuberta, Universitat de Barcelona, Barcelona, Spain John Greenlees, University of Warwick, Coventry, UK Angus MacIntyre, Queen Mary University of London, London, UK Claude Sabbah, École Polytechnique, CNRS, Université Paris-Saclay, Palaiseau, France Endre Süli, University of Oxford, Oxford, UK

Universitext is a series of textbooks that presents material from a wide variety of mathematical disciplines at master’s level and beyond. The books, often well classtested by their author, may have an informal, personal, or even experimental approach to their subject matter. Some of the most successful and established books in the series have evolved through several editions, always following the evolution of teaching curricula, into very polished texts. Thus as research topics trickle down into graduate-level teaching, first textbooks written for new, cutting-edge courses may find their way into Universitext.

Junjiro Noguchi

Basic Oka Theory in Several Complex Variables

Junjiro Noguchi Graduate School of Mathematical Sciences The University of Tokyo Tokyo, Japan

ISSN 0172-5939 ISSN 2191-6675 (electronic) Universitext ISBN 978-981-97-2055-2 ISBN 978-981-97-2056-9 (eBook) https://doi.org/10.1007/978-981-97-2056-9 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore Paper in this product is recyclable.

Preface

This book provides a comprehensive self-contained account of Oka theory, mainly concerned with the proofs of the Three Big Problems of approximation, Cousin and pseudoconvexity (Hartogs, Levi) stated below, which were solved by Kiyoshi Oka and form the basics of complex analysis in several variables. It is the purpose to present a textbook of course lectures that follow on from complex function theory of one variable. The presentation is intended to be readable, enjoyable and self-contained for those from beginners in mathematics to researchers interested in complex analysis in several variables and complex geometry. The nature of the present book should be evinced by the following two points: • We develop the theory by the method of the Oka Extension of holomorphic functions from a complex submanifold of a polydisk to the whole polydisk (Oka’s Joku-Iko Principle1); • We represent Oka’s original proofs, following his unpublished ve papers of 1943 and Oka IX (1953). In those unpublished ve papers (in Japanese), historically, the pseudoconvexity problem (Hartogs’ Inverse Problem, Levi’s Problem) was rst solved for domains of C𝑛 (𝑛 ≥ 2), and furthermore for unrami ed domains over C𝑛 as well (see [50], [44] which contains the English translation of the last one of the ve papers). We derive the Oka Extension of the Joku-Iko Principle from the coherence of the sheaf 𝒪C𝑛 of holomorphic functions on C𝑛 (Oka’s First Coherence Theorem), which is proved by Weierstrass’ Preparation Theorem; Weierstrass’ Preparation Theorem is shown by the residue theorem in one variable. In this way we use only elementary techniques, yet reach the core of the theory. The basis of analytic function theory of several variables or complex analysis in several variables was founded till the middle of the 1950s. Just afterwards new theories and generalizations were developed. Also the simpli cation of the theory 1 “Joku-Iko” is pronounced “dzóuku ikou” (not aiko). The Joku-Iko Principle is a term due to K. Oka himself, and the guiding methodological principle of Oka theory all through his works (cf. footnote 2).

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has been done, but the di culties of the introductory part from the theory of one variable to that of several variables has remained considerably for beginners. The present book aims to provide a smooth introduction for that part (cf. [39], [43], [2]). The Three Big Problems which were summarized by Behnke Thullen [4] in 1934 are stated as follows: (P1) Approximation Problem (problem of developments) (Runge’s Theorem in one variable). (P2) Cousin Problem (I and II) (Mittag-Le er and Weierstrass Theorems in one variable). (P3) Pseudoconvexity Problem (Hartogs’ Inverse Problem, Levi’s Problem) (the natural boundary problem of analytic continuation). K. Oka solved all of these problems in Oka I IX ([48], [49]); they are roughly classi ed into two groups: (G1) Oka I VI + IX. (G2) Oka VII, VIII, IX. Oka IX contains works belonging to both of the groups, and the Three Big Problems were solved by the rst group (G1); in the second group (G2) he proved his Three Coherence Theorems, aiming for development beyond the original problems. In the present book we restrict ourselves to the results of group (G1); this is the reason for the use of “Basic” in the title. Here we do not use: • Cohomology theory with coe cients in sheaves; • 𝐿 2 -𝜕¯ method. The solution of the Pseudoconvexity Problem (P3) is the culmination of the works (G1); Oka’s methods consist of: (i) (In VI, 1942; univalent domains of dim 𝑛 = 2) Cousin Problem & Weil’s integral formula & Fredholm integral equation of the second kind; (ii) (In VII XI, 1943, unpublished (cf. [44]); unramified multivalent domains over C𝑛 of general 𝑛 ≥ 2) Cousin Problem & “Primitive Coherence Theorem” & the Joku-Iko Principle 2 & Fredholm integral equation of the second kind type (a variant of the equation of the second kind combined with the Joku-Iko Principle);

2 A direct translation might be “a transfer (=Iko) to an upper space (=Joku)”. He found the principle in the study of Oka I (1936) and II (1937), and used it all through his works, till Oka IX (1953). It is an idea to solve the problem caused by the increased number of variables by increasing the number of variables more; one embeds the initial domains into simply shaped polydisks of higher dimensions, extends the problems over the polydisks, and then solves them by making use of the simplicity of polydisks. In T. Nishino [37] the term was translated to the “Lifting Principle”. As a matter of fact, the statement itself holds more generally for any subvarieties of Stein spaces, and so may be called an analytic extension or interpolation; then, however, the spirit of the wording will be lost, since the general case is proved through embeddings of analytic polyhedra into polydisks. So, here we prefer to use the original term as in [39].

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(iii) (In IX, 1953, published; unramified multivalent domains over C𝑛 of general 𝑛 ≥ 2) Cousin Problem & “Coherence Theorem” & the Joku-Iko Principle & Fredholm integral equation of the second kind type. After all, K. Oka proved the Pseudoconvexity Problem (P3) three times. It is noticed that in (ii) and (iii) above, Oka proved the Approximation Theorem and Cousin Problem for unrami ed domains, multivalent in general, over C𝑛 by a new method, which had been proved for univalent domains in his former papers (I III). The content of Oka IX ((iii) above) is essentially the same as that of (ii) except for the part of the “Coherence Theorem” obtained in (G2). In Oka VI ((i) above), he mentioned the validity of the result for general dimension 𝑛 ≥ 2 in a modest phrase at the end of the paper: L’auteur pense que cette conclusion sera aussi indépendante des nombres de variables complexes.

In fact, the method of (i) was later generalized to univalent domains of general dimension 𝑛 by S. Hitotsumatsu [28] 1949, H.J. Bremermann [8] 1954, and F. Norguet [45] 1954, independently. The method of (iii) was generalized for abstract complex spaces by T. Nishino [36] 1962 (cf. [37]), and A. Andreotti R. Narasimhan [3] 1964. In (ii) above, Oka formulated and proved a kind of “Primitive Coherence Theorem” with a certain condition, yet su cient for the purpose, and he used some type of the Fredholm integral equation of the second kind. In Oka IX, he replaced the “Primitive Coherence Theorem” with his Coherence Theorems. The present book, hopefully, presents an easy comprehensive account of that theory. H. Cartan has written ([49], p. XII): ............. Mais il faut avouer que les aspects techniques de ses démonstrations et le mode de présentation de ses résultats rendent difficile la tâche du lecteur, et que ce n’est qu’au prix d’un réel effort que l’on parvient à saisir la portée de ses résultats, qui est considérable. C’est pourquoi il est peut-être encore utile aujourd’hui, en hommage au grand créateur que fut Kiyoshi Oka, de présenter l’ensemble de son œuvre. .................

In English (by Noguchi), ............. But we must admit that the technical aspects of his proofs and the mode of presentation of his results make it difficult to read, and that it is possible only at the cost of a real effort to grasp the scope of its results, which is considerable. This is why it is perhaps still useful today, for the homage of the great creator that was Kiyoshi Oka, to present the collection of his work. .................

It is interesting that, looking for an easier introduction of analytic function theory of several variables, we came back to Oka’s original method. To the best of the author’s knowledge, there is no book or monograph presenting Oka’s original method except for Nishino [37], while there are many for the developments or other proofs obtained after Oka’s works. The author hopes that the present book lls the gap even a little, and is useful to recognize Oka’s original ideas.

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It should be worthwhile for students and researchers to look into the original work of K. Oka, which may still contain some new ideas. Therefore the prerequisites of the present book are minimized to the contents from standard complex analysis in one variable (cf., e.g., [38]), which range from Cauchy’s integral formula to Riemann’s mapping theorem. We explain the necessary contents of topology, rings and modules; if they are not su cient, it may su ce to confer any nearby books on those elementary materials. We avoided the general notion of manifolds. Now we brie y describe the contents of the present book. In Chapter 1 we begin with the de nition of holomorphic or analytic functions of several variables, and convergent power series. We then explain Hartogs’ phenomenon, which was the starting point of analytic function theory of several variables. For the preparation of the chapters in the sequel we show the Runge approximation theorem on convex cylinder domains, and explain the Cousin integral and analytic subsets. Chapter 2 describes the notion of analytic sheaves and the coherence. Analytic sheaves will be de ned just as sets or as collections of rings or modules without topology. We then show Weierstrass’ Preparation Theorem by making use of the residue theorem in one variable. We then prove Oka’s First Coherence Theorem, Cartan’s Matrix Lemma and then, Oka’s Syzygy Lemma, with which we nally derive the Oka Extension of the Joku-Iko Principle. Chapter 3 is devoted to the theory of domains of holomorphy and holomorphically convex domains. We prove the foundational Cartan Thullen Theorem, which asserts the equivalence of those two domains in the univalent (schlicht) case. Then an analytic polyhedron is introduced, and the Oka Weil Approximation Theorem is proved as a solution of the First Big Problem (P1) above by means of the JokuIko Principle. As a special case of one variable, we show Runge’s Approximation Theorem. Subsequently the Cousin Problem (the Second Big Problem (P2)) is dealt with. Here we formulate the Continuous Cousin Problem by which we unify the treatment ¯ of the Cousin I, II Problems and the problem of 𝜕-equation for functions, where the Oka Principle is included. We then solve the Continuous Cousin Problem on holomorphically convex domains, equivalently on domains of holomorphy in the univalent case. We discuss the applications to the case of one variable, proving the Mittag-Le er and Weierstrass Theorems. As an application of the Continuous Cousin Problem we prove the Hartogs extension of holomorphic functions over a compact subset of a domain of C𝑛 . By a similar method of the proof of the Continuous Cousin Problem, we solve the interpolation problem for complex submanifolds of univalent domains of holomorphy. At the end of Chapter 3 we introduce the notion of multivalent domains over C𝑛 , which are here assumed to be unrami ed. We de ne the envelope of holomorphy of such a domain, and the notion of domains of holomorphy in the multivalent case. We introduce Stein domains in the multivalent case, and see that the results obtained for univalent domains of holomorphy remain valid for multivalent Stein domains. In Chapters 4 and 5 we deal with the Pseudoconvexity Problem (P3) for domains over C𝑛 , where domains are assumed to be multivalent in general. Chapter

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4 is devoted to the formulations and the reductions of Problem (P3). Firstly, we introduce the notion of plurisubharmonic functions. Using it, we prove Hartogs’ separate analyticity theorem; in the course, we explain the Baire Category Theorem. Then, several kinds of pseudoconvexities of domains are de ned. We discuss the equivalence and the relations of those pseudoconvexities, and formulate what is the pseudoconvexity problem. We then prove Oka’s Theorem of Boundary Distance Functions. This serves the rst important step toward the solution of Problem (P3). As an application we prove the Tube Theorem due to S. Bochner and K. Stein (𝑛 = 2). Finally, in Chapter 5, we solve the Pseudoconvexity Problem (P3), the last of the Three Big Problems, which is formulated in the previous chapter. To begin with, we introduce the notion of a semi-normed space and a Fréchet space, and prove Banach’s Open Mapping Theorem for mapppings from Fréchet spaces to Baire spaces. We then show the Oka Extension of the Joku-Iko Principle with estimate. We give two proofs of the Steinness of strongly pseudoconvex domains (Levi’s Problem); the rst is K. Oka’s and the second is the one due to H. Grauert; there is some similarity in the two proofs, which should be interesting for comparison. T. Nishino’s book [37] presents the proof of Oka IX (1953) in more generalized form for complex spaces. In the proof we formulate a variant of the Fredholm integral equation of the second kind combined with the Oka Extension; we call it a Fredholm integral equation of the second kind type. This is the key of the proof. It is solved by a successive approximation and the convergence is obtained by the method of majorants. It is rather surprising to see such a di cult problem being solved by such an elementary method. The second method due to Grauert is the well-known “bumping method” combined with L. Schwartz’s Fredholm Theorem,3 of which a short but complete proof is given; by making use of an idea of J.P. Demailly, we prove it in a slightly generalized form for Baire spaces as image spaces. For the purpose we introduce the rst cohomology 𝐻 1 (★,𝒪). With these preparations we nally prove Oka’s Pseudoconvexity Theorem that pseudoconvex domains unrami ed over C𝑛 are Stein. At the end of each chapter some historical comments are made from the author’s viewpoint and knowledge; it is expected to motivate the readers to confer other mentioned resources, but they are far from complete. The present book is an outcome of the author [43], largely rewritten with a number of additions. It does not aim to give the full exposition of the fundamentals of analytic function theory of several variables or complex analysis in several variables. But, the related topics are mentioned in various places with references, which readers are encouraged to confer. And some of them are presented in Exercises, which readers are expected to solve by themselves. It is also recommended for readers to consult those books and monographs referred to throughout. The author will be more than happy if readers get interested in the present subject through this book providing the elementary but self-contained proofs of the Three 3 This term is due to A. Andreotti, according to a personal communication with A. Huckleberry (cf. Jahresber. Dtsch. Math.Ver. 115 (2013), 21-45).

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Big Problems which form the basics of several complex variables, and if the original works of Kiyoshi Oka, full of creative ideas, are enjoyed and recognized more deeply. While the author was writing the present book, he gave several talks at the weekly seminar on complex analysis and geometry, the University of Tokyo, Komaba, Tokyo (so-called Monday Morning Seminar), having a number of discussions with the members, which were very helpful and encouraging. In May 2017, he was kindly invited by Professor Sachiko Hamano of Osaka City University then to give an intensive one-week lecture course, based on the rst draft of the book. In July of the same year, he gave a seminary talk on Oka’s original method and related topics at Tor Vergata, Rome at the invitation of Professor Filippo Bracci. In March 2019, he gave a talk on the book at the Japan Iceland Workshop, “Holomorphic Maps, Pluripotentials and Complex Geometry” at the invitation of Professor Masanori Adachi (Shizuoka University), and in May of that year he gave a series of talks at the invitation of Professor Steven Lu at Montreal; in July he gave a series of lectures based on this book at the Workshop, “Summer Program on Complex Geometry and Several Complex Variables” at Shanghai Center for Mathematical Sciences, Fudan University, Shanghai at the invitation of Professor Min Ru (University of Houston). The author learned a number of references on the pseudoconvexity problem from Professor Makoto Abe (Hiroshima University), and had many valuable suggestions and comments from Professors Hiroshi Yamaguchi (Shiga University, Emeritus; Nara Women University), Sachiko Hamano (Kyoto Sangyo University), Yohei Komori (Waseda Univerity, Tokyo), Shigeharu Takayama (University of Tokyo) and Jöel Merker (Université Paris-Saclay). Professor Viorel Vâjâitu (Université des Sciences et Technologies de Lille) kindly suggested useful information in Remarks at the end of Chap. 5. To all of them the author would like to express his sincere gratitude. The author acknowledges with many thanks the support of JSPS KAKENHI Grant Number JP19K03511, which has been always helpful in carrying out the present project. Kamakura Spring, 2024 Junjiro Noguchi

Contents

1

2

Holomorphic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Holomorphic Functions of Several Variables . . . . . . . . . . . . . . . . . . . . 1.1.1 Open Balls and Polydisks of C𝑛 . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 De nition of Holomorphic Functions . . . . . . . . . . . . . . . . . . . 1.1.3 Sequences and Series of Functions . . . . . . . . . . . . . . . . . . . . . . 1.1.4 Power Series of Several Variables . . . . . . . . . . . . . . . . . . . . . . . 1.1.5 Elementary Properties of Holomorphic Functions of Several Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Analytic Continuation and Hartogs’ Phenomenon . . . . . . . . . . . . . . . . 1.3 Runge Approximation on Convex Cylinder Domains . . . . . . . . . . . . . 1.3.1 Cousin Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Implicit and Inverse Function Theorems . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Analytic Subsets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 1 3 6 8 10 14 18 19 21 24 28

Coherent Sheaves and Oka’s Joku-Iko Principle . . . . . . . . . . . . . . . . . . . 2.1 Notion of Analytic Sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 De nitions of Rings and Modules . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Analytic Sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Coherent Sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Locally Finite Sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Coherent Sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Oka’s First Coherence Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Weierstrass’ Preparation Theorem . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Oka’s First Coherence Theorem . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Coherence of Ideal Sheaves of Complex Submanifolds . . . . . 2.4 Cartan’s Merging Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Matrices and Matrix-Valued Functions . . . . . . . . . . . . . . . . . . 2.4.2 Cartan’s Matrix Decomposition . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3 Cartan’s Merging Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

31 31 31 33 36 36 39 43 43 47 52 53 53 57 61

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2.5 Oka’s Joku-Iko Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Oka Syzygy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.2 Oka Extension of the Joku-Iko Principle . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

63 63 68 71

3

Domains of Holomorphy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 3.1 De nitions and Elementary Properties . . . . . . . . . . . . . . . . . . . . . . . . . 73 3.1.1 Relatively Compact Hull . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 3.1.2 Domain of Holomorphy and Holomorphic Convexity . . . . . . 75 3.2 Cartan Thullen Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 3.3 Analytic Polyhedron and Oka Weil Approximation . . . . . . . . . . . . . . 83 3.3.1 Analytic Polyhedron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 3.3.2 Oka Weil Approximation Theorem . . . . . . . . . . . . . . . . . . . . . 85 3.3.3 Runge Approximation Theorem (One Variable) . . . . . . . . . . . 87 3.4 Cousin Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 3.4.1 Cousin I Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 3.4.2 Continuous Cousin Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 3.4.3 Cousin I Problem continued . . . . . . . . . . . . . . . . . . . . . . . . . 97 3.4.4 Hartogs Extension over a Compact Subset . . . . . . . . . . . . . . . 98 3.4.5 Mittag-Le er Theorem (One Variable) . . . . . . . . . . . . . . . . . . 100 3.4.6 Cousin II Problem and Oka Principle . . . . . . . . . . . . . . . . . . . . 101 3.4.7 Weierstrass’ Theorem (One Variable) . . . . . . . . . . . . . . . . . . . 105 ¯ 3.4.8 𝜕-Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 3.5 Analytic Interpolation Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 3.6 Unrami ed Domains over C𝑛 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 3.7 Stein Domains over C𝑛 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 3.8 Supplement: Ideal Boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

4

Pseudoconvex Domains I — Problem and Reduction . . . . . . . . . . . . . . . 131 4.1 Plurisubharmonic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 4.1.1 Subharmonic Functions (One Variable) . . . . . . . . . . . . . . . . . . 131 4.1.2 Plurisubharmonic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 4.1.3 Smoothing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 4.2 Hartogs’ Separate Analyticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 4.2.1 Baire Category Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 4.2.2 Separate Analyticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 4.3 Pseudoconvexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 4.3.1 Pseudoconvexity Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 4.3.2 Bochner’s Tube Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 4.3.3 Pseudoconvex Boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 4.3.4 Levi Pseudoconvexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 4.3.5 Strongly Pseudoconvex Boundary Points and Stein Domains 166 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

Contents

5

xiii

Pseudoconvex Domains II — Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 5.1 The Oka Extension with Estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 5.1.1 Preparation from Topological Vector Spaces . . . . . . . . . . . . . . 175 5.1.2 The Oka Extension with Estimate . . . . . . . . . . . . . . . . . . . . . . 178 5.2 Strongly Pseudoconvex Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 5.2.1 Oka’s Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 5.2.2 Grauert’s Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 5.3 Oka’s Pseudoconvexity Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206

Afterword — Historical Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

Conventions (i) Theorems, equations etc. are numbered consecutively. Here an equation is numbered as (1.1.1) with parentheses; the rst 1 stands for the chapter number and the second 1 for the section number. (ii) The standard notation of set theory is assumed. For example, maps, images and inverse images are used without speci c mention. Also, “∀𝑥” means “every or all 𝑥” and “∃ 𝑥” means “some 𝑥” or “existence of 𝑥”. (iii) The standard notion of the euclidean topology of R𝑛 (also of C𝑛  R2𝑛 ) is assumed. For example, open sets, closed sets, the connectedness, and the boundary 𝜕𝑈 of a subset 𝑈 ⊂ R𝑛 are used without speci c comments. (iv) By 𝐴 := 𝐵 we mean that 𝐴 is de ned to be 𝐵, or we put 𝐴 to be 𝐵. (v) A relation 𝑥 ∼ 𝑦 for all two elements 𝑥, 𝑦 ∈ 𝑆 of a set 𝑆 is called a equivalence relation if the following conditions are satis ed: (i) 𝑥 ∼ 𝑥 (∀𝑥 ∈ 𝑆) (re exive law); (ii) if two elements 𝑥, 𝑦 ∈ 𝑆 satisfy 𝑥 ∼ 𝑦, then 𝑦 ∼ 𝑥 (symmetric law); (iii) if three elements 𝑥, 𝑦, 𝑧 ∈ 𝑆 satisfy 𝑥 ∼ 𝑦 and 𝑦 ∼ 𝑧, then 𝑥 ∼ 𝑧 (transitive law). (vi) For a set 𝑆, |𝑆| denotes its cardinality. (vii) A map 𝑓 : 𝑋 → 𝑌 is said to be injective or an injection if 𝑓 (𝑥1 ) ≠ 𝑓 (𝑥 2 ) for every distinct 𝑥1 , 𝑥 2 ∈ 𝑋, and to be surjective or a surjection if 𝑓 (𝑋) = 𝑌 . If 𝑓 is injective and surjective, it is said to be bijective or called a bijection. The restriction of 𝑓 to a subset 𝐸 ⊂ 𝑋 is denoted by 𝑓 | 𝐸 . (viii) The set of natural numbers (positive integers) is denoted by N, the set of integers by Z, the set of rational numbers by Q, the set of real numbers by R, the set of complex numbers by C, and the imaginary unit by 𝑖, as usual. We write C∗ := C \ {0}. The set of non-negative integers (resp. numbers) is denoted by Z+ (resp. R+ ). (ix) For a complex number 𝑧 = 𝑥 + 𝑖𝑦 ∈ C we set ℜ𝑧 = 𝑥 and ℑ𝑧 = 𝑦. (x) Monotone increasing and monotone decreasing are used in the sense including the case of equality: e.g., a sequence of functions {𝜑 𝜈 (𝑥)}∞ 𝜈=1 is said to be monotone increasing if for every point 𝑥 of the de ning domain 𝜑 𝜈 (𝑥) ≤ 𝜑 𝜈+1 (𝑥) for all 𝜈 = 1, 2, . . .. (xi) For a (vector-valued) function 𝑓 on a subset 𝑈 ⊂ R𝑛 (more generally, on a topological space 𝑈) the support denoted by Supp 𝑓 is de ned as the (topological) closure of the set {𝑥 ∈ 𝑈 : 𝑓 (𝑥) ≠ 0} in 𝑈. (xii) A function 𝑓 de ned on an open subset 𝑈 ⊂ R𝑚 is said to be of 𝐶 𝑘 -class (1 ≤ 𝑘 ≤ ∞) if 𝑓 is 𝑘-times continuously di erentiable. 𝒞 𝑘 (𝑈) denotes the set of all functions of 𝐶 𝑘 -class on 𝑈. 𝒞0𝑘 (𝑈) stands for the set of all 𝑓 ∈ 𝒞 𝑘 (𝑈) with compact supports. (xiii) For subsets 𝐴 ⊂ 𝐵 ⊂ C𝑛 (or R𝑛 ), “𝐴 ⋐ 𝐵” means that the closure 𝐴¯ is compact and 𝐴¯ ⊂ 𝐵; in this case, 𝐴 is said to be relatively compact (in 𝐵). (xiv) A continuous map 𝑓 : 𝑉 → 𝑊 from a set 𝑉 ⊂ C𝑚 (or R𝑚 ) to another 𝑊 ⊂ C𝑛 (or R𝑛 ) (more generally, between locally compact topological spaces) is said

xv

xvi

Conventions

to be proper if for every compact subset 𝐾 ⊂ 𝑊, the inverse image 𝑓 −1 𝐾 is compact, too. (xv) (Landau symbol) With a small variable 𝑟 > 0 and a constant 𝛼 ≥ 0, 𝑜(𝑟 𝛼 ) stands for a term or a quantity such that lim𝑟→+0 𝑜(𝑟 𝛼 )/𝑟 𝛼 = 0. In particular, if 𝛼 = 0, lim𝑟→+0 𝑜(1) = 0.

Chapter 1

Holomorphic Functions

We define holomorphic functions of 𝑛 complex variables and present the elementary properties. A characteristic caused by increasing the number 𝑛 of variables more than one is the so-called Hartogs phenomenon in the analytic continuation, such that all holomorphic functions in a domain of C𝑛 (𝑛 ≥ 2) are simultaneously analytically continued over a strictly larger domain; such a phenomenon never occurs in one variable. We also prepare some basic notion necessary in the later chapters.

1.1 Holomorphic Functions of Several Variables 1.1.1 Open Balls and Polydisks of C𝒏 Let 𝑛 ∈ N. We denote by C𝑛 the complex vector space of the 𝑛-product of the complex plane C and by 𝑧 = (𝑧1 , . . . , 𝑧 𝑛 ) ∈ C𝑛 the natural coordinate system. We de ne the euclidean norm by p (1.1.1) ∥𝑧∥ = |𝑧1 | 2 + · · · + |𝑧 𝑛 | 2 (≥ 0). Put (1.1.2)

B(𝑎;𝑟) = {𝑧 ∈ C𝑛 : ∥𝑧 − 𝑎∥ < 𝑟 },

B(𝑟) = B(0;𝑟),

which is called an open ball or simply a ball with center 𝑎 ∈ C𝑛 and radius 𝑟 (> 0). In particular, B(1) is called the unit ball. In the case of 𝑛 = 1, we call it a disk and write (1.1.3)

𝛥(𝑎;𝑟) = {𝑧 ∈ C : |𝑧 − 𝑎| < 𝑟 },

𝛥(𝑟) = 𝛥(0;𝑟).

¯ We call 𝛥(1) the unit disk. The closure of 𝛥(𝑎;𝑟) (resp. 𝛥(𝑟)) is denoted by 𝛥(𝑎;𝑟) ¯ (resp. 𝛥(𝑟)) and called a closed disk. For a point 𝑎 = (𝑎 1 , . . . , 𝑎 𝑛 ) ∈ C𝑛 and positive numbers 𝑟 𝑗 > 0, 1 ≤ 𝑗 ≤ 𝑛, we put © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 J. Noguchi, Basic Oka Theory in Several Complex Variables, Universitext, https://doi.org/10.1007/978-981-97-2056-9_1

1

2

(1.1.4)

1 Holomorphic Functions

P𝛥(𝑎; (𝑟 𝑗 )) = {𝑧 = (𝑧 𝑗 ) ∈ C𝑛 : |𝑧 𝑗 − 𝑎 𝑗 | < 𝑟 𝑗 , 1 ≤ 𝑗 ≤ 𝑛},

which is called a polydisk of polyradius (𝑟 𝑗 ) with center at 𝑎; for 𝑎 = 0 we write P𝛥((𝑟 𝑗 )) = P𝛥(0; (𝑟 𝑗 )). With all 𝑟 𝑗 = 1 (1 ≤ 𝑗 ≤ 𝑛), P𝛥((1, . . . , 1)) is called the unit polydisk. We use the above notation all through the present book. A connected open set 𝑈 ⊂ C𝑛 is called a domain. The closure 𝑈¯ of 𝑈 is called a closed domain. For example, a polydisk P𝛥(𝑎; (𝑟 𝑗 )) is a domain, and its closure P𝛥(𝑎; (𝑟 𝑗 )) = {𝑧 = (𝑧 𝑗 ) ∈ C𝑛 : |𝑧 𝑗 − 𝑎 𝑗 | ≤ 𝑟 𝑗 , 1 ≤ 𝑗 ≤ 𝑛} is a closed domain, called a closed polydisk. A subset 𝐴 ⊂ C𝑛 is called a cylinder, if there are subsets 𝐴 𝑗 ⊂ C (1 ≤ 𝑗 ≤ 𝑛) Î such that 𝐴 = 𝑛𝑗=1 𝐴 𝑗 ; in particular, if every 𝐴 𝑗 is a domain, 𝐴 is called a cylinder domain, A subset 𝐵 ⊂ R𝑚 of the real 𝑚-dimensional vector space R𝑚 is said to be (a ne) convex if for arbitrary two points 𝑥, 𝑦 ∈ 𝐵 the line segment connecting them is contained in 𝐵, i.e., (1 − 𝑡)𝑥 + 𝑡𝑦 ∈ 𝐵, 0 ≤ 𝑡 ≤ 1. The smallest convex set containing 𝐵 is called the convex hull of 𝐵 (the existence is clear), and denoted by ch(𝐵). A subset 𝐴 ⊂ C𝑛 is said to be convex if with the natural identi cation C𝑛  R2𝑛 , 𝐴 is convex in R2𝑛 . Proposition 1.1.5. Let 𝐸 ⋐ C be a compact convex subset with a neighborhood 𝑈 ⋑ 𝐸. Then there is an open convex polygon 𝐺 such that 𝐸 ⋐ 𝐺 ⋐ 𝑈. Proof. Without loss of generality, 𝑈 may be assumed to be bounded. Since 𝐸 is convex, there is a unique line segment ℓ0 of the shortest distance connecting a point 𝑝 ∈ 𝜕𝑈 and 𝐸 (i.e., points of 𝐸). Let ℓ1 be the line passing through the middle point of ℓ0 and orthogonal to ℓ0 . Since 𝐸 ∩ ℓ1 = ∅, ℓ1 divides the plane C to two half-planes, one of which contains 𝐸, and the other has no intersection with 𝐸 (cf. Fig. 1.1). The latter is denoted by 𝐻, so that 𝐻 ∩ 𝜕𝑈 ∋ 𝑝. Since 𝜕𝑈 is compact, there are nitely

Fig. 1.1 Convex polygon neighborhood.

1.1 Holomorphic Functions of Several Variables

3

many such half-planes 𝐻1 , . . . , 𝐻 𝑘 satisfying 𝜕𝑈 ⋐

𝑘 Ø

𝐻ℎ ,

𝐸⋐

ℎ=1

Then we get 𝐺 :=

Ñ𝑘

ℎ=1 (C \ 𝐻 ℎ )

¯

𝑘 Ù

(C \ 𝐻¯ ℎ ) ⋐ 𝑈.

ℎ=1

with 𝐸 ⋐ 𝐺 ⋐ 𝑈.

⊓ ⊔

1.1.2 Definition of Holomorphic Functions We consider a function 𝜑 : 𝐴 → C de ned on a subset 𝐴 ⊂ C𝑛 . We say that 𝜑 is continuous if for every point 𝑎 ∈ 𝐴 and every 𝜀 > 0 there is a polyradius (𝛿 𝑗 ) satisfying |𝜑(𝑧) − 𝜑(𝑎)| < 𝜀, ∀ 𝑧 ∈ P𝛥(𝑎; (𝛿 𝑗 )) ∩ 𝐴. Let 𝑥 𝑗 = ℜ𝑧 𝑗 and 𝑦 𝑗 = ℑ𝑧 𝑗 be the real and imaginary parts of the complex variables 𝑧 𝑗 = 𝑥 𝑗 + 𝑖𝑦 𝑗 , 1 ≤ 𝑗 ≤ 𝑛, respectively. Then the holomorphic partial differential operators and anti-holomorphic partial differential operators are de ned respectively by   𝜕 1 𝜕 1 𝜕 (1.1.6) = + , 1 ≤ 𝑗 ≤ 𝑛, 𝜕𝑧 𝑗 2 𝜕𝑥 𝑗 𝑖 𝜕𝑦 𝑗   𝜕 1 𝜕 1 𝜕 = − , 1 ≤ 𝑗 ≤ 𝑛. 𝜕 𝑧¯ 𝑗 2 𝜕𝑥 𝑗 𝑖 𝜕𝑦 𝑗 A function 𝜑(𝑧1 , . . . , 𝑧 𝑛 ) is said to be of 𝐶 𝑙 -class or a 𝐶 𝑙 function (𝑙 ∈ N ∪ {∞}) when it is of 𝐶 𝑙 -class as a function of the real and imaginary parts, (𝑥1 , 𝑦 1 , . . . , 𝑥 𝑛 , 𝑦 𝑛 ). Further, if 𝑧 𝑗 (𝜉) = 𝑧 𝑗 (𝜉1 , . . . , 𝜉 𝑚 ) are 𝐶 1 functions of 𝑚 complex variables 𝜉 = (𝜉1 , . . . , 𝜉 𝑚 ), we have (1.1.7)

 𝑛  𝜕𝑧 𝑗 𝜕 𝑧¯ 𝑗 𝜕𝜑(𝑧(𝜉)) Õ 𝜕𝜑 𝜕𝜑 = (𝑧(𝜉)) · (𝜉) + (𝑧(𝜉)) · (𝜉) , 𝜕𝜉 𝑘 𝜕𝑧 𝑗 𝜕𝜉 𝑘 𝜕 𝑧¯ 𝑗 𝜕𝜉 𝑘 𝑗=1  𝑛  𝜕𝑧 𝑗 𝜕 𝑧¯ 𝑗 𝜕𝜑(𝑧(𝜉)) Õ 𝜕𝜑 𝜕𝜑 = (𝑧(𝜉)) · (𝜉) + (𝑧(𝜉)) · (𝜉) , 𝜕𝑧 𝑗 𝜕 𝑧¯ 𝑗 𝜕 𝜉¯𝑘 𝜕 𝜉¯𝑘 𝜕 𝜉¯𝑘 𝑗=1

which follow from the formula of the di erentiation of composed functions of real variables. For a multi-index 𝛼 = (𝛼1 , . . . , 𝛼𝑛 ) ∈ Z+𝑛 we set |𝛼| =

𝑛 Õ 𝑗=1

𝛼𝑗,

𝛼! =

𝑛 Ö 𝑗=1

𝛼𝑗!.

4

1 Holomorphic Functions

We de ne the holomorphic partial di erential operator of (multi-) order 𝛼 by (1.1.8)

𝜕 𝛼 = 𝜕𝑧𝛼 =

𝜕 | 𝛼| . 𝜕𝑧 1𝛼1 · · · 𝜕𝑧 𝑛𝛼𝑛

Let 𝛺 ⊂ C𝑛 be an open subset. Definition 1.1.9 (Holomorphic function). (i) A holomorphic function 𝑓 : 𝛺 → C on 𝛺 is a function of 𝐶 1 -class satisfying the so-called Cauchy–Riemann equations, (1.1.10)

𝜕𝑓 (𝑧) = 0, 𝜕 𝑧¯ 𝑗

1 ≤ 𝑗 ≤ 𝑛.

(ii) In general, a function 𝑓 : 𝛺 → C is separately holomorphic or separately analytic if in a neighborhood of every point 𝑎 = (𝑎 1 , . . . , 𝑎 𝑛 ) ∈ 𝛺, the function 𝑧 𝑗 ↦−→ 𝑓 (𝑎 1 , . . . , 𝑎 𝑗 −1 , 𝑧 𝑗 , 𝑎 𝑗+1 , . . . , 𝑎 𝑛 ) of one variable 𝑧 𝑗 with xed others is holomorphic in a neighborhood of 𝑧 𝑗 = 𝑎 𝑗. Proposition 1.1.11. If 𝜑(𝑧) and 𝑧 𝑗 (𝜉) in (1.1.7) are holomorphic, then the composed function 𝜑(𝑧(𝜉)) is holomorphic in 𝜉, and satisfies 𝑛 𝜕𝑧 𝑗 𝜕𝜑(𝑧(𝜉)) Õ 𝜕𝜑 = (𝑧(𝜉)) · (𝜉), 𝜕𝜉 𝑘 𝜕𝑧 𝜕𝜉 𝑗 𝑘 𝑗=1

Proof. It is immediate by (1.1.7) and (1.1.10).

1 ≤ 𝑘 ≤ 𝑚. ⊓ ⊔

Example 1.1.12. A polynomial 𝑃(𝑧) in variables 𝑧 = (𝑧 1 , 𝑧 2 , . . . , 𝑧 𝑛 ) with coe cients in C de nes naturally a holomorphic function in C𝑛 . We denote by C[(𝑧 𝑗 )] := C[𝑧1 , . . . , 𝑧 𝑛 ] the set of of all polynomials in 𝑧 with complex coe cients. As shown in what follows, holomorphic functions are expanded to convergent power series in a neighborhood of each point where they are de ned (Theorem 1.1.34), and hence analytic in all variables. Therefore, holomorphic functions are often called analytic functions; here we use mainly “holomorphic” functions. We denote by 𝒪(𝛺) the set of all holomorphic functions on 𝛺. We then have the natural inclusions: C[𝑧1 , . . . , 𝑧 𝑛 ] ⊂ 𝒪(C𝑛 ) ⊂ 𝒪(𝛺). When a point 𝑎 ∈ C𝑛 is contained in the a set 𝐴 where a function 𝑓 is de ned, 𝑓 being “holomorphic at 𝑎” means that there are a neighborhood 𝑉 (⊂ C𝑛 ) of 𝑎 and 𝑔 ∈ 𝒪(𝑉) with 𝑔| 𝐴∩𝑉 = 𝑓 | 𝐴∩𝑉 . Theorem 1.1.13 (Cauchy’s integral formula). Let 𝑓 (𝑧) be a continuous function in a neighborhood of a closed polydisk P𝛥(𝑎; (𝑟 𝑗 )). If 𝑓 (𝑧) is separately holomorphic in P𝛥(𝑎; (𝑟 𝑗 )), then the following integral formula holds:

1.1 Holomorphic Functions of Several Variables

 (1.1.14)

𝑓 (𝑧 1 , . . . , 𝑧 𝑛 ) =

1 2𝜋𝑖

𝑛

5

∫

∫ ···

| 𝜁1 −𝑎1 |=𝑟1

| 𝜁𝑛 −𝑎𝑛 |=𝑟𝑛

𝑓 (𝜁1 , . . . , 𝜁 𝑛 ) 𝑑𝜁1 · · · 𝑑𝜁 𝑛 , (𝜁1 − 𝑧 1 ) · · · (𝜁 𝑛 − 𝑧 𝑛 )

𝑧 ∈ P𝛥(𝑎; (𝑟 𝑗 )).

Proof. Repeat Cauchy’s integral formula of one variable. The function

⊓ ⊔

1 (𝜁1 − 𝑧1 ) · · · (𝜁 𝑛 − 𝑧 𝑛 )

in (1.1.14) is called the Cauchy kernel (of 𝑛 variables). Theorem 1.1.15. Let 𝛺 ⊂ C𝑛 be an open set. (i) A continuous function 𝑓 : 𝛺 → C is holomorphic if and only if it is separately holomorphic. (ii) Any holomorphic function 𝑓 ∈ 𝒪(𝛺) is of 𝐶 ∞ -class. (iii) Let 𝑓 ∈ 𝒪(𝛺) and let 𝑧 ∈ P𝛥(𝑎; (𝑟 𝑗 )) ⋐ 𝛺. For a multi-index 𝛼 = (𝛼 𝑗 ) we have  𝑛 ∫ ∫ 1 (1.1.16) 𝜕 𝛼 𝑓 (𝑧1 , . . . , 𝑧 𝑛 ) = 𝛼! ··· 2𝜋𝑖 | 𝜁1 −𝑎1 |=𝑟1

| 𝜁𝑛 −𝑎𝑛 |=𝑟𝑛

𝑓 (𝜁1 , . . . , 𝜁 𝑛 ) 𝑑𝜁1 · · · 𝑑𝜁 𝑛 . (𝜁1 − 𝑧1 ) 𝛼1 +1 · · · (𝜁 𝑛 − 𝑧 𝑛 ) 𝛼𝑛 +1 Proof. (i), (ii): These are immediate from Theorem 1.1.13. (iii): From integral formula (1.1.14) it follows, e.g., that 𝜕 𝑓 (𝑧 1 , . . . , 𝑧 𝑛 ) 𝜕𝑧 1  𝑛 ∫ 1 = 2𝜋𝑖

∫ ···

| 𝜁1 −𝑎1 |=𝑟1

| 𝜁𝑛 −𝑎𝑛 |=𝑟𝑛

𝑓 (𝜁1 , . . . , 𝜁 𝑛 ) 𝑑𝜁1 · · · 𝑑𝜁 𝑛 . (𝜁1 − 𝑧1 ) 2 · · · (𝜁 𝑛 − 𝑧 𝑛 ) ⊓ ⊔

Repeating this, we obtain (1.1.16).

Remark 1.1.17. (i) In (i) of the above theorem the continuity condition for 𝑓 is, in fact, unnecessary, but the proof is not as easy as above (see Hartogs’ Theorem 4.2.9). (ii) In the category of real analytic functions, the separate analyticity does not imply even the continuity. Consider the following function of (𝑥, 𝑦) ∈ R2 : 𝑓 (𝑥, 𝑦) =

    0,

𝑥𝑦   𝑥2 + 𝑦2 , 

(𝑥, 𝑦) = (0, 0), (𝑥, 𝑦) ≠ (0, 0).

6

1 Holomorphic Functions

Then, 𝑓 (𝑥, 𝑦) is bounded and analytic in one variable with any xed other, but it is not continuous at (0, 0). For lim 𝑥→0 𝑓 (𝑥, 𝑘𝑥) = 𝑘/(1 + 𝑘 2 ) with 𝑦 = 𝑘𝑥 (𝑘 ∈ R), while 𝑓 (0, 0) = 0 by de nition. (iii) Furthermore, in the category of real analytic functions, the continuity and the separate analyticity do not imply the analyticity in general; for example, we take  (𝑥, 𝑦) = (0, 0),   0,  𝑔(𝑥, 𝑦) = 𝑥 2 𝑦 2   2 2 , (𝑥, 𝑦) ≠ (0, 0). 𝑥 +𝑦 Then 𝑔(𝑥, 𝑦) is continuous in R2 and separately analytic. But it is not analytic at the origin. For, if 𝑔(𝑥, 𝑦) is analytic at (0, 0), then 𝑔(0, 𝑦) = 0. We would have 𝑔(𝑥, 𝑦) = 𝑥𝑔1 (𝑥, 𝑦), where 𝑔1 (𝑥, 𝑦) should be analytic (same in the sequel). Since 𝑔1 (𝑥, 0) = 0, we get similarly 𝑔1 (𝑥, 𝑦) = 𝑦𝑔2 (𝑥, 𝑦). Thus, 𝑔(𝑥, 𝑦) = 𝑥𝑦𝑔2 (𝑥, 𝑦), and hence 𝑔2 (𝑥, 𝑦) = 𝑥𝑦/(𝑥 2 + 𝑦 2 ). A similar argument would imply 𝑔2 (𝑥, 𝑦) = 𝑥𝑦𝑔3 (𝑥, 𝑦), so that 𝑔3 (𝑥, 𝑦) = 1/(𝑥 2 + 𝑦 2 ) should be analytic; this is absurd.

1.1.3 Sequences and Series of Functions 𝑛 For a sequence of functions { 𝑓 𝜈 }∞ 𝜈=0 de ned on a subset 𝐴 ⊂ C with

𝑓 𝜈 : 𝐴 → C,

𝜈 = 0, 1, . . . ,

we de ne the notions of point-convergence, Cauchy sequence, (locally) uniform convergence, (locally) uniform Cauchy sequence in the same way as in the case of one variable; when 𝐴 is open, the locally uniform convergence is equivalent to the convergence uniform on every compact subset of 𝐴. In particular we state: Theorem 1.1.18. If a series { 𝑓 𝜈 }∞ 𝜈=0 of holomorphic functions defined on an open subset 𝐴 ⊂ C𝑛 converges locally uniformly, then the limit 𝑓 (𝑧) = lim𝜈→∞ 𝑓 𝜈 (𝑧) is a holomorphic function on 𝐴. Proof. Firstly, 𝑓 (𝑧) is continuous. For an arbitrary point 𝑎 ∈ 𝐴, we take a polydisk neighborhood P𝛥(𝑎; (𝑟 𝑗 )) ⋐ 𝐴 and apply (1.1.14) for 𝑓 𝜈 (𝑧). Since { 𝑓 𝜈 (𝑧)}∞ 𝜈=0 converges to 𝑓 (𝑧) uniformly on P𝛥(𝑎; (𝑟 𝑗 )), 𝑓 (𝑧) also satis es  𝑛 ∫ 1 𝑓 (𝑧1 , . . . , 𝑧 𝑛 ) = 2𝜋𝑖 | 𝜁1 −𝑎1 |=𝑟1 ∫ 𝑓 (𝜁1 , . . . , 𝜁 𝑛 ) ··· 𝑑𝜁1 · · · 𝑑𝜁 𝑛 , | 𝜁𝑛 −𝑎𝑛 |=𝑟𝑛 (𝜁1 − 𝑧 1 ) · · · (𝜁 𝑛 − 𝑧 𝑛 ) 𝑧 ∈ P𝛥(𝑎; (𝑟 𝑗 )).

1.1 Holomorphic Functions of Several Variables

7

The integrand of the right-hand side of the equation above is holomorphic in 𝑧 ∈ P𝛥(𝑎; (𝑟 𝑗 )), and so is 𝑓 (𝑧). ⊓ ⊔ With a given sequence of functions { 𝑓 𝜈 (𝑧)}∞ 𝜈=0 on 𝐴, the formal sum ∞ Õ

𝑓 𝜈 (𝑧)

𝜈=0

is called a series of functions on 𝐴, and for 𝑁 ∈ Z+ 𝑠 𝑁 (𝑧) =

𝑁 Õ

𝑓 𝜈 (𝑧),

𝑧∈𝐴

𝜈=0 ∞ is called the (𝑁th) partial sum. When Í∞ the sequence {𝑠 𝑁 (𝑧)} 𝑁 =0 of the partial sums converges, the series of functions 𝜈=0 𝑓 𝜈 (𝑧) is said to be convergent, and the limit is written as ∞ Õ 𝑓 𝜈 (𝑧) = lim 𝑠 𝑁 (𝑧), 𝑧 ∈ 𝐴. 𝑁 →∞

𝜈=0

The (locally) uniform convergence is de ned similarly. If Í {𝑠 𝑁 (𝑧)}∞ 𝑁 =0 is a (uniform) Cauchy sequence, we say that the series of functions ∞ 𝑓 (𝑧) satis es the 𝜈 𝜈=0 (uniform) Cauchy condition. Í We say that ∞ 𝜈=0 𝑓 𝜈 (𝑧) converges absolutely, if ∞ Õ

| 𝑓 𝜈 (𝑧)|,

𝑧∈𝐴

𝜈=0

converges; if so, the original series of functions converges and the limit is independent from the Íchoice of orders of terms. Let ∞ 𝜈=0 𝑓 𝜈 (𝑧) (𝑧 ∈ 𝐴) be a series of functions. If non-negative constants 𝑀𝜈 ∈ R+ , 𝜈 = 0, 1, . . . satisfy | 𝑓 𝜈 (𝑧)| ≤ 𝑀𝜈 ,

∀ 𝑧 ∈ 𝐴, 𝜈 = 0, 1, . . . ,

Í∞ Í∞ Í∞ the series Í 𝜈=0 𝑀𝜈 is called a majorant of 𝜈=0 𝑓 𝜈 (𝑧). If 𝜈=0 𝑀𝜈 < ∞ (convergent), we say that ∞ 𝜈=0 𝑓 𝜈 (𝑧) has a convergent majorant. Theorem 1.1.19. If a series of functions has a convergent majorant, it converges absolutely and uniformly. Proof. Use the uniform Cauchy condition.

⊓ ⊔

8

1 Holomorphic Functions

1.1.4 Power Series of Several Variables With a point 𝑎 = (𝑎 𝑗 ) ∈ C𝑛 and a multi-index 𝛼 = (𝛼1 , . . . , 𝛼𝑛 ) ∈ Z+𝑛 we set (𝑧 − 𝑎) 𝛼 =

(1.1.20)

𝑛 Ö

(𝑧 𝑗 − 𝑎 𝑗 ) 𝛼 𝑗 .

𝑗=1

For a polyradius 𝑟 = (𝑟 1 , . . . , 𝑟 𝑛 ) (𝑟 𝑗 > 0) we get 𝑟 𝛼 = 𝑟 1𝛼1 · · · 𝑟 𝑛𝛼𝑛 .

(1.1.21) A series of functions of type

Õ

𝑓 (𝑧) =

(1.1.22)

𝛼∈Z+𝑛

𝑐 𝛼 (𝑧 − 𝑎) 𝛼 ,

𝑐𝛼 ∈ C

is called a power series (of 𝑛 variables) with center 𝑎. While the convergence of (1.1.22) does not make sense unless the order of terms is given, it is noticed that the absolute convergence is well-de ned, independently from the choice of orders of terms. For a moment, we set 𝑎 = 0. Í Lemma 1.1.23. (i) If there is a point 𝑧 = (𝑧 𝑗 ) ∈ (C∗ ) 𝑛 such that 𝛼∈Z+𝑛 𝑐 𝛼 𝑧 𝛼 converges with respect to some order, then {𝑐 𝛼 𝑧 𝛼 : 𝛼 ∈ Z+𝑛 } is bounded. Í (ii) If {𝑐 𝛼 𝑤 𝛼 : 𝛼 ∈ Z+𝑛 } is bounded for a point 𝑤 = (𝑤 𝑗 ) ∈ (C∗ ) 𝑛 , then 𝛼∈Z+𝑛 𝑐 𝛼 𝑧 𝛼 converges absolutely and locally uniformly in the polydisk P𝛥((|𝑤 𝑗 |)) = {(𝑧 𝑗 ) : |𝑧 𝑗 | < |𝑤 𝑗 |, 1 ≤ 𝑗 ≤ 𝑛}. Proof. (i) is clear. (ii) By the assumption there is an 𝑀 > 0 such that |𝑐 𝛼 𝑤 𝛼 | < 𝑀,

∀𝛼 ∈ Z+𝑛 .

With 0 < 𝜃 < 1 we set |𝑧 𝑗 | ≤ 𝜃|𝑤 𝑗 | (1 ≤ 𝑗 ≤ 𝑛). Then, Õ Õ |𝑐 𝛼 𝑧 𝛼 | ≤ |𝑐 𝛼 𝑤 𝛼 | · 𝜃 | 𝛼| 𝛼∈Z+𝑛

𝛼∈Z+𝑛

≤𝑀

Õ

𝛼1 ≥0,..., 𝛼𝑛 ≥0

Hence by Theorem 1.1.19 in P𝛥((|𝑤 𝑗 |)).

Í

𝛼∈Z+𝑛

𝜃 𝛼1 · · · 𝜃 𝛼𝑛 = 𝑀



1 1−𝜃

𝑛

< ∞.

𝑐 𝛼 𝑧 𝛼 converges absolutely and locally uniformly ⊓ ⊔

If a power series (1.1.22) satis es the condition of Lemma 1.1.23 (i), 𝑓 (𝑧) is called a convergent power series: In this case we set

1.1 Holomorphic Functions of Several Variables

(1.1.24)

9

𝛺 ∗ ( 𝑓 ) = {𝑟 = (𝑟 𝑗 ) ∈ (R+ \ {0}) 𝑛 : |𝑐 𝛼 𝑟 𝛼 |, 𝛼 ∈ Z+𝑛 , are bounded}◦ , 𝛺( 𝑓 ) = {(𝑧 𝑗 ) ∈ C𝑛 : ∃ (𝑟 𝑗 ) ∈ 𝛺 ∗ ( 𝑓 ), |𝑧 𝑗 | < 𝑟 𝑗 , 1 ≤ 𝑗 ≤ 𝑛},

log 𝛺 ∗ ( 𝑓 ) = {(log𝑟 𝑗 ) ∈ R𝑛 : (𝑟 𝑗 ) ∈ 𝛺 ∗ ( 𝑓 )}.

Here {·}◦ denotes the subset of the interior points. We call 𝛺( 𝑓 ) the domain of convergence of the power series 𝑓 (𝑧). Í Theorem 1.1.25. Let 𝑓 (𝑧) = 𝛼∈Z+𝑛 𝑐 𝛼 𝑧 𝛼 be a convergent power series. Then: (i) 𝑓 (𝑧) is holomorphic in 𝛺( 𝑓 ). (ii) (Fabry; logarithmic convexity) log 𝛺 ∗ ( 𝑓 ) is a convex set. (iii) 𝑓 (𝑧) is termwise partially differentiable in 𝛺( 𝑓 ), and satisfies (1.1.26)

𝜕𝑓 (𝑧) = 𝜕𝑧 𝑗 𝛼

Õ

1 ≥0,..., 𝛼 𝑗 ≥1,..., 𝛼𝑛 ≥0

(1.1.27)

𝛺( 𝑓 ) ⊂ 𝛺



𝛼 −1

𝛼 𝑗 𝑐 𝛼1 ... 𝛼 𝑗 ... 𝛼𝑛 𝑧 1𝛼1 · · · 𝑧 𝑗 𝑗

· · · 𝑧 𝑛𝛼𝑛 ,

 𝜕𝑓 . 𝜕𝑧 𝑗

Proof. (i) Note that 𝑐 𝛼 𝑧 𝛼 is holomorphic. It follows from Theorem 1.1.18 and Lemma 1.1.23 (ii) that 𝑓 (𝑧) is holomorphic in 𝛺( 𝑓 ). (ii) Take two points (log𝑟 𝑗 ), (log 𝑠 𝑗 ) ∈ log 𝛺 ∗ ( 𝑓 ). Since they are interior points, there exist some 𝑟 ′𝑗 > 𝑟 𝑗 , 𝑠′𝑗 > 𝑠 𝑗 (1 ≤ 𝑗 ≤ 𝑛) and 𝑀 > 0 such that with 𝑟 ′ = (𝑟 ′𝑗 ) and 𝑠′ = (𝑠′𝑗 ) |𝑐 𝛼 𝑟 ′ 𝛼 | ≤ 𝑀, |𝑐 𝛼 𝑠′ 𝛼 | ≤ 𝑀, ∀𝛼 ∈ Z+𝑛 . It su ces to show that for every 𝜃 with 0 < 𝜃 < 1 𝜃 (log𝑟 𝑗 ) + (1 − 𝜃) (log 𝑠 𝑗 ) ∈ log 𝛺 ∗ ( 𝑓 ).

(1.1.28) For 𝑟 ′ and 𝑠′ we have

|𝑐 𝛼 𝑟 ′ 𝜃 𝛼 𝑠′ (1− 𝜃 ) 𝛼 | = (|𝑐 𝛼 |𝑟 ′ 𝛼 ) 𝜃 · (|𝑐 𝛼 |𝑠′ 𝛼 ) 1− 𝜃 ≤ 𝑀 𝜃 · 𝑀 1− 𝜃 = 𝑀. Therefore, (1.1.28) follows. (iii) The proof of termwise partial di erentiability is the same as in the case of one variable (cf., e.g., [38] Theorem (3.1.10)). We show (1.1.27). Take an arbitrary point (𝑧 𝑘 ) ∈ 𝛺( 𝑓 ). By de nition there is some (𝑟 𝑘 ) ∈ 𝛺 ∗ ( 𝑓 ) and a constant 𝑀 > 0 such that |𝑧 𝑘 | < 𝑟 𝑘 , 1 ≤ 𝑘 ≤ 𝑛, and |𝑐 𝛼 𝑟 𝛼 | ≤ 𝑀,

∀𝛼 ∈ Z+𝑛 .

Take (𝑡 𝑘 ) so that |𝑧 𝑘 | < 𝑡 𝑘 < 𝑟 𝑘 (1 ≤ 𝑘 ≤ 𝑛). For every 𝛼 = (𝛼 𝑘 ) ∈ Z+𝑛 , 𝛼 𝑗 ≥ 1, we have   𝛼 𝑗 𝑡 𝑗 𝛼𝑗 𝛼 −1 |𝛼 𝑗 𝑐 𝛼 |𝑡1𝛼1 · · · 𝑡 𝑗 𝑗 · · · 𝑡 𝑛𝛼𝑛 ≤ |𝑐 𝛼 |𝑟 𝛼 . 𝑡𝑗 𝑟𝑗

10

1 Holomorphic Functions

Because 0 < 𝑡 𝑗 /𝑟 𝑗 < 1, lim 𝛼 𝑗 →∞ 𝛼 𝑗 (𝑡 𝑗 /𝑟 𝑗 ) 𝛼 𝑗 = 0. Thus there is some 𝐿 > 0 such that  𝛼𝑗

𝑡𝑗 𝑟𝑗

 𝛼𝑗

Hence

≤ 𝐿,

𝛼 −1

|𝛼 𝑗 𝑐 𝛼 |𝑡1𝛼1 · · · 𝑡 𝑗 𝑗 It follows that (𝑡 𝑘 ) ∈ 𝛺 ∗



𝜕𝑓 𝜕𝑧 𝑗



∀𝛼 𝑗 ≥ 1.

· · · 𝑡 𝑛𝛼𝑛 ≤

, and (𝑧 𝑘 ) ∈ 𝛺



𝜕𝑓 𝜕𝑧 𝑗



𝐿𝑀 . 𝑡𝑗 ⊓ ⊔

.

Because of the property (ii) above we say that 𝛺 ∗ ( 𝑓 ) is logarithmically convex.

1.1.5 Elementary Properties of Holomorphic Functions of Several Variables Let 𝛺 be a domain and let 𝑓 ∈ 𝒪(𝛺). We take arbitrarily a closed polydisk P𝛥(𝑎; (𝑟 𝑗 )) ⋐ 𝛺. For the sake of simplicity we assume 𝑎 = 0 by a parallel translation of the coordinates. By the Cauchy integral formula (1.1.14),  𝑛 ∫ ∫ 1 𝑓 (𝜁1 , . . . , 𝜁 𝑛 ) (1.1.29) 𝑓 (𝑧) = ··· 𝑑𝜁1 · · · 𝑑𝜁 𝑛 , 2𝜋𝑖 (𝜁1 − 𝑧1 ) · · · (𝜁 𝑛 − 𝑧 𝑛 ) | 𝜁1 |=𝑟1

| 𝜁𝑛 |=𝑟𝑛

𝑧 = (𝑧 1 , . . . , 𝑧 𝑛 ) ∈ P𝛥((𝑟 𝑗 )). We expand the Cauchy kernel in the integrand above as follows:

(1.1.30)

1 1    =  (𝜁1 − 𝑧1 ) · · · (𝜁 𝑛 − 𝑧 𝑛 ) 𝜁 1 − 𝑧1 · · · 𝜁 1 − 𝑧𝑛 1 𝑛 𝜁1 𝜁𝑛   𝛼1   𝛼𝑛 Õ 1 𝑧1 1 𝑧𝑛 = ··· . 𝜁 𝜁 𝜁 1 1 𝑛 𝜁𝑛 𝛼 ≥0,..., 𝛼 ≥0 𝑛

1

Since |𝑧 𝑗 /𝜁 𝑗 | = |𝑧 𝑗 |/𝑟 𝑗 < 1 (1 ≤ 𝑗 ≤ 𝑛), the power series (1.1.30) converges absolutely, and locally uniformly in P𝛥((𝑟 𝑗 )). Together with (1.1.29) we obtain Õ (1.1.31) 𝑓 (𝑧) = 𝑐 𝛼 𝑧 𝛼, 𝛼∈Z+𝑛



(1.1.32)

1 𝑐𝛼 = 2𝜋𝑖

𝑛 ∫ | 𝜁1 |=𝑟1

∫ ···

𝑓 (𝜁1 , . . . , 𝜁 𝑛 )

𝜁 𝛼1 +1 · · · 𝜁 𝑛𝛼𝑛 +1 | 𝜁𝑛 |=𝑟𝑛 1

𝛼 = (𝛼 𝑗 ) ∈ Z+𝑛 .

𝑑𝜁1 · · · 𝑑𝜁 𝑛 ,

1.1 Holomorphic Functions of Several Variables

11

For a holomorphic partial di erential operator 𝜕 𝛼 we have (1.1.33)

𝜕 𝛼 𝑓 (0) = 𝛼! 𝑐 𝛼 ,

𝛼 ∈ Z+𝑛 .

Therefore the coe cients 𝑐 𝛼 are uniquely determined by 𝑓 (𝑧). Theorem 1.1.34 (Power Series Expansion (or Development)). Let 𝑓 ∈ 𝒪(𝛺) and P𝛥(𝑎; (𝑟 𝑗 )) ⊂ 𝛺. Then 𝑓 (𝑧) is uniquely expanded in P𝛥(𝑎; (𝑟 𝑗 )) to an absolutely and locally uniformly convergent power series Õ (1.1.35) 𝑓 (𝑧) = 𝑐 𝛼 (𝑧 − 𝑎) 𝛼 , 𝛼∈Z+𝑛

𝜕 𝛼 𝑓 (𝑎) = 𝛼! 𝑐 𝛼 ,

𝛼 ∈ Z+𝑛 .

Proof. Take arbitrarily 0 < 𝑟 ′𝑗 < 𝑟 𝑗 (1 ≤ 𝑗 ≤ 𝑛). Then, P𝛥(𝑎; (𝑟 ′𝑗 )) ⊂ 𝛺, and so by (1.1.31) and (1.1.33) the theorem holds on P𝛥(𝑎; (𝑟 ′𝑗 )). Since the coe cients 𝑐 𝛼 of the power series expansion are unique and independent from the choices of 𝑟 ′𝑗 , we deduce the theorem on P𝛥(𝑎; (𝑟 𝑗 )) by letting 𝑟 ′𝑗 ↗ 𝑟 𝑗 . ⊓ ⊔ We present (1.1.35) as a series of homogeneous polynomials as follows: Õ 𝑃 𝜈 (𝑧 − 𝑎) = 𝑐 𝛼 (𝑧 − 𝑎) 𝛼 , 𝛼∈Z+𝑛 , | 𝛼|=𝜈

(1.1.36)

𝑓 (𝑧) =

∞ Õ

𝑃 𝜈 (𝑧 − 𝑎),

𝜈=0

which is called a homogeneous polynomial expansion of 𝑓 (𝑧). Remark 1.1.37. A homogeneous polynomial expansion is a series with a given order: It is noted that the domain of convergence of (1.1.36) can be di erent from that of (1.1.35) (cf. Exercise 4 at the end of this chapter). A holomorphic function in C𝑛 is called an entire function. By Theorem 1.1.34 we immediately see: Corollary 1.1.38. An entire function 𝑓 (𝑧) is developed in C𝑛 to Õ (1.1.39) 𝑓 (𝑧) = 𝑐 𝛼 𝑧 𝛼, 𝛼∈Z+𝑛

where the convergence is absolute and locally uniform. Theorem 1.1.40. Let 𝛺 be a domain. Let 𝐾 ⋐ 𝛺 be a compact subset and let 𝑈 be an open subset with 𝐾 ⋐ 𝑈 ⋐ 𝛺. Then there is a positive constant 𝐶 depending only on 𝐾,𝑈 and 𝛼 ∈ Z+𝑛 such that |𝜕 𝛼 𝑓 (𝑎)| ≤ 𝐶 sup | 𝑓 (𝑧)|, 𝑧 ∈𝑈

∀ 𝑎 ∈ 𝐾, ∀ 𝑓 ∈ 𝒪(𝛺).

12

1 Holomorphic Functions

Proof. Take a su ciently small polyradius 𝑟 = (𝑟 𝑗 ) so that P𝛥(𝑎; (𝑟 𝑗 )) ⋐ 𝑈,

∀ 𝑎 ∈ 𝐾.

With 𝑎 ∈ 𝐾, (1.1.33) and (1.1.32) imply  ∫ 1 𝑛 ∫ 𝛼 (1.1.41) |𝜕 𝑓 (𝑎)| = 𝛼! ··· 2𝜋𝑖 | 𝜁1 −𝑎1 |=𝑟1 | 𝜁𝑛 −𝑎𝑛 |=𝑟𝑛

𝑓 (𝜁1 , . . . , 𝜁 𝑛 ) 𝑑𝜁 · · · 𝑑𝜁 1 𝑛 (𝜁1 − 𝑎 1 ) 𝛼1 +1 · · · (𝜁 𝑛 − 𝑎 𝑛 ) 𝛼𝑛 +1

≤

𝛼! sup | 𝑓 (𝑧)|, 𝑟 𝛼 𝑧 ∈𝑈

Therefore it su cient to set 𝐶 =

𝑟 𝛼 = 𝑟 1𝛼1 · · · 𝑟 𝑛𝛼𝑛 .

𝛼! 𝑟𝛼 .

⊓ ⊔

Theorem 1.1.42 (Liouville). A bounded entire function is a constant. Proof. By Corollary 1.1.38 the entire function 𝑓 (𝑧) is developed in C𝑛 to an absolutely and locally uniformly convergent power series: Õ (1.1.43) 𝑓 (𝑧) = 𝑐𝛼𝑧𝛼. 𝛼∈Z+𝑛

If | 𝑓 (𝑧)| ≤ 𝑀 (𝑧 ∈ C𝑛 ) with a constant 𝑀 (> 0), (1.1.33) and (1.1.41) imply (1.1.44)

|𝑐 𝛼 | ≤

𝑀 , 𝑟 1𝛼1 · · · 𝑟 𝑛𝛼𝑛

where P𝛥((𝑟 𝑗 )) ⊂ C𝑛 is arbitrary. For any multi-index 𝛼 (|𝛼| > 0) there is an index 𝑗 with 𝛼 𝑗 > 0, and then by letting 𝑟 𝑗 ↗ ∞ we see that 𝑐 𝛼 = 0. Therefore, 𝑓 (𝑧) = 𝑐 0 , and hence 𝑓 (𝑧) is constant. ⊓ ⊔ Theorem 1.1.45 (Montel). If a sequence of holomorphic functions on 𝛺 is uniformly bounded, then it has a subsequence which converges locally uniformly in 𝛺. Proof. Use the Ascoli Arzelà Theorem and Theorem 1.1.40 with |𝛼| = 1 (similarly to the case of one variable: cf., e.g., [38] Theorem (6.4.2)). ⊓ ⊔ Theorem 1.1.46 (Identity Theorem). Let 𝛺 be a domain and let 𝑓 ∈ 𝒪(𝛺). Then the following three conditions are equivalent: (i) 𝑓 ≡ 0. (ii) There is a non-empty open set 𝑈 ⊂ 𝛺 such that 𝑓 |𝑈 ≡ 0. (iii) There is a point 𝑎 ∈ 𝛺 such that 𝜕 𝛼 𝑓 (𝑎) = 0 for all 𝛼 ∈ Z+𝑛 . Proof. The implication relations (i)⇒(ii)⇒(iii) (𝑎 ∈ 𝑈) will be clear. (iii)⇒(i) Take an arbitrary polydisk neighborhood P𝛥(𝑎; (𝑟 𝑗 )) ⊂ 𝛺 of 𝑎. Then Theorem 1.1.34 implies

1.1 Holomorphic Functions of Several Variables

𝑓 (𝑧) =

(1.1.47)

Õ 𝜕 𝛼 𝑓 (𝑎) 𝛼

𝛼!

(𝑧 − 𝑎) 𝛼 ≡ 0,

13

𝑧 ∈ P𝛥(𝑎; (𝑟 𝑗 )).

We set 𝑉 = {𝑧 ∈ 𝛺 : ∃ P𝛥(𝑧; (𝑠 𝑗 )) ⊂ 𝛺, 𝑓 | P𝛥(𝑧;(𝑠 𝑗 ) ) ≡ 0}. It follows from the de nition and (1.1.47) that 𝑉 is open and 𝑉 ≠ ∅. On the other hand, by (1.1.47) we write Ù 𝑉= {𝑧 ∈ 𝛺 : 𝜕 𝛼 𝑓 (𝑧) = 0}, 𝛼∈Z+𝑛

so that 𝑉 is also closed. The connectedness of 𝛺 implies 𝑉 = 𝛺.

⊓ ⊔

Theorem 1.1.48 (Maximum Principle). Let 𝛺 be a domain and let 𝑓 ∈ 𝒪(𝛺). If | 𝑓 (𝑧)| takes a maximal value at a point 𝑎 ∈ 𝛺 (in particular, the maximum value there), 𝑓 is constant. Proof. For the sake of simplicity, we assume 𝑎 = 0 by a parallel translation. With a su ciently small polydisk P𝛥((𝑟 𝑗 )) ⋐ 𝛺, | 𝑓 (0)| is the maximum value in P𝛥((𝑟 𝑗 )). In a neighborhood of P𝛥((𝑟 𝑗 )), 𝑓 (𝑧) is expanded to a power series: Õ 𝑓 (𝑧) = 𝑐 𝛼 𝑧 1𝛼1 · · · 𝑧 𝑛𝛼𝑛 . ( 𝛼 𝑗 ) ∈Z+𝑛

With 𝑧 𝑗 = 𝑟 𝑗 𝑒 𝑖 𝜃 𝑗 we have  (1.1.49)

1 2𝜋

=

𝑛 ∫ Õ

2𝜋

∫ 𝑑𝜃 1 · · ·

0

( 𝛼 𝑗 ) ∈Z+𝑛

0

2𝜋

2 Õ 𝛼1 𝑖 𝛼1 𝜃1 𝛼𝑛 𝑖 𝛼𝑛 𝜃𝑛 𝑑𝜃 𝑛 𝑐 𝛼𝑟1 𝑒 · · · 𝑟𝑛 𝑒 ( 𝛼 𝑗 ) ∈Z+𝑛

|𝑐 𝛼 | 2 𝑟 12𝛼1 · · · 𝑟 𝑛2𝛼𝑛 ≥ |𝑐 (0,...,0) | 2 = | 𝑓 (0)| 2 .

On the other hand, since | 𝑓 (0)| 2 is the maximum of | 𝑓 (𝑧)| 2 in 𝑧 ∈ P𝛥((𝑟 𝑗 )), Õ |𝑐 𝛼 | 2 𝑟 12𝛼1 · · · 𝑟 𝑛2𝛼𝑛 ≤ | 𝑓 (0)| 2 . ( 𝛼 𝑗 ) ∈Z+𝑛

Therefore it follows that 𝑐 𝛼 = 0,

∀𝛼 ∈ Z+𝑛 , |𝛼| > 0.

By the Identity Theorem 1.1.46, 𝑓 (𝑧) ≡ 𝑓 (0) on 𝛺.

⊓ ⊔

14

1 Holomorphic Functions

1.2 Analytic Continuation and Hartogs’ Phenomenon Let 𝛺 be a domain of C𝑛 . Definition 1.2.1 (Analytic Continuation). (i) Let 𝑓 ∈ 𝒪(𝛺). Let 𝑉 ⊄ 𝛺 be a domain such that 𝛺 ∩ 𝑉 ≠ ∅. Assume that there are an element 𝑔 ∈ 𝒪(𝑉) and a connected component 𝑊 of 𝛺 ∩ 𝑉 such that 𝑓 | 𝑊 = 𝑔| 𝑊 on 𝑊. Then we say that 𝑓 is analytically continued over 𝑉 (to 𝑔) (through 𝑊). Also 𝑔 is called an analytic continuation of 𝑓 (through 𝑊). (ii) The notion of analytic continuation along curves is de ned in the same way as in the case of one variable (cf., e.g., [38] 5.2). Remark 1.2.2. The above analytic continuation 𝑔 of 𝑓 , if it exists, is unique (Theorem 1.1.46). Also, the above 𝑉 may have in general a non-empty intersection with 𝛺 other than 𝑊, and 𝑔(𝑧) may have di erent values to those of the original 𝑓 (𝑧). Allowing multivalues, we may de ne a multivalued function 𝑓˜ to be 𝑓 on 𝛺, and 𝑔 on 𝑉; 𝑓˜ is called an analytic continuation of 𝑓 , too. Remark 1.2.3. Let 𝑛 = 1. Given 𝑉,𝑊 as above, we take a boundary point 𝑏 ∈ 𝜕𝛺 ∩𝑊. Then 𝑓 (𝑧) = 1/(𝑧 − 𝑏) ∈ 𝒪(𝛺), so that there is no analytic continuation 𝑔 ∈ 𝒪(𝑉) of 𝑓. In the case of 𝑛 ≥ 2, however, the issue of analytic continuation appears to be totally di erent. In fact, according to the shape of 𝜕𝛺 the following may happen: Definition 1.2.4 (Hartogs’ phenomenon). We say that Hartogs’ phenomenon occurs for 𝛺 if there exist a domain 𝑉 ⊄ 𝛺 with 𝑉 ∩ 𝛺 ≠ ∅ and a connected component 𝑊 of 𝑉 ∩ 𝛺 such that all holomorphic functions in 𝛺 are analytically continued over 𝑉 through 𝑊. We observe such Hartogs’ phenomena in the sequel, assuming 𝑛 ≥ 2. (a) Let 𝑡 𝑗 > 0 (1 ≤ 𝑗 ≤ 𝑛) and 0 < 𝑠 𝑗 < 𝑡 𝑗 ( 𝑗 = 1, 2) be given. We consider the polydisk P𝛥(𝑎; (𝑡 𝑗 )) of polyradius (𝑡 𝑗 ) with center at 𝑎 = (𝑎 𝑗 ), and the closed subset  (1.2.5) 𝐹 = (𝑧1 , 𝑧 2 ,𝑧 ′′ ) ∈ P𝛥(𝑎; (𝑡 𝑗 )) : |𝑧 𝑗 − 𝑎 𝑗 | ≤ 𝑠 𝑗 , 𝑗 = 1, 2, 𝑧 ′′ = (𝑧 3 , . . . , 𝑧 𝑛 ) ∈ P𝛥(𝑎 ′′ ;𝑟 ′′ ) , where 𝑎 ′′ = (𝑎 3 , . . . , 𝑎 𝑛 ) and 𝑟 ′′ = (𝑟 3 , . . . , 𝑟 𝑛 ). Then 𝐹 contains 𝑎 as an interior point. Lemma 1.2.6 (Hartogs). Every holomorphic function in P𝛥(𝑎; (𝑡 𝑗 )) \ 𝐹 is necessarily analytically continued on P𝛥(𝑎; (𝑡 𝑗 )). Proof. We may assume 𝑎 = 0 by a parallel translation. Take any element 𝑓 ∈ 𝒪(P𝛥((𝑡 𝑗 )) \ 𝐹). For a point 𝑧 = (𝑧 𝑗 ) ∈ P𝛥((𝑡 𝑗 )) we choose 𝜌1 with max{|𝑧 1 |, 𝑠1 } < 𝜌1 < 𝑡 1 and set ∫ 1 𝑓 (𝜁1 , 𝑧 ′ ) 𝑓˜(𝑧 1 , 𝑧 ′ ) = 𝑑𝜁1 , 𝑧 ′ = (𝑧2 , . . . , 𝑧 𝑛 ). 2𝜋𝑖 | 𝜁1 |=𝜌1 𝜁1 − 𝑧1

1.2 Analytic Continuation and Hartogs’ Phenomenon

15

It is easy to see that this is independent of the choice of 𝜌1 , and so 𝑓˜ ∈ 𝒪(P𝛥((𝑡 𝑗 ))). For 𝑧2 with 𝑠2 < |𝑧2 | < 𝑡 2 , 𝑓˜(𝑧1 , 𝑧 2 , 𝑧 ′′ ) = 𝑓 (𝑧1 , 𝑧 2 , 𝑧 ′′ ). The uniqueness of analytic continuation implies that 𝑓˜ = 𝑓 on P𝛥((𝑡 𝑗 )) \ 𝐹. ⊓ ⊔ Even from this simple theorem the following facts which feature di erences of 𝑛 (≥ 2) variables and one variable are deduced: Theorem 1.2.7. Set 𝑆 = {(𝑧 𝑗 ) ∈ C𝑛 : 𝑧 1 = 𝑧2 = 0}. Then every holomorphic function in P𝛥((𝑟 𝑗 )) \ 𝑆 is analytically continued on P𝛥((𝑟 𝑗 )). Proof. Use Lemma 1.2.6.

Î𝑛

⊓ ⊔

Theorem 1.2.8. Let 𝛺 be a cylinder domain P𝛥(𝑎; (𝑟 𝑗 )) = 𝑗=1 𝛥(𝑎 𝑗 ;𝑟 𝑗 ) (⊂ C𝑛 ) with allowing some 𝑟 𝑗 = ∞, for which 𝛥(𝑎 𝑗 ; ∞) = C. Let 𝐾 ⋐ 𝛺 be a compact subset such that 𝛺 \ 𝐾 is connected. Then every holomorphic function in 𝛺 \ 𝐾 is analytically continued to a unique holomorphic function on 𝛺. Proof. It is easily reduced to the case of a (bounded) polydisk 𝛺 = P𝛥(𝑎; (𝑟 𝑗 )) ⋑ 𝐾. We choose a polydisk P𝛥(𝑎; (𝑠 𝑗 )) such that 𝐾 ⋐ P𝛥(𝑎; (𝑠 𝑗 )) ⋐ P𝛥(𝑎; (𝑟 𝑗 )). Take any function 𝑓 ∈ 𝒪(P𝛥(𝑎; (𝑟 𝑗 )) \ 𝐾). Then 𝑓 is holomorphic in P𝛥(𝑎; (𝑟 𝑗 )) \ P𝛥(𝑎; (𝑠 𝑗 )). It follows from Lemma 1.2.6 that 𝑓 is analytically continued to a unique holomorphic function 𝑓˜ in P𝛥(𝑎; (𝑟 𝑗 )). Since P𝛥(𝑎; (𝑟 𝑗 )) \ 𝐾 is connected, 𝑓˜| P𝛥(𝑎;(𝑟 𝑗 ) )\𝐾 = 𝑓 by the uniqueness of analytic continuation. ⊓ ⊔ This will be extended to general domains of C𝑛 (see 3.4.4). Corollary 1.2.9. The zero set {𝑧 ∈ 𝛺 : 𝑓 (𝑧) = 0} of a holomorphic function 𝑓 in a domain 𝛺 contains no isolated point. Proof. Set 𝛴 = {𝑧 ∈ 𝛺 : 𝑓 (𝑧) = 0}. Suppose that 𝑎 ∈ 𝛴 is isolated. We take a small polydisk P𝛥(𝑎; (𝑟 𝑗 )) ⊂ 𝛺 such that P𝛥(𝑎; (𝑟 𝑗 )) ∩ 𝛴 = {𝑎}. Then 𝑔 := 1/ 𝑓 ∈ 𝒪(P𝛥(𝑎; (𝑟 𝑗 )) \ {𝑎}). It follows from Theorem 1.2.8 with 𝐾 = {𝑎} that 𝑔 ∈ 𝒪(P𝛥(𝑎; (𝑟 𝑗 ))) and 𝑓 (𝑧) · 𝑔(𝑧) = 1 on P𝛥(𝑎; (𝑟 𝑗 )). At 𝑧 = 𝑎, 𝑓 (𝑎) · 𝑔(𝑎) = 1, which contradicts 𝑓 (𝑎) = 0. ⊓ ⊔ Remark 1.2.10. In function theory of one variable it is a well-known fact that the zero set of a non-constant holomorphic function consists only of isolated points. And in the case of real 𝑛 (≥ 1) variables the function 𝑓 (𝑥 1 , . . . , 𝑥 𝑛 ) = 𝑥1𝜈 + · · · + 𝑥 𝑛𝜈 with even natural number 𝑛 has the zero set {𝑥 = (𝑥 𝑗 ) ∈ R𝑛 : 𝑓 (𝑥) = 0} = {0}, and so the zero point of 𝑓 is isolated. The corollary above shows that the gure of the zero set or a constant surface { 𝑓 = 𝑐} (𝑐 ∈ C) of an analytic function 𝑓 of complex variables more than one is considerably di erent to that of the one-variable case or the real analytic case.

16

1 Holomorphic Functions

(b) In the above (a), the case of 𝑛 = 2 is essential, and in that case the set over which functions are analytically continued is contained in the outside larger domain as a relatively compact subset. In what follows we give an example such that the analytic continuation takes place on a relatively non-compact subset. We keep 𝑛 ≥ 2. Let 0 < 𝑠 𝑗 < 𝑡 𝑗 , 1 ≤ 𝑗 ≤ 𝑛, be given. The center can be an arbitrary point of C𝑛 , to say, 0. We put (cf. Fig. 1.2) (1.2.11)

𝛺 1 = {(𝑧 𝑗 ) : |𝑧1 | < 𝑡 1 , |𝑧 𝑗 | < 𝑠 𝑗 , 2 ≤ 𝑗 ≤ 𝑛}, 𝛺 2 = {(𝑧 𝑗 ) : 𝑠1 < |𝑧1 | < 𝑡 1 , |𝑧 𝑗 | < 𝑡 𝑗 , 2 ≤ 𝑗 ≤ 𝑛}, 𝛺H = 𝛺1 ∪ 𝛺2 .

The domain 𝛺 H is classically known as a Hartogs domain. Write 𝑧 ′ = (𝑧2 , . . . , 𝑧 𝑛 ) and let 𝑓 (𝑧 1 , 𝑧 ′ ) ∈ 𝒪(𝛺 H ). By Cauchy’s integral formula we have ∫ 1 𝑓 (𝜁, 𝑧 ′ ) 𝑓 (𝑧 1 , 𝑧 ′ ) = 𝑑𝜁 2𝜋𝑖 | 𝜁 |=𝑟1 𝜁 − 𝑧 1 for (𝑧 1 , 𝑧 ′ ) ∈ 𝛺 1 , where |𝑧1 | < 𝑟 1 < 𝑡 1 ; furthermore, since 𝑟 1 can be chosen arbitrarily close to 𝑡 1 , we see by the integral formula that 𝑓 (𝑧1 , 𝑧 ′ ) is analytically continued on P𝛥((𝑡 𝑗 )). Hence we obtain: Theorem 1.2.12. Every holomorphic function in 𝛺 H is analytically continued on the whole polydisk P𝛥((𝑡 𝑗 )). (c) Let 𝛺 be a domain. For a point 𝑎 = (𝑎 𝑗 ) ∈ 𝛺 we put 𝑎 + R𝑛 = {𝑧 = (𝑧 𝑗 ) ∈ C𝑛 : ℑ(𝑧 𝑗 − 𝑎 𝑗 ) = 0, 1 ≤ 𝑗 ≤ 𝑛}, which is called a totally real subspace of 𝛺 through 𝑎. Theorem 1.2.13 (Removability of totally real subspaces). With the notation above, every 𝑓 ∈ 𝒪(𝛺 \ (𝑎 + R𝑛 )) is analytically continued on 𝛺. Proof. It is su cient to prove the analytic continuation in a neighborhood of an arbitrarily given point 𝑐 ∈ 𝛺 ∩ (𝑎 + R𝑛 ). Since 𝑎 + R𝑛 = 𝑐 + R𝑛 , we may put 𝑐 = 𝑎. By

Fig. 1.2 Hartogs domain 𝛺H .

1.2 Analytic Continuation and Hartogs’ Phenomenon

17

translations and multiplications (by non-zero numbers) of the coordinates we may assume that P𝛥 = P𝛥((2, . . . , 2)) ⊂ 𝛺, P𝛥 𝑗 = {(𝑧 1 , . . . , 𝑧 𝑛 ) ∈ P𝛥 : |𝑧 𝑗 | < 1}, 𝑛 Ø 𝜔= P𝛥 𝑗 , 𝑗=1

𝑎 = (𝜌𝑖, . . . , 𝜌𝑖) ∈ P𝛥,

√ 1 < 𝜌 < 2.

If (𝑤 𝑗 ) ∈ 𝑎 +R𝑛 , then |𝑤 𝑗 | ≥ 𝜌 > 1 and so 𝜔 ∩ (𝑎 +R𝑛 ) = ∅. Take a function 𝑓 ∈ 𝒪(𝛺 \ (𝑎 +R𝑛 )). Since P𝛥((1, . . . , 1)) ⊂ 𝜔, 𝑓 (𝑧) is developed in the polydisk P𝛥((1, . . . , 1)) to a power series Õ (1.2.14) 𝑓 (𝑧) = 𝑐𝛼𝑧𝛼. 𝛼∈Z+𝑛

The domain of convergence satis es 𝛺( 𝑓 ) ⊃ 𝜔. Set 𝜔∗ = {(𝑧 𝑗 ) ∈ 𝜔 : 𝑧 𝑗 ≠ 0, 1 ≤ 𝑗 ≤ 𝑛},

log 𝜔∗ = {(log |𝑧 𝑗 |) ∈ R𝑛 : (𝑧 𝑗 ) ∈ 𝜔∗ },

𝑈 = {(𝑧 𝑗 ) ∈ P𝛥 : 0 < |𝑧 1 | · · · |𝑧 𝑛 | < 2𝑛−1 }, log𝑈 = {(log |𝑧 𝑗 |) : (𝑧 𝑗 ) ∈ 𝑈}, log 𝑎 = (log 𝜌, . . . , log 𝜌) ∈ R𝑛 . Note that log𝑈 is the convex hull of log 𝜔∗ (cf. Fig. 1.3). Since the domain 𝛺 ∗ ( 𝑓 ) is logarithmically convex by Theorem 1.1.25 (ii), we see that

Fig. 1.3 log 𝜔 ∗ ⊂ log𝑈 (the case of 𝑛 = 2)

18

1 Holomorphic Functions

𝑈 = {(𝑧 𝑗 ) ∈ P𝛥 : 0 < |𝑧1 | · · · |𝑧 𝑛 | < 2𝑛−1 } ⊂ 𝛺 ∗ ( 𝑓 ). For 𝑎 we have

|𝑎 1 | · · · |𝑎 𝑛 | = 𝜌 𝑛 < 2𝑛/2 ≤ 2𝑛−1

(𝑛 ≥ 2).

Therefore, 𝑎 ∈ 𝛺 ∗ ( 𝑓 ), so that (1.2.14) converges absolutely and uniformly in a neighborhood of 𝑎; i.e., 𝑓 (𝑧) is analytically continued on a neighborhood of 𝑎. ⊓ ⊔

1.3 Runge Approximation on Convex Cylinder Domains In general, we de ne the sup-norm (supremum norm) of a function 𝑓 : 𝐴 → C on 𝐴 ⊂ C𝑛 by ∥ 𝑓 ∥ 𝐴 = sup{| 𝑓 (𝑧)| : 𝑧 ∈ 𝐴}.

(1.3.1)

Theorem 1.3.2. Let 𝛺 ⊂ C𝑛 be a bounded convex cylinder domain. Then a holomorphic function 𝑓 (𝑧) in a neighborhood of the closure 𝛺¯ is approximated uniformly on 𝛺¯ by polynomials, i.e., for every 𝜀 > 0 there is a polynomial 𝑃(𝑧) such that ∥ 𝑓 − 𝑃∥ 𝛺¯ < 𝜀. Î Proof. Let 𝛺 = 𝑛𝑗=1 𝛺 𝑗 with convex open sets 𝛺 𝑗 ⋐ C, 1 ≤ 𝑗 ≤ 𝑛. We take convex open polygons 𝐸 𝑗 ⋑ 𝛺 𝑗 so that 𝑓 (𝑧) is holomorphic in a neighborhood of the convex Î Ð𝑙 𝑗 closed cylinder 𝑛𝑗=1 𝐸¯ 𝑗 (Proposition 1.1.5). The boundary 𝜕𝐸 𝑗 = 𝐶 𝑗 = 𝑘=1 𝐶 𝑗𝑘 of each 𝐸 𝑗 consists of a nite number of line segments 𝐶 𝑗 𝑘 . By Cauchy’s integral formula we have  𝑛 ∫ ∫ 1 𝑓 (𝜁1 , . . . , 𝜁 𝑛 ) (1.3.3) 𝑓 (𝑧) = 𝑑𝜁1 · · · 𝑑𝜁 𝑛 2𝜋𝑖 (𝜁 − 1 𝑧 1 ) · · · (𝜁 𝑛 − 𝑧 𝑛 ) 𝐶1 𝐶𝑛 ∫ Õ  1 𝑛 ∫ 𝑓 (𝜁1 , . . . , 𝜁 𝑛 ) = 𝑑𝜁1 · · · 𝑑𝜁 𝑛 2𝜋𝑖 (𝜁1 − 𝑧 1 ) · · · (𝜁 𝑛 − 𝑧 𝑛 ) 𝐶1𝑘1 𝐶𝑛𝑘𝑛 𝑘 ,...,𝑘 1

𝑛

¯ where the summation runs over 1 ≤ 𝑘 1 ≤ 𝑙1 , . . . , 1 ≤ 𝑘 𝑛 ≤ 𝑙 𝑛 . Take a point for 𝑧 ∈ 𝛺, 𝜉 𝑗 𝑘 𝑗 on the line passing through the middle point of each line segment 𝐶 𝑗 𝑘 𝑗 , vertical to it in the same side as 𝛺 𝑗 and su ciently far from 𝐶 𝑗 𝑘 𝑗 (cf. Fig. 1.4). Then there is a constant 𝜃 > 0 such that 𝑧 𝑗 − 𝜉 𝑗𝑘𝑗 (1.3.4) 𝜁 𝑗 − 𝜉 𝑗 𝑘 < 𝜃 < 1, 𝜁 𝑗 ∈ 𝐶 𝑗 𝑘 𝑗 , 𝑧 𝑗 ∈ 𝛺 𝑗 , 1 ≤ 𝑘 𝑗 ≤ 𝑙 𝑗 , 1 ≤ 𝑗 ≤ 𝑛. 𝑗 For these 𝑧 𝑗 and 𝜁 𝑗 we have

1.3 Runge Approximation on Convex Cylinder Domains

19

Fig. 1.4 Convex domain 𝛺 𝑗 .

(1.3.5)

∞ Õ (𝑧 𝑗 − 𝜉 𝑗 𝑘 𝑗 ) 𝛼 𝑗 1 1 = = . 𝜁 𝑗 − 𝑧 𝑗 𝜁 𝑗 − 𝜉 𝑗 𝑘 𝑗 − (𝑧 𝑗 − 𝜉 𝑗 𝑘 𝑗 ) 𝛼 =0 (𝜁 𝑗 − 𝜉 𝑗 𝑘 𝑗 ) 𝛼 𝑗 +1 𝑗

By (1.3.4) the series of the right-hand side above is of majorant convergence. It follows from (1.3.3) and (1.3.5) that Õ Õ  1 𝑛 ∫ (1.3.6) 𝑓 (𝑧) = 𝑑𝜁1 · · · 2𝜋𝑖 𝐶1𝑘1 𝑛 𝑘 ,...,𝑘 ( 𝛼 𝑗 ) ∈Z+

1

𝑛

∫

···

𝐶𝑛𝑘𝑛

𝑑𝜁 𝑛 𝑓 (𝜁)

𝑛 Ö (𝑧 𝑗 − 𝜉 𝑗 𝑘 𝑗 ) 𝛼 𝑗 𝑗=1

(𝜁 𝑗 − 𝜉 𝑗 𝑘 𝑗 ) 𝛼 𝑗 +1

,

¯ 𝑧 ∈ 𝛺.

Î Since 𝑓 (𝜁) (𝜁 ∈ 𝑗 𝐸¯ 𝑗 ) is bounded, we see by (1.3.4) that the right-hand side of (1.3.6) is of majorant convergence. Hence for every 𝜀 > 0 there is a su ciently large number 𝑁 ∈ N such that the polynomial Õ Õ  1 𝑛 ∫ 𝑃(𝑧) = 𝑑𝜁1 · · · 2𝜋𝑖 𝐶1𝑘1 𝑘 ,...,𝑘 | ( 𝛼𝑗 ) | ≤ 𝑁

1

···

∫

𝑛

𝐶𝑛𝑘𝑛

𝑑𝜁 𝑛 𝑓 (𝜁)

𝑛 Ö (𝑧 𝑗 − 𝜉 𝑗 𝑘 𝑗 ) 𝛼 𝑗 𝑗=1

(𝜁 𝑗 − 𝜉 𝑗 𝑘 𝑗 ) 𝛼 𝑗 +1

of degree at most 𝑁 satis es ∥ 𝑓 (𝑧) − 𝑃(𝑧) ∥ 𝛺¯ < 𝜀.

⊓ ⊔

1.3.1 Cousin Integral In C𝑛 (∋ 𝑧 = (𝑧 1 , . . . , 𝑧 𝑛 )) we consider a cuboid 𝐹 (always bounded) containing a part of the boundary:

20

1 Holomorphic Functions

𝐹 = 𝐹 ′ × {𝑧 𝑛 ∈ C : 𝑎 < ℜ𝑧 𝑛 < 𝑏, |ℑ𝑧 𝑛 | ≤ 𝑐}, 𝐹 ◦ = 𝐹 ′ × {𝑧 𝑛 ∈ C : 𝑎 < ℜ𝑧 𝑛 < 𝑏, |ℑ𝑧 𝑛 | < 𝑐}, where 𝑐 > 0, 𝐹 ′ is an open cuboid in C𝑛−1 (∋ 𝑧 ′ = (𝑧 1 , . . . , 𝑧 𝑛−1 )) and 𝐹 ◦ is the interior of 𝐹. Let 𝜑(𝑧) be a continuous function on 𝐹, holomorphic in 𝐹 ◦ . With 𝑡 = (𝑎 + 𝑏)/2 we denote by ℓ the oriented line segment in 𝑧 𝑛 -plane from 𝑧 𝑛 = 𝑡 − 𝑖𝑐 to 𝑧 𝑛 = 𝑡 + 𝑖𝑐. The following path integral of 𝜑(𝑧) is called the Cousin integral: ∫ 1 𝜑(𝑧 ′ , 𝜁) (1.3.7) 𝛷(𝑧 ′ , 𝑧 𝑛 ) = 𝑑𝜁 . 2𝜋𝑖 ℓ 𝜁 − 𝑧 𝑛 We rst consider it as 𝛷(𝑧 ′ , 𝑧 𝑛 ) ∈ 𝒪(𝐹 ′ × {𝑧 𝑛 : ℜ𝑧 𝑛 < 𝑡, |ℑ𝑧 𝑛 | < 𝑐}). With 𝑡 < 𝑏 ′ < 𝑏 and 0 < 𝑐 ′ < 𝑐 the part of the oriented boundary of the domain {𝑧 𝑛 : 𝑡 < ℜ𝑧 𝑛 < 𝑏 ′ , |ℑ𝑧 𝑛 | < 𝑐 ′ } which is a part of ℓ is denoted by −ℓ ′ , and the other part of the boundary is denoted by ℓ ′′ (cf. Fig. 1.5). By Cauchy’s integral formula, ∫ ∫ 1 𝜑(𝑧 ′ , 𝜁) 1 𝜑(𝑧 ′ , 𝜁) 𝑑𝜁 = 𝑑𝜁 2𝜋𝑖 ℓ ′ 𝜁 − 𝑧 𝑛 2𝜋𝑖 ℓ ′′ 𝜁 − 𝑧 𝑛 for ℜ𝑧 𝑛 < 𝑡, |ℑ𝑧 𝑛 | < 𝑐 and 𝑧 ′ ∈ 𝐹 ′ . Therefore ∫ ∫ 1 𝜑(𝑧 ′ , 𝜁) 1 𝜑(𝑧 ′ , 𝜁) (1.3.8) 𝛷(𝑧 ′ , 𝑧 𝑛 ) = 𝑑𝜁 + 𝑑𝜁 . 2𝜋𝑖 ℓ\ℓ ′ 𝜁 − 𝑧 𝑛 2𝜋𝑖 ℓ ′′ 𝜁 − 𝑧 𝑛 The right-hand side above is holomorphic in 𝐹 ′ × {𝑧 𝑛 : ℜ𝑧 𝑛 < 𝑏 ′ , |ℑ𝑧 𝑛 | < 𝑐 ′ }, and so 𝛷(𝑧 ′ , 𝑧 𝑛 ) is analytically continued on 𝐹 ′ × {𝑧 𝑛 : ℜ𝑧 𝑛 < 𝑏 ′ , |ℑ𝑧 𝑛 | < 𝑐 ′ }. Letting 𝑏 ′ → 𝑏 and 𝑐 ′ → 𝑐, we see that 𝛷(𝑧 ′ , 𝑧 𝑛 ) is analytically continued on 𝑈1 := 𝐹 ′ × {𝑧 𝑛 : ℜ𝑧 𝑛 < 𝑏, |ℑ𝑧 𝑛 | < 𝑐},

Fig. 1.5 Cousin integral.

1.4 Implicit and Inverse Function Theorems

21

and write 𝛷1 (𝑧 ′ , 𝑧 𝑛 ) ∈ 𝒪(𝑈1 ) for it. Next, we consider (1.3.7) in 𝐹 ′ × {𝑧 𝑛 : ℜ𝑧 𝑛 > 𝑡, |ℑ𝑧 𝑛 | < 𝑐}. With 𝑎 < 𝑎 ′ < 𝑡, 0 < ′ 𝑐 < 𝑐, the part of the oriented boundary of the domain {𝑧 𝑛 : 𝑎 ′ < ℜ𝑧 𝑛 < 𝑡, |ℑ𝑧 𝑛 | < 𝑐 ′ } contained in ℓ is denoted by ℓ ′ , and the other part of the boundary is denoted by ℓ ′′′ (cf. Fig. 1.5). In the same way as in the case of 𝛷1 , we have ∫ ∫ 1 𝜑(𝑧 ′ , 𝜁) 1 𝜑(𝑧 ′ , 𝜁) 𝑑𝜁 = − 𝑑𝜁, 2𝜋𝑖 ℓ ′ 𝜁 − 𝑧 𝑛 2𝜋𝑖 ℓ ′′′ 𝜁 − 𝑧 𝑛 ∫ ∫ 1 𝜑(𝑧 ′ , 𝜁) 1 𝜑(𝑧 ′ , 𝜁) (1.3.9) 𝛷(𝑧 ′ , 𝑧 𝑛 ) = 𝑑𝜁 − 𝑑𝜁 2𝜋𝑖 ℓ\ℓ ′ 𝜁 − 𝑧 𝑛 2𝜋𝑖 ℓ ′′′ 𝜁 − 𝑧 𝑛 for ℜ𝑧 𝑛 > 𝑡, |ℑ𝑧 𝑛 | < 𝑐 and 𝑧 ′ ∈ 𝐹 ′ . The right-hand side above is holomorphic in 𝐹 ′ × {𝑧 𝑛 : ℜ𝑧 𝑛 > 𝑎 ′ , |ℑ𝑧 𝑛 | < 𝑐 ′ }, and so 𝛷(𝑧 ′ , 𝑧 𝑛 ) is analytically continued on 𝐹 ′ × {𝑧 𝑛 : ℜ𝑧 𝑛 > 𝑎 ′ , |ℑ𝑧 𝑛 | < 𝑐 ′ }. As 𝑎 ′ → 𝑎 and 𝑐 ′ → 𝑐, 𝛷(𝑧 ′ , 𝑧 𝑛 ) is analytically continued on 𝑈2 := 𝐹 ′ × {𝑧 𝑛 : ℜ𝑧 𝑛 > 𝑎, |ℑ𝑧 𝑛 | < 𝑐}. We denote it by 𝛷2 (𝑧 ′ , 𝑧 𝑛 ) ∈ 𝒪(𝑈2 ). Lemma 1.3.10 (Cousin decomposition). 𝛷 𝑗 (𝑧 ′ , 𝑧 𝑛 ) ∈ 𝒪(𝑈 𝑗 ) ( 𝑗 = 1, 2) satisfy (1.3.11)

𝜑(𝑧 ′ , 𝑧 𝑛 ) = 𝛷1 (𝑧 ′ , 𝑧 𝑛 ) −𝛷2 (𝑧 ′ , 𝑧 𝑛 ),

(𝑧 ′ , 𝑧 𝑛 ) ∈ 𝑈1 ∩ 𝑈2 (= 𝐹 ◦ ).

Proof. For (𝑧 ′ , 𝑧 𝑛 ) ∈ 𝑈1 ∩ 𝑈2 we take 𝑎 ′ , 𝑏 ′ , 𝑐 ′ so that 𝑎 < 𝑎 ′ < min{𝑡, ℜ𝑧 𝑛 }, max{𝑡, ℜ𝑧 𝑛 } < 𝑏 ′ < 𝑏, |ℑ𝑧 𝑛 | < 𝑐 ′ < 𝑐. Then by Cauchy’s integral formula ∫ ∫ 1 𝜑(𝑧 ′ , 𝜁) 1 𝜑(𝑧 ′ , 𝜁) 𝜑(𝑧 ′ , 𝑧 𝑛 ) = 𝑑𝜁 + 𝑑𝜁 . 2𝜋𝑖 ℓ ′′ 𝜁 − 𝑧 𝑛 2𝜋𝑖 ℓ ′′′ 𝜁 − 𝑧 𝑛 From this together with (1.3.8) and (1.3.9) we obtain (1.3.11).

⊓ ⊔

1.4 Implicit and Inverse Function Theorems Here, readers may consult any textbook ready to hand on the implicit function theorem of real functions of real variables. We consider simultaneous equations with holomorphic functions (1.4.1)

𝑓 𝑗 (𝑧1 , . . . , 𝑧 𝑛 , 𝑤1 , . . . , 𝑤𝑚 ) = 0,

1 ≤ 𝑗 ≤ 𝑚.

The complex Jacobi matrix and complex Jacobian for 𝑓 = ( 𝑓1 , . . . , 𝑓𝑚 ) are de ned respectively by

22

1 Holomorphic Functions

 (1.4.2)

𝜕 𝑓𝑗 𝜕𝑤 𝑘





1≤ 𝑗,𝑘 ≤𝑚

𝜕 𝑓𝑗 𝜕𝑓 := det 𝜕𝑤 𝜕𝑤 𝑘

,

 1≤ 𝑗,𝑘 ≤𝑚

.

For the real and imaginary parts of 𝑓 𝑗 and 𝑤 𝑘 we write respectively 𝑓 𝑗 = 𝑓 𝑗1 + 𝑖 𝑓 𝑗2 ,

𝑤 𝑘 = 𝑤 𝑘1 + 𝑖𝑤 𝑘2 .

Then (1.4.1) is equivalent to the following: (1.4.3)

𝑓 𝑗1 (𝑧1 , . . . , 𝑧 𝑛 , 𝑤11 , 𝑤12 , . . . , 𝑤𝑚1 , 𝑤𝑚2 ) = 0,

1 ≤ 𝑗 ≤ 𝑚,

𝑓 𝑗2 (𝑧1 , . . . , 𝑧 𝑛 , 𝑤11 , 𝑤12 , . . . , 𝑤𝑚1 , 𝑤𝑚2 ) = 0,

1 ≤ 𝑗 ≤ 𝑚.

The real Jacobian of (1.4.3) is

(1.4.4)

𝜕 𝑓11 𝜕𝑤11 𝜕 ( 𝑓 𝑗1 , 𝑓 𝑗2 ) 𝜕 𝑓12 = 𝜕 (𝑤 𝑘1 , 𝑤 𝑘2 ) 𝜕𝑤.11 ..

𝜕 𝑓11 𝜕𝑤12 𝜕 𝑓12 𝜕𝑤12

.. .

𝜕 𝑓11 𝜕𝑤21 𝜕 𝑓12 𝜕𝑤21

.. .

𝜕 𝑓11 𝜕𝑤22 𝜕 𝑓12 𝜕𝑤22

.. .

· · · · · · .

Lemma 1.4.5. The real Jacobian and the complex Jacobian associated with holomorphic functions 𝑓 𝑗 (𝑧, 𝑤), 1 ≤ 𝑗 ≤ 𝑚, with 𝑧 = (𝑧 1 , . . . , 𝑧 𝑛 ) and 𝑤 = (𝑤1 , . . . , 𝑤𝑚 ) are related by   𝜕 ( 𝑓 𝑗1 , 𝑓 𝑗2 ) 𝜕 𝑓 𝑗 2 = det . 𝜕 (𝑤 𝑘1 , 𝑤 𝑘2 ) 𝜕𝑤 𝑘 ⊓ ⊔

Proof. Cf. [39] Lemma 1.2.39.

Theorem 1.4.6 (Implicit function). In a neighborhood of a pint (𝑎, 𝑏) ∈ C𝑛 × C𝑚 we consider simultaneous equations (1.4.1) defined by holomorphic functions 𝑓 𝑗 (𝑎, 𝑏) = 0, 1 ≤ 𝑗 ≤ 𝑚. Assume that   𝜕 𝑓𝑗 (1.4.7) det (𝑎, 𝑏) ≠ 0. 𝜕𝑤 𝑘 1≤ 𝑗,𝑘 ≤𝑚 Then there are uniquely holomorphic solutions (1.4.1) in a neighborhood of (𝑎, 𝑏) (𝑤 𝑗 ) = (𝑔 𝑗 (𝑧1 , . . . , 𝑧 𝑛 )),

𝑏 = (𝑔 𝑗 (𝑎))

(1 ≤ 𝑗 ≤ 𝑚). ⊓ ⊔

Proof. Cf. [39] Theorem 1.2.41. A map from an open set 𝑈 ⊂

C𝑛

to

C𝑚

𝑓 : 𝑧 ∈ 𝑈 → ( 𝑓1 (𝑧), . . . , 𝑓𝑚 (𝑧)) ∈ C𝑚 is called a holomorphic map (or mapping) if all elements 𝑓 𝑗 (𝑧) are holomorphic functions.

1.4 Implicit and Inverse Function Theorems

23

Theorem 1.4.8 (Inverse function). If a holomorphic map between neighborhoods 𝑈,𝑉 of the origin of C𝑛 𝑓 : 𝑧 = (𝑧 𝑘 ) ∈ 𝑈 → ( 𝑓 𝑗 (𝑧)) ∈ 𝑉,

𝑓 (0) = 0

has a non-zero complex Jacobian 𝜕𝜕𝑧𝑓 (0) ≠ 0 at 0, then 𝑓 with 𝑈 and 𝑉 shrunk smaller if necessary, has the holomorphic inverse 𝑓 −1 : 𝑉 → 𝑈. ⊓ ⊔

Proof. Cf. [39] Theorem 1.2.43.

If a holomorphic map 𝑓 : 𝑈 → 𝑉 between two open sets 𝑈 and 𝑉 has the holomorphic inverse 𝑓 −1 : 𝑉 → 𝑈, 𝑓 is called a biholomorphic map; in this case, we say that 𝑈 and 𝑉 are holomorphically isomorphic and write 𝑈  𝑉. In the case of 𝑛 = 1 the following theorem which standardizes simply connected domains is well known. Theorem 1.4.9 (Riemann’s Mapping Theorem). Let 𝛺 ⊂ C be a simply connected domain. If 𝜕𝛺 ≠ ∅, then there exists a biholomorphic mapping 𝑓 : 𝛺 → 𝛥(1) (unit disk). Furthermore, for an arbitrarily fixed 𝑎 ∈ 𝛺, 𝑓 is unique with 𝑓 (𝑎) = 0 and 𝑓 ′ (𝑎) > 0. ⊓ ⊔

Proof. Cf. [38] Theorem (6.4.4).

There is no such simple standardization of topologically simple domains in several variables; this was proved by H. Poincaré in 1907. A polydisk and an open ball of C𝑛 (𝑛 ≥ 2) are the natural generalizations of a disk of C, which are topologically cells and homeomorphic to each other. But, they are not biholomorphic: Theorem 1.4.10 (Poincaré). Let 𝑛 ≥ 2. Then there is no biholomorphic map 𝑓 : 𝛥(1) 𝑛 → B(1) from the unit polydisk 𝛥(1) 𝑛 onto the unit ball B(1) in C𝑛 . Proof. We write (𝑧, 𝑤, 𝑣) ∈ 𝛥(1) × 𝛥(1) × 𝛥(1) 𝑛−2 for the variables. Fix 𝑣 = 𝑣0 ∈ 𝛥(1) 𝑛−2 arbitrarily. We consider a proper map 𝑔(𝑧, 𝑤) := 𝑓 (𝑧, 𝑤, 𝑣0 ) from 𝛥(1) 2 into B(1). Let 𝑤𝜆 ∈ 𝛥(1) (𝜆 = 1, 2, . . .) be any sequence with lim𝜆→∞ 𝑤𝜆 ∈ 𝜕 𝛥(1). By Montel’s Theorem 1.1.45 the sequence {𝑔(𝑧, 𝑤𝜆 )}𝜆 (𝑧 ∈ 𝛥(1)) of uniformly bounded (vector-valued) functions has a subsequence {𝑔(𝑧, 𝑤𝜆 𝜇 )} 𝜇 which converges ¯ ¯ locally uniformly to ℎ : 𝛥(1) → B(1), where B(1) denotes the closure of B(1). But 𝜕2 2 2 ′ 2 in fact, ℎ( 𝛥(1)) ⊂ 𝜕B(1). Therefore, ∥ℎ(𝑧) ∥ ≡ 1. Since 𝜕𝑧𝜕 𝑧¯ ∥ℎ(𝑧)∥ = ∥ℎ (𝑧)∥ , ℎ′ (𝑧) ≡ 0. We consider a holomorphic function

{ 𝜕𝑔 𝜕𝑧 (𝑧, 𝑤𝜆 𝜇 )} 𝜇

𝜕𝑔 𝜕𝑧 (𝑧, 𝑤)

and see that the sequence

of partial derivatives converges locally uniformly to ℎ′ (𝑧) = 0. 𝜕𝑔 𝜕𝑧 (𝑧, 𝑤)

is holomorphic in (𝑧, 𝑤) ∈ ¯ 𝛥(1) × 𝛥(1) and extends continuously over 𝛥(1) × 𝛥(1) so that 𝜕𝑔 𝜕𝑧 (𝑧, 𝑤) = 0 for It follows from the above arguments that

𝜕𝑔 𝜕𝑧 (𝑧, 𝑤) ≡ 0. Since 𝜕𝑓 𝜕𝑧 (𝑧, 𝑤, 𝑣) ≡ 0. Then

|𝑧| < 1 and |𝑤| = 1. By the Maximum Principle (Theorem 1.1.48), 𝜕𝑔 𝜕𝑧 (𝑧, 𝑤)

= 𝜕𝜕𝑧𝑓 (𝑧, 𝑤, 𝑣0 ) and 𝑣0 ∈ 𝛥(1) 𝑛−2 is arbitrary, we see that the complex Jacobian of 𝑓 must be identically 0: Contradiction.

⊓ ⊔

24

1 Holomorphic Functions

1.5 Analytic Subsets We describe the de nition of analytic subsets and the preliminary properties. Let 𝑈 ⊂ C𝑛 be an open set. Definition 1.5.1. A subset 𝐴 ⊂ 𝑈 is called an analytic (sub)set if for every point 𝑎 ∈ 𝑈 there are a neighborhood 𝑉 ⊂ 𝑈 of 𝑎 and nitely many holomorphic functions 𝑓 𝑗 ∈ 𝒪(𝑉), 1 ≤ 𝑗 ≤ 𝑙, satisfying 𝐴 ∩𝑉 = {𝑧 ∈ 𝑉 : 𝑓1 (𝑧) = · · · = 𝑓𝑙 (𝑧) = 0}. In particular, if at every 𝑎 ∈ 𝐴, 𝑙 = 1 and 𝐴 ∩𝑉 ≠ 𝑉 for a connected neighborhood 𝑉 of 𝑎, 𝐴 is called a complex hypersurface. By de nition an analytic subset of 𝑈 is closed in 𝑈 (cf. Exercise 9 at the end of the present chapter). Theorem 1.5.2. If an analytic subset 𝐴 of a domain 𝛺 contains an interior point, then 𝐴 = 𝛺; hence, if 𝐴 ≠ 𝛺, 𝐴 is a nowhere dense closed subset of 𝛺. Proof. Let 𝐴◦ be the set of interior points of 𝐴. By the assumption, 𝐴◦ ≠ ∅. Take a point 𝑎 ∈ 𝐴◦ ∩ 𝛺. Then there are a connected neighborhood 𝑉 of 𝑎 in 𝛺, and nitely many 𝑓 𝑗 ∈ 𝒪(𝑉), 1 ≤ 𝑗 ≤ 𝑙, such that 𝐴 ∩𝑉 = { 𝑓1 = · · · = 𝑓𝑙 = 0}. For a point 𝑏 ∈ 𝑉 ∩ 𝐴◦ , there is a neighborhood 𝑊 ⊂ 𝐴 ∩ 𝑉 of 𝑏 with 𝑊 ∩ 𝐴 = 𝑊; i.e., 𝑓 𝑗 | 𝑊 (𝑧) ≡ 0, 1 ≤ 𝑗 ≤ 𝑙. By the Identity Theorem 1.1.46, 𝑓 𝑗 (𝑧) ≡ 0, 1 ≤ 𝑗 ≤ 𝑙. Therefore 𝑉 ∩ 𝐴 = 𝑉 and 𝑎 ∈ 𝐴◦ . We see that 𝐴◦ (⊂ 𝛺) is open and closed. Since 𝛺 is connected, 𝐴◦ = 𝛺, so that 𝐴 = 𝛺. ⊓ ⊔ Remark 1.5.3. In the case 𝑛 = 1, an analytic subset of 𝑈 without interior point is the same as a closed discrete subset of 𝑈 (i.e., a discrete subset of 𝑈 with no accumulation point in 𝑈). We consider a holomorphic function 𝑓 ∈ 𝒪(P𝛥(𝑎; (𝑟 𝑗 ))) in a polydisk P𝛥(𝑎; (𝑟 𝑗 )). Assume that 𝑓 ≠ 0 ( 𝑓 (𝑧) . 0). Then 𝑓 (𝑧) is expanded to a series of homogeneous polynomials as follows: (1.5.4)

𝑓 (𝑧) =

Õ

∞ Õ

𝑐 𝜆 (𝑧 − 𝑎) 𝜆 =

𝑃 𝜈 (𝑧 − 𝑎),

𝜈=𝜈0

𝜆

𝑃 𝜈 (𝑧 − 𝑎) =

Õ

𝑐 𝜆 (𝑧 − 𝑎) 𝜆 ,

|𝜆|=𝜈

The order of zero of 𝑓 at 𝑎 is de ned by (1.5.5)

ord𝑎 𝑓 := 𝜈0 .

𝑃 𝜈0 (𝑧 − 𝑎) . 0.

1.5 Analytic Subsets

25

For 𝑓 = 0 we set ord𝑎 𝑓 = ∞. Similarly, for a holomorphic function 𝑔 ∈ 𝒪(P𝛥(𝑎; (𝑟 𝑗 ))) with 𝑔 ≠ 0 we have the Í homogeneous polynomial expansion 𝑔(𝑧) = ∞ 𝜈=𝜈1 𝑄 𝜈 (𝑧 − 𝑎) with 𝑄 𝜈1 (𝑧 − 𝑎) . 0. Therfore, it follows that
(1.5.6) 𝑓 (𝑧)𝑔(𝑧) = 𝑃 𝜈0 (𝑧 − 𝑎)𝑄 𝜈1 (𝑧 − 𝑎) + term of order ≥ (𝜇1 + 𝜇2 + 1) , 𝑃 𝜈0 (𝑧 − 𝑎)𝑄 𝜈1 (𝑧 − 𝑎) ≠ 0, and so ord𝑎 ( 𝑓 𝑔) = ord𝑎 𝑓 + ord𝑎 𝑔.

(1.5.7)

For the sake of simplicity we set 𝑎 = 0 by a parallel translation, and assume 𝑓 (0) = 0 (𝜈0 ≥ 1). We take a vector 𝑣 ∈ C𝑛 \ {0} with 𝑃 𝜈0 (𝑣) ≠ 0. Then for small 𝜁 ∈ C we have 𝑓 (𝜁 𝑣) =

∞ Õ

𝜁 𝜈 𝑃 𝜈 (𝑣) = 𝜁 𝜈0 (𝑃 𝜈0 (𝑣) + 𝜁 𝑃 𝜈0 +1 (𝑣) + · · · ).

𝜈=𝜈0

By a linear transform of the coordinates we choose a new coordinate system 𝑧 = (𝑧1 , . . . , 𝑧 𝑛 ) with 𝑣 = (0, . . . , 0, 1). With a polyradius 𝑟 = (𝑟 1 , . . . , 𝑟 𝑛 ) we write (1.5.8)

P𝛥((𝑟 𝑗 )) = P𝛥𝑛−1 × 𝛥(0;𝑟 𝑛 ) ⊂ C𝑛−1 × C, P𝛥𝑛−1 = {𝑧 ′ = (𝑧 1 , . . . , 𝑧 𝑛−1 ) ∈ C𝑛−1 : |𝑧 𝑗 | < 𝑟 𝑗 , 1 ≤ 𝑗 ≤ 𝑛 − 1}

for the polydisks and for the coordinates 𝑧 = (𝑧 ′ , 𝑧 𝑛 ) ∈ P𝛥𝑛−1 × 𝛥(𝑟 𝑛 ),

0 = (0, 0).

Since 𝑓 (0, 0) = 0 and 𝑓 (0, 𝑧 𝑛 ) . 0, for a small 𝑟 𝑛 > 0 there is a 𝛿 > 0 such that {𝑧 𝑛 : |𝑧 𝑛 | ≤ 𝑟 𝑛 , 𝑓 (0, 𝑧 𝑛 ) = 0} = {0}, | 𝑓 (0, 𝑧 𝑛 )| > 𝛿,

|𝑧 𝑛 | = 𝑟 𝑛 .

Therefore with su ciently small 𝑟 𝑗 > 0, 1 ≤ 𝑗 ≤ 𝑛 − 1, | 𝑓 (𝑧 ′ , 𝑧 𝑛 )| > 𝛿,

𝑧 ′ ∈ P𝛥𝑛−1 , |𝑧 𝑛 | = 𝑟 𝑛 .

Summarizing the above, we have: Lemma 1.5.9. For a holomorphic function 𝑓 (𝑧) (. 0) in a connected neighborhood of 0 ∈ C𝑛 , there are a coordinate system 𝑧 = (𝑧1 , 𝑧 2 , . . . , 𝑧 𝑛 ) = (𝑧 ′ , 𝑧 𝑛 ) and a polydisk P𝛥((𝑟 𝑗 )) = P𝛥𝑛−1 × 𝛥(𝑟 𝑛 ) satisfying the following: (i) 𝑓 (𝑧) is holomorphic on the closed polydisk P𝛥((𝑟 𝑗 )), and the homogeneous Í polynomial expansion 𝑓 (𝑧) = ∞ 𝜈=𝜈0 𝑃 𝜈 (𝑧) satisfies that 𝑃 𝜈0 (0, 1) ≠ 0 and

26

1 Holomorphic Functions

𝑓 (0, 𝑧 𝑛 ) = 𝑧 𝑛𝜈0 (𝑃 𝜈0 (0, 1) + 𝑧 𝑛 𝑃 𝜈0 +1 (0, 1) + · · · ). (ii) For a sufficiently small 𝑟 𝑛 > 0, {|𝑧 𝑛 | ≤ 𝑟 𝑛 : 𝑓 (0, 𝑧 𝑛 ) = 0} = {0}. (iii) For sufficiently small 𝑟 1 , . . . , 𝑟 𝑛−1 > 0 dependent on 𝑟 𝑛 > 0, the roots 𝑧 𝑛 of 𝑓 (𝑧 ′ , 𝑧 𝑛 ) = 0 with every fixed 𝑧 ′ ∈ P𝛥𝑛−1 , are contained in 𝛥(𝑟 𝑛 ). In particular, | 𝑓 (𝑧 ′ , 𝑧 𝑛 )| > 0 for all (𝑧 ′ , 𝑧 𝑛 ) ∈ P𝛥𝑛−1 × {|𝑧 𝑛 | = 𝑟 𝑛 }. Definition 1.5.10. We call the above P𝛥((𝑟 𝑗 )) = P𝛥𝑛−1 × 𝛥(0;𝑟 𝑛 ) the standard polydisk of 𝑓 , and 𝑧 = (𝑧 ′ , 𝑧 𝑛 ) the standard coordinate system of 𝑓 . Theorem 1.5.11 (Riemann’s Extension). Let 𝛺 be a domain and let 𝐴 & 𝛺 be an analytic set. If a holomorphic function 𝑓 ∈ 𝒪(𝛺 \ 𝐴) is locally bounded about every point 𝑎 ∈ 𝐴, that is, there is a neighborhood 𝑉 of 𝑎 such that the restriction 𝑓 | 𝑉\𝐴 is bounded, then 𝑓 is analytically continued uniquely on 𝛺. Proof. In the case of 𝑛 = 1, the theorem follows from Remark 1.5.3 and Riemann’s Extension Theorem of one variable ([38] Theorem (5.1.1)). Let 𝑛 ≥ 2, and let 𝑎 ∈ 𝐴. By a parallel translation we may set 𝑎 = 0. By the assumption there are a neighborhood 𝑉 of 0 and 𝜙 ∈ 𝒪(𝑉) \ {0} such that 𝐴 ∩ 𝑉 ⊂ {𝜙 = 0}. For 𝑉 we take the standard polydisk P𝛥 = P𝛥𝑛−1 × 𝛥(𝑟 𝑛 ) of 𝜙. Then, 𝜙(𝑧 ′ , 𝑧 𝑛 ) ≠ 0 for all 𝑧 ′ ∈ P𝛥𝑛−1 and |𝑧 𝑛 | = 𝑟 𝑛 , i.e., (P𝛥𝑛−1 × {|𝑧 𝑛 | = 𝑟 𝑛 }) ∩ 𝐴 = ∅. With a xed 𝑧 ′ ∈ P𝛥𝑛−1 , 𝜙(𝑧 ′ , 𝑧 𝑛 ) = 0 has at most nitely many zeros in 𝛥(𝑟 𝑛 ), and 𝑓 (𝑧 ′ , 𝑧 𝑛 ) is locally bounded about those zeros. By the case of 𝑛 = 1 above, 𝑓 (𝑧 ′ , 𝑧 𝑛 ) is holomorphic in 𝑧 𝑛 ∈ 𝛥(𝑟 𝑛 ). Therefore we have ∫ 1 𝑓 (𝑧 ′ , 𝜁 𝑛 ) 𝑓 (𝑧 ′ , 𝑧 𝑛 ) = 𝑑𝜁 𝑛 , |𝑧 𝑛 | < 𝑟 𝑛 . 2𝜋𝑖 | 𝜁𝑛 |=𝑟𝑛 𝜁 𝑛 − 𝑧 𝑛 Since 𝑓 (𝑧 ′ , 𝜁 𝑛 ) with |𝜁 𝑛 | = 𝑟 𝑛 is holomorphic in 𝑧 ′ , the integral expression above implies 𝑓 (𝑧 ′ , 𝑧 𝑛 ) being holomorphic in P𝛥. ⊓ ⊔ Theorem 1.5.12. Let 𝛺 be a domain and let 𝐴 & 𝛺 be an analytic set. Then 𝛺 \ 𝐴 is a domain, too. Proof. Suppose that 𝛺 \ 𝐴 is not connected. There are non-empty open subsets 𝑉1 ,𝑉2 of 𝛺 \ 𝐴 such that 𝛺 \ 𝐴 = 𝑉1 ∪𝑉2 , 𝑉1 ∩𝑉2 = ∅. Put 𝑓 ∈ 𝒪(𝛺 \ 𝐴) as follows: ( 𝑓 (𝑧) =

0, 1,

𝑧 ∈ 𝑉1 , 𝑧 ∈ 𝑉2 .

It follows from Theorem 1.5.11 that 𝑓 is analytically continued to a unique 𝑓˜ ∈ 𝒪(𝛺). Since 𝑓˜| 𝑉1 ≡ 0, the Identity Theorem 1.1.46 implies 𝑓˜(𝑧) ≡ 0; this is absurd. ⊓ ⊔ Let 𝑈 ⊂ C𝑛 be an open set and let 𝐴 ⊂ 𝑈 be an analytic set.

1.5 Analytic Subsets

27

Definition 1.5.13 ((non-)singular point). A point 𝑎 ∈ 𝐴 is called a non-singular point or a smooth point of 𝐴 if the following property holds: There are a neighborhood 𝑉 (⊂ 𝑈) of 𝑎 and nitely many holomorphic functions 𝑓 𝑗 ∈ 𝒪(𝑉) (1 ≤ 𝑗 ≤ 𝑙) such that (1.5.14)

𝐴 ∩𝑉 = {𝑧 ∈ 𝑉 : 𝑓 𝑗 (𝑧) = 0, 1 ≤ 𝑗 ≤ 𝑙},

and (1.5.15)

 rank

𝜕 𝑓𝑗 (𝑎) 𝜕𝑧 𝑘

 1≤ 𝑗 ≤𝑙,1≤ 𝑘 ≤𝑛

= 𝑙.

Necessarily, 𝑙 ≤ 𝑛, and (1.5.15) holds for all 𝑧 ∈ 𝑉 after shrinking 𝑉 if necessary. A point of 𝐴 which is not non-singular is called a singular point. The set 𝛴 ( 𝐴) of all singular points of 𝐴 is closed in 𝑈 by de nition, but moreover it is known that 𝛴 ( 𝐴) is an analytic subset (this is non-trivial; cf., e.g., [39] Theorem 6.5.10). Definition 1.5.16 (complex submanifold). An analytic set without singular point is called a complex submanifold. N.B. The connectedness is not assumed in the de nition. Theorem 1.5.17. Let 𝑆 ⊂ 𝑈 be a complex submanifold. For every point 𝑎 ∈ 𝑆 there are a neighborhood 𝑉 (resp. 𝑊) of 𝑎 (resp. 0 ∈ C𝑛 ) and a biholomorphic map 𝜑 : 𝑧 ∈ 𝑉 → 𝑤 = 𝑓 (𝑧) ∈ 𝑊 such that (1.5.18)

𝜑(𝑆 ∩𝑉) = {𝑤 = (𝑤1 , . . . , 𝑤𝑙 , 𝑤𝑙+1 , . . . 𝑤𝑛 ) ∈ 𝑊 : 𝑤1 = · · · = 𝑤𝑙 = 0}.

Proof. We may set 𝑎 = 0 by a parallel transformation. After changing the order of coordinates we have by (1.5.15)   𝜕 𝑓𝑗 (1.5.19) rank (𝑧) = 𝑙, 𝑧 ∈ 𝑉 . 𝜕𝑧 𝑘 1≤ 𝑗,𝑘 ≤𝑙 We then consider the following holomorphic map: (1.5.20)

𝜑 : 𝑧 ∈ 𝑉 → ( 𝑓1 (𝑧), . . . , 𝑓𝑙 (𝑧), 𝑧𝑙+1 , . . . , 𝑧 𝑛 ) = 𝑤 ∈ C𝑛 .

It follows from (1.5.19) that the Jacobian of (1.5.20) satis es 𝜕𝜑 𝜕𝑧 (0) ≠ 0. By the inverse function Theorem 1.4.8 we see with suitably chosen neighborhoods 𝑉 of 𝑎 and 𝑊 of 0 that 𝜑 : 𝑉 → 𝑊 is biholomorphic. By the de nition, (1.5.18) holds. ⊓ ⊔ Through (1.5.18) we may identify 𝑆 ∩ 𝑉 with 𝜑(𝑆 ∩ 𝑉). Then a point of 𝑆 ∩ 𝑉 is parameterized by (𝑤𝑙+1 , . . . , 𝑤𝑛 ), where (0, . . . , 0, 𝑤𝑙+1 , . . . , 𝑤𝑛 ) ∈ 𝑊; in this sense we call (𝑤𝑙+1 , . . . , 𝑤𝑛 ) a holomorphic local coordinate system (of 𝑆) in 𝑆 ∩ 𝑉 or simply about 𝑎.

28

1 Holomorphic Functions

Definition 1.5.21. If a function 𝑔 : 𝑆 → C on a complex submanifold 𝑆 (⊂ 𝑈) is holomorphic in a holomorphic local coordinate system (𝑤𝑙+1 , . . . , 𝑤𝑛 ) about every 𝑎 ∈ 𝑆, 𝑔 is called a holomorphic function on 𝑆. This property is independent of the choice of the holomorphic local coordinate system, and we write 𝒪(𝑆) for the set of all holomorphic functions on 𝑆. Remark 1.5.22. Let 𝑔 : 𝑆 → C be a holomorphic function on a complex submanifold 𝑆 (⊂ 𝑈). For every point 𝑎 ∈ 𝑆 there are a neighborhood 𝑉 (⊂ 𝑈) of 𝑎 and 𝑔˜ ∈ 𝒪(𝑉) with 𝑔| ˜ 𝑆∩𝑉 = 𝑔| 𝑆∩𝑉 ; 𝑔˜ is called a local extension of 𝑔. The local extension is, of course, not unique. With the notation of (1.5.14), if 𝑔˜ is a local extension of 𝑔, then 𝑔(𝑧) ˜ +

𝑙 Õ

𝑐 𝑗 (𝑧) 𝑓 𝑗 (𝑧)

(𝑐 𝑗 (𝑧) ∈ 𝒪(𝑉))

𝑗=1

is also a local extension of 𝑔. Note. In the beginning of the 19th century, Theorem 1.1.25 (ii) was proved by E. Fabry [15] (1902); it was an important nding in the early time of the theory of analytic functions of several variables that the shape of singularities of analytic functions of variables more than one is not arbitrary and its nature is di erent to the case of one variable. The method of Lemma 1.3.10 to express a holomorphic function 𝜑(𝑧) de ned in the overlapped part 𝑈1 ∩ 𝑈2 = 𝐹 ◦ of two adjacent open sets 𝑈1 and 𝑈2 by the di erence of elements of 𝒪(𝑈1 ) and 𝒪(𝑈2 ) was used by P. Cousin [12] (1895), when he resolved the problem that carries his name for cylinder domains, and also used in Oka Theory repeatedly. Therefore it is convenient to put a name to it; here, we call equation (1.3.11) the Cousin decomposition of 𝜑(𝑧). H. Poincaré [52] (1907) proved Theorem 1.4.10 in the case of 𝑛 = 2 by studying the holomorphic automorphism groups of the domains: In the proof, a boundary regularity was assumed, and it was removed later by K. Reinhardt [54] (1921) (𝑛 = 2). The present proof is found in R. Range [53], Chap. I 2; the technique seems to be rather old (cf. ibid., p. 41).

Exercises 1. Show (1.1.7). 2. (i) Expand 𝑒 𝑧1 𝑒 𝑧2 to a homogeneous polynomial series. (ii) Expand sin 𝑧1 cos 𝑧2 to a homogeneous polynomial series. 3. Draw the gure of Fig. 1.3 in the case of 𝑛 = 3. 4. a. What is the domain 𝛺( 𝑓 ) of convergence of the series

Exercises

29

𝑓 (𝑧, 𝑤) =

 ∞  Õ 𝜈+𝜇 𝜈 𝜇 𝑧 𝑤 𝜈 𝜈, 𝜇=0

in two variables 𝑧, 𝑤? Í 𝜈 b. Let 𝑓 (𝑧, 𝑤) = ∞ 𝜈=0 (𝑧 + 𝑤) be the expansion of 𝑓 (𝑧, 𝑤) to a series of homogeneous polynomials. What is the domain of convergence of the series of homogeneous polynomials? 5. If a domain 𝛺 ⊂ C𝑛 is invariant by poly-rotations (𝑧 𝑗 ) ↦→ (𝑒 𝑖 𝜃 𝑗 𝑧 𝑗 ) (𝜃 𝑗 ∈ R, 1 ≤ 𝑗 ≤ 𝑛) (i.e., (𝑒 𝑖 𝜃 𝑗 𝑧 𝑗 ) ∈ 𝛺 for all (𝑧 𝑗 ) ∈ 𝛺 and (𝜃 𝑗 ) ∈ R𝑛 ), 𝛺 is called a Reinhardt domain. Let 𝛺 be a Reinhardt domain containing the origin 0. Show that every 𝑓 ∈ 𝒪(𝛺) is expanded to a series of polynomials, 𝑓 (𝑧) = Í 𝛼 𝑛 𝛼∈Z+ 𝑐 𝛼 𝑧 (𝑧 ∈ 𝛺). (Cf., e.g., [39] 5.2.) 6. Let 𝑛, 𝑚, 𝑙 ∈ N such that 𝑛 = 𝑚 + 𝑙 ≥ 2. Let B𝑛 (1), B𝑚 (1) and B𝑙 (1) be the unit balls of C𝑛 , C𝑚 and C𝑙 , respectively. Show that B𝑚 (1) × B𝑙 (1) is not biholomorphic to B𝑛 (1). 7. Let 𝛺 ⊂ C𝑛 be a domain, let 𝑎 = (𝑎 𝑗 ) ∈ 𝛺 and let 𝑓 ∈ 𝒪(𝛺). Assume that there is a neighborhood 𝑈 of 𝑎 such that 𝑓 (𝑧) = 0,

∀ 𝑧 ∈ 𝑈 ∩ {(𝑧 𝑗 ) ∈ C𝑛 : ℑ𝑧 𝑗 = ℑ𝑎 𝑗 , 1 ≤ 𝑗 ≤ 𝑛}.

Then, show that 𝑓 (𝑧) ≡ 0 on 𝛺. 8. (Schwarz’s Lemma in several variables) a. With the notation of (1.1.2) we consider a bounded holomorphic function 𝑓 (𝑧) in B(1) such that | 𝑓 (𝑧)| ≤ 𝑀 (𝑧 ∈ B(1)) and 𝑓 (0) = 0. Show that | 𝑓 (𝑧)| ≤ 𝑀 ∥𝑧∥ (𝑧 ∈ B(1)). b. With |𝑧| max = max1≤ 𝑗 ≤𝑛 |𝑧 𝑗 | for 𝑧 = (𝑧1 , . . . , 𝑧 𝑛 ) ∈ C𝑛 we write P𝛥1 = {|𝑧| max < 1} for the unit polydisk. Let 𝑓 (𝑧) be a bounded holomorphic function in P𝛥1 such that | 𝑓 (𝑧)| ≤ 𝑀 (𝑧 ∈ P𝛥1 ) and 𝑓 (0) = 0. Show that | 𝑓 (𝑧)| ≤ 𝑀 |𝑧| max (𝑧 ∈ P𝛥1 ). 9. Prove that an analytic subset of an open set 𝑈 of C𝑛 is closed in 𝑈. 10. Let 𝛺 ⊂ C𝑛 be a domain, and let 𝑓 𝑗 ∈ 𝒪(𝛺) (1 ≤ 𝑗 ≤ 𝑚) be nitely many holomorphic functions. Show that the graph 𝛴 = {(𝑧, (𝑤 𝑗 )) ∈ 𝛺 × C𝑚 : 𝑤 𝑗 = 𝑓 𝑗 (𝑧), 1 ≤ 𝑗 ≤ 𝑚} is a complex submanifold of 𝛺 × C𝑚 .

Chapter 2

Coherent Sheaves and Oka’s Joku-Iko Principle

Beginning with the definition of analytic sheaves, we introduce the notion of coherence. We prove the coherence of 𝒪C𝑛 (Oka’s First Coherence Theorem), and then prove the coherence of ideal sheaves of complex submanifolds (a special case of Oka’s Second Coherence Theorem). We then prove Cartan’s Merging Lemma, the Oka Syzygy for coherent sheaves, and the Oka Extension of the Joku-Iko Principle, which is the key to Oka Theory.

2.1 Notion of Analytic Sheaves 2.1.1 Definitions of Rings and Modules We begin with the de nitions of algebraic terminologies of rings and modules. Those who know the de nitions of these terminologies may skip this section. Let 𝑅 be a set. Assume that for every two elements 𝑎, 𝑏 ∈ 𝑅 there is associated the third element 𝑎 + 𝑏 ∈ 𝑅 satisfying condition (1) below; we call this an algebraic operation. If an algebraic operation satis es condition (2) below, it is called an addition. Furthermore, if it satis es (3) and (4), 𝑅 is called an additive group (or commutative group, also abelian group): (1) For every 𝑐 ∈ 𝑅, (𝑎 + 𝑏) + 𝑐 = 𝑎 + (𝑏 + 𝑐) (associative law). (2) 𝑎 + 𝑏 = 𝑏 + 𝑎 (commutative). (3) There is a special element 0 ∈ 𝑅, called a zero (element) satisfying 𝑎 + 0 = 𝑎 for all 𝑎 ∈ 𝑅. (4) For every element 𝑎 ∈ 𝑅 there exists a unique element −𝑎 ∈ 𝑅 called the inverse of 𝑎 such that 𝑎 + (−𝑎) = 0. For example, Z is an additive group by the natural addition; N carries the natural addition, but is not an additive group. Assume that for every two elements 𝑎, 𝑏 ∈ 𝑅, there is the associated third element 𝑎 · 𝑏 ∈ 𝑅, and the following condition is satis ed:

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 J. Noguchi, Basic Oka Theory in Several Complex Variables, Universitext, https://doi.org/10.1007/978-981-97-2056-9_2

31

32

2 Coherent Sheaves and Oka’s Joku-Iko Principle

(5) For every 𝑐 ∈ 𝑅, (𝑎 · 𝑏) · 𝑐 = 𝑎 · (𝑏 · 𝑐) (associative law). The operation to associate 𝑎 · 𝑏 with 𝑎 and 𝑏 is called a multiplication. If there exists an element 1 ∈ 𝑅 with 𝑎 · 1 = 1 · 𝑎 = 𝑎 (∀ 𝑎 ∈ 𝑅), 1 is called a unit element for multiplication, which is unique if it exists. Definition 2.1.1 (ring). If a set 𝑅 carries an addition “+” and a multiplication “·” and the following conditions are satis ed, then 𝑅 is called a ring: (i) 𝑅 has a unit element 1 for multiplication. (ii) For arbitrary three elements 𝑎, 𝑏, 𝑐 ∈ 𝑅, the so-called distributive laws hold: 𝑎 · (𝑏 + 𝑐) = (𝑎 · 𝑏) + (𝑎 · 𝑐) (= 𝑎 · 𝑏 + 𝑎 · 𝑐,

so written),

(𝑏 + 𝑐) · 𝑎 = (𝑏 · 𝑎) + (𝑐 · 𝑎) (= 𝑏 · 𝑎 + 𝑐 · 𝑎). In particular, if the multiplication is commutative (𝑎 · 𝑏 = 𝑏 · 𝑎), 𝑅 is called a commutative ring The multiplication 𝑎 · 𝑏 is often simpli ed to 𝑎𝑏 without “·”. Remark 2.1.2. Here we deal with only commutative rings, and so a ring is always assumed to be commutative all through the present textbook. Let 𝑅 be a ring, and let 𝑎 ∈ 𝑅. If there is an element 𝑏 ∈ 𝑅 with 𝑎𝑏 = 1, 𝑎 is called a unit and 𝑏 is denoted by 𝑎 −1 . A subset 𝐼 of 𝑅 is called an ideal if 𝑎 · (𝑏 + 𝑐) ∈ 𝐼,

∀ 𝑎 ∈ 𝑅, ∀ 𝑏, 𝑐 ∈ 𝐼.

In particular, 0 ∈ 𝐼, and −𝑎 ∈ 𝐼 for all 𝑎 ∈ 𝐼. Definition 2.1.3 (module). Let 𝑅 be a ring, and let 𝑀 be an additive group. If for arbitrary elements 𝑎 ∈ 𝑅 and 𝑥 ∈ 𝑀, there is associated an element 𝑎 · 𝑥 ∈ 𝑀, and the following conditions are satis ed, then 𝑀 is called a module over 𝑅 or a 𝑅-module : that is, for 𝑎, 𝑏 ∈ 𝑅, 𝑥, 𝑦 ∈ 𝑀: (i) 1 · 𝑥 = 𝑥; (ii) (𝑎𝑏) · 𝑥 = 𝑎 · (𝑏 · 𝑥); (iii) (𝑎 + 𝑏) · 𝑥 = 𝑎 · 𝑥 + 𝑏 · 𝑥; (iv) 𝑎 · (𝑥 + 𝑦) = 𝑎 · 𝑥 + 𝑎 · 𝑦. A subset 𝑀 ′ ⊂ 𝑀 is called a submodule, if 𝑀 ′ forms a module over 𝑅 by itself with respect to the algebraic operations induced from those of 𝑀. Example 2.1.4. (i) Any additive group may be regarded as a module over Z. (ii) The 𝑞th product Z𝑞 with 𝑞 ∈ N forms an additive group by the element-wise addition, and by the element-wise multiplication: 𝑎 · 𝑥 = (𝑎𝑥1 , . . . , 𝑎𝑥 𝑞 ) ∈ Z𝑞 ,

𝑎 ∈ Z, 𝑥 = (𝑥 1 , . . . , 𝑥 𝑞 ) ∈ Z𝑞 .

Z𝑞 is a module over Z. (iii) Let 𝑝 ∈ N and let 𝑝Z be the set of all multiples of 𝑝. Then, 𝑝Z is an ideal of Z.

2.1 Notion of Analytic Sheaves

33

(iv) The set 𝒪(𝑈) of all holomorphic functions in an open subset 𝑈 ⊂ C𝑛 is a ring by the natural addition and multiplication. As in (ii), 𝒪(𝑈) 𝑞 is an 𝒪(𝑈)module. (v) Let 𝑋 ⊂ 𝑈 be a subset of an open subset 𝑈 ⊂ C𝑛 . Let 𝐼 (𝑋) denote the set of all 𝑓 ∈ 𝒪(𝑈) such that 𝑓 | 𝑋 ≡ 0. Then, 𝐼 (𝑋) is an ideal of 𝒪(𝑈). (vi) In general, with a ring 𝑅 given and with nitely many indeterminates 𝑋1 , . . . , 𝑋 𝑁 , we denote by 𝑅[𝑋1 , . . . , 𝑋 𝑁 ] the set of all polynomials in 𝑋 𝑗 (1 ≤ 𝑗 ≤ 𝑁) with coe cients in 𝑅. By the natural algebraic operations, 𝑅[𝑋1 , . . . , 𝑋 𝑁 ] forms a ring, called an 𝑅-polynomial ring (of 𝑁 variables), which is also an 𝑅-module.

2.1.2 Analytic Sheaves Here we leave the general treatment of the notion of sheaves to other books (e.g., [39] 1.3; more generally, cf. Godeman [20]), we restrict ourselves to dealing with necessary holomorphic (analytic) functions. We consider a holomorphic function 𝑓 (𝑧) in a connected neighborhood of 𝑎 ∈ C𝑛 . With a xed coordinate system 𝑧 = (𝑧 𝑗 )1≤ 𝑗 ≤𝑛 , 𝑓 (𝑧) is uniquely expressed by a convergent power series Õ (2.1.5) 𝑓 (𝑧) = 𝑐 𝜈 (𝑧 − 𝑎) 𝜈 . 𝜈 ∈Z+𝑛

We identify holomorphic functions about 𝑎 whose power series expansions (2.1.5) are the same, and then we denote by 𝒪𝑎 all of such equivalence classes. We write 𝑓 ∈ 𝒪𝑎 for an element of 𝒪𝑎 . An element with all 𝑐 𝜈 = 0 (𝜈 ∈ Z+𝑛 ) in (2.1.5) is denoted 𝑎 by 0𝑎 (often abbreviated to 0). An element with 𝑐 (0,...,0) = 1 and 𝑐 𝜈 = 0 (|𝜈| > 0) is denoted by 1𝑎 (often abbreviated to 1). 𝒪𝑎 forms a ring with the natural addition and multiplication, which is called an analytic local ring. Proposition 2.1.6 (Integral domain). If 𝑓 · 𝑔 = 0 for two elements 𝑓 , 𝑔 of 𝒪𝑎 , 𝑎 𝑎 𝑎 𝑎 then either 𝑓 = 0 or 𝑔 = 0. 𝑎

𝑎

⊓ ⊔

Proof. It is immediate by (1.5.6). For a domain 𝛺 ⊂ (2.1.7)

C𝑛

we put 𝒪𝛺 =

Ä

𝒪𝑎 ,

𝑎∈𝛺

where “⊔” stands for a disjoint union. We call 𝒪𝛺 the sheaf of holomorphic (or analytic) functions over 𝛺. A holomorphic function 𝑓 ∈ 𝒪(𝑈) on an open subset 𝑈 ⊂ 𝛺 induces a map (2.1.8)

𝑓 : 𝑎 ∈ 𝑈 −→ 𝑓 ∈ 𝒪𝑈 , 𝑎

34

2 Coherent Sheaves and Oka’s Joku-Iko Principle

which is called a section of 𝒪𝛺 (or 𝒪C𝑛 ) over 𝑈. We denote by 𝛤 (𝑈,𝒪𝛺 ) (or 𝛤 (𝑈,𝒪C𝑛 )) the set of all sections of 𝒪𝛺 over 𝑈. Then 𝛤 (𝑈,𝒪𝛺 ) is a ring with the natural algebraic structure of addition and multiplication. Remark 2.1.9. By de nition we can identify 𝒪(𝑈) and 𝛤 (𝑈,𝒪𝑈 ) through 𝑓 ∈ 𝒪(𝑈) −→ 𝑓 ∈ 𝛤 (𝑈,𝒪𝑈 ). It is noticed, however, that for 𝑎 ∈ 𝑈, 𝑓 (𝑎) is a value (complex number), and 𝑓 is a 𝑎 convergent power series about 𝑎. 𝑞

𝒪𝑎𝑞

Let 𝑞 ∈ N. Taking the 𝑞th direct product de ne the 𝑞th direct product sheaf of 𝒪𝛺 by Ä 𝑞 (2.1.10) 𝒪𝛺 = 𝒪𝑎𝑞 .

z }| { = 𝒪𝑎 × · · · × 𝒪𝑎 (𝑎 ∈ 𝛺) of 𝒪𝑎 , we

𝑎∈𝛺

𝑞

z }| { Note that it is not the 𝑞th direct product 𝒪𝛺 × · · · × 𝒪𝛺 of 𝒪𝛺 as sets. For 𝑓 ∈ 𝒪𝑎 𝑎 (𝑎 ∈ 𝛺) and (𝑔 𝑗 ), (ℎ 𝑗 ) ∈ 𝒪𝑎𝑞 (1 ≤ 𝑗 ≤ 𝑞) the following algebraic structure is 𝑎 𝑎 naturally introduced: (2.1.11)

𝑓 · (𝑔 𝑗 ) = ( 𝑓 𝑔 𝑗 ) ∈ 𝒪𝑎𝑞 , 𝑎 𝑎 𝑎 𝑎  𝑓 · (𝑔 𝑗 ) ± (ℎ 𝑗 ) = 𝑓 · (𝑔 𝑗 ) ± 𝑓 · (ℎ 𝑗 ). 

𝑎

𝑎

By the algebraic structure above,

𝑎

𝒪𝑎𝑞

𝑎

𝑎

𝑎

𝑎

is an 𝒪𝑎 -module.

𝑞 Definition 2.1.12 (Analytic sheaf). A subset ℱ ⊂ 𝒪𝛺 is called an analytic sheaf over 𝛺 if it has the following structure: 𝑞 (i) For every à 𝑎 ∈ 𝛺 an associated 𝒪𝑎 -submodule ℱ𝑎 ⊂ 𝒪𝑎 exists. (ii) ℱ = 𝑎∈𝛺 ℱ𝑎 . In particular, an analytic sheaf ℱ with 𝑞 = 1 is called a sheaf of ideals of 𝒪𝛺 . 𝑞 Naturally, 𝒪𝛺 itself is an analytic sheaf. Although it is a trivial example, ℱ with ℱ𝑎 = {0} (∀ 𝑎 ∈ 𝛺) is an analytic sheaf called the zero sheaf and written as 0. The restriction of an analytic sheaf ℱ to a subset 𝑈 ⊂ 𝛺 is de ned by Ä ℱ|𝑈 := ℱ𝑎 . 𝑎∈𝑈

Remark 2.1.13. It is common to de ne analytic sheaves by making use of a more abstract notion of sheaves (e.g., cf. [29], [26], [30], and [39], etc.). Here, for the purpose to solve the “Three Big Problems” it is su cient to deal with this simple case.

2.1 Notion of Analytic Sheaves

35

A 𝑞-vector-valued holomorphic function 𝑔 = (𝑔1 , . . . , 𝑔𝑞 ) : 𝑈 → C𝑞 on an open subset 𝑈 ⊂ 𝛺 induces a map     𝑞 (2.1.14) 𝑔 = 𝑔 𝑗 : 𝑎 ∈ 𝑈 −→ 𝑔 := 𝑔1 , . . . , 𝑔𝑞 ∈ 𝒪𝛺 . 𝑎

𝑞 of 𝒪𝛺

𝑎

𝑎

(or 𝒪C𝑞 𝑛 )

We call this a section over 𝑈, and the set of all of them is denoted by 𝑞 𝑞 𝛤 (𝑈,𝒪𝛺 ). For 𝑓 ∈ 𝒪(𝑈) ( 𝑓 ∈ 𝛤 (𝑈,𝒪𝛺 )) and 𝑔 ∈ 𝛤 (𝑈,𝒪𝛺 ) we put (2.1.15) so

𝑞 𝑓 · 𝑔 : 𝑎 ∈ 𝑈 −→ 𝑓 · 𝑔 ∈ 𝒪𝛺 , 𝑎

𝑎

𝑞 that 𝛤 (𝑈,𝒪𝛺 ) forms a module over the ring 𝒪(𝑈) (= 𝛤 (𝑈,𝒪𝛺 )). 𝑞 Let ℱ ⊂ 𝒪𝛺 be an analytic sheaf over 𝛺. If 𝑔 in (2.1.14) satis es

𝑔 ∈ ℱ𝑎 for all 𝑎 𝑎 ∈ 𝑈, 𝑔 is called a section of ℱ over 𝑈. We denote the section space of all of them by n o 𝑞 (2.1.16) 𝛤 (𝑈, ℱ) = 𝑔 ∈ 𝛤 (𝑈,𝒪𝛺 ) : 𝑔 ∈ ℱ𝑎 , ∀ 𝑎 ∈ 𝑈 . 𝑎

By (2.1.15), 𝛤 (𝑈, ℱ) is a module over the ring 𝒪(𝑈) (or, 𝛤 (𝑈,𝒪𝛺 )), and also a vector space over C. A section 𝑠 ∈ 𝛤 (𝑈, ℱ) may be regarded as a map from 𝑈 to ℱ, for which we write 𝑠 : 𝑎 ∈ 𝑈 −→ 𝑠(𝑎) ∈ ℱ. For a closed set 𝐸 we denote by 𝛤 (𝐸, ℱ) the set of all sections of ℱ over a neighborhood of 𝐸, where the neighborhood may vary. Remark 2.1.17. Throughout the present text we often consider holomorphic functions and sections of analytic sheaves over a neighborhood of a closed set, and the neighborhood will depend on each case. Therefore, by the term, holomorphic functions or sections of an analytic sheaf ℱ over 𝐸, we understand that they are de ned in some neighborhoods of 𝐸, and denote the set of all of them by 𝒪(𝐸) (or 𝛤 (𝐸, ℱ)); in cases where the neighborhood must be clari ed, we will state it precisely. Remark 2.1.18. If a knowledge of “general topology” is assumed, it is standard to de ne a sheaf with the topology such that the sections de ned above are continuous. Then, the de nition of analytic sheaves may take a more general abstract form. Here, we do not need a general theory of sheaves, so we prefer the style above: Cf., e.g., [39] 1.3.

36

2 Coherent Sheaves and Oka’s Joku-Iko Principle

2.2 Coherent Sheaves 2.2.1 Locally Finite Sheaves We begin with the de nition of the local niteness of an analytic sheaf. Let 𝑛, 𝑝 ∈ N. Let 𝛺 ⊂ C𝑛 be an open set and let ℱ ⊂ 𝒪𝛺𝑝 be an analytic sheaf. Definition 2.2.1. (i) Let 𝑈 ⊂ 𝛺 be an open subset. A nite family {𝜎 𝑗 }𝑙𝑗=1 ⊂ 𝛤 (𝑈, ℱ) is said to generate ℱ over 𝑈, if ℱ|𝑈 =

𝑙 Õ

𝒪𝑈 · 𝜎 𝑗 .

𝑗=1

That is, we have ℱ𝑧 =

𝑙 Õ

𝒪𝑧 · 𝜎 𝑗 , 𝑧

𝑗=1

∀ 𝑧 ∈ 𝑈.

In this case, {𝜎 𝑗 }𝑙𝑗=1 is called a finite generator system of ℱ over 𝑈. (ii) ℱ is said to be locally finite about a point 𝑎 ∈ 𝛺, if there is a neighborhood 𝑈 (⊂ 𝛺) of 𝑎 with a nite generator system {𝜎 𝑗 }𝑙𝑗=1 of ℱ over 𝑈. In this case,

{𝜎 𝑗 }𝑙𝑗=1 is called a locally finite generator system of ℱ about 𝑎. (iii) ℱ is said to be locally finite (in 𝛺), if ℱ is locally nite about every 𝑎 ∈ 𝛺.

Example 2.2.2. We give a simple example of an analytic sheaf which is not locally nite (hence not coherent as de ned in the next subsection). Let 𝑛 = 1, and let 𝛥(1) ⋐ C be the unit disk. We de ne an analytic sheaf ℱ ⊂ 𝒪C over C by ( 0, 𝑎 ∈ C \ 𝛥(1), ℱ𝑎 = 𝒪𝑎 , 𝑎 ∈ 𝛥(1). Here, 0 means a submodule consisting only of the zero element of 𝒪𝑎 , and equals {0𝑎 } as set.1 Over 𝛥(1), 1 ∈ 𝛤 ( 𝛥(1),𝒪C ) is a nite generator system of ℱ, and over ¯ ¯ C \ 𝛥(1), 0 ∈ 𝛤 (C \ 𝛥(1),𝒪 C ) is a generator system of ℱ. But, ℱ is not locally nite about any point of 𝜕 𝛥(1). In fact, if ℱ is locally nite about a point 𝑎 ∈ 𝜕 𝛥(1), then there would be a connected neighborhood 𝑈 ∋ 𝑎 and nitely many 𝑓 𝑗 ∈ 𝒪(𝑈) (1 ≤ 𝑗 ≤ 𝑁) such that ℱ𝑏 =

𝑁 Õ 𝑗=1

𝒪𝑏 · 𝑓 𝑗 , 𝑏

∀ 𝑏 ∈ 𝑈.

1 The zero module or a zero element is often simpli ed in this way; similarly, for instance, a vector space 𝐸 consisting only of the zero vector is written as 𝐸 = 0.

2.2 Coherent Sheaves

37

Since ℱ𝑏 = 0 for 𝑏 ∈ 𝑈 \ 𝛥(1), 𝑓 𝑗 = 0 for all 𝑗. Thus, 𝑓 𝑗 ≡ 0 (identically) in a 𝑏 su ciently small neighborhood of 𝑏. The Identity Theorem 1.1.46 would imply 𝑓 𝑗 ≡ 0 on 𝑈, and hence ℱ𝑏 = 0 (∀ 𝑏 ∈ 𝑈). On the other hand, ℱ𝑏 = 𝒪𝑏 for 𝑏 ∈ 𝑈 ∩ 𝛥(1) (≠ ∅); this is absurd. The above example is easily extended to the case of 𝑛 ≥ 2. Example 2.2.3. We introduce an example of an analytic sheaf due to Oka VII [48] which is not locally nite.2 We consider a complex hyperplane 𝑋 = {𝑧 = 𝑤} in C2 with variables 𝑧, 𝑤. Taking open balls B𝑖 = {|𝑧| 2 + |𝑤| 2 < 𝑟 𝑖2 } with 0 < 𝑟 1 < 𝑟 2 , we set 𝑋0 = 𝑋 ∩ B2 \ B1 . Let 𝛤1 = 𝜕B1 be the boundary sphere. Let ℬ𝑎 denote the set of all those power series 𝑓 about 𝑎 ∈ B2 of a holomorphic 𝑎 function 𝑓 (𝑧, 𝑤) in a neighborhood 𝑈 Ã of 𝑎 such that 𝑓 (𝑧, 𝑤)/(𝑧 − 𝑤) is holomorphic at every point of 𝑈 ∩ 𝑋0 , and set ℬ = 𝑎∈B2 ℬ𝑎 . Then, ℬ is an analytic subsheaf of 𝒪B2 . By the construction we have ( 𝒪𝑎 · (𝑧 − 𝑤) , 𝑎 ∈ 𝑋0 , 𝑎 (2.2.4) ℬ𝑎 = 𝒪𝑎 , 𝑎 ∈ B2 \ 𝑋0 . Now, ℬ is not locally nite at any point 𝑎 ∈ 𝑋0 ∩ 𝛤1 . If otherwise, there are a polydisk neighborhood 𝑈 of 𝑎 and nitely many holomorphic functions 𝑓 𝑗 ∈ 𝒪(𝑈), 𝑓 𝑗 ∈ 𝛤 (𝑈, ℬ), 1 ≤ 𝑗 ≤ 𝑁, such that ℬ𝑏 =

𝑁 Õ

𝒪𝑏 · 𝑓 𝑗 ,

𝑗=1

𝑏

∀ 𝑏 ∈ 𝑈.

However, since 𝑓 𝑗 (𝑧, 𝑧) ≡ 0, ℬ𝑏 ≠ 𝒪𝑏 at 𝑏 ∈ 𝑈 ∩ 𝑋 \ 𝑋0 ; this contradicts (2.2.4). Finally, although trivial, the zero sheaf 0 is coherent. Let 𝛺 ⊂ C𝑛 be an open set, and let 𝐴 ⊂ 𝛺 be a subset. For 𝑓 ∈ 𝒪𝑎 (𝑎 ∈ 𝛺) we 𝑎 write 𝑓 | 𝐴 = 0, if 𝑓 is holomorphic in a polydisk neighborhood P𝛥 of 𝑎, and 𝑓 | P𝛥∩𝐴 = 0, i.e., 𝑓 (𝑧) = 0, Put (2.2.5)

∀ 𝑧 ∈ P𝛥 ∩ 𝐴.

n o ℐ⟨𝐴⟩𝑎 = 𝑓 ∈ 𝒪𝑎 : 𝑓 | 𝐴 = 0 , 𝑎

ℐ⟨𝐴⟩ =

Ä

ℐ⟨𝐴⟩𝑎 .

𝑎∈𝛺

ℐ⟨𝐴⟩ is a sheaf of ideals of 𝒪𝛺 , and is called the ideal sheaf of the subset 𝐴.

2 H. Cartan put an emphasis on this counter-example in his comment on Oka VII [48].

38

2 Coherent Sheaves and Oka’s Joku-Iko Principle

It is an important case of ℐ⟨ 𝐴⟩ when 𝐴 is an analytic subset of 𝛺; in that case ℐ⟨ 𝐴⟩ is called the ideal sheaf of the analytic subset 𝐴.3 If 𝐴 is non-singular, i.e., a complex submanifold, we call ℐ⟨𝐴⟩ the ideal sheaf of the complex submanifold 𝐴. ¯ for any subset 𝐴 (⊂ 𝛺) and its topological closure 𝐴¯ In general, ℐ⟨𝐴⟩ = ℐ⟨ 𝐴⟩ in 𝛺. Lemman 2.2.6. Let 𝐴 ⊂ 𝛺 be a closed osubset. Assume that there is a finite generator system 𝛼 𝑗 ∈ 𝛤 (𝛺, ℐ⟨𝐴⟩) : 1 ≤ 𝑗 ≤ 𝐿 of ℐ⟨𝐴⟩ over 𝛺. Then 𝐴 = {𝑧 ∈ 𝛺 : 𝛼 𝑗 (𝑧) = 0, 1 ≤ 𝑗 ≤ 𝐿}.

(2.2.7)

In particular, 𝐴 is an analytic subset. Proof. We denote the right-hand side of (2.2.7) by 𝐵, which is an analytic subset by de nition. By the choice of 𝛼 𝑗 , 𝐵 ⊃ 𝐴. Let 𝑎 ∈ 𝛺 \ 𝐴 be an arbitrary point. Since 𝐴 is closed, there is a neighborhood 𝑈 ∋ 𝑎 with 𝐴 ∩𝑈 = ∅. Therefore ℐ⟨𝐴⟩𝑎 = 𝒪𝑎 ∋ 1𝑎 and then there are 𝑓 𝑗 ∈ 𝒪𝑎 (1 ≤ 𝑗 ≤ 𝐿) such that 𝑎

1 𝑎 = 𝑓 1 · 𝛼1 𝑎 + · · · + 𝑓 𝐿 · 𝛼 𝐿 𝑎 . 𝑎

𝑎

That is, there is a neighborhood 𝑉 of 𝑎 such that 𝑓 𝑗 , 𝛼 𝑗 are holomorphic in 𝑉, and 1 = 𝑓1 (𝑧) · 𝛼1 (𝑧) + · · · + 𝑓 𝐿 (𝑧) · 𝛼 𝐿 (𝑧),

𝑧 ∈ 𝑉.

Setting 𝑧 = 𝑎, we have some 𝛼 𝑗 (𝑎) ≠ 0. Therefore 𝑎 ∈ 𝛺 \ 𝐵, and 𝐴 ⊃ 𝐵, so that 𝐴 = 𝐵. ⊓ ⊔ The next theorem relates the local niteness of ℐ⟨ 𝐴⟩ with the analyticity of 𝐴, which is basic and a special case of a more general support theorem ([39] Theorem 6.9.10). Theorem 2.2.8. Let 𝐴 ⊂ 𝛺 be a closed set. If ℐ⟨ 𝐴⟩ is locally finite, then 𝐴 is an analytic subset. Proof. Let 𝑎 ∈ 𝛺 be an arbitrary point. Suppose that there are a neighborhood 𝑈 ⊂ 𝛺 of 𝑎 and a generator system 𝜎 𝑗 ∈ 𝛤 (𝑈, ℐ⟨ 𝐴⟩), 1 ≤ 𝑗 ≤ 𝑙, of ℐ⟨ 𝐴⟩ over 𝑈. It follows from Lemma 2.2.6 that 𝐴 ∩ 𝑈 = {𝑧 ∈ 𝑈 : 𝜎 𝑗 (𝑧) = 0, 1 ≤ 𝑗 ≤ 𝑙}. Therefore 𝐴 is an analytic subset.

⊓ ⊔

Remark 2.2.9. In fact, the converse of the above theorem holds by Oka’s Second Coherence Theorem mentioned in the next subsection.

3 Oka (VIII) called ℐ⟨ 𝐴⟩ “l’idéal géométrique de domaines indéterminés attaché à 𝐴”; so we may call it the geometric ideal sheaf of 𝐴.

2.2 Coherent Sheaves

39

2.2.2 Coherent Sheaves Let ℱ be an analytic sheaf over an open set 𝛺 ⊂ C𝑛 as above.

  Definition 2.2.10. A relation sheaf of ℱ is an analytic sheaf ℛ (𝜏 𝑗 )1≤ 𝑗 ≤𝑞 de ned as follows: (i) Let 𝑈 ⊂ 𝛺 be an open set. (ii) Let 𝜏 𝑗 ∈ 𝛤 (𝑈, ℱ), 1 ≤ 𝑗 ≤ 𝑞, be nitely many sections.   (iii) We consider an element 𝑎 1 𝑧 , . . . , 𝑎 𝑞 ∈ 𝒪𝑧𝑞 (𝑧 ∈ 𝑈) satisfying the linear 𝑧 relation (2.2.11)

𝑎 1 𝑧 · 𝜏1 𝑧 + · · · + 𝑎 𝑞 · 𝜏𝑞 = 0, 𝑧



and denote all of them by ℛ 𝜏1 , . . . , 𝜏𝑞 (2.2.12)

    ℛ 𝜏1 , . . . , 𝜏𝑞 = ℛ 𝜏 𝑗

𝑧

 𝑧

  = ℛ (𝜏 𝑗 )1≤ 𝑗 ≤𝑞 . We put 𝑧

 1≤ 𝑗 ≤𝑞

=

Ä 𝑧 ∈𝑈

ℛ

  𝜏𝑗

 1≤ 𝑗 ≤𝑞 𝑧

.

Definition 2.2.13 (Coherent sheaf). An analytic sheaf ℱ over 𝛺 is called a coherent analytic sheaf or simply a coherent sheaf if the following two conditions are satis ed: (i) ℱ is locally nite. (ii) Every relation sheaf of ℱ is locally nite. K. Oka proved three fundamental coherence theorems (Oka’s Three Coherence Theorems [48] Oka VII, VIII; cf. Noguchi [39]). The implication of the series of Oka’s Coherence Theorems is broad and deep, and it is impossible to explain the importance in a few lines. The rst is stated as follows: Oka’s First Coherence Theorem (Oka VII, 1948) 𝒪C𝑛 is coherent. We will prove this in the next section. When 𝑛 = 1, the proof is easy and left to the readers (Exercise 3 at the end of the chapter). Oka’s Second Coherence Theorem (Oka VIII, 1951, Cartan [10], 1950)4 The ideal sheaf of any analytic subset is coherent. This is the converse of Theorem 2.2.8, so that: Theorem 2.2.14. A closed subset 𝐴 ⊂ 𝛺 is analytic if and only if ℐ⟨ 𝐴⟩ is locally finite. The case of non-singular 𝐴 is easy and will be proved in 2.3.3 (cf. [39] 6.5 in general). 4 The main part of the proof was done in the paper of Oka’s First Coherence Theorem (Oka VII, 1948), and this result was announced there. In 1950, H. Cartan gave another proof of it by making use of Oka’s result (1948). Cf. [39] Chap. 9, On Coherence.

40

2 Coherent Sheaves and Oka’s Joku-Iko Principle

Since it requires a rather long preparation to state the third one, we have to allow ourselves to use some terminologies without precise de nitions for a moment. Let 𝐴 be an analytic subset of 𝛺. Then by Oka’s Second Coherence Theorem ℐ⟨ 𝐴⟩ is coherent, and the quotient sheaf 𝒪𝐴 := 𝒪𝛺 /ℐ⟨ 𝐴⟩ is de ned and gives rise to a coherent sheaf, which is called the structure sheaf of 𝐴 as a complex space. Then the so-called “normalization sheaf 𝒪ˆ 𝐴” is de ned: Oka’s Third Coherence Theorem (Oka VIII, 1951) The normalization sheaf 𝒪ˆ 𝐴 is coherent. Cf., e.g, [39] 6.10 for details. For general properties of coherent sheaves we have: Proposition 2.2.15. Let ℱ be a coherent sheaf over 𝛺. (i) An analytic subsheaf of ℱ is coherent if and only if it is locally finite. (ii) ℱ 𝑁 (𝑁 ∈ N) are coherent. That is, every simultaneous relation sheaf of ℱ is coherent.  𝐿  𝐿 (iii) Let 𝛾 𝑗 𝑗=1 be a finite family of sections of ℱ over 𝛺. If 𝛾 𝑗 𝑗=1 generates 𝑎 ℱ𝑎 at a point 𝑎 ∈ 𝛺, then there is a neighborhood 𝑉 (⊂ 𝛺) of 𝑎 such that  𝐿 𝛾 𝑗 𝑗=1 generates ℱ𝑧 at every point 𝑧 ∈ 𝑉. 𝑧

Proof. (i) It remains to show the local niteness of relation sheaves, but they are so since ℱ is coherent. (ii) We shall proceed by induction on 𝑁. The case of 𝑁 = 1 is the assumption. Let 𝑁 ≥ 2, and suppose that it holds for 𝑁 − 1. The local niteness of ℱ 𝑁 follows from that of ℱ. Let 𝑈 ⊂ 𝛺 be an open subset, and let 𝐹𝑖 ∈ 𝛤 (𝑈, ℱ 𝑁 ), 1 ≤ 𝑖 ≤ 𝑞, be a nite number of sections. It su ces to show the local niteness of the relation sheaf 𝑞 n o Õ ℛ = (𝑎 𝑖 ) ∈ 𝒪𝑧𝑞 : 𝑎 𝑖 𝐹𝑖 (𝑧) = 0, 𝑧 ∈ 𝑈 ⊂ 𝒪𝑈𝑞 . 𝑖=1

With the expressions 𝐹𝑖 = (𝐹𝑖1 , . . . , 𝐹𝑖 𝑁 ), ℛ (⊂ (𝒪|𝑈 ) 𝑞 ) is determined by (𝑎 𝑖 ) ∈ 𝒪𝑧𝑞 ,

𝑞 Õ

𝑎 𝑖 𝐹𝑖 𝑗 = 0, 𝑧

𝑖=1

1 ≤ 𝑗 ≤ 𝑁, 𝑧 ∈ 𝑈.

We rst consider the case for 𝑗 = 1. Denote by ℛ1 ⊂ (𝒪|𝑈 ) 𝑞 the relation sheaf de ned by 𝑞 Õ (𝑎 𝑖 ) ∈ 𝒪𝑧𝑞 , 𝑎 𝑖 𝐹𝑖1 𝑧 = 0, 𝑧 ∈ 𝑈. 𝑖=1

Then ℛ ⊂ ℛ1 , and since ℱ is coherent, ℛ1 is locally nite. For every point 𝑎 ∈ 𝑈 𝐿 there are a neighborhood 𝑉 ⊂ 𝑈 of 𝑎 and a locally nite generator system {𝜙 (𝜆) }𝜆=1 of ℛ1 | 𝑉 with 𝜙 (𝜆) ∈ 𝛤 (𝑉, ℛ1 ). Set 𝜙 (𝜆) = (𝜙𝑖(𝜆) )1≤𝑖 ≤𝑞 . At every point 𝑧 ∈ 𝑉 an element of ℛ1𝑧 , Õ  (𝑎 𝑖 ) = 𝑐 𝜆 𝑧 · 𝜙𝑖(𝜆) (𝑧) , 𝑐 𝜆 𝑧 ∈ 𝒪𝑧 𝜆

2.2 Coherent Sheaves

41

belongs to ℛ𝑧 if and only if ÕÕ (2.2.16) 𝑐 𝜆 𝑧 · 𝜙𝑖(𝜆) (𝑧) · 𝐹𝑖 𝑗 = 0, 𝑖

𝑧

𝜆

1 ≤ 𝑗 ≤ 𝑁.

  We consider this as a linear relation on 𝑐 𝜆 𝑧 . For 𝑗 = 1 it is already satis ed

because of the choice of 𝜙𝑖(𝜆) . Therefore, simultaneous relation (2.2.16), in fact, consists of 𝑁 − 1 relations.   The induction hypothesis implies that in a neighborhood 𝑊 (⊂ 𝑉) of 𝑎 such 𝑐 𝜆 𝑧 is written by a linear sum of nitely many sections 𝛾 (𝜈) =   𝛾𝜆(𝜈) with 𝛾𝜆(𝜈) ∈ 𝛤 (𝑊,𝒪). Therefore, the sections 

 Õ 𝑎 𝑖(𝜈) = 𝛾𝜆(𝜈) · 𝜙𝑖(𝜆) 𝜆

generate ℛ𝑧 at every 𝑧 ∈ 𝑊.  𝑁 (iii) Because of the coherence of ℱ there is a nite generator system 𝜎ℎ ℎ=1 of ℱ over a neighborhood 𝑈 (⊂ 𝛺) of 𝑎. By the assumption, there are elements 𝑓 ℎ 𝑗 ∈ 𝒪𝑎 with 𝑎

𝜎ℎ 𝑎 =

𝐿 Õ

𝑓ℎ 𝑗 𝛾 𝑗 , 𝑎

𝑗=1

𝑎

1 ≤ ℎ ≤ 𝑁.

It follows that there is a neighborhood 𝑉 ⊂ 𝑈 of 𝑎, where 𝜎ℎ (𝑧) =

𝐿 Õ

𝑓 ℎ 𝑗 (𝑧) 𝛾 𝑗 (𝑧),

∀ 𝑧 ∈ 𝑉,

1 ≤ ℎ ≤ 𝑁.

𝑗=1

Hence, 𝜎ℎ 𝑧 =

𝐿 Õ 𝑗=1

𝑓ℎ 𝑗 𝛾 𝑗 , 𝑧

𝑧

∀ 𝑧 ∈ 𝑉,

1 ≤ ℎ ≤ 𝑁.

 𝐿 Therefore, 𝛾 𝑗 𝑗=1 generates ℱ over 𝑉.

⊓ ⊔

Proposition 2.2.17. Let ℱ be a coherent sheaf over 𝛺. If ℱ𝑖 ⊂ ℱ, 𝑖 = 1, 2, are coherent analytic subsheaves of ℱ over 𝛺, so is the intersection ℱ1 ∩ ℱ2 . Proof. For every point 𝑎 ∈ 𝛺 there are a neighborhood 𝑈 (⊂ 𝛺) of 𝑎 and locally nite generator systems of ℱ𝑖 , 𝑖 = 1, 2, 𝛼 𝑗 ∈ 𝛤 (𝑈, ℱ1 ),

1 ≤ 𝑗 ≤ 𝑙,

𝛽 𝑘 ∈ 𝛤 (𝑈, ℱ2 ),

1 ≤ 𝑘 ≤ 𝑚.

At any point 𝑧 ∈ 𝑈, 𝛾 ∈ ℱ𝑧 belongs to ℱ1𝑧 ∩ ℱ2𝑧 if and only if it is written as

42

2 Coherent Sheaves and Oka’s Joku-Iko Principle

𝛾=

Õ

𝑎 𝑗 𝛼 𝑗 (𝑧) =

𝑗

Õ

𝑏 𝑘 𝛽 𝑘 (𝑧),

𝑎 𝑗 , 𝑏 𝑘 ∈ 𝒪𝑧 .

𝑘

This is equivalent to Õ

(2.2.18)

𝑗

𝛾=

𝑎 𝑗 𝛼 𝑗 (𝑧) + Õ

Õ

𝑏 𝑘 (−𝛽 𝑘 (𝑧)) = 0,

𝑘

𝑎 𝑗 𝛼 𝑗 (𝑧).

𝑗

The above expression de nes a relation sheaf of ℱ with (𝑎 𝑗 , 𝑏 𝑘 ) being unknowns, which is denoted by ℛ := ℛ(. . . , 𝛼 𝑗 , . . . , −𝛽 𝑘 , . . .). Since ℱ is coherent, ℛ is locally nite. Taking a smaller 𝑈 ∋ 𝑎 if necessary, we may assume that  ℛ|𝑈 isgenerated by a (ℎ) nite number of 𝜂 (ℎ) ∈ 𝛤 (𝑈, ℱ), 1 ≤ ℎ ≤ 𝐿. With 𝜂 (ℎ) := 𝑎 (ℎ) , (ℱ1 ∩ ℱ2 )|𝑈 𝑗 , 𝑏𝑘 is generated by Õ Õ 𝜉 (ℎ) := 𝑎 (ℎ) 𝑏 𝑘(ℎ) 𝛽 𝑘 ∈ 𝛤 (𝑈, ℱ1 ∩ ℱ2 ), 1 ≤ ℎ ≤ 𝐿. ⊓ ⊔ 𝑗 𝛼𝑗 = 𝑗

𝑘

Example 2.2.19. The analytic sheaf ℬ of Example 2.2.3 is, of course, not coherent. We give an example of a “relation sheaf” by non-analytic sections which is not locally nite. We take an open ball 𝐵 of C𝑛 with center at the origin. Denote by 𝛤 its boundary. Let 𝜒(𝑎) denote the characteristic function of 𝐵; that is, on 𝐵, 𝜒 = 1, and 𝜒 = 0 on C𝑛 \ 𝐵. Set n o ℛ = 𝑓 ∈ 𝒪𝑛,𝑎 : 𝑓 · 𝜒 = 0, 𝑎 ∈ C𝑛 . 𝑎

Then,

𝑎

( ℛ𝑎 =

0, 𝒪𝑛,𝑎 ,

𝑎

𝑎 ∈ 𝐵 ∪ 𝛤, 𝑎 ∉ 𝐵 ∪ 𝛤.

Therefore ℛ is not coherent about 𝑎 ∈ 𝛤. In the above example, the length of the relation is one. To make the length two, we set 𝜙(𝑎) = 1 − 𝜒(𝑎) and n o 𝒮 = 𝑓 ⊕ 𝑔 ∈ 𝒪𝑛,𝑎 ⊕ 𝒪𝑛,𝑎 ; 𝑓 · 𝜒 + 𝑔 · 𝜙 = 0, 𝑎 ∈ C𝑛 ⊂ 𝒪𝑛2 . 𝑎

Then,

𝑎

𝑎

  0 ⊕ 𝒪𝑛,𝑎 ,    𝒮𝑎 = 0 ⊕ 0,   𝒪𝑛,𝑎 ⊕ 0, 

𝑎

𝑎

𝑎

𝑎 ∈ 𝐵, 𝑎 ∈ 𝛤, 𝑎 ∉ 𝐵 ∪ 𝛤.

Therefore, 𝒮 is not locally nite about any point of 𝛤. If one requires the di erentiability for 𝜒 and 𝜙, it su ces to take a 𝐶 ∞ function such that

2.3 Oka’s First Coherence Theorem

𝜒(𝑎) > 0,

43

𝑎 ∈ 𝐵; 𝜒(𝑎) = 0,

𝑎 ∉ 𝐵.

Then, the same conclusion is obtained.

2.3 Oka’s First Coherence Theorem 2.3.1 Weierstrass’ Preparation Theorem We rst prove the titled theorem by the residue theorem in one variable and the relations of roots and coe cients of algebraic equations. Let 𝑓 ∈ 𝒪C𝑛 ,𝑎 . For a moment we assume that 𝑎 = 0 and 𝑓 ≠ 0. Then 𝑓 is a 𝑎 0 holomorphic function in a neighborhood of 0. By Lemma 1.5.9 we may take the standard polydisk P𝛥(0;𝑟) = P𝛥𝑛−1 × 𝛥(0;𝑟 𝑛 ) (⊂ C𝑛−1 × C) with 𝑓 ∈ 𝒪(P𝛥(0;𝑟)) and with the standard coordinate system 𝑧 = (𝑧 ′ , 𝑧 𝑛 ). We call P𝛥(0;𝑟) = P𝛥𝑛−1 × 𝛥(0;𝑟 𝑛 ) the standard polydisk of 𝑓 , and 𝑧 = (𝑧 ′ , 𝑧 𝑛 ) the standard coordinate system 0 of 𝑓 (see De nition 1.5.10). 0 We note: (i) The standard polydisks of 𝑓 form a basis of neighborhoods about 0 0, because 𝑟 𝑛 > 0 can be chosen arbitrarily small and then, depending on it, 𝑟 𝑗 , 1 ≤ 𝑗 ≤ 𝑛 − 1, are chosen arbitrarily small. (ii) Let 𝑃 𝜈0 (. 0) be the rst term in the homogeneous polynomial expansion of 𝑓 about 0 (see (1.5.4)). Since {𝑣 ∈ C𝑛 : 𝑃 𝜈0 (𝑣) = 0} contains no interior point, the standard coordinate system and the standard polydisk can be chosen to be the same for nitely many 𝑓 𝑘 ∈ 𝒪C𝑛 ,0 \ {0}, 1 ≤ 𝑘 ≤ 𝑙.

2.3.1.

0

Theorem 2.3.2 (Weierstrass’ Preparation Theorem). Let 𝑓 ∈ 𝒪C𝑛 ,0 \ {0} with 𝑝 = 0 ord0 𝑓 > 0. Let P𝛥 = P𝛥𝑛−1 × 𝛥(0;𝑟 𝑛 ) (∋ 𝑧 = (𝑧 ′ , 𝑧 𝑛 )) be the standard polydisk of 𝑓 . 0
(i) There exist unique holomorphic functions, 𝑎 𝜈 ∈ 𝒪 P𝛥𝑛−1 with 𝑎 𝜈 (0) = 0, 1 ≤
𝜈 ≤ 𝑝, and zero-free 𝑢 ∈ 𝒪 P𝛥 such that ! 𝑝 Õ 𝑝 ′ ′ 𝑝−𝜈 (2.3.3) 𝑓 (𝑧) = 𝑓 (𝑧 , 𝑧 𝑛 ) = 𝑢(𝑧) 𝑧 𝑛 + 𝑎 𝜈 (𝑧 )𝑧 𝑛 , 𝜈=1 ′

¯ 𝑧 = (𝑧 , 𝑧 𝑛 ) ∈ P𝛥𝑛−1 × 𝛥(0;𝑟 𝑛 ). (ii) For every 𝜑 ∈ 𝒪(P𝛥) there are unique holomorphic functions, 𝑎 ∈ 𝒪(P𝛥) and 𝑏 𝜈 ∈ 𝒪(P𝛥𝑛−1 ), 1 ≤ 𝜈 ≤ 𝑝, satisfying (2.3.4)

𝜑(𝑧) = 𝑎 𝑓 +

𝑝 Õ 𝜈=1

Proof. (i) For 𝑘 ∈ Z+ we set

𝑏 𝜈 (𝑧 ′ )𝑧 𝑛𝑝−𝜈 ,

𝑧 = (𝑧 ′ , 𝑧 𝑛 ) ∈ P𝛥𝑛−1 × 𝛥(0;𝑟 𝑛 ).

44

(2.3.5)

2 Coherent Sheaves and Oka’s Joku-Iko Principle

𝜎𝑘 (𝑧 ′ ) =

1 2𝜋𝑖

∫ | 𝑧𝑛 |=𝑟𝑛

𝑧 𝑛𝑘

𝜕𝑓 𝜕𝑧𝑛

(𝑧 ′ , 𝑧 𝑛 )

𝑓 (𝑧 ′ , 𝑧 𝑛 )

𝑑𝑧 𝑛 ,

𝑧 ′ ∈ P𝛥𝑛−1 .


It follows that 𝜎𝑘 ∈ 𝒪 P𝛥𝑛−1 . By the argument principle 𝜎0 (𝑧 ′ ) ≡ 𝑝 ∈ N. Therefore, for every 𝑧 ′ ∈ P𝛥𝑛−1 , the number of roots of 𝑓 (𝑧 ′ , 𝑧 𝑛 ) = 0 with counting multiplicities is identically 𝑝. We write 𝜁1 (𝑧 ′ ), . . . , 𝜁 𝑝 (𝑧 ′ ) for them with counting multiplicities. From the residue theorem we get 𝜎𝑘 (𝑧 ′ ) =

𝑝 Õ

(𝜁 𝑗 (𝑧 ′ )) 𝑘 ,

𝑘 = 0, 1, . . . .

𝑗=1

We set the elementary symmetric polynomial of degree 𝜈 in 𝜁1 (𝑧 ′ ), . . . , 𝜁 𝑝 (𝑧 ′ ), multiplied by (−1) 𝜈 , Õ 𝑎 𝜈 (𝑧 ′ ) = (−1) 𝜈 𝜁 𝑗1 (𝑧 ′ ) · · · 𝜁 𝑗𝜈 (𝑧 ′ ). 1≤ 𝑗1 𝑟 𝑛 su ciently close to 𝑟 𝑛 , and with every 𝑧 ′ ∈ P𝛥𝑛−1 xed, the roots of 𝑊 (𝑧 ′ , 𝑧 𝑛 ) = 0 and 𝑓 (𝑧 ′ , 𝑧 𝑛 ) = 0 in |𝑧 𝑛 | < 𝑠 𝑛 are the same with counting multiplicities, so that 𝑢(𝑧 ′ , 𝑧 𝑛 ) :=

𝑓 (𝑧 ′ , 𝑧 𝑛 ) 𝑊 (𝑧 ′ , 𝑧 𝑛 )

is zero-free on P𝛥. We have the following integral formula: ∫ 𝑓 (𝑧 ′ , 𝜁 𝑛 ) 𝑑𝜁 𝑛 𝑢(𝑧 ′ , 𝑧 𝑛 ) = · . ′ | 𝜁𝑛 |=𝑠𝑛 𝑊 (𝑧 , 𝜁 𝑛 ) 𝜁 𝑛 − 𝑧 𝑛
It follows that 𝑢 ∈ 𝒪 P𝛥𝑛−1 × {|𝑧 𝑛 | ≤ 𝑟 𝑛 } (zero-free), and that ! 𝑝 Õ 𝑝 𝑝−𝜈 𝑓 (𝑧 ′ , 𝑧 𝑛 ) = 𝑢(𝑧) 𝑧 𝑛 + 𝑎 𝜈 (𝑧 ′ )𝑧 𝑛 = 𝑢(𝑧)𝑊 (𝑧 ′ , 𝑧 𝑛 ), 𝜈=1

2.3 Oka’s First Coherence Theorem

45


where 𝑎 𝜈 ∈ 𝒪 P𝛥𝑛−1 and 𝑎 𝜈 (0) = 0. We con rm that 𝑢(𝑧) and 𝑊 (𝑧 ′ , 𝑧 𝑛 ) are uniquely determined as elements of 𝒪0 . Suppose that     Í𝑝 Í𝑝 𝑓 = 𝑢 0 · 𝑧 𝑛𝑝 + 𝜈=1 𝑎 𝜈 (𝑧 ′ )𝑧 𝑛𝑝−𝜈 = 𝑢˜ 0 · 𝑧 𝑛𝑝 + 𝜈=1 𝑎˜ 𝜈 (𝑧 ′ )𝑧 𝑛𝑝−𝜈 . 0

0

0

f 𝑛−1 × 𝛥(0; 𝑟˜𝑛 ) be a standard polydisk for which the above expressions make Let P𝛥 f 𝑛−1 , the roots of two equations sense. For each xed 𝑧 ′ ∈ P𝛥 𝑧 𝑛𝑝 +

𝑝 Õ

𝑎 𝜈 (𝑧 ′ )𝑧 𝑛𝑝−𝜈 = 0,

𝑧 𝑛𝑝 +

𝜈=1

𝑝 Õ

𝑎˜ 𝜈 (𝑧 ′ )𝑧 𝑛𝑝−𝜈 = 0

𝜈=1

are identical with counting multiplicities, and then 𝑎 𝜈 (𝑧 ′ ) = 𝑎˜ 𝜈 (𝑧 ′ ),

1 ≤ 𝜈 ≤ 𝑝.

Hence, 𝑢(𝑧) = 𝑢(𝑧) ˜ follows. (ii) We may assume that 𝑓 (𝑧 ′ , 𝑧 𝑛 ) = 𝑊 (𝑧 ′ , 𝑧 𝑛 ) = 𝑧 𝑛𝑝 + =

𝑝 Õ

𝑝 Õ

𝑎 𝜈 (𝑧 ′ )𝑧 𝑛𝑝−𝜈

𝜈=1


𝑎 𝜈 (𝑧 ′ )𝑧 𝑛𝑝−𝜈 ∈ 𝒪 P𝛥𝑛−1 [𝑧 𝑛 ].

𝜈=0

Here, we put 𝑎 0 (𝑧 ′ ) = 1. For 𝜑 ∈ 𝒪(P𝛥𝑛−1 × 𝛥(0;𝑟 𝑛 )) we set ∫ 1 𝜑(𝑧 ′ , 𝜁 𝑛 ) 𝑑𝜁 𝑛 (2.3.7) 𝑎(𝑧 ′ , 𝑧 𝑛 ) = , (𝑧 ′ , 𝑧 𝑛 ) ∈ P𝛥𝑛−1 × 𝛥(0;𝑟 𝑛 ), 2𝜋𝑖 | 𝜁𝑛 |=𝑡𝑛 𝑊 (𝑧 ′ , 𝜁 𝑛 ) 𝜁 𝑛 − 𝑧 𝑛 where |𝑧 𝑛 | < 𝑡 𝑛 < 𝑟 𝑛 . Since 𝑎(𝑧 ′ , 𝑧 𝑛 ) is independent of the choice of 𝑡 𝑛 close to 𝑟 𝑛 , 𝑎(𝑧 ′ , 𝑧 𝑛 ) ∈ 𝒪(P𝛥𝑛−1 × 𝛥(0;𝑟 𝑛 )) is determined. For 𝑧 ′ ∈ P𝛥𝑛−1 and |𝑧 𝑛 | < 𝑡 𝑛 we write (2.3.8)

𝜑(𝑧 ′ , 𝑧 𝑛 ) − 𝑎(𝑧 ′ , 𝑧 𝑛 )𝑊 (𝑧 ′ , 𝑧 𝑛 ) ∫ ∫ 1 𝑑𝜁 𝑛 𝑊 (𝑧 ′ , 𝑧 𝑛 ) 𝜑(𝑧 ′ , 𝜁 𝑛 ) 𝑑𝜁 𝑛 ′ = 𝜑(𝑧 , 𝜁 𝑛 ) − ′ 2𝜋𝑖 | 𝜁𝑛 |=𝑡𝑛 𝜁𝑛 − 𝑧𝑛 2𝜋𝑖 | 𝜁𝑛 |=𝑡𝑛 𝑊 (𝑧 , 𝜁 𝑛 ) 𝜁 𝑛 − 𝑧 𝑛   ∫ ′ 1 𝑊 (𝑧 , 𝑧 𝑛 ) 𝑑𝜁 𝑛 = 𝜑(𝑧 ′ , 𝜁 𝑛 ) 1 − ′ 2𝜋𝑖 | 𝜁𝑛 |=𝑡𝑛 𝑊 (𝑧 , 𝜁 𝑛 ) 𝜁 𝑛 − 𝑧 𝑛 Í 𝑝−1 ∫ 𝑎 𝜈 (𝑧 ′ ) (𝜁 𝑛𝑝−𝜈 − 𝑧 𝑛𝑝−𝜈 ) 1 = 𝜑(𝑧 ′ , 𝜁 𝑛 ) 𝜈=0 ′ 𝑑𝜁 𝑛 2𝜋𝑖 | 𝜁𝑛 |=𝑡𝑛 𝑊 (𝑧 , 𝜁 𝑛 )(𝜁 𝑛 − 𝑧 𝑛 ) (continued)

46

2 Coherent Sheaves and Oka’s Joku-Iko Principle

1 = 2𝜋𝑖

∫ | 𝜁𝑛 |=𝑡𝑛

 𝑝−1 𝜑(𝑧 ′ , 𝜁 𝑛 ) Õ 𝑎 𝜈 (𝑧 ′ )(𝜁 𝑛𝑝−𝜈−1 + 𝜁 𝑛𝑝−𝜈−2 𝑧 𝑛 + · · · 𝑊 (𝑧 ′ , 𝜁 𝑛 ) 𝜈=0  𝑝−𝜈−1 + 𝑧𝑛 ) 𝑑𝜁 𝑛

= 𝑏 1 (𝑧 ′ )𝑧 𝑛𝑝−1 + 𝑏 2 (𝑧 ′ )𝑧 𝑛𝑝−2 + · · · + 𝑏 𝑝 (𝑧 ′ ), where 𝑏 𝜈 (𝑧 ′ ) are given by (2.3.9)

1 𝑏 𝜈 (𝑧 ) = 2𝜋𝑖 ′

∫ | 𝜁𝑛 |=𝑡𝑛

! 𝜈−1 𝜑(𝑧 ′ , 𝜁 𝑛 ) Õ ′ 𝜈−1−ℎ 𝑎 ℎ (𝑧 )𝜁 𝑛 𝑑𝜁 𝑛 . 𝑊 (𝑧 ′ , 𝜁 𝑛 ) ℎ=0

It follows from this expression that 𝑏 𝜈 (𝑧 ′ ) ∈ 𝒪(P𝛥𝑛−1 ), 1 ≤ 𝜈 ≤ 𝑝 (independent of 𝑡 𝑛 ). Therefore we obtain (2.3.10)

𝜑(𝑧 ′ , 𝑧 𝑛 ) = 𝑎(𝑧 ′ , 𝑧 𝑛 )𝑊 (𝑧 ′ , 𝑧 𝑛 ) +

𝑝 Õ

𝑏 𝜈 (𝑧 ′ )𝑧 𝑛𝑝−𝜈 .

𝜈=1

Next, we show the uniqueness. Suppose that (2.3.11)

𝜑(𝑧 ′ , 𝑧 𝑛 ) = 𝑎(𝑧 ˜ ′ , 𝑧 𝑛 )𝑊 (𝑧 ′ , 𝑧 𝑛 ) +

𝑝 Õ

𝑏˜ 𝜈 (𝑧 ′ )𝑧 𝑛𝑝−𝜈 .

𝜈=1

Subtracting both sides of (2.3.10) and (2.3.11) and shifting terms, we assume that (𝑎(𝑧 ′ , 𝑧 𝑛 ) − 𝑎(𝑧 ˜ ′ , 𝑧 𝑛 )) 𝑊 (𝑧 ′ , 𝑧 𝑛 ) =

𝑝 Õ


𝑏˜ 𝜈 (𝑧 ′ ) − 𝑏 𝜈 (𝑧 ′ ) 𝑧 𝑛𝑝−𝜈 . 0.

𝜈=1

f 𝑛−1 the left-hand side has at least 𝑝 roots with counting Then for a xed 𝑧 ′ ∈ P𝛥 multiplicities. The right-hand side has at most 𝑝 −1 roots with counting multiplicities; this is absurd. Hence, 𝑏˜ 𝜈 (𝑧 ′ ) = 𝑏 𝜈 (𝑧 ′ ),

𝑎(𝑧 ˜ ′ , 𝑧 𝑛 ) = 𝑎(𝑧 ′ , 𝑧 𝑛 ).

Definition 2.3.12. (i) Letting
P𝛥𝑛−1 ⊂ coe cients in 𝒪 P𝛥𝑛−1 𝑊 (𝑧 ′ , 𝑧 𝑛 ) = 𝑧 𝑛𝑝 +

𝑝 Õ

C𝑛−1 ,

𝑎 𝜈 (𝑧 ′ ) · 𝑧 𝑛𝑝−𝜈 ,

we call the 𝑧 𝑛 -polynomial with
𝑎 𝜈 ∈ 𝒪 P𝛥𝑛−1 , 𝑎 𝜈 (0) = 0,

𝜈=1

a Weierstrass polynomial (in 𝑧 𝑛 ). Considering the induced germ 𝑊 = 𝑧 𝑛𝑝 +

𝑝 Õ 𝜈=1

⊓ ⊔

𝑎 𝜈 0 · 𝑧 𝑛𝑝−𝜈 ∈ 𝒪P𝛥𝑛−1 ,0 [𝑧 𝑛 ],

2.3 Oka’s First Coherence Theorem

47

we also call this a Weierstrass polynomial (in 𝑧 𝑛 ). (ii) Write (2.3.3) as 𝑓 (𝑧 ′ , 𝑧 𝑛 ) = 𝑢𝑊 (𝑧 ′ , 𝑧 𝑛 ) with unit 𝑢 (i.e., ∃ 𝑢 −1 ) and Weierstrass polynomial 𝑊 (𝑧 ′ , 𝑧 𝑛 ). We call 𝑓 (𝑧 ′ , 𝑧 𝑛 ) = 𝑢𝑊 (𝑧 ′ , 𝑧 𝑛 ) the Weierstrass decomposition of 𝑓 at 0, which is unique. Lemma 2.3.13. Let 𝑄(𝑧 ′ , 𝑧 𝑛 ) ∈ 𝒪𝑛−1,0 [𝑧 𝑛 ] be a Weierstrass polynomial, and let 𝑅 ∈ 𝒪𝑛−1,0 [𝑧 𝑛 ]. If 𝑅 = 𝑄 · 𝑔 with 𝑔 ∈ 𝒪𝑛,0 , then 𝑔 ∈ 𝒪𝑛−1,0 [𝑧 𝑛 ]. 0

0

0

Proof. Assume that all the functions above are holomorphic in a neighborhood of a closed polydisk P𝛥. Since the leading coe cient of 𝑄(𝑧 ′ , 𝑧 𝑛 ) as a polynomial in 𝑧 𝑛 is 1, the Euclid division algorithm implies that (2.3.14)

𝑅 = 𝜑 𝑄 + 𝜓,

𝜑, 𝜓 ∈ 𝒪𝑛−1,0 [𝑧 𝑛 ],

deg𝑧𝑛 𝜓 < 𝑝 = deg𝑧𝑛 𝑄. We may assume that P𝛥 = P𝛥𝑛−1 × 𝛥 (𝑛) (𝛥 (𝑛) = {|𝑧 𝑛 | < 𝑟 𝑛 }) is a standard polydisk for 𝑄. With every 𝑧 ′ ∈ P𝛥𝑛−1 xed, 𝑄(𝑧 ′ , 𝑧 𝑛 ) = 0 has 𝑝 zeros with counting multiplicities. Therefore, 𝑅 has at least 𝑝 zeros with counting multiplicities. By (2.3.14) 𝜓 has at least 𝑝 zeros with counting multiplicities, too. Since deg 𝜓 < 𝑝, 𝜓 ≡ 0. Therefore we obtain 𝑅 = 𝜑𝑄 = 𝑔𝑄, (𝜑 − 𝑔)𝑄 = 0,

𝑄 ≠ 0.

Since 𝒪𝑛,0 is a ring of integral domain, 𝜑 − 𝑔 = 0. Thus we see that 𝑔 = 𝜑 ∈ 𝒪𝑛−1,0 [𝑧 𝑛 ]. ⊓ ⊔ Remark 2.3.15. The following properties of the integral domain 𝒪𝑎 follow from Weierstrass’ Preparation Theorem 2.3.2: (i) 𝒪𝑎 is a unique factorization domain (cf., e.g., [39] Theorem 2.2.12). (ii) 𝒪𝑎 is a noetherian ring (cf., e.g., [39] Theorem 2.2.20).

2.3.2 Oka’s First Coherence Theorem Theorem 2.3.16 (Oka’s First Coherence). The sheaf 𝒪C𝑛 is coherent; hence, all 𝒪C𝑁𝑛 (𝑁 ≥ 1) are coherent. Proof. We proceed by induction on 𝑛 ≥ 0 with general 𝑁 ≥ 1. We write 𝒪C𝑛 = 𝒪𝑛 . (a) 𝑛 = 0: In this case, it is a matter of a nite-dimensional vector space over C. 𝑁 (b) 𝑛 ≥ 1: Suppose that 𝒪𝑛−1 is coherent for every 𝑁 ≥ 1. By Proposition 2.2.15 (ii) it su ces to prove the coherence of 𝒪𝑛 . The problem is local and it is su cient to prove De nition 2.2.13 (ii). Taking an open subset 𝛺 ⊂ C𝑛 and 𝜏 𝑗 ∈ 𝒪(𝛺)  𝛤 (𝛺,𝒪𝑛 ), 1 ≤ 𝑗 ≤ 𝑞, we consider the relation sheaf ℛ(𝜏1 , . . . , 𝜏𝑞 ) de ned by

48

(2.3.17)

2 Coherent Sheaves and Oka’s Joku-Iko Principle

𝑓1 𝜏1 𝑧 + · · · + 𝑓𝑞 𝜏𝑞 = 0, 𝑧

𝑧

𝑓 𝑗 ∈ 𝒪𝑛,𝑧 , 𝑧 ∈ 𝛺.

𝑧

𝑧

What we want to show is: Claim 2.3.18. For every point 𝑎 ∈ 𝛺 there are a neighborhood 𝑉 ⊂ 𝛺 of 𝑎 and finitely many sections 𝑠 𝑘 ∈ 𝛤 (𝑉, ℛ(𝜏1 , . . . , 𝜏𝑞 )), 1 ≤ 𝑘 ≤ 𝑙, such that ℛ(𝜏1 , . . . , 𝜏𝑞 ) 𝑏 =

𝑙 Õ

𝒪𝑛,𝑏 · 𝑠 𝑘 (𝑏),

∀𝑏 ∈ 𝑉.

𝑘=1

For the proof we may assume that 𝑎 = 0. The case of 𝑞 = 1 is trivial, and so we assume 𝑞 ≥ 2. If an element 𝜏 𝑗 = 0, then the 𝑗 th component of ℛ(𝜏1 , . . . , 𝜏𝑞 )| 𝑉 ⊂ 0 (O𝑉 ) 𝑞 is just O𝑉 in a neighborhood 𝑉 of 0; so we may assume 𝜏 𝑗 ≠ 0, 1 ≤ 𝑗 ≤ 𝑞. 0 We set (2.3.19)


𝑗 th 𝑖 th 𝑇𝑖, 𝑗 = 0, . . . , 0, −𝜏 𝑗 , 0, . . . , 0, 𝜏𝑖 , 0, . . . , 0 ,

1 ≤ 𝑖 < 𝑗 ≤ 𝑞,

which trivially satisfy (2.3.17), and call 𝑇𝑖, 𝑗 the trivial solutions. If there is an element 𝜏 𝑗 , to say, 𝜏1 with 𝜏(0) ≠ 0, then (2.3.17) is solved as 𝑓1 = −

(2.3.20)

𝑏

𝑞 𝜏𝑗 Õ 𝑗=2

𝑏

𝜏1 𝑏

𝑓𝑗 ; 𝑏

that is, ℛ is generated by {𝑇1, 𝑗 } 𝑞𝑗=2 about 0. So, we assume all 𝜏 𝑗 (0) = 0 (1 ≤ 𝑗 ≤ 𝑞). Let 𝑝 𝑗 be the order of zero of 𝜏 𝑗 at 0, and set 𝑝 = max 𝑝 𝑗 , 1≤ 𝑗 ≤𝑞

𝑝 ′ = min 𝑝 𝑗 ≥ 0. 1≤ 𝑗 ≤𝑞

After reordering the indices, we may assume that 𝑝 ′ = 𝑝 1 . To avoid the notational complications we use 𝑓 , 𝑧 𝑛 etc. for their sections 𝑓 , 𝑧 𝑛 etc., unless confusion occurs. Take a common standard polydisk P𝛥 = P𝛥𝑛−1 × 𝛥 (𝑛) with 𝛥 (𝑛) = {|𝑧 𝑛 | < 𝑟 𝑛 } for all 𝜏 𝑗 . By Weierstrass’s Preparation Theorem 2.3.2 at 0 one can transfer a unit factor of 𝜏 𝑗 to 𝑓 𝑗 in (2.3.17) so that all 𝜏 𝑗 may be assumed to be Weierstrass polynomials: (2.3.21)

′

𝜏 𝑗 (𝑧) = 𝑃 𝑗 (𝑧 , 𝑧 𝑛 ) =

𝑝𝑗 Õ 𝜈=0

𝑎 𝑗 𝜈 (𝑧 ′ )𝑧 𝑛𝜈 =

𝑎 𝑗 𝜈 (0) = 0 (𝜈 < 𝑝 𝑗 ),

𝑝 Õ 𝜈=0

𝑎 𝑗 𝜈 (𝑧 ′ )𝑧 𝑛𝜈 ∈ 𝒪(P𝛥𝑛−1 ) [𝑧 𝑛 ],

𝑎 𝑗 𝑝 𝑗 = 1,

𝑎 𝑗 𝜈 = 0 ( 𝑝 𝑗 < 𝜈 ≤ 𝑝).

Now we have ℛ = ℛ(𝑃1 , . . . , 𝑃𝑞 ) de ned by the equation (2.3.22) The trivial solutions are

𝑓1 𝑃1 + · · · + 𝑓𝑞 𝑃𝑞 = 0.

2.3 Oka’s First Coherence Theorem

49

𝑗 th 𝑖 th
𝑇𝑖, 𝑗 = 0, . . . , 0, −𝑃 𝑗 , 0, . . . , 0, 𝑃𝑖 , 0, . . . , 0 ,

(2.3.23)

1 ≤ 𝑖 < 𝑗 ≤ 𝑞,

We perform a kind of division algorithm for an unknown 𝑞-tuple 𝛼 = (𝛼 𝑗 ) ∈ ℛ with respect to the trivial solutions 𝑇𝑖, 𝑗 (cf. (2.3.29), (2.3.30)). Take an arbitrary point 𝑏 = (𝑏 ′ , 𝑏 𝑛 ) ∈ P𝛥𝑛−1 × 𝛥 (𝑛) . We call an element of 𝒪𝑛−1,𝑏′ [𝑧 𝑛 ] a 𝑧 𝑛 -polynomial-like germ. In the same way, we call 𝛼 = (𝛼1 , . . . , 𝛼𝑞 ) ∈ 𝑞 𝒪𝑛,𝑏 consisting of 𝑧 𝑛 -polynomial-like germs 𝛼 𝑗 a polynomial-like element, and 𝑓 = ( 𝑓 𝑗 ) with ( 𝑓 𝑗 )1≤ 𝑗 ≤𝑞 ∈ (𝒪(P𝛥𝑛−1 ) [𝑧 𝑛 ]) 𝑞 a 𝑧 𝑛 -polynomial-like section. We set deg 𝛼 = deg𝑧𝑛 𝛼 = max deg𝑧𝑛 𝛼 𝑗 , 𝑗

deg 𝑓 = deg𝑧𝑛 𝑓 = max deg𝑧𝑛 𝑓 𝑗 . 𝑗

Then we have: 2.3.24. The trivial solutions 𝑇𝑖, 𝑗 are 𝑧 𝑛 -polynomial-like sections of deg𝑇𝑖, 𝑗 ≤ 𝑝. We now show the following: Lemma 2.3.25 (Degree structure). Let the notation be as above. Then an element of ℛ𝑏 is written as a finite linear sum of the trivial solutions, 𝑇1, 𝑗 (2 ≤ 𝑗 ≤ 𝑞), and a 𝑧 𝑛 -polynomial-like element 𝛼 = (𝛼1 , 𝛼2 , . . . , 𝛼𝑞 ) of ℛ𝑏 with coefficients in 𝒪𝑛,𝑏 such that deg 𝛼 𝑗 < 𝑝 ′

deg 𝛼1 < 𝑝,

(2 ≤ 𝑗 ≤ 𝑞).

∵ ) By making use of Weierstrass’ Preparation Theorem 2.3.2 at 𝑏 = (𝑏 ′ , 𝑏 𝑛 ) we decompose 𝑃1 into a unit 𝑢 and a Weierstrass polynomial 𝑄 at 𝑏: (2.3.26)

𝑃1 (𝑧 ′ , 𝑧 𝑛 ) = 𝑢 · 𝑄(𝑧 ′ , 𝑧 𝑛 − 𝑏 𝑛 ),

deg 𝑄 = 𝑑 ≤ 𝑝 1 ,

where 𝑄 ∈ 𝒪𝑛−1,𝑏′ [𝑧 𝑛 − 𝑏 𝑛 ] = 𝒪𝑛−1,𝑏′ [𝑧 𝑛 ]. Lemma 2.3.13 implies that 𝑢 ∈ 𝒪𝑛−1,𝑏′ [𝑧 𝑛 − 𝑏 𝑛 ] = 𝒪𝑛−1,𝑏′ [𝑧 𝑛 ]. It follows that (2.3.27)


deg𝑧𝑛 𝑢 = 𝑝 1 − 𝑑.

Take an arbitrary 𝑓 = 𝑓1 , . . . , 𝑓𝑞 ∈ ℛ𝑏 . By Weierstrass’ Preparation Theorem 2.3.2 (ii) we have (2.3.28)

𝑓𝑖 =𝑐 𝑖 𝑄 + 𝛽𝑖 ,

𝑐 𝑖 ∈ 𝒪𝑛,𝑏 ,

deg𝑧𝑛 𝛽𝑖 ≤ 𝑑 − 1

𝛽𝑖 ∈ 𝒪𝑛−1,𝑏′ [𝑧 𝑛 − 𝑏 𝑛 ] = 𝒪𝑛−1,𝑏′ [𝑧 𝑛 ],

(1 ≤ 𝑖 ≤ 𝑞).

Since 𝑢 ∈ 𝒪𝑛,𝑏 is a unit, with 𝑐˜𝑖 := 𝑐 𝑖 𝑢 −1 we get (2.3.29)

𝑓𝑖 = 𝑐˜𝑖 𝑃1 + 𝛽𝑖 ,

1 ≤ 𝑖 ≤ 𝑞.

50

2 Coherent Sheaves and Oka’s Joku-Iko Principle

By making use of this we perform the following calculation: (2.3.30)

( 𝑓1 , . . . , 𝑓𝑞 ) − 𝑐˜2𝑇2 − · · · − 𝑐˜𝑞 𝑇𝑞 = ( 𝑐˜1 𝑃1 + 𝛽1 , 𝑐˜2 𝑃1 + 𝛽2 , . . . , 𝑐˜𝑞 𝑃1 + 𝛽𝑞 ) + ( 𝑐˜2 𝑃2 , −𝑐˜2 𝑃1 , 0, . . . , 0) + · · · + ( 𝑐˜𝑞 𝑃𝑞 , 0, . . . , 0, −𝑐˜𝑞 𝑃1 ) ! 𝑞 Õ = 𝑐˜𝑖 𝑃𝑖 + 𝛽1 , 𝛽2 , . . . , 𝛽𝑞 𝑖=1


= 𝑔1 , 𝛽2 , . . . , 𝛽 𝑞 , Í𝑞 where 𝑔1 := 𝑖=1 𝑐˜𝑖 𝑃𝑖 + 𝛽1 ∈ 𝒪𝑛,𝑏 ; note that it is unknown if
𝑔1 ∈ 𝒪𝑛−1,𝑏′ [𝑧 𝑛 ]. It is noticed that 𝛽𝑖 ∈ 𝒪𝑛−1,𝑏′ [𝑧 𝑛 ], 2 ≤ 𝑖 ≤ 𝑞. Since 𝑔1 , 𝛽2 , . . . , 𝛽𝑞 ∈ ℛ𝑏 , (2.3.31)

𝑔1 𝑃1 = −𝛽2 𝑃2 − · · · − 𝛽𝑞 𝑃𝑞 ∈ 𝒪𝑛−1,𝑏′ [𝑧 𝑛 ].

It follows from the expression of the right-hand side of (2.3.31) that deg𝑧𝑛 𝑔1 𝑃1 ≤ max deg𝑧𝑛 𝛽𝑖 + max deg𝑧𝑛 𝑃𝑖 ≤ 𝑑 + 𝑝 − 1. 2≤𝑖 ≤𝑞

2≤𝑖 ≤𝑞

On the other hand, 𝑔1 𝑃1 = (𝑔1 𝑢)𝑄 and 𝑄 is a Weierstrass polynomial at 𝑏. Again by Lemma 2.3.13 we see that 𝛼1 := 𝑔1 𝑢 ∈ 𝒪𝑛−1,𝑏′ [𝑧 𝑛 ], deg𝑧𝑛 𝛼1 = deg𝑧𝑛 𝑔1 𝑃1 − deg𝑧𝑛 𝑄

(2.3.32)

≤ 𝑑 + 𝑝 − 1 − 𝑑 = 𝑝 − 1. Setting 𝛼𝑖 = 𝑢𝛽𝑖 (∈ 𝒪𝑛−1,𝑏′ [𝑧 𝑛 ]) for 2 ≤ 𝑖 ≤ 𝑞, we have by (2.3.27) and (2.3.28) that (2.3.33)

deg𝑧𝑛 𝛼𝑖 ≤ 𝑝 1 − 𝑑 + 𝑑 − 1 = 𝑝 1 − 1 = 𝑝 ′ − 1,

2 ≤ 𝑖 ≤ 𝑞,

and then by (2.3.30) that 𝑓=

(2.3.34)

𝑞 Õ

𝑐˜𝑖 𝑇𝑖 + 𝑢 −1 (𝛼1 , 𝛼2 , . . . , 𝛼𝑞 ).

𝑖=2

△

Until now we have not used the induction hypothesis. Now we are going to use it to prove the existence of a locally nite generator system of those (𝛼1 , . . . , 𝛼𝑞 ) appearing in (2.3.34). We write (2.3.35)

𝛼1 =

𝑝−1 Õ 𝜈=0

𝑐 1𝜈 (𝑧 ′ ) ′ 𝑧 𝑛𝜈 , 𝑏

𝑐 1𝜈 (𝑧 ′ )

𝑏′

∈ 𝒪𝑛−1,𝑏′ ,

2.3 Oka’s First Coherence Theorem

𝛼𝑖 =

′ −1 𝑝Õ

𝜈=0

51

𝑐 𝑖𝜈 (𝑧 ′ ) ′ 𝑧 𝑛𝜈 ,

𝑐 𝑖𝜈 (𝑧 ′ )

𝑏

𝑏′

∈ 𝒪𝑛−1,𝑏′ , 2 ≤ 𝑖 ≤ 𝑞.

By 𝒮 we denote the sheaf of all (𝛼1 , . . . , 𝛼𝑞 ) over P𝛥 = P𝛥𝑛−1 × 𝛥 (𝑛) written as in (2.3.35), which satisfy 𝛼1 𝑃1 + 𝛼2 𝑃2 + · · · + 𝛼𝑞 𝑃𝑞 = 0.

(2.3.36)

The left-hand side above is a 𝑧 𝑛 -polynomial-like element of degree at most 𝑝 + 𝑝 ′ − 1, and relation (2.3.36) is equivalent to the nullity of all 𝑝 + 𝑝 ′ coe cients. With the expression in (2.3.21) we have (2.3.37)

𝑞 Õ Õ′ 𝑖=1 𝑘+ℎ=𝜈

𝑐 𝑖ℎ (𝑧 ′ ) ′ · 𝑎 𝑖𝑘 (𝑧 ′ ) 𝑏

𝑏′

= 0 ∈ 𝒪𝑛−1,𝑏′ ,

0 ≤ 𝜈 ≤ 𝑝 + 𝑝 ′ − 1.

Í′

Here, stands for the sum over those indices ℎ, 𝑘 to which some elements 𝑎 𝑖𝑘 (𝑧 ′ ) ′ , 𝑏 𝑐 𝑖ℎ (𝑧 ′ ) ′ correspond. Then (2.3.37) de nes a ( 𝑝 + 𝑝 ′ )-simultaneous relation sheaf 𝑏 𝑝+ 𝑝 ′ (𝑞−1) 𝒮˜ in 𝒪P𝛥 with 𝑝 + 𝑝 ′ (𝑞 − 1) unknowns, 𝑐 𝑖ℎ ’s. The induction hypothesis on 𝑛 𝑛−1 implies that 𝒮˜ is coherent, and hence there is a locally nite generator system of 𝒮˜ f 𝑛−1 ⊂ P𝛥𝑛−1 of 0. We set over a polydisk neighborhood P𝛥 f = P𝛥 f 𝑛−1 × 𝛥 (𝑛) ⊂ P𝛥𝑛−1 × 𝛥 (𝑛) = P𝛥. P𝛥 Together with (2.3.35)
we then infer that 𝒮 has a locally nite generator system f 𝑛−1 [𝑧 𝑛 ]) over P𝛥. f {𝜋 𝜇 } 𝑀 (⊂ 𝒪 P𝛥 𝜇=1 f Thus, the nite system {𝑇1, 𝑗 } 𝑞𝑗=2 ∪ {𝜋 𝜇 } 𝑀 ⊓ ⊔ 𝜇=1 generates ℛ over P𝛥. Once the coherence of 𝒪𝑛 is proved, it is immediate but important to see: Corollary 2.3.38. Every relation sheaf of a coherent sheaf is coherent. Remark 2.3.39. (i) In the proof of Theorem 2.3.16 it is essential to convert the expression of 𝑓𝑖 in (2.3.28) with Weierstrass’ polynomial 𝑄 at 𝑏 to that in (2.3.29) with 𝑃1 after multiplying the unit 𝑢 to 𝑄. If we keep the expression with 𝑄, we could not advance the arguments, nor have an expression e ective about 0. To see it, we consider a simple example, 𝑃1 (𝑧1 , 𝑧 2 ) = 𝑧22 − 𝑧 1 in (𝑧1 , 𝑧 2 ) ∈ C2 . Let √ 𝑏 = (𝑏 1 , 𝑏√2 ) ∈ C2 such that 𝑃1 (𝑏 1 , 𝑏 2 ) = 0 and 𝑏 ≠ 0. Let 𝑧1 be a branch about √ √ √ 𝑏 1 with 𝑏 1 = 𝑏 2 . Then 𝑃1 𝑏 = (𝑧 2 − 𝑧1 ) · (𝑧2 + 𝑧1 ) and 𝑢 = (𝑧2 + 𝑧1 ) 𝑏 𝑏 𝑏 is a unit of 𝒪2,𝑏 . We have the decomposition 𝑃1 𝑏 = 𝑢𝑄 (cf. (2.3.26)) with Weierstrass’ polynomial at 𝑏, p √ 𝑄(𝑧1 , 𝑧 2 − 𝑏 2 ) = (𝑧2 − 𝑏 2 ) − 𝑧1 + 𝑏 1 ∈ 𝒪1,𝑏1 [𝑧 2 − 𝑏 2 ] = 𝒪1,𝑏1 [𝑧 2 ].

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2 Coherent Sheaves and Oka’s Joku-Iko Principle

This expression of 𝑄 does not make sense about 0. After multiplying 𝑢 to 𝑄 again, we obtain 𝑃1 = 𝑢𝑄, which provides an expression e ective about 0. (ii) In Lemma 2.3.25 the role of the minimum degree 𝑝 ′ was observed in [40]. In the former arguments, the minimum degree 𝑝 ′ was not used and only the maximum one 𝑝 was used (cf. Oka [48] VII, Cartan [10], Hörmander [30] 6.4). The proof of Lemma 2.3.25 works even in the case of 𝑝 ′ = 0, when the argument is reduced to the simplest case of (2.3.20); this reduction is not implied by the arguments only with the maximum degree 𝑝.

2.3.3 Coherence of Ideal Sheaves of Complex Submanifolds Theorem 2.3.40. The ideal sheaf of a complex submanifold is coherent. Proof. Let 𝑆 ⊂ 𝛺 be a submanifold of an open set 𝛺 ⊂ C𝑛 . By Proposition 2.2.15 (i) it su ces to show the local niteness of ℐ⟨ 𝑆⟩. Take a point 𝑎 ∈ 𝛺. Case of 𝑎 ∉ 𝑆: Since 𝑆 is a closed set, there is a neighborhood 𝑈 ⊂ 𝛺 of 𝑎 such that 𝑈 ∩ 𝑆 = ∅. Since ℐ⟨𝑆⟩𝑧 = 𝒪𝑧 = 1 · 𝒪𝑧 , ∀ 𝑧 ∈ 𝑈, {1} is a nite generator system of ℐ⟨𝑆⟩ over 𝑈. Case of 𝑎 ∈ 𝑆: There is a neighborhood 𝑈 of 𝑎 with a local coordinate system 𝑧 = (𝑧1 , . . . , 𝑧 𝑛 ) such that (2.3.41)

𝑎 = (0, . . . , 0) ∈ 𝑈 = P𝛥((𝑟 𝑗 )), 𝑆 ∩ 𝑈 = {𝑧 = (𝑧 𝑗 ) ∈ 𝑈 : 𝑧1 = · · · = 𝑧 𝑞 = 0}

(1 ≤ ∃ 𝑞 ≤ 𝑛).

We take arbitrarily an element 𝑓 ∈ ℐ⟨𝑆⟩𝑏 (𝑏 ∈ 𝑈 ∩ 𝑆). Set 𝑏 = (𝑏 𝑗 ) in coordinates 𝑏 (𝑧 𝑗 ). Then, 𝑏 = (0, . . . , 0, 𝑏 𝑞+1 , . . . , 𝑏 𝑛 ), and 𝑓 is uniquely expanded to a power series Õ 𝑓 (𝑧) = 𝑐 𝜈 (𝑧 − 𝑏) 𝜈 . 𝜈=(𝜈1 ,𝜈 ′ ) ∈Z+𝑛 , | 𝜈 |>0

We decompose this as follows: Õ 𝑓 (𝑧) = 𝑐 𝜈 (𝑧 − 𝑏) 𝜈 + 𝜈=(𝜈1 ,𝜈 ′ ) ∈Z+𝑛 ,𝜈1 >0

Õ

𝑐 𝜈 (𝑧 − 𝑏) 𝜈

𝜈=(𝜈1 ,𝜈 ′ ) ∈Z+𝑛 ,𝜈1 =0

Õ Õ ′ª ′ © =­ 𝑐 𝜈 𝑧1𝜈1 −1 (𝑧 ′ − 𝑏 ′ ) 𝜈 ® 𝑧1 + 𝑐 0𝜈 ′ (𝑧 ′ − 𝑏 ′ ) 𝜈 , 𝑛 ′ 𝜈 ′ ∈Z+𝑛−1 «𝜈=(𝜈1 ,𝜈 ) ∈Z+ ,𝜈1 >0 ¬ where 𝜈 ′ = (𝜈2 , . . . , 𝜈 𝑛 ), 𝑧 ′ = (𝑧 2 , . . . , 𝑧 𝑛 ), and 𝑏 ′ = (𝑏 2 , . . . , 𝑏 𝑛 ). Setting

2.4 Cartan’s Merging Lemma

53

Õ ′ª © ℎ1 (𝑧 1 , 𝑧 ′ ) = ­ 𝑐 𝜈 𝑧1𝜈1 −1 (𝑧 ′ − 𝑏 ′ ) 𝜈 ® , 𝑛 ′ 1 ,𝜈 ) ∈Z+ ,𝜈1 >0 «𝜈=(𝜈 ¬ Õ ′ ′ ′ 𝜈′ ′ 𝑔1 (𝑧 ) = 𝑐 0𝜈 (𝑧 − 𝑏 ) , 𝜈 ′ ∈Z+𝑛−1

we write (2.3.42)

𝑓 (𝑧1 , 𝑧 ′ ) = ℎ1 (𝑧 1 , 𝑧 ′ ) · 𝑧 1 + 𝑔1 (𝑧 ′ ).

Considering a similar decomposition of 𝑔1 (𝑧 ′ ) with respect to variable 𝑧2 , we get 𝑔1 (𝑧 ′ ) = ℎ2 · 𝑧 2 + 𝑔2 (𝑧 ′′ ),

𝑧 ′′ = (𝑧3 , . . . , 𝑧 𝑛 ).

Repeating this procedure, we obtain 𝑓 (𝑧) =

𝑞 Õ

ℎ 𝑗 (𝑧) · 𝑧 𝑗 + 𝑔𝑞 (𝑧 𝑞+1 , . . . , 𝑧 𝑛 ).

𝑗=1

Since 𝑓 (𝑧) = 0 for 𝑧1 = · · · = 𝑧 𝑞 = 0, we deduce 𝑔𝑞 (𝑧 𝑞+1 , . . . , 𝑧 𝑛 ) = 0. It follows that 𝑓 (𝑧) =

𝑞 Õ

ℎ 𝑗 (𝑧) · 𝑧 𝑗 .

𝑗=1

Therefore we have a nite generation (2.3.43)

ℐ⟨𝑆⟩|𝑈 =

𝑞 Õ

𝒪𝑈 · 𝑧 𝑗 .

𝑗=1

⊓ ⊔

2.4 Cartan’s Merging Lemma Let ℱ → 𝛺 be a locally nite analytic sheaf over a domain 𝛺 ⊂ C𝑛 . For given nite generator systems of ℱ over adjacent closed subdomains 𝐸 ′ , 𝐸 ′′ (⋐ 𝛺), we would like to construct a nite generator system of ℱ over 𝐸 ′ ∪ 𝐸 ′′ . We begin with the elementary facts of matrices.

2.4.1 Matrices and Matrix-Valued Functions We describe some basic properties of sequences, series, in nite products of matrices, and matrix-valued functions, which will be necessary in the forthcoming discussions.

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2 Coherent Sheaves and Oka’s Joku-Iko Principle

In general, we may consider two norms for a 𝑝 (∈ N)th (order) square (complex) matrix 𝐴 = (𝑎 𝑖 𝑗 ): ∥ 𝐴∥ ∞ = max{|𝑎 𝑖 𝑗 |}, 𝑖, 𝑗

∥ 𝐴∥ = max{∥ 𝐴𝜉 ∥ : 𝜉 ∈ C 𝑝 , ∥𝜉 ∥ = 1}. We call ∥ 𝐴∥ the operator norm of 𝐴. By taking 𝜉 = 𝑡 (0, . . . , 0, 1, 0, . . . , 0) we easily see that ∥ 𝐴∥ ∞ ≤ ∥ 𝐴∥ ≤ 𝑝 ∥ 𝐴∥ ∞ . Hence the convergence is the same with respect to either of them, so that the norms are complete. As for multiplication, ∥ 𝐴∥ has a better property than ∥ 𝐴∥ ∞ , and henceforth we use ∥ 𝐴∥. If 𝐴 = 𝐴(𝑧) is a square matrix valued function on a subset 𝐸 ⊂ C𝑛 , we write ∥ 𝐴∥ 𝐸 = sup{∥ 𝐴(𝑧) ∥ : 𝑧 ∈ 𝐸 }. Let 1 𝑝 denote the unit square matrix of order 𝑝. Proposition 2.4.1. Let 𝐴 denote a 𝑝th square matrix or a 𝑝th square matrix valued function 𝐴(𝑧) in 𝑧 ∈ 𝐸 (⊂ C𝑛 ). Let 𝐵 be another 𝑝th square matrix. Then the following hold: (i) ∥ 𝐴 + 𝐵∥ ≤ ∥ 𝐴∥ + ∥𝐵∥. (ii) ∥ 𝐴𝐵∥ ≤ ∥ 𝐴∥ · ∥𝐵∥; in particular, ∥ 𝐴 𝑘 ∥ ≤ ∥ 𝐴∥ 𝑘 (𝑘 ∈ N). (iii) If ∥ 𝐴∥ 𝐸 ≤ 𝜀 < 1 for 𝐴 = 𝐴(𝑧) (𝑧 ∈ 𝐸), (1 𝑝 − 𝐴(𝑧)) is invertible and (1 𝑝 − 𝐴(𝑧)) −1 = 1 𝑝 + 𝐴(𝑧) + 𝐴(𝑧) 2 + · · · , where the right-hand side converges uniformly on 𝐸 with respect to the operator norm, and ∥ (1 𝑝 − 𝐴) −1 ∥ 𝐸 ≤ 1−1 𝜀 ; in particular, ∥(1 𝑝 − 𝐴) −1 ∥ 𝐸 ≤ 2 with 𝜀 = 12 . (iv) Assume that sequences of 0 < 𝜀 𝑘 < 1 (𝑘 = 0, 1, . . .) and 𝑝th square matrix valued functions 𝐴 𝑘 (𝑧) in 𝑧 ∈ 𝐸 (⊂ C𝑛 ), are given and satisfy ∥ 𝐴 𝑘 ∥ 𝐸 ≤ 𝜀 𝑘 Í and ∞ 𝑘=0 𝜀 𝑘 < ∞. Then the two in nite products lim (1 𝑝 − 𝐴0 (𝑧)) · · · (1 𝑝 − 𝐴 𝑘 (𝑧)),

𝑘→∞

lim (1 𝑝 − 𝐴 𝑘 (𝑧)) · · · (1 𝑝 − 𝐴0 (𝑧))

𝑘→∞

converge uniformly on 𝐸, and the limits are both invertible matrix valued functions. Proof. (i) and (ii) are immediate from de nitions. (iii) follows from the following equations with 𝑘 → ∞:

2.4 Cartan’s Merging Lemma

55

(1 𝑝 − 𝐴(𝑧)) (1 𝑝 + 𝐴(𝑧) + 𝐴(𝑧) 2 + · · · + 𝐴(𝑧) 𝑘 ) = 1 𝑝 − 𝐴(𝑧) 𝑘+1 , ∥1 𝑝 + 𝐴(𝑧) + 𝐴(𝑧) 2 + · · · + 𝐴(𝑧) 𝑘 ∥ 𝐸 ≤

𝑘 Õ 𝑗=0

𝑗

∥ 𝐴∥ 𝐸 ≤

𝑘 Õ

𝜀𝑗 =

𝑗=0

1 − 𝜀 𝑘+1 . 1−𝜀

(iv) The proofs of the two are similar, and so we prove the rst. We set 𝐺 𝑘 (𝑧) = (1 𝑝 − 𝐴0 (𝑧)) · · · (1 𝑝 − 𝐴 𝑘 (𝑧)) =

𝑘 Ö

(1 𝑝 − 𝐴 𝑗 (𝑧)),

𝑗=0

𝑘 = 0, 1, . . . . ∞ It su ces to show that {𝐺 𝑘 }∞ Cauchy sequence, and that {𝐺 −1 𝑘=0 is a uniform 𝑘 } 𝑘=0 Í converges uniformly. Setting 𝐶0 = exp( ∞ 𝜀 ), we have 𝑘=0 𝑘

∥𝐺 𝑘 ∥ 𝐸 ≤

𝑘 Ö 𝑗=0

∥1 𝑝 − 𝐴 𝑗 ∥ 𝐸 ≤

𝑘 Ö

(1 + ∥ 𝐴 𝑗 ∥ 𝐸 ) ≤

𝑗=0

𝑘 Ö

(1 + 𝜀 𝑗 )

𝑗=0

𝑘 𝑘 ©Õ ª ©Õ ª = exp ­ log(1 + 𝜀 𝑗 ) ® < exp ­ 𝜀 𝑗 ® < 𝐶0 . « 𝑗=0 ¬ « 𝑗=0 ¬

Making use of the equations above, we have for 𝑙 > 𝑘 > 0

∥𝐺 𝑙 − 𝐺 𝑘 ∥ 𝐸 ≤ ∥𝐺 𝑘 ∥ 𝐸 · (1 𝑝 − 𝐴 𝑘+1 )(1 𝑝 − 𝐴 𝑘+2 ) · · · (1 𝑝 − 𝐴𝑙 ) − 1 𝑝 𝐸 ≤ 𝐶0 ∥ − 𝐴 𝑘+1 − 𝐴 𝑘+2 − · · · − 𝐴𝑙 + 𝐴 𝑘+1 𝐴 𝑘+2 + · · · + (−1) 𝑙−𝑘 𝐴 𝑘+1 · · · 𝐴𝑙 ∥ 𝐸 ≤ 𝐶0 (∥ 𝐴 𝑘+1 ∥ 𝐸 + ∥ 𝐴 𝑘+2 ∥ 𝐸 + · · · + ∥ 𝐴𝑙 ∥ 𝐸 + ∥ 𝐴 𝑘+1 ∥ 𝐸 · ∥ 𝐴 𝑘+2 ∥ 𝐸 + · · · + ∥ 𝐴 𝑘+1 ∥ 𝐸 · · · ∥ 𝐴𝑙 ∥ 𝐸 ) 𝑙 𝑙 ©Ö ª ©Ö ª = 𝐶0 ­ (1 + ∥ 𝐴 𝑗 ∥ 𝐸 ) − 1® ≤ 𝐶0 ­ (1 + 𝜀 𝑗 ) − 1® « 𝑗=𝑘+1 ¬ « 𝑗=𝑘+1 ¬ 𝑙 Õ © © ª ª ≤ 𝐶0 ­exp ­ 𝜀 𝑗 ® − 1® −→ 0 (𝑙 > 𝑘 → ∞). « « 𝑗=𝑘+1 ¬ ¬ Î0 −1 −1 For 𝐺 −1 𝑗=𝑘 (1 𝑝 − 𝐴 𝑗 ) , we set 𝐵 𝑘 = −𝐴 𝑘 (1 𝑝 − 𝐴 𝑘 ) . It follows that 𝑘 =

(1 𝑝 − 𝐴 𝑘 ) −1 = 1 𝑝 − 𝐵 𝑘 , so that by (iii) above ∥𝐵 𝑘 ∥ 𝐸 ≤ ∥ 𝐴 𝑘 ∥ 𝐸 · ∥ (1 𝑝 − 𝐴 𝑘 ) −1 ∥ 𝐸 ≤

𝜀𝑘 . 1 − 𝜀𝑘

56

2 Coherent Sheaves and Oka’s Joku-Iko Principle

Setting 0 < 𝜃 := max 𝑘 {𝜀 𝑘 } < 1, we get 𝜀𝑘 . 1−𝜃

∥𝐵 𝑘 ∥ 𝐸 ≤ Therefore there is some 𝑘 0 ∈ N such that 𝜀𝑘 < 1, 1−𝜃

∀ 𝑘 ≥ 𝑘0.

∞ Since for all 𝑘 ≥ 𝑘 0 , 𝐵 𝑘 ful ll the conditions which 𝐴 𝑘 satis es, {𝐺 −1 𝑘 } 𝑘=0 converges uniformly on 𝐸. ⊓ ⊔

Let 𝑆,𝑇 be 𝑝th square matrices. Assuming the existences of the inverses (1 𝑝 − 𝑆) −1 , (1 𝑝 − 𝑇) −1 , we put (2.4.2)

𝑀 (𝑆,𝑇) = (1 𝑝 − 𝑆) −1 (1 𝑝 − 𝑆 − 𝑇) (1 𝑝 − 𝑇) −1 , 𝑁 (𝑆,𝑇) = 1 𝑝 − 𝑀 (𝑆,𝑇).

The next lemma is a key in the later arguments on convergences. Lemma 2.4.3. Assume that max{∥𝑆∥, ∥𝑇 ∥} ≤ 𝜀 with 0 < 𝜀 < 1. Then 

1 ∥𝑁 (𝑆,𝑇) ∥ ≤ 1−𝜀 In particular, with 𝜀 =

1 2

2 (max{∥𝑆∥, ∥𝑇 ∥}) 2 .

we have ∥𝑁 (𝑆,𝑇)∥ ≤ 22 (max{∥𝑆∥, ∥𝑇 ∥}) 2 .

Proof. Noting that (1 𝑝 −𝑇) −1 = 1 𝑝 +𝑇 (1 𝑝 −𝑇) −1 = 1 𝑝 +𝑇 +𝑇 2 (1 𝑝 −𝑇) −1 , we have 𝑀 (𝑆,𝑇) = (1 𝑝 − 𝑆) −1 (1 𝑝 − 𝑆 − 𝑇)(1 𝑝 − 𝑇) −1 = (1 𝑝 − (1 𝑝 − 𝑆) −1𝑇)(1 𝑝 − 𝑇) −1 = 1 𝑝 + 𝑇 + 𝑇 2 (1 𝑝 − 𝑇) −1 − (1 𝑝 + 𝑆(1 𝑝 − 𝑆) −1 )𝑇 (1 𝑝 + 𝑇 (1 𝑝 − 𝑇) −1 ) = 1 𝑝 + 𝑇 + 𝑇 2 (1 𝑝 − 𝑇) −1 − 𝑇 − 𝑇 2 (1 𝑝 − 𝑇) −1 − 𝑆(1 𝑝 − 𝑆) −1𝑇 (1 𝑝 − 𝑇) −1 = 1 𝑝 − 𝑆(1 𝑝 − 𝑆) −1𝑇 (1 𝑝 − 𝑇) −1 , 𝑁 (𝑆,𝑇) = 𝑆(1 𝑝 − 𝑆) −1𝑇 (1 𝑝 − 𝑇) −1 . It follows from Proposition 2.4.1 (iii) that 

    2 1 1 1 ∥𝑁 (𝑆,𝑇)∥ ≤ ∥𝑆∥ · · ∥𝑇 ∥ · ≤ (max{∥𝑆∥, ∥𝑇 ∥}) 2 . 1−𝜀 1−𝜀 1−𝜀

⊓ ⊔

2.4 Cartan’s Merging Lemma

57

2.4.2 Cartan’s Matrix Decomposition We assume the following setting: 2.4.4 (closed cuboid). Here a closed cuboid or a closed rectangle is bounded and the edges are parallel to the real and imaginary axes, and the case of some degenerate edges with width 0 is included. Let 𝐸 ′ , 𝐸 ′′ ⋐ 𝛺 be closed cuboids as follows: There are a closed cuboid 𝐹 ⋐ C𝑛−1 and two adjacent closed rectangles 𝐸 𝑛′ , 𝐸 𝑛′′ ⋐ C sharing an edge ℓ, such that 𝐸 ′ = 𝐹 × 𝐸 𝑛′ ,

𝐸 ′′ = 𝐹 × 𝐸 𝑛′′ ,

ℓ = 𝐸 𝑛′ ∩ 𝐸 𝑛′′

(cf. Fig. 2.1).

Fig. 2.1 Adjacent closed cuboids.

Let 𝐺 𝐿 ( 𝑝; C) be the general linear group of all invertible 𝑝th square matrices. Lemma 2.4.5 (Cartan’s matrix decomposition). Let the notation be as above. Then there is a neighborhood 𝑉0 ⊂ 𝐺 𝐿 ( 𝑝; C) of 1 𝑝 such that for a matrix-valued holomorphic function 𝐴 : 𝑈 → 𝑉0 in a neighborhood 𝑈 of 𝐹 × ℓ, there is a matrixvalued holomorphic function 𝐴′ : 𝑈 ′ → 𝐺 𝐿( 𝑝; C) (resp. 𝐴′′ : 𝑈 ′′ → 𝐺 𝐿 ( 𝑝; C)) in a neighborhood 𝑈 ′ (resp. 𝑈 ′′ ) of 𝐸 ′ (resp. 𝐸 ′′ ) satisfying 𝐴 = 𝐴′ · 𝐴′′ in a neighborhood of 𝐹 × ℓ. Proof. We widen each edge of 𝐹, 𝐸 𝑛′ , 𝐸 𝑛′′ by the same length 𝛿 > 0 outward and denote ˜ 𝐸˜ ′ and 𝐸˜ ′′ , respectively. the resulting closed cube and closed rectangles by 𝐹, 𝑛(1) 𝑛(1) With su ciently small 𝛿 > 0 we have ′ ′′ 𝐹 × ℓ ⊂ 𝐹˜ × ( 𝐸˜ 𝑛(1) ∩ 𝐸˜ 𝑛(1) ) ⋐ 𝑈. ′ ′′ We set the boundary of 𝐸˜ 𝑛(1) ∩ 𝐸˜ 𝑛(1) with positive orientation as in Fig. 2.2, divided by ℓ to the right-hand side and the left-hand:   ′ ′′ (2.4.6) 𝜕 𝐸˜ 𝑛(1) ∩ 𝐸˜ 𝑛(1) = 𝛾 (1) = 𝛾 ′(1) + 𝛾 ′′(1) . ′ Similarly, keeping the inner 2𝛿 of the width 𝛿 as 𝐸 𝑛′ is widened to 𝐸˜ 𝑛(1) , we 𝛿 ′ ˜ successively shrink inward by dividing the outer 2 in half. That is, 𝐸 𝑛(2) denotes ′ ′ the closed cuboid shrunk inward by 4𝛿 from 𝐸˜ 𝑛(1) . Assuming 𝐸˜ 𝑛(𝑘 determined, we )

58

2 Coherent Sheaves and Oka’s Joku-Iko Principle

Fig. 2.2 𝛿-closed neighborhoods of adjacent closed rectangles.

Fig. 2.3

𝛿 2𝑘

′ -closed neighborhood of 𝐸˜ 𝑛(𝑘) .

′ denote by 𝐸˜ 𝑛(𝑘+1) the closed cuboid shrunk inward by

Since

𝛿 4

+

𝛿 8

𝛿 2 𝑘+1

′ from 𝐸˜ 𝑛(𝑘 (cf. Fig. 2.3). )

+ · · · = 2𝛿 , we see that ∞ Ù

𝛿 ′ ′ 𝐸˜ 𝑛(𝑘 ) = the closed cuboid widened from 𝐸 𝑛 by 2 . 𝑘=1

′′ , similarly. As (2.4.6) we write We set 𝐸˜ 𝑛(𝑘 )

(2.4.7) Let

  ′ ′ ′′ ˜ ′′ 𝜕 𝐸˜ 𝑛(𝑘 ∩ 𝐸 ) 𝑛(𝑘 ) = 𝛾 (𝑘 ) = 𝛾 (𝑘 ) + 𝛾 (𝑘 ) . ′ 𝐸˜ ′(𝑘 ) = 𝐹˜ × 𝐸˜ 𝑛(𝑘 ),

˜ ˜ ′′ 𝐸˜ ′′ (𝑘 ) = 𝐹 × 𝐸 𝑛(𝑘 )

be the closed cuboid neighborhoods of 𝐸 ′ and 𝐸 ′′ , respectively. We set 𝐵1 (𝑧) = 1 𝑝 − 𝐴(𝑧) for 𝑧 ∈ 𝐸˜ ′(1) ∩ 𝐸˜ ′′ . Here we apply the idea of the Cousin (1) ′ ′ ˜ integral discussed in 1.3.1. For (𝑧 , 𝑧 𝑛 ) ∈ 𝐸 (2) ∩ 𝐸˜ ′′ , we have by Cauchy’s integral (2) formula

2.4 Cartan’s Merging Lemma

59

𝐵1 (𝑧 ′ , 𝑧 𝑛 ) =

(2.4.8)

1 2𝜋𝑖

1 = 2𝜋𝑖 =

∫ 𝛾 (1)

∫

′ 𝛾 (1)

𝐵1 (𝑧 ′ , 𝜁) 𝑑𝜁 𝜁 − 𝑧𝑛 𝐵1 (𝑧 ′ , 𝜁) 1 𝑑𝜁 + 𝜁 − 𝑧𝑛 2𝜋𝑖

∫ ′′ 𝛾 (1)

𝐵1 (𝑧 ′ , 𝜁) 𝑑𝜁 𝜁 − 𝑧𝑛

𝐵1′ (𝑧 ′ , 𝑧 𝑛 ) + 𝐵1′′ (𝑧 ′ , 𝑧 𝑛 ).

Here, 𝐵1′ (𝑧 ′ , 𝑧 𝑛 ) (resp. 𝐵1′′ (𝑧 ′ , 𝑧 𝑛 )) is holomorphic in (𝑧 ′ , 𝑧 𝑛 ) ∈ 𝐸˜ ′(2) (resp. (𝑧 ′ , 𝑧 𝑛 ) ∈ 𝐸˜ ′′ ), and (2) 𝛿 |𝑧 𝑛 − 𝜁 | ≥ , 4

(2.4.9)

∀ (𝑧 ′ , 𝑧 𝑛 ) ∈ 𝐸˜ ′(2) ,

∀ 𝜁 ∈ 𝛾 ′(1) .

Letting 𝐿 be the length of 𝛾 ′(1) , we get 𝐿 = the length of 𝛾 ′′(1) ≥ the length of 𝛾 ′(𝑘 ) (also, the length of 𝛾 ′′(𝑘 ) ), 𝑘 = 1, 2, . . . . It follows from (2.4.8) and (2.4.9) that for (𝑧 ′ , 𝑧 𝑛 ) ∈ 𝐸˜ ′(2) ∥𝐵1′ (𝑧 ′ , 𝑧 𝑛 ) ∥ ≤

1 4 · 𝐿 · max ∥𝐵1 (𝑧 ′ , 𝜁)∥. 𝛾 (1) 2𝜋 𝛿

Therefore ∥𝐵1′ ∥ 𝐸˜ ′ ≤

2𝐿 ∥𝐵1 ∥ 𝐸˜ ′ ∩𝐸˜ ′′ . (1) (1) 𝜋𝛿

∥𝐵1′′ ∥ 𝐸˜ ′′ ≤

2𝐿 ∥𝐵1 ∥ 𝐸˜ ′ ∩𝐸˜ ′′ . (1) (1) 𝜋𝛿

(2)

Similarly, we have (2)

We set (2.4.10)

 n o  2𝐿 𝜀 1 = max ∥𝐵1′ ∥ 𝐸˜ ′ , ∥𝐵1′′ ∥ 𝐸˜ ′′ ≤ ∥𝐵1 ∥ 𝐸˜ ′ ∩𝐸˜ ′′ . (2) (2) (1) (1) 𝜋𝛿

Choose 𝛿 > 0, smaller if necessary so that ∥𝐵1 ∥ 𝐸˜ ′

𝜋𝛿 25 𝐿

′′ ∩ 𝐸˜ (1) (1)

≤

≤ 12 . Suppose that 𝜋2 𝛿2 . 26 𝐿 2

Then (2.4.11)

𝜀1 ≤

𝜋𝛿 1 ≤ , 25 𝐿 2

60

2 Coherent Sheaves and Oka’s Joku-Iko Principle

𝐴(𝑧) = (1 𝑝 − 𝐵1 (𝑧)) = (1 𝑝 − 𝐵1′ (𝑧))(1 𝑝 − 𝑁 (𝐵1′ (𝑧), 𝐵1′′ (𝑧))) · (1 𝑝 − 𝐵1′′ (𝑧)), 𝑧 ∈ 𝐸˜ ′(2) ∩ 𝐸˜ ′′ (2) .

(2.4.12)

Here it follows from Lemma 2.4.3 that ∥𝑁 (𝐵1′ (𝑧), 𝐵1′′ (𝑧)) ∥ 𝐸˜ ′

(2)

′′ ∩ 𝐸˜ (2)

≤ 22 𝜀 12 ;

here the estimate “ 𝜀 12 ” is the point. In what follows, we proceed inductively; for 𝑗 = 1, . . . , 𝑘 (∈ N) we choose 𝑝th square matrix-valued holomorphic functions 𝐵′𝑗 (𝑧) (𝑧 ∈ 𝐸˜ ′( 𝑗+1) ),

𝐵′′𝑗 (𝑧) (𝑧 ∈ 𝐸˜ ′′ ( 𝑗+1) )

satisfying (2.4.13) (2.4.14)

n 𝜀 𝑗 := max ∥𝐵′𝑗 ∥ 𝐸˜ ′

, ∥𝐵′′𝑗 ∥ 𝐸˜ ′′

o

≤

𝜋𝛿

 ≤

 1 , 1 ≤ 𝑗 ≤ 𝑘, 2𝑗

2 𝑗+4 𝐿 𝐴(𝑧) = (1 𝑝 − 𝐵1′ (𝑧)) · · · (1 𝑝 − 𝐵′𝑘 (𝑧)) · (1 𝑝 − 𝑁 (𝐵′𝑘 (𝑧), 𝐵′′𝑘 (𝑧))) · (1 𝑝 − 𝐵′′𝑘 (𝑧)) · · · (1 𝑝 − 𝐵1′′ (𝑧)), 𝑧 ∈ 𝐸˜ ′(𝑘+1) ∩ 𝐸˜ ′′ (𝑘+1) . ( 𝑗+1)

( 𝑗+1)

The case of 𝑘 = 1 holds by (2.4.11) and (2.4.12). We set 𝐵 𝑘+1 (𝑧) = 𝑁 (𝐵′𝑘 (𝑧), 𝐵′′𝑘 (𝑧)) for 𝑧 ∈ 𝐸˜ ′(𝑘+2) ∩ 𝐸˜ ′′ (cf. (2.4.2)). With (𝑘+2) ′ 𝛾 (𝑘+1) , 𝛾 ′′(𝑘+1) de ned by (2.4.7) we set 𝐵′𝑘+1 (𝑧 ′ , 𝑧 𝑛 ) = 𝐵′′𝑘+1 (𝑧 ′ , 𝑧 𝑛 ) Since |𝜁 − 𝑧 𝑛 | ≥ 2.4.3 that

𝛿 2 𝑘+2

1 2𝜋𝑖

1 = 2𝜋𝑖

∫ ′ 𝛾 (𝑘+1)

∫

′′ 𝛾 (𝑘+1)

𝐵 𝑘+1 (𝑧 ′ , 𝜁) 𝑑𝜁, 𝜁 − 𝑧𝑛

(𝑧 ′ , 𝑧 𝑛 ) ∈ 𝐸˜ ′(𝑘+2) ,

𝐵 𝑘+1 (𝑧 ′ , 𝜁) 𝑑𝜁, 𝜁 − 𝑧𝑛

(𝑧 ′ , 𝑧 𝑛 ) ∈ 𝐸˜ ′′ (𝑘+2) .

in the integrand above, it follows from (2.4.13) and Lemma

𝐿 2 𝑘+2 ∥𝑁 (𝐵′𝑘 , 𝐵′′𝑘 )∥ 𝐸˜ ′ ∩𝐸˜ ′′ (𝑘+1) (𝑘+1) 2𝜋 𝛿 𝐿 2 𝑘+2 2 2 1 𝜋𝛿 ≤ 2 𝜀 𝑘 ≤ 𝜀 𝑘 ≤ 𝑘+5 , 2𝜋 𝛿 2 2 𝐿 1 𝑝 − 𝑁 (𝐵′𝑘 (𝑧), 𝐵′′𝑘 (𝑧)) = (1 𝑝 − 𝐵′𝑘+1 (𝑧)) (1 𝑝 − 𝑁 (𝐵′𝑘+1 (𝑧), 𝐵′′𝑘+1 (𝑧))) · (1 𝑝 − 𝐵′′ (𝑧)), 𝑧 ∈ 𝐸˜ ′ ∩ 𝐸˜ ′′ . 𝜀 𝑘+1 ≤

𝑘+1

(𝑘+2)

(𝑘+2)

Thus we deduce (2.4.13) and (2.4.14) for 𝑘 + 1. By (2.4.13) and Proposition 2.4.1 (iv), the in nite products 𝐴′ (𝑧) = lim (1 𝑝 − 𝐵1′ (𝑧)) · · · (1 𝑝 − 𝐵′𝑘 (𝑧)), 𝑘→∞

𝑧 ∈ 𝐸˜ ′ :=

Ù∞ 𝑘=1

𝐸˜ ′(𝑘 ) ,

2.4 Cartan’s Merging Lemma

61

𝐴′′ (𝑧) = lim (1 𝑝 − 𝐵′′𝑘 (𝑧)) · · · (1 𝑝 − 𝐵1′′ (𝑧)), 𝑘→∞

𝑧 ∈ 𝐸˜ ′′ :=

Ù∞ 𝑘=1

𝐸˜ ′′ (𝑘 )

converge uniformly in each de ning domain, and the limits are 𝑝th square invertible matrix-valued holomorphic functions. It follows from (2.4.13) and Lemma 2.4.3 that for 𝑧 ∈ 𝐸˜ ′ ∩ 𝐸˜ ′′ ∥𝑁 (𝐵′𝑘 (𝑧), 𝐵′′𝑘 (𝑧)) ∥ ≤ 22 𝜀 2𝑘 ≤

1 −→ 0 22𝑘−2

(𝑘 → ∞),

and so 𝐴(𝑧) = 𝐴′ (𝑧) 𝐴′′ (𝑧) by (2.4.14).

⊓ ⊔

2.4.3 Cartan’s Merging Lemma It is the aim to prove Cartan’s Merging Lemma: ′ ′′ Lemma 2.4.15. and let ℱ be an analytic sheaf n Let 𝐸 , 𝐸 ⋐ 𝛺 be as in oLemma 2.4.5, n o ′ ′ ′ ′′ over 𝛺. Let 𝜎 𝑗 ∈ 𝛤 (𝐸 , ℱ) : 1 ≤ 𝑗 ≤ 𝑝 (resp. 𝜎𝑘 ∈ 𝛤 (𝐸 ′′ , ℱ) : 1 ≤ 𝑘 ≤ 𝑝 ′′ ) be

a finite generator system of ℱ over 𝐸 ′ (resp. 𝐸 ′′ ). Assume further that there are 𝑎 𝑗 𝑘 , 𝑏 𝑘 𝑗 ∈ 𝒪(𝐸 ′ ∩ 𝐸 ′′ ), 1 ≤ 𝑗 ≤ 𝑝 ′ , 1 ≤ 𝑘 ≤ 𝑝 ′′ , satisfying ′′

𝜎 ′𝑗

=

𝑝 Õ 𝑘=1

′

𝑎 𝑗 𝑘 · 𝜎𝑘′′ ,

𝜎𝑘′′

=

𝑝 Õ 𝑗=1

𝑏 𝑘 𝑗 · 𝜎 ′𝑗

(on 𝐸 ′ ∩ 𝐸 ′′ ).

 Then, there exists a finite generator system 𝜎𝑙 ∈ 𝛤 (𝐸 ′ ∪ 𝐸 ′′ , ℱ) : 1 ≤ 𝑙 ≤ 𝑝 ′ + 𝑝 ′′ of ℱ over 𝐸 ′ ∪ 𝐸 ′′ . Proof. Set column vectors and matrices, 𝜎′ © 1ª ­ . ® 𝜎 ′ = ­ .. ® , ­ ′ ® 𝜎 ′ « 𝑝 ¬

𝜎 ′′ © 1 ª ­ . ® 𝜎 ′′ = ­ .. ® , ­ ′′ ® 𝜎 ′′ « 𝑝 ¬

and 𝐴 = (𝑎 𝑗 𝑘 ),

𝐵 = (𝑏 𝑘 𝑗 ).

Unless confusion occurs, 𝐴 (resp. 𝐵) stands for (𝑎 𝑗 𝑘 ) (resp. (𝑏 𝑘 𝑗 )): (2.4.16)

𝜎 ′ = 𝐴 𝜎 ′′ ,

𝜎 ′′ = 𝐵 𝜎 ′ .

Adding 0 to 𝜎 ′ and 𝜎 ′′ to have the same size, we set 𝑝 (= 𝑝 ′ + 𝑝 ′′ )th column vectors as follows:

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2 Coherent Sheaves and Oka’s Joku-Iko Principle

𝜎′ © .1 ª ­ .. ® ­ ® ­ 𝜎′ ® ′ ­ ® 𝜎 ˜ ′ = ­ 𝑝 ®, ­ 0 ® ­ . ® ­ . ® ­ . ®

0 © . ­ .. ­ ­ 0 ­ 𝜎 ˜ ′′ = ­ ′′ ­ 𝜎1 ­ . ­ . ­ .

ª ® ® ® ® ®. ® ® ® ® ′′ « 𝜎𝑝′′ ¬

« 0 ¬ Also we set 𝐴˜ =

!

𝐴

1 𝑝′

.

−𝐵 1 𝑝′′ − 𝐵𝐴

By making use of 𝐵𝐴 𝜎 ′′ = 𝜎 ′′ by (2.4.16), we get 𝜎 ˜ ′ = 𝐴˜ 𝜎 ˜ ′′ .

(2.4.17)

We take the compositions of fundamental transformations, 𝑃=

!

1 𝑝′ 𝐴

0 1 𝑝′′

(2.4.18) 𝑄=

1 𝑝′

,

𝑃 −1

,

𝑄 −1

1 𝑝′ −𝐴

=

𝐵 1 𝑝′′

,

0 1 𝑝′′

!

0

!

1 𝑝′

=

0

!

−𝐵 1 𝑝′′

.

We transform 𝐴˜ from right and left: 𝑄 𝐴˜ 𝑃 −1 = 1 𝑝 . Since 𝐴˜ = 𝑄 −1 𝑃, with 𝑅 := 𝑃 −1 𝑄 we get (2.4.19)

𝑅=

1 𝑝′ −𝐴

0 1 𝑝′′

!

1 𝑝′

0

𝐵 1 𝑝′′

! ,

˜ = 1𝑝. 𝐴𝑅

Because of the form, 𝑅 is invertible for any choices of 𝐴 and 𝐵. Since the elements 𝑎 𝑗 𝑘 , 𝑏 𝑘 𝑗 of 𝐴 and 𝐵 are holomorphic in a neighborhood of 𝐸 ′ ∩ 𝐸 ′′ = 𝐹 × ℓ, it follows from Theorem 1.3.2 that 𝑎 𝑗 𝑘 , 𝑏 𝑗 𝑘 are uniformly approximated by polynomials 𝑎˜ 𝑗 𝑘 , 𝑏˜ 𝑘 𝑗 on a neighborhood 𝑊0 ⋑ 𝐸 ′ ∩ 𝐸 ′′ with 𝑊0 ⋐ 𝑈 ′ ∩ 𝑈 ′′ . We de ne 𝑅˜ by (2.4.19) with those polynomials 𝑎˜ 𝑗 𝑘 , 𝑏˜ 𝑘 𝑗 . Taking the su ciently enough approximations, we deduce that for the neighborhood 𝑉 of 1 𝑝 in Lemma 2.4.5, (2.4.20)

ˆ ˜ 𝑅(𝑧) ˜ 𝐴(𝑧) = 𝐴(𝑧) ∈ 𝑉0 ,

𝑧 ∈ 𝑊0 .

Then by Lemma 2.4.5 there is a 𝑝th invertible matrix valued function 𝐴ˆ ′ (resp. 𝐴ˆ ′′ ) on 𝐸 ′ (resp. 𝐸 ′′ ), such that (2.4.21)

𝐴ˆ = 𝐴ˆ ′ 𝐴ˆ ′′

2.5 Oka’s Joku-Iko Principle

63

on 𝐸 ′ ∩ 𝐸 ′′ . From this and (2.4.20) we deduce 𝐴˜ = 𝐴ˆ ′ 𝐴ˆ ′′ 𝑅˜ −1 , and so from (2.4.17) (2.4.22)

𝐴ˆ ′−1 𝜎 ˜ ′ = 𝐴ˆ ′′ 𝑅˜ −1 𝜎 ˜ ′′

on 𝐸 ′ ∩ 𝐸 ′′ . Therefore 𝜏ℎ ∈ 𝛤 (𝐸 ′ ∪ 𝐸 ′′ , ℱ), 1 ≤ ℎ ≤ 𝑝, are well-de ned by ( 𝜏 © 1ª 𝐴ˆ ′−1 𝜎 ˜ ′, ­ .. ® ­ . ® = ˆ ′′ −1 ′′ ­ ® 𝐴 𝑅˜ 𝜎 ˜ , 𝜏𝑝 « ¬

on 𝐸 ′ , on 𝐸 ′′ .

 Since 𝐴ˆ ′−1 and 𝐴ˆ ′′ 𝑅˜ −1 are invertible, the nite system 𝜏ℎ : 1 ≤ ℎ ≤ 𝑝 generates ℱ over 𝐸 ′ ∪ 𝐸 ′′ . ⊓ ⊔   ′  ′′ The above obtained 𝜏ℎ is called a merged system from 𝜎 𝑗 and 𝜎𝑘 .

2.5 Oka’s Joku-Iko Principle Let 𝑆 ⊂ P𝛥 denote a complex submanifold of a polydisk P𝛥 ⊂ C 𝑁 . Oka’s JokuIko Principle is a methodological principle to reduce a problem on 𝑆 to that of a polydisk P𝛥 of even higher dimension by embedding the original domain under the consideration onto 𝑆 ⊂ P𝛥 and extending the problem over P𝛥, to solve the problem by virtue of the simple shape of P𝛥, and then to get the solution on 𝑆 by restriction. It is the innovative point of Oka’s idea to solve a problem caused by the increase of numbers of variables by increasing more the number of variables.

2.5.1 Oka Syzygy Let 𝐸 ⋐ C𝑛 be a closed cuboid (cf. 2.4.4). The number of the edges of 𝐸 with positive length is called the dimension of 𝐸, and denoted by dim 𝐸, where 0 ≤ dim 𝐸 ≤ 2𝑛. We consider an analytic sheaf ℱ ⊂ 𝒪𝑈𝑞 over a neighborhood 𝑈 of 𝐸. The two claims of the next lemma are the most fundamental properties of coherent sheaves. Lemma 2.5.1 (Oka Syzygy). Let 𝐸 ⋐ C𝑛 be a closed cuboid as above, and let ℱ be a coherent sheaf over 𝐸. Then we have: (i) ℱ has  a finite generator system over 𝐸. (ii) Let 𝜎 𝑗 1≤ 𝑗 ≤ 𝑁 be a finite generator system of ℱ over 𝐸. Then, for every 𝜎 ∈ 𝛤 (𝐸, ℱ) there are holomorphic functions 𝑎 𝑗 ∈ 𝒪(𝐸), 1 ≤ 𝑗 ≤ 𝑁, such that 𝑁 Õ (2.5.2) 𝜎= 𝑎 𝑗 · 𝜎 𝑗 (on 𝐸). 𝑗=1

64

2 Coherent Sheaves and Oka’s Joku-Iko Principle

Proof. We prove (i) and (ii) simultaneously by the double induction on 𝜈 := dim 𝐸. Writing respectively (i) 𝜈 and (ii) 𝜈 for the statements (i) and (ii) in the case of dim 𝐸 = 𝜈, we shall proceed as [(i) 𝜈−1 + (ii) 𝜈−1 ] =⇒ (i) 𝜈 =⇒ (ii) 𝜈 . (a) The case of 𝜈 = 0: Since 𝐸 is a singleton, the both are immediate by the assumptions. (b) Suppose 𝜈 ≥ 1, and that each of (i)𝜈−1 and (ii)𝜈−1 holds for any coherent sheaf over an arbitrary closed cuboid of dimension 𝜈 − 1. (Here it is noted that the induction hypothesis is not claiming the validities of (i)𝜈−1 and (ii)𝜈−1 for the same coherent sheaf, but the validity of each of (i)𝜈−1 and (ii)𝜈−1 for arbitrary coherent sheaves: Either, 𝐸 is not xed.) (i) Let 𝐸 be a closed cuboid of dimension 𝜈, and let ℱ be a coherent sheaf over a neighborhood of 𝐸. Let 𝑧 = (𝑧 𝑗 ) ∈ C𝑛 be the coordinates. By parallel transitions, changes of the order of the coordinates and the multiplications to the coordinates by the imaginary unit 𝑖, we may assume without loss of generality that 𝐸 is expressed as (2.5.3)

𝐸 = 𝐹 × {𝑧 𝑛 : 0 ≤ ℜ𝑧 𝑛 ≤ 𝑇, |ℑ𝑧 𝑛 | ≤ 𝜃}, 𝑇 > 0, 𝜃 ≥ 0.

Here, in the case 𝜃 = 0 (resp. 𝜃 > 0), 𝐹 is a closed cuboid of dimension 𝜈 − 1 (resp. 𝜈 − 2). Take arbitrarily a point 𝑡 ∈ [0,𝑇] and set 𝐸 𝑡 := 𝐹 × {𝑧 𝑛 : ℜ𝑧 𝑛 = 𝑡, |ℑ𝑧 𝑛 | ≤ 𝜃}. Since 𝐸 𝑡 is a closed cuboid of dimension 𝜈 − 1, it follows from (i)𝜈−1 that ℱ has a nite generator system over a neighborhood of 𝐸 𝑡 . By Heine Borel’s Theorem there is a nite partition 0 = 𝑡0 < 𝑡1 < · · · < 𝑡 𝐿 = 𝑇

(2.5.4)

 𝑁𝛼 with nite generator systems 𝜎𝛼 𝑗 𝑗=1 of ℱ over 𝐸 𝛼 := 𝐹 × {𝑧 𝑛 : 𝑡 𝛼−1 ≤ ℜ𝑧 𝑛 ≤ 𝑡 𝛼 , |ℑ𝑧 𝑛 | ≤ 𝜃}

(1 ≤ 𝛼 ≤ 𝐿).

Since 𝐸 𝛼 ∩ 𝐸 𝛼+1 = 𝐸 𝑡 𝛼 is a closed cuboid of dimension 𝜈 − 1, the induction hypothesis (ii)𝜈−1 implies the existence of holomorphic functions 𝑎 𝑗 𝑘 , 𝑏 𝑘 𝑗 on 𝐸 𝛼 ∩ 𝐸 𝛼+1 satisfying Õ Õ 𝜎𝛼 𝑗 = 𝑎 𝑗 𝑘 · 𝜎𝛼+1𝑘 , 𝜎𝛼+1𝑘 = 𝑏 𝑘 𝑗 · 𝜎𝛼 𝑗 . 𝑘

𝑗

Applying Cartan’s Merging Lemma 2.4.15, we obtain a nite generator system of ℱ over 𝐸 𝛼 ∪ 𝐸 𝛼+1 . Merging rst the nite generator systems of ℱ over 𝐸 1 and over 𝐸 2 , we obtain a merged nite generator system of ℱ over 𝐸 1 ∪ 𝐸 2 (cf. Fig. 2.4). Similarly, from the nite generator system above over 𝐸 1 ∪ 𝐸 2 and the one over 𝐸 3 , we obtain a merged

2.5 Oka’s Joku-Iko Principle

65

Fig. 2.4 𝐸 𝛼 = 𝐹 × [𝑡 𝛼−1 , 𝑡 𝛼 ].

Ð nite generator system of ℱ over 3𝛼=1 𝐸 𝛼 . Repeating the process, we have a nite Ð𝐿 generator system  of ℱ over 𝛼=1 𝐸 𝛼 = 𝐸. (ii) Let 𝜎 𝑗 1≤ 𝑗 ≤ 𝑁 , and 𝜎 be the given ones. We take a closed cuboid 𝐸 as in (2.5.3), and use the same notation there. For every point 𝑡 ∈ [0,𝑇], 𝐸 𝑡 is a closed cuboid of dimension 𝜈 − 1, and hence it follows from (ii)𝜈−1 that there are holomorphic functions 𝑎 𝑡 𝑗 ∈ 𝒪(𝐸 𝑡 ) such that 𝜎=

𝑁 Õ

𝑎𝑡 𝑗 · 𝜎𝑗

(on 𝐸 𝑡 ).

𝑗=1

By the same arguments as in (i) there is a nite partition (2.5.4) (cf. Fig. 2.4) with holomorphic functions 𝑎 𝛼 𝑗 on 𝐸 𝛼 satisfying Õ (2.5.5) 𝜎= 𝑎 𝛼 𝑗 · 𝜎 𝑗 (on 𝐸 𝛼 ). 𝑗

 
Let ℛ := ℛ (𝜎 𝑗 ) 𝑗 denote the relation sheaf given by 𝜎 𝑗 𝑗 . By Corollary 2.3.38, ℛ is coherent. If 𝐸 𝛼+1 is adjacent to 𝐸 𝛼 , we have Õ (2.5.6) (𝑎 𝛼 𝑗 − 𝑎 𝛼+1 𝑗 ) · 𝜎 𝑗 = 0 𝑗

over 𝐸 𝛼 ∩ 𝐸 𝛼+1 . Therefore

(2.5.7) 𝑏 𝛼 𝑗 𝑗 := 𝑎 𝛼 𝑗 − 𝑎 𝛼+1 𝑗 𝑗 ∈ 𝛤 (𝐸 𝛼 ∩ 𝐸 𝛼+1 , ℛ). Since ℛis coherent, it follows from the above result (i)𝜈 that ℛ has a nite generator system 𝜏ℎ over a neighborhood of 𝐸. Since 𝐸 𝛼 ∩ 𝐸 𝛼+1 is a closed cuboid of dimension 𝜈 − 1, it follows again from (ii)𝜈−1 that there exist 𝑐 𝛼ℎ ∈ 𝒪(𝐸 𝛼 ∩ 𝐸 𝛼+1 ) such that

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2 Coherent Sheaves and Oka’s Joku-Iko Principle

 (2.5.8)

𝑏𝛼𝑗

 𝑗

=

Õ

𝑐 𝛼ℎ · 𝜏ℎ

(on 𝐸 𝛼 ∩ 𝐸 𝛼+1 ).

ℎ

Note that 𝐸 𝛼 ∩ 𝐸 𝛼+1 = 𝐹 × {𝑧 𝑛 : ℜ𝑧 𝑛 = 𝑡 𝛼 , |ℑ𝑧 𝑛 | ≤ 𝜃}. For a su ciently small 𝛿 > 0, (2.5.8) holds over 𝐹 × {𝑧 𝑛 : ℜ𝑧 𝑛 = 𝑡 𝛼 , |ℑ𝑧 𝑛 | ≤ 𝜃 + 𝛿}. We consider the Cousin integral (1.3.7) of 𝑐 𝛼ℎ along the line segment ℓ 𝛼 = {𝑧 𝑛 : ℜ𝑧 𝑛 = 𝑡 𝛼 , |ℑ𝑧 𝑛 | ≤ 𝜃 + 𝛿} in 𝑧 𝑛 -plane with orientation as ℑ𝑧 𝑛 increases. Let the Cousin decomposition be 𝑑 𝛼ℎ ∈ 𝒪(𝐸 𝛼 ), (2.5.9)

𝑑 𝛼+1ℎ ∈ 𝒪(𝐸 𝛼+1 ),

𝑐 𝛼ℎ = 𝑑 𝛼ℎ − 𝑑 𝛼+1ℎ

(on 𝐸 𝛼 ∩ 𝐸 𝛼+1 ).

(Cf. Lemma 1.3.10.) It hence follows from (2.5.7) (2.5.9) that    Õ 𝑎 𝛼 𝑗 − 𝑎 𝛼+1 𝑗 = 𝑑 𝛼ℎ − 𝑑 𝛼+1ℎ 𝜏ℎ . 𝑗

ℎ

With 𝜏ℎ = (𝜏ℎ 𝑗 ) expressed by the components 𝜏ℎ 𝑗 ∈ 𝒪(𝐸), we have 𝑎𝛼𝑗 −

Õ ℎ

𝑑 𝛼ℎ 𝜏ℎ 𝑗 = 𝑎 𝛼+1 𝑗 −

Õ

𝑑 𝛼+1ℎ 𝜏ℎ 𝑗

ℎ

over 𝐸 𝛼 ∩ 𝐸 𝛼+1 . The left-hand side is an element of 𝒪(𝐸 𝛼 ), and the right-hand side is that of 𝒪(𝐸 𝛼+1 ), and so it de nes an element 𝑎˜ 𝛼 𝑗 ∈ 𝒪(𝐸 𝛼 ∪ 𝐸 𝛼+1 ). Since 𝜏ℎ is a section of ℛ, Õ (2.5.10) 𝜎= 𝑎˜ 𝛼 𝑗 𝜎 𝑗 (on 𝐸 𝛼 ∪ 𝐸 𝛼+1 ). 𝑗

We thus patched together the expression (2.5.5) of 𝜎 over 𝐸 𝛼 and that over 𝐸 𝛼+1 to obtain an expression (2.5.10) of 𝜎 over 𝐸 𝛼 ∪ 𝐸 𝛼+1 . Beginning with 𝛼 = 1, we patch together the expression of 𝜎 over 𝐸 1 and that over 𝐸 2 to obtain an expression of 𝜎 over 𝐸 1 ∪ 𝐸 2 . We then patch together it and that over 𝐸 3 to obtain an expression of 𝜎 over 𝐸 1 ∪ 𝐸 2 ∪ 𝐸 3 . Repeating this procedure up to 𝛼 = 𝐿 − 1, we obtain an expression (2.5.2) of 𝜎 over 𝐸 = 𝐸 1 ∪ · · · ∪ 𝐸 𝐿 . Thus, the proof by induction is completed. ⊓ ⊔ Applying the Oka Syzygy Lemma 2.5.1 for a geometric ideal sheaf ℱ = ℐ⟨ 𝑆⟩ (cf. footnote at p. 38) with a complex submanifold 𝑆, we immediately have by Theorem 2.3.40: Lemma 2.5.11 (Geometric Syzygy). Let 𝑆 be a complex submanifold of a neighborhood of a closed cuboid 𝐸 ⋐ C𝑛 . Then we have: (i) ℐ⟨𝑆⟩ has a finite generator system over 𝐸.

2.5 Oka’s Joku-Iko Principle

67

 (ii) If 𝜎 𝑗 1≤ 𝑗 ≤ 𝑁 (𝜎 𝑗 ∈ 𝒪(𝐸), 𝜎 𝑗 | 𝑆 = 0) is a finite generator system of ℐ⟨𝑆⟩ over 𝐸, then every section 𝜎 ∈ 𝛤 (𝐸, ℐ⟨𝑆⟩) (𝜎 ∈ 𝒪(𝐸), 𝜎| 𝑆 = 0) is written as 𝑁 Õ (2.5.12) 𝜎= 𝑎 𝑗 · 𝜎 𝑗 (on 𝐸), 𝑗=1

where 𝑎 𝑗 ∈ 𝒪(𝐸), 1 ≤ 𝑗 ≤ 𝑁. Remark 2.5.13. (i) In the proof of the Oka Syzygy Lemma 2.5.1 the induction on the cuboid dimension combined with the Cousin integral worked very e ciently. Since the method will be used repeatedly henceforth, we call it the induction on cuboid  𝑙 dimension for convenience. (ii) (Syzygy) Let 𝜎 𝑗 𝑗=1 be a nite generator system obtained in the Oka Syzygy Lemma 2.5.1 (i). Then the map 𝜙 : (𝑎 𝑗 ) ∈ 𝒪𝑧𝑙 ⊂ 𝒪𝐸𝑙 −→ 𝑧

𝑙 Õ

𝑎 𝑗 · 𝜎 𝑗 ∈ ℱ𝑧 ⊂ ℱ| 𝐸 𝑧

𝑗=1

𝑧

(𝑧 ∈ 𝐸)

is surjective, and the condition of the image to be 0 is given by 𝑙 Õ

𝑎 𝑗 · 𝜎𝑗 = 0 𝑧

𝑗=1

(𝑧 ∈ 𝐸),

𝑧

which yields a relation sheaf ℛ𝑉 over a neighborhood 𝑉 of 𝐸. With the inclusion map 𝜄 : ℛ𝑉 ↩→ 𝒪𝑉𝑙 , we have 𝜄(ℛ𝑉 ) = Ker 𝜙 := 𝜙 −1 0, called the kernel of the above surjection; the sequence of maps 𝜙

𝜄

ℛ𝑉 −→ 𝒪𝑉𝑙 −→ ℱ| 𝑉 is called an exact sequence. Let ℱ| 𝑉 → 0 be the zero map. Since 𝜙 is surjective, the sequence of maps 𝜙

𝒪𝑉𝑙 −→ ℱ| 𝑉 −→ 0 is exact. Therefore we obtain the following composite of those two exact sequences: 𝜄

𝜙

ℛ𝑉 −→ 𝒪𝑉𝑙 −→ ℱ| 𝑉 −→ 0, which is also said to be exact: This exact sequence is called a syzygy of ℱ (the term “syzygy” is due to D. Hilbert).

68

2 Coherent Sheaves and Oka’s Joku-Iko Principle

2.5.2 Oka Extension of the Joku-Iko Principle The following is the key of Oka’s Joku-Iko Principle. Theorem 2.5.14 (Oka Extension). Let 𝑆 ⊂ P𝛥 be a complex submanifold. Then, for every holomorphic function 𝑓 ∈ 𝒪(𝑆) there exists a holomorphic function 𝐹 ∈ 𝒪(P𝛥) with 𝐹 | 𝑆 = 𝑓 ; i.e., the following is an exact sequence: 𝒪(P𝛥) ∋ 𝐹 ↦−→ 𝐹 | 𝑆 ∈ 𝒪(𝑆) −→ 0. Proof. By Riemann’s mapping Theorem 1.4.9 we may assume that P𝛥 is an open cuboid P ⋐ C𝑛 . (1) We rst consider the problem quasi-globally; that is, we are going to solve it in a neighborhood of an arbitrary compact subset of P. We will employ the induction on cuboid dimension. Lemma 2.5.15 (Extension). Let 𝐸 ⋐ P be an arbitrary closed cuboid. Then there is a neighborhood 𝑊 of 𝐸 such that for every 𝑔 ∈ 𝒪(𝑊 ∩ 𝑆) there exists 𝐺 ∈ 𝒪(𝐸) with the restriction to 𝐸 ∩ 𝑆 (⋐ 𝑆) satisfying 𝐺 | 𝐸∩𝑆 = 𝑔| 𝐸∩𝑆 , where the equality means the validity of the equation in a neighborhood of 𝐸 ∩ 𝑆 in 𝑆. We call 𝐺 a solution on 𝐸. ∵ ) (a) When dim 𝐸 = 0, 𝐸 consists of one point 𝑎 ∈ P, and so we choose a neighborhood 𝑊 ∋ 𝑎 and a local coordinate system such as (2.3.41). For a holomorphic function 𝑔(𝑤 𝑘+1 , . . . , 𝑤𝑛 ) ∈ 𝒪(𝑊 ∩ 𝑆) we set 𝐺 (𝑤1 , . . . , 𝑤 𝑘+1 , . . . , 𝑤𝑛 ) = 𝑔(𝑤 𝑘+1 , . . . , 𝑤𝑛 ) ∈ 𝒪(𝑊), which satis es the required property. (b) Suppose that the claimed statement holds in the case of dim 𝐸 = 𝜈 − 1 with 𝜈 ≥ 1. Let dim 𝐸 = 𝜈. By the Geometric Syzygy Lemma 2.5.11 (i) there is a nite  𝑁 generator system 𝜎 𝑗 𝑗=1 of ℐ⟨𝑆⟩ over a neighborhood 𝑊 of 𝐸. Without loss of generality we may assume that 𝐸 is of the form (2.5.3). In the sequel, the symbols such as 𝐸 𝑡 , 𝐸 𝛼 stand for the same as those used after (2.5.3). Because of dim 𝐸 𝑡 = 𝜈 − 1 the induction hypothesis implies the existence of a solution 𝐺 𝑡 on 𝐸 𝑡 , so that 𝐺 𝑡 | 𝑆∩𝐸𝑡 = 𝑔| 𝑆∩𝐸𝑡 . By the Heine Borel Theorem there are a partition (2.5.4) and holomorphic functions 𝐺 𝛼 ∈ 𝒪(𝐸 𝛼 ) on each 𝐸 𝛼 such that 𝐺 𝛼 | 𝑆∩𝐸 𝛼 = 𝑔| 𝑆∩𝐸 𝛼 . Therefore we deduce that 𝐺 𝛼+1 − 𝐺 𝛼 ∈ 𝛤 (𝐸 𝛼 ∩ 𝐸 𝛼+1 , ℐ⟨𝑆⟩). By the Geometric Syzygy Lemma 2.5.11 (ii) there are 𝑎 𝛼 𝑗 ∈ 𝒪(𝐸 𝛼 ∩ 𝐸 𝛼+1 ) (1 ≤ 𝑗 ≤ 𝑁) such that

2.5 Oka’s Joku-Iko Principle

69

𝐺 𝛼+1 − 𝐺 𝛼 =

(2.5.16)

𝑁 Õ

𝑎 𝛼 𝑗 𝜎𝑗

(on 𝐸 𝛼 ∩ 𝐸 𝛼+1 ).

𝑗=1

Taking the Cousin decomposition (1.3.11) of 𝑎 𝛼 𝑗 , we write (2.5.17)

𝑎 𝛼 𝑗 = 𝑏 𝛼 𝑗 − 𝑏 𝛼+1 𝑗 ,

𝑏 𝛼 𝑗 ∈ 𝒪(𝐸 𝛼 ), 𝑏 𝛼+1 𝑗 ∈ 𝒪(𝐸 𝛼+1 ).

Then it follows that (2.5.18)

𝐺𝛼 +

𝑁 Õ

𝑏 𝛼 𝑗 𝜎 𝑗 = 𝐺 𝛼+1 +

𝑗=1

𝑁 Õ

𝑏 𝛼+1 𝑗 𝜎 𝑗

(on 𝐸 𝛼 ∩ 𝐸 𝛼+1 ).

𝑗=1

Since the left-hand side above is a solution on 𝐸 𝛼 and the right-hand side is a solution on 𝐸 𝛼+1 , we obtain a solution on 𝐸 𝛼 ∪ 𝐸 𝛼+1 , which we call a solution patched together with the solutions 𝐺 𝛼 on 𝐸 𝛼 and 𝐺 𝛼+1 on 𝐸 𝛼+1 . Beginning with 𝛼 = 1, we patch together 𝐺 1 and 𝐺 2 to obtain a solution 𝐻2 on 𝐸 1 ∪ 𝐸 2 , and then we patch together 𝐻2 and 𝐺 3 to obtain a solution Ð 𝐻3 on 𝐸 1 ∪ 𝐸 2 ∪ 𝐸 3 . Repeating this up to 𝛼 = 𝐿 − 1, We have a solution 𝐻 𝐿 on 𝐸 = 𝐿𝛼=1 𝐸 𝛼 . It su ces to set 𝐺 = 𝐻 𝐿 . △ (2) We take a covering of P by increasing open cuboids P1 ⋐ P2 ⋐ · · · ⋐ P 𝜇 ⋐ · · · ,

∞ Ø

P 𝜇 = P.

𝜇=1

By the Geometric Syzygy Lemma 2.5.11 (i) there is a  𝑁𝜇 𝜎𝜇 𝑗 𝑗=1 of ℐ⟨𝑆⟩ on each closure P 𝜇 .

nite generator system

For a given 𝑓 ∈ 𝒪(𝑆) there is a solution 𝐺 𝜇 on each P 𝜇 by Extension Lemma 2.5.15.

Set 𝐹1 = 𝐺 1 (∈ 𝒪 P1 ). Assume that solutions 𝐹 𝑗 ∈ 𝒪 P 𝑗 , 1 ≤ 𝑗 ≤ 𝜇, are determined so that (2.5.19)

∥𝐹 𝑗 −1 − 𝐹 𝑗 ∥ P 𝑗 −1
∥ 𝑓 ∥ 𝐴. Let 𝑧 𝑗 = 𝑥 𝑗 + 𝑖𝑦 𝑗 (1 ≤ 𝑗 ≤ 𝑛) be the natural coordinates. By the convexity assumption there is a real linear functional 𝐿(𝑧) such that 𝐿(𝑧) = 𝐿 (𝑥1 , 𝑦 1 , . . . , 𝑥 𝑛 , 𝑦 𝑛 ) =

𝑛 Õ

(𝑎 𝑗 𝑥 𝑗 + 𝑏 𝑗 𝑦 𝑗 ),

𝑗=1

(3.1.15) Then 𝐿 0 (𝑧) :=

𝐿 (𝑏) > max{𝐿 (𝑧) : 𝑧 ∈ 𝐴}. Í𝑛

𝑗=1 (𝑎 𝑗

− 𝑖𝑏 𝑗 )𝑧 𝑗 is holomorphic and satis es 𝐿 (𝑧) = ℜ 𝐿 0 (𝑧),

|𝑒 𝐿0 (𝑧) | = 𝑒 𝐿 (𝑧) .

𝑎 𝑗 , 𝑏 𝑗 ∈ R,

3.1 De nitions and Elementary Properties

77

With 𝑓 (𝑧) = 𝑒 𝐿0 (𝑧) ∈ 𝒪(C𝑛 ) it follows from (3.1.15) that | 𝑓 (𝑏)| > ∥ 𝑓 ∥ 𝐴. (iv) This is immediate from (ii). (v) This is clear, too. (vi) Let 𝑈 ⊂ 𝛺 \ 𝐾 be a connected component. Suppose that 𝑈 ⋐ 𝛺. Then, 𝜕𝑈 ⊂ 𝐾. It follows from the Maximum Principle (Theorem 1.1.48) that for any point 𝑎 ∈ 𝑈 and any function 𝑓 ∈ 𝒪(𝛺) | 𝑓 (𝑎)| ≤ ∥ 𝑓 ∥ 𝜕𝑈 ≤ ∥ 𝑓 ∥ 𝐾 . b𝛺 , and so 𝑈 ⊂ 𝐾 b𝛺 ; consequently, 𝐾 ★ ⊂ 𝐾 b𝛺 . Thus 𝑎 ∈ 𝐾 𝛺 The second inclusion relation is clear. (vii) Let 𝛺 ⊂ C be a domain. If 𝛺 = C, 𝜕𝛺 = ∅ and 𝛺 is holomorphically convex (see Example 3.1.12). So, we assume 𝛺 ≠ C. Let 𝐾 ⋐ 𝛺 be a compact subset. With 𝑏 ∈ 𝜕𝛺 we consider 𝑓 (𝑧) = 1/(𝑧 − 𝑏) ∈ 𝒪(𝛺), and then deduce that | 𝑓 (𝑧)| ≤ ∥ 𝑓 ∥ 𝐾 ,

b𝛺 , ∀𝑧 ∈ 𝐾

b𝛺 . |𝑧 − 𝑏| ≥ 𝑑 (𝑏, 𝐾), ∀ 𝑧 ∈ 𝐾 b𝛺 , 𝜕𝛺} ≥ 𝑑 (𝐾, 𝛺). Because 𝐾 ⊂ 𝐾 ★ ⊂ 𝐾 b𝛺 by (3.1.14), Therefore, 𝑑 ( 𝐾 𝛺 b𝛺 , 𝛺) = 𝑑 (𝐾 ★ , 𝜕𝛺) = 𝑑 (𝐾, 𝜕𝛺) > 0; 𝑑 (𝐾 𝛺 b𝛺 ⋐ 𝛺. therefore, 𝐾 (viii) This follows from (vii) and (v).

⊔ ⊓

Corollary 3.1.16. Let 𝐾 be a compact subset of a domain 𝛺 ⊂ C𝑛 . Then there is no b𝛺 which is relatively compact in 𝛺. connected component of 𝛺 \ 𝐾 ★ =𝐾 b𝛺 holds in Proposition 3.1.13 (vi) (see Remark 3.1.17. If 𝑛 = 1, the equality 𝐾𝛺 Lemma 3.3.20).

Example 3.1.18. (i) An open ball B(𝑎;𝑟) is polynomially convex (Proposition 3.1.13 (iii)). (ii) Let 𝛺 = 𝛥(1) be the unit disk, and let 𝐾 = {𝑧 ∈ 𝛥(1) : |𝑧| = 𝑟 } with 0 < 𝑟 < 1. Then by the Maximum Principle, it is easy to have ¯ ˆ b 𝐾★ 𝛥(1) = 𝛥(𝑟) = 𝐾 𝛥 = 𝛥poly . (iii) Let 𝛺 = P𝛥 = 𝛥(1) × 𝛥(1) ⊂ C2 be the 2-dimensional unit polydisk, and let 𝐾 = {(𝑧, 𝑤) ∈ P𝛥 : |𝑧| = |𝑤| = 𝑟} with 0 < 𝑟 < 1. Then P𝛥 \ 𝐾 is connected and ¯ × 𝛥(𝑟) ¯ by the Maximum Principle; hence, bP𝛥 = 𝐾 bpoly = 𝛥(𝑟) so 𝐾 ★ = 𝐾, but 𝐾 P𝛥

★ bP𝛥 = 𝐾 bpoly , 𝐾 = 𝐾P𝛥 $𝐾 ★ bP𝛥 , 𝜕P𝛥) = 𝑑 ( 𝐾 bpoly , 𝜕P𝛥). 𝑑 (𝐾, 𝜕P𝛥) = 𝑑 (𝐾P𝛥 , 𝜕P𝛥) = 𝑑 ( 𝐾

78

3 Domains of Holomorphy

3.2 Cartan–Thullen Theorem We prove that a domain of holomorphy and a holomorphically convex domain are the same. This result was proved by a paper of Cartan and Thullen in 1932 before K. Oka started to work on the Three Big Problems.2 The rst two problems of approximation (development of functions) and Cousin had been posed for domains of holomorphy. Oka, in fact, proved them for holomorphically convex domains. Therefore the contents of this section are not necessary, if Oka’s solutions of the rst two problems are stated for holomorphically convex domains. It is the third Pseudoconvexity Problem where the results of the present section are used in full (cf. Chaps. 4 and 5). Let 𝛺 ⊂ C𝑛 be a domain, and let B(1) be the unit ball of C𝑛 . Then the boundary distance function 𝑑 (𝑧, 𝜕𝛺) de ned by (3.1.3) is written as 𝑑 (𝑧, 𝜕𝛺) = 𝛿B(1) (𝑧, 𝜕𝛺) := sup{𝑠 > 0 : 𝑧 + 𝑠B(1) ⊂ 𝛺}, where 𝑠B(1) = {𝑠𝑧 : 𝑧 ∈ B(1)}. With B(𝑅) (𝑅 > 0) we de ne 𝛿B(𝑅) (𝑧, 𝜕𝛺) similarly to 𝛿B(1) (𝑧, 𝜕𝛺), and then have 𝛿B(𝑅) (𝑧, 𝜕𝛺) =

1 𝛿B(1) (𝑧, 𝜕𝛺). 𝑅

Let 𝑟 = (𝑟 𝑗 ) = (𝑟 1 , . . . , 𝑟 𝑛 ) with 𝑟 𝑗 > 0 be a polyradius, and x a polydisk P𝛥 = P𝛥((𝑟 𝑗 )) of polyradius 𝑟 with center at the origin. For 𝑠 > 0 we put 𝑠P𝛥 = 𝑠 · P𝛥 = P𝛥((𝑠𝑟 𝑗 )). Similarly to 𝛿B(𝑅) (𝑧, 𝜕𝛺), we de ne: (3.2.1)

𝛿P𝛥 (𝑧, 𝜕𝛺) = sup{𝑠 > 0 : 𝑧 + 𝑠P𝛥 ⊂ 𝛺}(> 0), ∥𝑧∥ P𝛥 = inf{𝑠 > 0 : 𝑧 ∈ 𝑠P𝛥} ≥ 0,

𝑧 ∈ 𝛺, 𝑛

𝑧∈C .

An easy computation yields that ∥𝑧∥ P𝛥 satis es the axioms of a norm (cf. Exercise 2 at the end of this chapter) and (3.2.2)

|𝛿P𝛥 (𝑧, 𝜕𝛺) − 𝛿P𝛥 (𝑧 ′ , 𝜕𝛺)| ≤ ∥𝑧 − 𝑧 ′ ∥ P𝛥 ,

𝑧, 𝑧 ′ ∈ 𝛺.

We call 𝛿P𝛥 (𝑧, 𝜕𝛺) the boundary distance function of 𝛺 with respect to P𝛥. Let ∥𝑧∥ be the euclidean norm de ned by (1.1.1). Then there is a constant 𝐶 > 0 such that 𝐶 −1 ∥𝑧∥ ≤ ∥𝑧∥ P𝛥 ≤ 𝐶 ∥𝑧∥. Thus 𝛿P𝛥 (𝑧, 𝜕𝛺) is a Lipschitz continuous function by (3.2.2). 2 Cf. the beginning of Preface.

3.2 Cartan Thullen Theorem

79

For a subset 𝐴 ⊂ 𝛺 we put 𝛿P𝛥 ( 𝐴, 𝜕𝛺) = inf{𝛿P𝛥 (𝑧, 𝜕𝛺) : 𝑧 ∈ 𝐴}.

(3.2.3)

If 𝐴 ⋐ 𝛺, then 𝛿P𝛥 ( 𝐴, 𝜕𝛺) > 0. Lemma 3.2.4. Let 𝑓 ∈ 𝒪(𝛺) and let 𝐾 ⋐ 𝛺 be compact. Assume that | 𝑓 (𝑧)| ≤ 𝛿P𝛥 (𝑧, 𝜕𝛺),

(3.2.5)

𝑧 ∈ 𝐾.

b𝛺 : Expand an arbitrary 𝑢 ∈ 𝒪(𝛺) to a power series about a point 𝜉 ∈ 𝐾 𝑢(𝑧) =

(3.2.6)

Õ 𝜕 𝛼 𝑢(𝜉) 𝛼

𝛼!

(𝑧 − 𝜉) 𝛼 .

Then this power series converges at 𝑧 ∈ 𝜉 + | 𝑓 (𝜉)| · P𝛥. Proof. Fix 0 < 𝑡 < 1 and set 𝛺 𝑡 = {(𝑧 𝑗 ) : ∃ 𝑤 ∈ 𝐾, |𝑧 𝑗 − 𝑤 𝑗 | ≤ 𝑡𝑟 𝑗 | 𝑓 (𝑤)|, 1 ≤ 𝑗 ≤ 𝑛} o Øn ⊂ (𝑧 𝑗 ) : (𝑧 𝑗 ) ∈ (𝑤 𝑗 ) + 𝑡𝛿P𝛥 (𝑤, 𝜕𝛺) · P𝛥 . 𝑤∈𝐾

Then 𝛺 𝑡 is compact in 𝛺. There is a constant 𝑀 > 0 with |𝑢(𝑧)| ≤ 𝑀, 𝑧 ∈ 𝛺 𝑡 . From this we obtain the estimates of partial derivatives: Let 𝛼 = (𝛼 𝑗 ) ∈ Z+𝑛 and 𝑤 ∈ 𝐾. Keeping 𝜌 𝑗 > 0 so that the appearing variables stay in 𝛺, we have  𝑛 ∫ ∫ 1 𝑢(𝜉) Î 𝑢(𝑧) = ··· 𝑑𝜉1 · · · 𝑑𝜉 𝑛 , 2𝜋𝑖 (𝜉 𝑗 − 𝑧 𝑗) 𝑗 𝜕 𝛼 𝑢(𝑧) =



1 2𝜋𝑖

𝑛

| 𝜉 𝑗 −𝑤 𝑗 |=𝜌 𝑗

∫

𝛼!

∫

··· | 𝜉 𝑗 −𝑤 𝑗 |=𝜌 𝑗

𝑢(𝜉) (𝜉1 − 𝑧1

) 𝛼1 +1 · · · (𝜉

𝑛 − 𝑧𝑛)

𝛼𝑛 +1

𝑑𝜉1 · · · 𝑑𝜉 𝑛 .

Suppose that 𝑓 (𝑤) ≠ 0. Set 𝑧 = 𝑤, 𝜌 𝑗 = 𝑡𝑟 𝑗 | 𝑓 (𝑤)| and 𝜌 = (𝜌 𝑗 ). Then 𝛼!𝑀 𝛼!𝑀 = | 𝛼| , 𝛼 𝜌 𝑡 | 𝑓 (𝑤)| | 𝛼| 𝑟 𝛼 |𝜕 𝛼 𝑢(𝑤)|𝑡 | 𝛼| | 𝑓 (𝑤)| | 𝛼| 𝑟 𝛼 ≤ 𝑀, 𝑤 ∈ 𝐾. 𝛼!

|𝜕 𝛼 𝑢(𝑤)| ≤

The last inequality holds trivially in the case 𝑓 (𝑤) = 0. Therefore we get 𝛼! · 𝑀 𝑓 (𝑤) | 𝛼| 𝜕 𝛼 𝑢(𝑤) ≤ | 𝛼| 𝛼 , 𝑡 𝑟

𝑤 ∈ 𝐾.

b𝛺 that Since 𝑓 (𝑤) | 𝛼| 𝜕 𝛼 𝑢(𝑤) ∈ 𝒪(𝛺), we deduce from the de nition of 𝐾

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3 Domains of Holomorphy

𝛼!𝑀 𝑓 (𝑤) | 𝛼| 𝜕 𝛼 𝑢(𝑤) ≤ | 𝛼| 𝛼 , 𝑡 𝑟

b𝛺 . 𝑤∈𝐾

b𝛺 , (3.2.6) converges at 𝑧 ∈ 𝜉 + | 𝑓 (𝜉)|𝑡 · P𝛥. As 𝑡 ↗ 1, we see that With 𝑤 = 𝜉 ∈ 𝐾 (3.2.6) converges at 𝑧 ∈ 𝜉 + | 𝑓 (𝜉)| · P𝛥. ⊓ ⊔ Lemma 3.2.7 (Cartan Thullen). Let 𝛺 ⊂ C𝑛 be a domain of holomorphy, let 𝑓 ∈ 𝒪(𝛺), and let 𝐾 ⋐ 𝛺 be compact. By 𝛿∗ (𝑧, 𝜕𝛺) we denote either 𝛿P𝛥 (𝑧, 𝜕𝛺) or 𝛿B(𝑅) (𝑧, 𝜕𝛺). If (3.2.8)

| 𝑓 (𝑧)| ≤ 𝛿∗ (𝑧, 𝜕𝛺),

𝑧 ∈ 𝐾,

then (3.2.9)

| 𝑓 (𝑧)| ≤ 𝛿∗ (𝑧, 𝜕𝛺),

b𝛺 . 𝑧∈𝐾

In particular, with 𝑓 ≡ 𝛿∗ (𝐾, 𝜕𝛺) (constant) we get   b𝛺 , 𝜕𝛺 . (3.2.10) 𝛿∗ (𝐾, 𝜕𝛺) = 𝛿∗ 𝐾 Proof. (a) Case of 𝛿P𝛥 (𝑧, 𝜕𝛺): It follows from Lemma 3.2.4 that for arbitrary b𝛺 , 𝑢 is holomorphic in 𝑧 + | 𝑓 (𝑧)| · P𝛥. The assumption of 𝛺 𝑢 ∈ 𝒪(𝛺) and 𝑧 ∈ 𝐾 being a domain of holomorphy implies that 𝑧 + | 𝑓 (𝑧)| · P𝛥 ⊂ 𝛺. Therefore | 𝑓 (𝑧)| ≤ 𝛿P𝛥 (𝑧, 𝜕𝛺),

b𝛺 . 𝑧∈𝐾

In particular, we take 𝑓 ≡ 𝛿P𝛥 (𝐾, 𝜕𝛺). Then   b𝛺 , 𝜕𝛺 . 𝛿P𝛥 (𝐾, 𝜕𝛺) ≤ 𝛿P𝛥 𝐾 b𝛺 , and so the equality holds. The opposite inequality follows from 𝐾 ⊂ 𝐾 (b) Case of 𝛿B(𝑅) (𝑧, 𝜕𝛺): By the multiplication of a positive constant to coordinates we may assume 𝑅√= 1, and write B(1) = B. We take a special polydisk P𝛥 = {(𝑧 𝑗 ) ∈ C𝑛 : |𝑧 𝑗 | ≤ 1/ 𝑛}. Then, P𝛥 ⊂ B. We denote by 𝑈 (𝑛) all unitary transformations of the coordinates. Then Ø B= 𝐴P𝛥. 𝐴∈𝑈 (𝑛)

Note that what was proved in (a) above remains valid for the changes of coordinates, in particular after replacing P𝛥 with 𝐴P𝛥 ( 𝐴 ∈ 𝑈 (𝑛)). Suppose that | 𝑓 (𝑧)| ≤ 𝛿B (𝑧, 𝜕𝛺),

𝑧 ∈ 𝐾.

Since 𝛿B (𝑧, 𝜕𝛺) ≤ 𝛿 𝐴P𝛥 (𝑧, 𝜕𝛺), | 𝑓 (𝑧)| ≤ 𝛿 𝐴P𝛥 (𝑧, 𝜕𝛺),

b𝛺 , ∀ 𝐴 ∈ 𝑈 (𝑛). 𝑧∈𝐾

3.2 Cartan Thullen Theorem

81

Therefore | 𝑓 (𝑧)| ≤ 𝛿B (𝑧, 𝜕𝛺),

b𝛺 . 𝑧∈𝐾

⊓ ⊔

Theorem 3.2.11 (Cartan–Thullen). The following three conditions are equivalent for a domain 𝛺 ⊂ C𝑛 : (i) 𝛺 is a domain of holomorphy. (ii) There is a holomorphic function 𝑓 ∈ 𝒪(𝛺) whose domain of existence is 𝛺. (iii) 𝛺 is a holomorphically convex domain. b𝛺 Proof. (i)⇒(iii): We take an arbitrary compact subset 𝐾 ⋐ 𝛺. By the de nition, 𝐾 is bounded and closed in 𝛺. Since 𝛺 is a domain of holomorphy, (3.2.10) implies   b𝛺 , 𝜕𝛺 = 𝛿P𝛥 (𝐾, 𝜕𝛺) > 0. 𝛿P𝛥 𝐾 b𝛺 ⋐ 𝛺. Therefore, 𝐾 (iii)⇒(ii): We take a discrete sequence {𝑎 𝑗 }∞ 𝑗=1 of points of 𝛺 such that it has no accumulation point in 𝛺 and all points of 𝜕𝛺 are the accumulation points of it. Set 𝐷 𝑗 = 𝑎 𝑗 + 𝛿P𝛥 (𝑎 𝑗 , 𝜕𝛺) · P𝛥 ⊂ 𝛺. Let 𝐾 𝑗 , 𝑗 = 1, 2, . . ., be an increasing sequence of compact subsets of 𝛺 such that ∞ Ø

𝐾 𝑗 ⋐ 𝐾 ◦𝑗+1 ,

𝑗=1

𝐾 ◦𝑗 = 𝛺,

c𝑗 ) ≠ ∅ for all 𝑗 ≥ 1. where 𝐾 ◦𝑗 denotes the interior of 𝐾 𝑗 . By the choice, 𝐷 𝑗 ∩ (𝛺 \ 𝐾 𝛺 c𝑗 there is an 𝑓 𝑗 ∈ 𝒪(𝛺) satisfying It follows that for a point 𝑧 𝑗 ∈ 𝐷 𝑗 \ 𝐾 𝛺

max | 𝑓 𝑗 | < | 𝑓 𝑗 (𝑧 𝑗 )|. 𝐾𝑗

Dividing 𝑓 𝑗 by 𝑓 𝑗 (𝑧 𝑗 ), we may assume 𝑓 𝑗 (𝑧 𝑗 ) = 1 and that max | 𝑓 𝑗 | < | 𝑓 𝑗 (𝑧 𝑗 )| = 1. 𝐾𝑗

Taking a power 𝑓 𝑗𝜈 with su ciently large 𝜈 and denoting it again by 𝑓 𝑗 , we have max | 𝑓 𝑗 | < 𝐾𝑗

Since

Í

𝑗 𝑗 2𝑗

1 , 2𝑗

𝑓 𝑗 (𝑧 𝑗 ) = 1.

< ∞, the in nite product 𝑓 (𝑧) =

∞ Ö 𝑗=1

(1 − 𝑓 𝑗 (𝑧)) 𝑗

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3 Domains of Holomorphy

converges uniformly on compact subsets of 𝛺 (cf. [38] Chap. 2 6). Necessarily, 𝑓 . 0. We show that 𝛺 is a domain of existence of 𝑓 (𝑧). If it were not the case, there should exist a domain 𝑉 ⊄ 𝛺 with 𝑉 ∩ 𝛺 ≠ ∅, an element 𝑔 ∈ 𝒪(𝑉) and a connected component 𝑊 ⊂ 𝑉 ∩ 𝛺 such that 𝑓 | 𝑊 = 𝑔| 𝑊 . For a boundary point 𝑏 ∈ 𝜕𝛺 ∩ 𝜕𝑊 ∩𝑉 there is a subsequence {𝑎 𝑗𝜈 } converging to 𝑏. Since 𝛿P𝛥 (𝑎 𝑗𝜈 , 𝜕𝛺) → 0 (𝜈 → ∞), {𝑧 𝑗𝜈 } also converges to 𝑏. Note that 𝑓 (𝑧) has a zero of order 𝑗 𝜈 at 𝑧 = 𝑧 𝑗𝜈 . Thus for an arbitrary partial derivative 𝜕 𝛼 with |𝛼| ≤ 𝑗 𝜈 𝜕 𝛼 𝑓 (𝑧 𝑗𝜈 ) = 0. With an arbitrary xed 𝜕 𝛼 (|𝛼| > 0) 𝜕 𝛼 𝑓 (𝑧 𝑗𝜈 ) = 0 for 𝜈 ≫ 1, and then 𝜕 𝛼 𝑓 (𝑧 𝑗𝜈 ) → 𝜕 𝛼 𝑓 (𝑏), It thus follows that

𝜕 𝛼 𝑓 (𝑏) = 0,

𝜈 → ∞.

∀𝛼.

The Identity Theorem 1.1.46 implies 𝑓 ≡ 0, which is absurd. (ii)⇒(i): Since 𝑓 itself cannot be analytically continued over a properly larger domain than 𝛺, 𝛺 is a domain of holomorphy. Thus the proof is completed. ⊓ ⊔ Corollary 3.2.12. Let {𝛺 𝛾 } 𝛾 ∈𝛤 be a family ofÑdomains of holomorphy. Then every connected component of the interior points of 𝛾 ∈𝛤 𝛺 𝛾 is a domain of holomorphy. Proof. Let 𝛺 be such a component. Let 𝐾 ⋐ 𝛺 be a compact subset. Then, 𝐾 ⊂ b𝛺 ⊂ 𝐾 b𝛺𝛾 . Because of 𝛺 𝛾 being a domain of holomorphy, (3.2.10) implies 𝐾 b𝛺𝛾 , 𝜕𝛺 𝛾 ). 𝛿0 := 𝛿P𝛥 (𝐾, 𝜕𝛺) ≤ 𝛿P𝛥 (𝐾, 𝜕𝛺 𝛾 ) = 𝛿P𝛥 ( 𝐾 The inclusion relations lead to b𝛺 , 𝜕𝛺 𝛾 ) ≥ 𝛿P𝛥 ( 𝐾 b𝛺𝛾 , 𝜕𝛺 𝛾 ). 𝛿P𝛥 (𝐾, 𝜕𝛺 𝛾 ) ≥ 𝛿P𝛥 ( 𝐾 b𝛺 , 𝜕𝛺 𝛾 ) = 𝛿P𝛥 (𝐾, 𝜕𝛺 𝛾 ) ≥ 𝛿0 > 0. Thus, for every 𝑎 ∈ 𝐾 b𝛺 we It follows that 𝛿P𝛥 ( 𝐾 see 𝑎 + 𝛿0 P𝛥 ⊂ 𝛺 𝛾 ,

∀ 𝛾 ∈ 𝛤.

b𝛺 ⋐ 𝛺; hence, 𝛺 is Since 𝑎 + 𝛿0 P𝛥 is connected, 𝑎 + 𝛿0 P𝛥 ⊂ 𝛺. It follows that 𝐾 holomorphically convex. By Theorem 3.2.11 𝛺 is a domain of holomorphy. ⊓ ⊔ Remark 3.2.13. In the proof of Theorem 3.2.11, Lemmata 3.2.4 and 3.2.7 were used only with conditions (3.2.5) and (3.2.8) restricted to 𝑓 = constant. There may be some readers who wonder why general 𝑓 is considered. In fact, the matter remains the same until the solution of the Cousin Problem. It will make sense to deal with a general holomorphic function 𝑓 in those conditions after going into the Pseudoconvexity

3.3 Analytic Polyhedron and Oka Weil Approximation

83

Problem; it plays an essential role in the proof of Oka’s Theorem 4.3.1 of Boundary Distance Functions. It leads to the solution of the Pseudoconvexity Problem (see Chap. 5).

3.3 Analytic Polyhedron and Oka–Weil Approximation 3.3.1 Analytic Polyhedron Let 𝛺 ⊂ C𝑛 be a domain. Definition 3.3.1 (Analytic polyhedron). With holomorphic functions 𝜑 𝑗 ∈ 𝒪(𝛺) and positive numbers 𝜌 𝑗 > 0 (1 ≤ 𝑗 ≤ 𝑙) given, a union P (⋐ 𝛺) of a nite number of connected components of the open set (3.3.2)

{𝑧 ∈ 𝛺 : |𝜑 𝑗 (𝑧)| < 𝜌 𝑗 , 1 ≤ 𝑗 ≤ 𝑙},

which are relatively compact in 𝛺, is called an 𝒪(𝛺)-analytic polyhedron or simply analytic polyhedron of 𝛺. We say that P is de ned by 𝜑 𝑗 (1 ≤ 𝑗 ≤ 𝑁) of (3.3.2). In particular, if 𝜑 𝑗 ∈ 𝒪(𝛺) (1 ≤ 𝑗 ≤ 𝑙) are polynomials, P is called a polynomial polyhedron. A connected analytic (resp. polynomial) polyhedron is called an analytic (resp. polynomial) polyhedral domain. Let P ⋐ 𝛺 be an analytic polyhedron de ned by (3.3.2). Since P is bounded, there is a polydisk P𝛥((𝑟 𝑗 )) ⊃ P. We consider a holomorphic map (3.3.3)

𝜑 : 𝑧 ∈ P → (𝑧, 𝜑1 (𝑧), . . . , 𝜑𝑙 (𝑧)) ∈ P𝛥, P𝛥 := P𝛥((𝑟 𝑗 )) × P𝛥((𝜌 𝑗 )) (⊂ C𝑛 × C𝑙 ).

We call 𝜑 the Oka map of P. We denote by C[𝑧, 𝜑] (⊂ 𝒪(𝛺)) the set of all polynomials in 𝑧1 , . . . , 𝑧 𝑛 , 𝜑1 , . . . , 𝜑𝑙 with complex coe cients. Lemma 3.3.4. The Oka map 𝜑 : P → P𝛥 is proper, and the image 𝜑(P) is a complex submanifold of P𝛥. Proof. As 𝑧 (∈ P) → 𝜕P (𝑧 approaches 𝜕P), 𝜑(𝑧) → 𝜕P𝛥; therefore, 𝜑 is proper, and the image 𝜑(P) is a closed subset of P𝛥. The rest is clear. ⊓ ⊔ Lemma 3.3.5. Let P ⋐ 𝛺 be an analytic polyhedron defined by 𝜑 𝑗 ∈ 𝒪(𝛺) (1 ≤ 𝑗 ≤ 𝑙). Let 𝐾 ⋐ P be compact and let 𝑔 ∈ 𝒪(P). Then 𝑔 is approximated uniformly on 𝐾 by elements of C[𝑧, 𝜑]. Proof. Let 𝜑 : P → P𝛥 be the Oka map and set 𝑆 = 𝜑(P). We may consider 𝑔 as a holomorphic function on 𝑆 and 𝐾 ⋐ 𝑆. It follows from the Oka Extension Theorem 2.5.14 that there is an element 𝐺 ∈ 𝒪(P𝛥) with 𝐺 | 𝑆 = 𝑔. We expand 𝐺 to a power series:

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3 Domains of Holomorphy

𝐺 (𝑤) =

Õ

𝑐 𝛽 𝑤𝛽 ,

𝑤 ∈ P𝛥.

𝛽 ∈Z+𝑛+𝑙

For every 𝜀 > 0 there is a su ciently large 𝑁 ∈ N such that with 𝐺 𝑁 (𝑤) := Í 𝛽 |𝛽| ≤ 𝑁 𝑐𝛽 𝑤 , |𝐺 (𝑤) − 𝐺 𝑁 (𝑤)| < 𝜀,

(3.3.6)

𝑤 ∈ 𝐾.

Setting 𝑔 𝑁 (𝑧) := 𝐺 𝑁 (𝜑(𝑧)) ∈ C[𝑧, 𝜑] ⊂ 𝒪(𝛺), we have by (3.3.6) that ∥𝑔 − 𝑔 𝑁 ∥ 𝐾 < 𝜀.

⊓ ⊔

Remark 3.3.7. If 𝜑 𝑗 (𝑧) (1 ≤ 𝑗 ≤ 𝑙) in Lemma 3.3.5 are bounded, then elements of C[𝑧, 𝜑] are also bounded on P. Proposition 3.3.8. Let 𝛺 ⊂ C𝑛 be a holomorphically convex domain, and let P ⋐ 𝛺 be an analytic polyhedron. Then the closure P is holomorphically convex. Proof. Assume that b P𝛺 % P. By the assumption, b P𝛺 ⋐ 𝛺.

(3.3.9)

We take a point 𝑏 ∈ b P𝛺 \ P. Let P be de ned by (3.3.2). With su ciently small 𝜀 > 0 we denote by P 𝜀 the nite union of connected components of (3.3.10)

{𝑧 ∈ 𝛺 : |𝜑 𝑗 (𝑧)| < 𝜌 𝑗 + 𝜀, 1 ≤ 𝑗 ≤ 𝑙},

which are relatively compact in 𝛺 and contain points of P; P 𝜀 is an analytic polyhedron and satis es 𝑏 ∉ P 𝜀 . With more 𝜑 𝑗 if necessary, it follows from (3.3.9) that the connected component 𝑄 of (3.3.10) containing 𝑏 is relatively compact in 𝛺. We consider the analytic polyhedron e P 𝜀 = P 𝜀 ∪ 𝑄 ⋑ P ∪ {𝑏}, We de ne 𝑔 ∈ 𝒪(e P 𝜀 ) by

( 𝑔(𝑧) =

0, 1,

P 𝜀 ∩ 𝑄 = ∅.

𝑧 ∈ P𝜀, 𝑧 ∈ 𝑄.

By Lemma 3.3.5 𝑔(𝑧) is uniformly approximated on P ∪ {𝑏} by elements of 𝒪(𝛺). Therefore there is an element 𝑓 ∈ 𝒪(𝛺) such that ∥ 𝑓 ∥P < However, this contradicts 𝑏 ∈ b P𝛺 .

1 < | 𝑓 (𝑏)|. 2 ⊓ ⊔

3.3 Analytic Polyhedron and Oka Weil Approximation

85

Lemma 3.3.11. A holomorphically convex compact subset 𝐾 ⋐ 𝛺 has a fundamental neighborhood system of analytic polyhedra; that is, for an arbitrary neighborhood 𝑈 ⊃ 𝐾 there is an 𝒪(𝛺)-analytic polyhedron P such that 𝐾 ⋐ P ⋐ 𝑈. b𝛺 , for each 𝑏 ∈ 𝜕𝑈 there is a function Proof. We may assume 𝑈 ⋐ 𝛺. Since 𝐾 = 𝐾 𝑓 ∈ 𝒪(𝛺) such that ∥ 𝑓 ∥ 𝐾 < 1 < | 𝑓 (𝑏)|. Then 𝑉 𝑓 (𝑏) = {𝑧 ∈ 𝛺 : | 𝑓 (𝑧)| > 1} is a neighborhood of 𝑏. Since 𝜕𝑈 is compact, we can take nitely many such 𝑉 𝑓 𝑗 (𝑏 𝑗 ), 1 ≤ 𝑗 ≤ 𝑙, so that 𝜕𝑈 ⊂

𝑙 Ø

𝑉 𝑓 𝑗 (𝑏 𝑗 ).

𝑗=1

Let P be the nite union of connected components of 𝑊 = {𝑧 ∈ 𝛺 : | 𝑓 𝑗 (𝑧)| < 1, 1 ≤ 𝑗 ≤ 𝑙} which contain the points of 𝐾. Then P ⊃ 𝐾; since 𝑊 ∩ 𝜕𝑈 = ∅, P ⋐ 𝑈. ⊓ ⊔

3.3.2 Oka–Weil Approximation Theorem Now we solve the Approximation Problem (P1).3 Approximation Theorem 1.3.2 of Runge type is generalized to: Theorem 3.3.12 (Oka–Weil Approximation). Let 𝐾 ⋐ 𝛺 be a holomorphically convex set in 𝛺. Then a holomorphic function on 𝐾 is uniformly approximated on 𝐾 by elements of 𝒪(𝛺). Proof. This follows from Lemmata 3.3.5 and 3.3.11.

⊓ ⊔

bC𝑛 (= 𝐾 bpoly ), a holomorphic function on 𝐾 is Corollary 3.3.13 (Weil). If 𝐾 = 𝐾 approximated uniformly on 𝐾 by polynomials. Proof. It is immediate from Theorem 3.3.12 with 𝛺 = C𝑛 , since holomorphic functions in C𝑛 are expanded to power series. ⊓ ⊔ Corollary 3.3.14. A holomorphic function on a compact convex subset 𝐾 of C𝑛 is approximated uniformly on 𝐾 by polynomials. Proof. Use Proposition 3.1.13 (iii) and Corollary 3.3.13.

⊓ ⊔

Remark 3.3.15. It is noted that Oka’s Joku-Iko Principle enables us to get rid of the “cylinder” condition relative to the coordinate system for the domain in Corollary 3.3.14. Definition 3.3.16. We consider a pair of open sets 𝛺 1 ⊂ 𝛺 2 ⊂ C𝑛 . Assume that every connected component of 𝛺 𝑗 ( 𝑗 = 1, 2) is holomorphically convex. If every 3 Cf. Preface.

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3 Domains of Holomorphy

holomorphic function 𝑓 ∈ 𝒪(𝛺 1 ) is approximated uniformly on compact subsets of 𝛺 1 by elements of 𝒪(𝛺 2 ), (𝛺 1 , 𝛺 2 ) is called a Runge pair. Lemma 3.3.17. Let 𝛺 be a holomorphically convex domain and let P be an analytic polyhedron of 𝛺. Then (P, 𝛺) is a Runge pair. ⊓ ⊔

Proof. Use Lemma 3.3.5. (⊂ C𝑛 )

Theorem 3.3.18. Let 𝛺 1 ⊂ 𝛺 2 be a pair of holomorphically convex domains. The following four conditions are equivalent: (i) (𝛺 1 , 𝛺 2 ) is a Runge pair. b𝛺1 = 𝐾 b𝛺2 . (ii) For every compact subset 𝐾 ⋐ 𝛺 1 , 𝐾 b𝛺2 ⋐ 𝛺 1 . (iii) For every compact subset 𝐾 ⋐ 𝛺 1 , 𝐾 b𝛺2 ∩ 𝛺 1 ⋐ 𝛺 1 . (iv) For every compact subset 𝐾 ⋐ 𝛺 1 , 𝐾 b𝛺1 ⋐ 𝛺 1 . We take arbitrarily its neighborhood 𝑈 Proof. (i)⇒(ii): For 𝐾 given, 𝐾 b𝛺1 ⋐ 𝑈 ⋐ 𝛺 1 . It follows from Lemma 3.3.11 that there is an 𝒪(𝛺 1 )-analytic with 𝐾 polyhedron P1 satisfying 𝐾 ⋐ P1 ⋐ 𝑈. Let P1 be de ned by |𝜑 𝑗 | < 1, 𝜑 𝑗 ∈ 𝒪(𝛺 1 ) (1 ≤ 𝑗 ≤ 𝑙). By the assumption, 𝜑 𝑗 can be approximated uniformly on P¯ 1 by functions 𝜓 𝑗 ∈ 𝒪(𝛺 2 ), so that Q := {𝑧 ∈ 𝛺 2 : |𝜓 𝑗 (𝑧)| < 1, 1 ≤ 𝑗 ≤ 𝑙} ⊃ 𝐾,

P2 ⋐ 𝑈,

where P2 denotes the nite union of connected components of Q which contain points of 𝐾. From Proposition 3.3.8 it follows that b𝛺1 ⊂ 𝐾 b𝛺2 ⊂ P¯b2𝛺 = P¯ 2 ⊂ 𝑈. 𝐾 2 b𝛺1 is arbitrary, 𝐾 b𝛺1 = 𝐾 b𝛺2 . Since 𝑈 ⋑ 𝐾 (ii)⇒(iii)⇒(iv): These are clear. (iv)⇒(i): Take an element 𝑓 ∈ 𝒪(𝛺 1 ) and a compact subset 𝐾 ⋐ 𝛺 1 . By the assumption there is an 𝒪(𝛺 2 )-analytic polyhedron P such that 𝐾 ⋐ P ⋐ 𝛺 1 . It follows from Lemma 3.3.17 that 𝑓 can be approximated uniformly on 𝐾 by elements of ⊓ ⊔ 𝒪(𝛺 2 ). Therefore, (𝛺 1 , 𝛺 2 ) is a Runge pair. Proposition 3.3.19. A holomorphically convex domain 𝛺 carries an increasing open covering by 𝒪(𝛺)-analytic polyhedral domains P𝜈 (𝜈 ∈ N) such that P1 ⋐ P2 ⋐ · · · ⋐ P𝜈 ⋐ · · · ,

∞ Ø

P𝜈 = 𝛺.

𝜈=1

Proof. If 𝛺 = C𝑛 , it su ces to take polydisks P𝜈 (𝜈 = 1, 2, . . .) of monotone increasing polyradii diverging to in nity. Suppose that 𝛺 ≠ C𝑛 , 𝜕𝛺 ≠ ∅. Fix arbitrarily a point 𝑎 0 ∈ 𝛺. With the boundary distance function 𝑑 (𝑧, 𝜕𝛺) we have

3.3 Analytic Polyhedron and Oka Weil Approximation



87



𝑈1 = 𝑧 ∈ 𝛺 : ∥𝑧∥ < 𝑟 0 , 𝑑 (𝑧, 𝜕𝛺) >

1 ∋ 𝑎0 𝑟0

for a large 𝑟 0 > 0. Let 𝑉1 be the connected component of 𝑈1 containing 𝑎 0 . Similarly, let 𝑉𝜈 be the connected components of   1 𝑈𝜈 = 𝑧 ∈ 𝛺 : ∥𝑧∥ < 𝜈𝑟 0 , 𝑑 (𝑧, 𝜕𝛺) > , 𝜈 = 1, 2, . . . , 𝜈𝑟 0 Ð which contain 𝑎 0 . It follows that ∞ 𝜈=1 𝑈 𝜈 = 𝛺. For every point 𝑧 ∈ 𝛺 there is a curve connecting 𝑎 0 and 𝑧. Since 𝐶 is compact Ð with 𝐶 ⊂ 𝑈𝜈0 . Because of the connectedness, and 𝐶 ⊂ ∞ 𝜈=1 𝑈 𝜈 , there is a number Ð 𝜈0 𝐶 ⊂ 𝑉𝜈0 , so that 𝑧 ∈ 𝑉𝜈0 . Thus, ∞ 𝜈=1 𝑉𝜈 = 𝛺. We obtain an increasing open covering of 𝛺 by ∞ Ø 𝑉𝜈 ⋐ 𝑉𝜈+1 , 𝑉𝜈 = 𝛺. 𝜈=1

¯1𝛺 be the holomorphically convex hull of 𝑉¯1 . Since 𝛺 is holomorphically Let 𝑉b ¯1𝛺 is compact. It follows from Lemma 3.3.11 that there is an 𝒪(𝛺)-analytic convex, 𝑉b ¯1𝛺 ⋐ P1 ⋐ 𝛺. We rewrite P1 for the connected component of polyhedron P1 with 𝑉b P1 containing 𝑉1 . We next take 𝑉𝜈2 such that 𝑉𝜈2 ⊃ P¯ 1 ∪ 𝑉¯2 . Repeating the same argument as above again to 𝑉¯ 𝜈2 , we obtain an 𝒪(𝛺)-analytic polyhedral domain P2 such that 𝑉𝜈2 ⋐ P2 ⋐ 𝛺. Repeating this, we obtain the required analytic polyhedral domains P𝜈 , 𝜈 = 1, 2, . . .. ⊓ ⊔

3.3.3 Runge Approximation Theorem (One Variable) In this subsection we prove the classical Runge Approximation Theorem on a domain of C. The Oka Weil Approximation Theorem 3.3.12 restricted to the case 𝑛 = 1 is very close to Runge’s, but not exactly the same. In one variable we will see that the holomorphic convex hull is determined by the topology. Let 𝑛 = 1 and let 𝛺 ⊂ C be a domain. Let 𝐾 ⋐ 𝛺 be a compact subset. Let ★ be the relatively compact hull de ned by (3.1.2). In general, 𝐾 ★ is compact 𝐾𝛺 𝛺 (Proposition 3.1.7). In one variable the rst inclusion relation in (3.1.14) holds in fact with equality: ★ =𝐾 b𝛺 . Lemma 3.3.20. With the notation above, 𝐾𝛺 ★ . It su ces to obtain an element 𝑓 ∈ 𝒪(𝛺) such that Proof. Take a point 𝑎 ∈ 𝛺 \ 𝐾𝛺 | 𝑓 (𝑎)| > ∥ 𝑓 ∥ 𝐾𝛺★ . ★ containing 𝑎. In the case when Let 𝜔 be the connected component of 𝛺 \ 𝐾𝛺 𝜔 is bounded in C, we take a point 𝑐 ∈ 𝜕𝜔 ∩ 𝜕𝛺 (≠ ∅): By the linear transform 𝑤 = 1/(𝑧 −𝑐), we can reduce 𝜔 to an unbounded domain. Henceforth, 𝜔 is unbounded.

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3 Domains of Holomorphy

★ ∪ {𝑎}. Taking a point 𝑏 ∈ 𝜔 \ 𝛥(𝑅), Let 𝛥(𝑅) be a large disk such that 𝛥(𝑅) ⋑ 𝐾𝛺 we connect 𝑎 (the initial point) and 𝑏 (the terminal point) by a curve 𝐶 ⊂ 𝜔. Set ★ } > 0. 𝛿0 := min{|𝜁 − 𝑧| : 𝜁 ∈ 𝐶, 𝑧 ∈ 𝐾𝛺

If 𝑏 ∈ 𝛥(𝑎; 𝛿0 /2), we let 𝑏 1 = 𝑏. Otherwise, we consider a moving point 𝜁 on 𝐶 starting from 𝑎; then, 𝜁 crosses the boundary 𝜕 𝛥(𝑎; 𝛿0 /2). We take the last crossing point denoted by 𝑏 1 ∈ 𝐶 ∩ 𝜕 𝛥(𝑎; 𝛿0 /2). We set 𝑔(𝑧) = 1/(𝑧 − 𝑏 1 ), which is a rational function with a pole only at 𝑏 1 , and |𝑔(𝑎)| ≥

(3.3.21)

2 1 > = ∥𝑔∥ 𝐾𝛺★ . 𝛿0 𝛿0

We next consider 𝛥(𝑏 1 ; 𝛿0 /4). If 𝑏 ∈ 𝛥(𝑏 1 ; 𝛿0 /4), we set 𝑏 2 = 𝑏. Otherwise, the moving point 𝜁 ∈ 𝐶 crosses 𝜕 𝛥(𝑏 1 ; 𝛿0 /4), and the last point is denoted by 𝑏 2 ∈ 𝐶 ∩ 𝜕 𝛥(𝑏 1 ; 𝛿0 /4). Repeating this procedure, we can take a nite sequence of points on 𝐶 such that (3.3.22)

𝐶 ∋ 𝑏 1 , 𝑏 2 , . . . , 𝑏 𝑙 = 𝑏, |𝑧 − 𝑏 𝑗 | ≥

𝛿0 , 2

|𝑏 𝑗 − 𝑏 𝑗+1 | ≤

𝛿0 , 1 ≤ 𝑗 ≤ 𝑙 − 1, 4

★ 𝑧 ∈ 𝐾𝛺 ∪ {𝑎}, 1 ≤ 𝑗 ≤ 𝑙.

1 −𝑏2 If |𝑧 − 𝑏 2 | > 𝛿0 /4, then 𝑏𝑧−𝑏 < 1, and 2 Õ (𝑏 1 − 𝑏 2 ) 𝜈 1 1 · = . 𝑧 − 𝑏 2 1 − 𝑏1 −𝑏2 𝜈=0 (𝑧 − 𝑏 2 ) 𝜈+1 𝑧−𝑏2 ∞

𝑔(𝑧) =

Here the convergence is uniform on compact subsets of |𝑧 − 𝑏 2 | > 𝛿0 /4 and in ★ ∪ {𝑎}. That is, 𝑔(𝑧) can be approximated particular, by (3.3.22) it is uniform on 𝐾𝛺 ★ ∪ {𝑎} by rational functions with only one pole at 𝑏 . uniformly on 𝐾𝛺 2 1 Applying the same argument to 𝑧−𝑏 , we have 2 Õ (𝑏 2 − 𝑏 3 ) 𝜈 1 = , 𝑧 − 𝑏 2 𝜈=0 (𝑧 − 𝑏 3 ) 𝜈+1 ∞

1 where the convergence is uniform on compact subsets of |𝑧 − 𝑏 3 | > 𝛿0 /4. Thus 𝑧−𝑏 2 ★ can be approximated uniformly on 𝐾𝛺 ∪ {𝑎} by rational functions with a pole only at 𝑏 3 . Combining the above two steps, we see that 𝑔(𝑧) can be approximated uniformly ★ ∪ {𝑎} by rational functions with a pole only at 𝑏 . Repeating the process up on 𝐾𝛺 3 ★ ∪ {𝑎} by rational to 𝑏 𝑙 = 𝑏, we see that 𝑔(𝑧) can be approximated uniformly on 𝐾𝛺 functions with a pole only at 𝑏. ★ ∪ {𝑎} ⊂ 𝛥(𝑅) and |𝑏| > 𝑅, we have an expansion Now, since 𝐾𝛺

3.4 Cousin Problem

89

Õ 𝑧𝜈 1 = − , 𝑧 − 𝑏 𝜈=0 𝑏 𝜈+1 ∞

𝑧 ∈ 𝛥(𝑅),

★ ∪ {𝑎}. Therefore 𝑔(𝑧) can be approximated where the convergence is uniform on 𝐾𝛺 ★ uniformly on 𝐾𝛺 ∪ {𝑎} by polynomials. It follows from (3.3.21) that there is a polynomial 𝑓 (𝑧) satisfying | 𝑓 (𝑎)| > ∥ 𝑓 ∥ 𝐾𝛺★ ; here, needless to say, 𝑓 ∈ 𝒪(𝛺). ⊓ ⊔

Theorem 3.3.23 (Runge Approximation). Let 𝛺 ⊂ C be a domain and let 𝐾 ⋐ 𝛺 ★ = 𝐾. Then a holomorphic function on 𝐾 is be a compact subset such that 𝐾𝛺 approximated uniformly on 𝐾 by functions of 𝒪(𝛺). Proof. Use Lemma 3.3.20 and the Oka Weil Approximation Theorem 3.3.12.

⊓ ⊔

Remark 3.3.24. It is a specialty of one variable that the possibility of the uniform approximation on 𝐾 can be characterized by the topological property. There is a more elementary proof of Theorem 3.3.23 by means of only the Cauchy integral formula and Lemma 3.3.20 (see Exercise 5 at the end of this chapter).

3.4 Cousin Problem We solve the Cousin Problem (P2).4 The Cousin Problem consists of I and II; we shall formulate a Continuous Cousin Problem and solve I and II simultaneously. We begin with the de nition of meromorphic functions. Let 𝛺 ⊂ C𝑛 be an open set. Definition 3.4.1. A pair ( 𝑓 , 𝛺 ′ ) of 𝑓 ∈ 𝒪(𝛺 ′ ) and an open dense subset 𝛺 ′ ⊂ 𝛺 is said to be meromorphic in 𝛺 if for every point 𝑎 ∈ 𝛺 there are a connected neighborhood 𝑈 of 𝑎 and 𝑔, ℎ ∈ 𝒪(𝑈) with 𝑔 . 0 satisfying (3.4.2)

𝑓 (𝑧) =

ℎ(𝑧) , 𝑔(𝑧)

𝑧 ∈ 𝛺 ′ ∩ 𝑈 \ {𝑔 = 0}.

Two meromorphic pairs ( 𝑓 𝑗 , 𝛺 ′𝑗 ) ( 𝑗 = 1, 2) in 𝛺 are equivalent if there is an open dense subset 𝛺 ′′ ⊂ 𝛺 such that (3.4.3)

𝛺 ′′ ⊂ 𝛺 1′ ∩ 𝛺 2′ , 𝑓1 (𝑧) = 𝑓2 (𝑧),

𝑓 𝑗 | 𝛺 ′′ ∈ 𝒪(𝛺 ′′ ),

𝑧 ∈ 𝛺 ′′ .

It is easy to check that this relation is in fact an equivalence relation (cf. Exercise 4 at the end of this chapter); the equivalence class of ( 𝑓 , 𝛺 ′ ) is called a meromorphic function in 𝛺, denoted simply by 𝑓 or 𝑓 (𝑧). Let ℳ(𝛺) denote the set of all meromorphic functions in 𝛺. If an element 𝑓 ∈ ℳ(𝛺) has a representative ( 𝑓0 , 𝛺) with 𝑓0 ∈ 𝒪(𝛺), then 𝑓0 is uniquely determined, so that we have the natural inclusion 𝒪(𝛺) ⊂ ℳ(𝛺). 4 Cf. Preface.

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3 Domains of Holomorphy

If 𝑔(𝑧) in (3.4.2) can be chosen to satisfy 𝑔(𝑎) ≠ 0, after shrinking 𝑈 smaller if necessary, we have 1/𝑔 ∈ 𝒪(𝑈). In this case, 𝑓 ∈ 𝒪(𝑈), and so 𝑓 is said to be holomorphic at 𝑎. The set 𝛺 0 of all points where 𝑓 is holomorphic is an open set, and 𝑍∞ := 𝛺 \ 𝛺 0 is a closed set; A point of 𝑍∞ is called a pole of 𝑓 , and 𝑍∞ is called the polar set of 𝑓 . Let 𝑓 ∈ ℳ(𝛺). For a connected component 𝛺 ′ of 𝛺 we have either 𝑓 | 𝛺 ′ ≡ 0 or 𝑓 | 𝛺 ′ . 0. In the rst case, the zero set of 𝑓 in 𝛺 ′ is 𝛺 ′ itself. In the latter case, 1/( 𝑓 | 𝛺 ′ ) ∈ ℳ(𝛺 ′ ), and the zero set of 𝑓 in 𝛺 ′ is de ned to be the polar set of 1/( 𝑓 | 𝛺 ′ ) in 𝛺 ′ ; the zero set of 𝑓 is the union 𝑍0 of all those zero sets in the connected components of 𝛺. Theorem 3.4.4. Let 𝛺 be a domain and let 𝑓 ∈ ℳ(𝛺) be a non-zero meromorphic function. Let 𝑍 = 𝑍∞ ∪ 𝑍0 with the polar set 𝑍∞ and the zero set 𝑍0 of 𝑓 . (i) 𝑍 is a nowhere dense closed subset. (ii) 𝛺 \ 𝑍 is a domain. (iii) If 𝑛 ≥ 2, 𝑍 contains no isolated point. (iv) About every point 𝑎 ∈ 𝑍∞ , 𝑓 is unbounded; i.e., for any neighborhood 𝑈 of 𝑎 in 𝛺, 𝑓 |𝑈\𝑍∞ is unbounded; equivalently, if 𝑓 is bounded in a dense open subset of a neighborhood of a point 𝑎 ∈ 𝛺, then 𝑓 is holomorphic about 𝑎. Proof. (i), (ii) By de nition there are a connected neighborhood 𝑈 of every 𝑎 ∈ 𝛺 and a proper analytic subset 𝑌 of 𝑈 such that 𝑍 ∩ 𝑈 ⊂ 𝑌 . By Theorem 1.5.2 𝑈 \ 𝑌 is a connected open dense subset of 𝑈, and hence so is 𝑈 \ 𝑍 in 𝑈. Therefore 𝑍 is a nowhere dense closed subset, and 𝛺 \ 𝑍 is connected if so is 𝛺. (iii) This follows from Theorem 1.2.7. (iv) Suppose that for a point 𝑎 ∈ 𝛺 there is a neighborhood 𝑈 ∋ 𝑎 in 𝛺 such that 𝑓 |𝑈\𝑍∞ is bounded. By de nition, with 𝑈 chosen smaller if necessary, there are 𝑔, ℎ ∈ 𝒪(𝑈) such that 𝑔 . 0 and 𝑓 (𝑧) =

ℎ(𝑧) , 𝑔(𝑧)

𝑧 ∈ 𝑈 \ {𝑔 = 0} ⊂ 𝑈 \ 𝑍∞ .

Theorem 1.5.11 implies 𝑓 ∈ 𝒪(𝑈), and so 𝑎 ∉ 𝑍∞ .

⊓ ⊔

Assume that 𝛺 is a domain. It follows from the above theorem that if ℎ(𝑧) of (3.4.2) satis es ℎ ≠ 0 (i.e., ℎ(𝑧) . 0), then this holds for any other point of 𝛺. In that case, setting 1/ 𝑓 (𝑧) = 𝑔(𝑧)/ℎ(𝑧) locally in 𝑈, we may obtain a unique inverse element of 𝑓 . Thus ℳ(𝛺) naturally carries a structure of a eld. Remark 3.4.5. It may be helpful for understanding the notion of meromorphic functions to learn that the ring 𝒪𝑛,𝑎 is a unique factorization domain (see, e.g., [39] Theorem 2.2.12). Keeping the above notation, we suppose that 𝑓 . 0 and 𝑍∞ ≠ ∅. The following is known for 𝑍∞ : For every pole 𝑎 ∈ 𝑍∞ there are a neighborhood 𝑈 of 𝑎 and functions 𝑔, ℎ ∈ 𝒪(𝑈) satisfying 𝑓=

ℎ , 𝑔

𝑍∞ ∩ 𝑈 = {𝑔 = 0},

𝑍0 ∩ 𝑈 = {ℎ = 0}

3.4 Cousin Problem

91

(cf. [39] 2.2.2). In particular, 𝑍∞ and 𝑍0 are complex hypersurfaces of 𝛺. But, we will not use this fact.

3.4.1 Cousin I Problem Problem 3.4.6 (Cousin I). Let {𝑈𝜆 }𝜆∈𝛬 be an open covering of a domain 𝛺 ⊂ C𝑛 and let 𝑓𝜆 ∈ ℳ(𝑈𝜆 ) be given so that (3.4.7)

𝑓𝜆 − 𝑓 𝜇 ∈ 𝒪(𝑈𝜆 ∩ 𝑈 𝜇 )

on 𝑈𝜆 ∩ 𝑈 𝜇 ,

∀𝜆, 𝜇 ∈ 𝛬.

Here, if 𝑈𝜆 ∩ 𝑈 𝜇 = ∅, we consider it to hold. Then, nd 𝐹 ∈ ℳ(𝛺) such that (3.4.8)

𝐹 − 𝑓𝜆 ∈ 𝒪(𝑈𝜆 ),

∀𝜆 ∈ 𝛬.

A family 𝒞𝐼 := {(𝑈𝜆 , 𝑓𝜆 )}𝜆∈𝛬 of pairs (𝑈𝜆 , 𝑓𝜆 ) satisfying (3.4.7) is called Cousin I data (also, a Cousin I distribution) on 𝛺. The above 𝐹, if it exists, is called a solution of 𝒞𝐼 . Also in this case, the polar sets 𝑍𝜆 of 𝑓𝜆 in 𝑈𝜆 satisfy 𝑍𝜆 ∩ (𝑈𝜆 ∩ 𝑈 𝜇 ) = 𝑍 𝜇 ∩ (𝑈𝜆 ∩ 𝑈 𝜇 ) for all 𝜈, 𝜇 ∈ 𝛬. Therefore the polar set 𝑍 of 𝒞𝐼 is well-de ned in 𝛺. By Theorem 3.4.4, 𝑍 is a nowhere dense closed subset of 𝛺, and if 𝑛 ≥ 2, 𝑍 contains no isolated point. Remark 3.4.9. In one variable (𝑛 = 1) the Cousin I Problem is known to hold as the Mittag-Le er Theorem ([38] Theorem (7.2.3)). The Cousin I Problem is a version of it in several variables. Remark 3.4.10 (Non-solvable example). In the case of 𝑛 ≥ 2 there is a non-solvable example of 𝒞𝐼 due to the shape of 𝛺. For example, we consider a Hartogs domain 𝛺 H in the space of two variables (𝑧, 𝑤) ∈ C2 : (3.4.11)

𝛺 1 = {(𝑧, 𝑤) ∈ C2 : |𝑧| < 3, |𝑤| < 1}, 𝛺 2 = {(𝑧, 𝑤) ∈ C2 : 2 < |𝑧| < 3, |𝑤| < 3}, 𝛺H = 𝛺1 ∪ 𝛺2 .

We consider Cousin I data 𝒞𝐼 de ned by 𝑓1 = 0 on 𝛺 1 and 𝑓2 = 1/(𝑧 − 𝑤) on 𝛺 2 (cf. Fig. 3.1). Suppose that 𝒞𝐼 has a solution 𝐹 ∈ ℳ(𝛺 H ). If we set 𝑔(𝑧, 𝑤) = (𝑧 − 𝑤)𝐹 (𝑧, 𝑤) ∈ 𝒪(𝛺 H ), we deduce from Hartogs’ Phenomenon (Theorem 1.2.12) that 𝑔 ∈ 𝒪(P𝛥((3, 3))). Then, 𝑔(𝑧, 𝑧) is holomorphic in 𝛥(3), and 𝑔(𝑧, 𝑧) = 0 in |𝑧| < 1, but 𝑔(𝑧, 𝑧) = 1 in 2 < |𝑧| < 3; this contradicts the Identity Theorem 1.1.46. Therefore, for the solvability of the Cousin I Problem in 𝑛 ≥ 2, the domain 𝛺 cannot be arbitrary, and it was asked for 𝛺 domain of holomorphy.

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3 Domains of Holomorphy

Fig. 3.1 Non-solvable Cousin I data.

3.4.2 Continuous Cousin Problem There is the Cousin II Problem, but before going into it we would like to prepare some topological setting and formulate the Continuous Cousin Problem in order to make the consideration easier. Let 𝛺 ⊂ C𝑛 be an open set. We set   1 (3.4.12) 𝑊 𝑗 = 𝑧 ∈ 𝛺 : ∥𝑧∥ < 𝑗, 𝑑 (𝑧, 𝜕𝛺) > , 𝑗 = 1, 2, . . . . 𝑗 Then, 𝑊 𝑗 ⋐ 𝑊 𝑗+1 ,

∞ Ø

𝑊 𝑗 = 𝛺.

𝑗=1

Definition 3.4.13 (Locally nite). Let {𝑈𝜆 }𝜆∈𝛬 be an arbitrary open covering of 𝛺. We say that {𝑈𝜆 }𝜆∈𝛬 is locally finite, if one of the following equivalent conditions holds: (i) For every point 𝑎 ∈ 𝛺 there is a neighborhood 𝑉 of 𝑎 such that there are only nitely many 𝑈𝜆 with 𝑈𝜆 ∩𝑉 ≠ ∅. (ii) For every compact subset 𝐾 ⋐ 𝛺 there are only nitely many 𝑈𝜆 such that 𝑈𝜆 ∩ 𝐾 ≠ ∅. Definition 3.4.14 (Re nement). Let 𝒰 = {𝑈𝜆 }𝜆∈𝛬 and 𝒱 = {𝑉𝛾 } 𝛾 ∈𝛤 be two open coverings of 𝛺. We say that 𝒱 is a refinement of 𝒰 (denoted by 𝒰 ≺ 𝒱) if for every 𝑉𝛾 there is a corresponding 𝑈𝜆 with 𝑉𝛾 ⊂ 𝑈𝜆 . If necessary, the correspondence is written as 𝜙 : 𝛤 → 𝛬 (or the map between the families is denoted by 𝜙 : 𝒱 → 𝒰), so that 𝑉𝛾 ⊂ 𝑈 𝜙 (𝛾) (or 𝑉𝛾 ⊂ 𝜙(𝑉𝛾 )). Proposition 3.4.15. For any open covering {𝑈𝜆 }𝜆∈𝛬 of 𝛺 there is its refinement 𝑁 {𝑉𝜇 } 𝜇=1 (𝑁 ≤ ∞) which is at most countable and locally finite. Proof. If 𝛺 is covered by nitely many 𝑈𝜆 (𝜆 ∈ 𝛬), it is su cient to put 𝑉𝜆 := 𝑈𝜆 for those 𝑈𝜆 . In the sequel we assume that 𝛺 cannot be covered by nitely many members of 𝑈𝜆 (𝜆 ∈ 𝛬).

3.4 Cousin Problem

93

Let 𝑊 𝑗 be as de ned in (3.4.12). The closure 𝑊¯ 𝑗 is compact. Since 𝑊¯ 1 is covered by nitely many 𝑈𝜆 , we denote them by 𝑈1 ,𝑈2 , . . . ,𝑈𝜈1 . In the same way as above, we take nitely many 𝑈𝜈1 +1 ,𝑈𝜈1 +2 , . . . ,𝑈𝜈2 which cover 𝑊¯ 2 \ 𝑊1 . Repeating this we take a countable open covering {𝑈 𝜇 }∞ 𝜈=1 out of {𝑈𝜆 } 𝜆∈𝛬 . Next, we set 𝑉𝜇 = 𝑈 𝜇 , 1 ≤ 𝜇 ≤ 𝜈1 , which covers 𝑊¯ 1 . We then set 𝑉𝜇 = 𝑈 𝜇 \ 𝑊¯ 1 ,

𝜈1 + 1 ≤ 𝜇 ≤ 𝜈2 .

Ð𝜈2

It follows that 𝑊¯ 2 ⊂ 𝜇=1 𝑉𝜇 . Repeating this procedure, we obtain {𝑉𝜇 }∞ 𝜇=1 . By the ∞ construction {𝑉𝜇 } 𝜇=1 is a locally nite open covering of 𝛺, and a re nement of {𝑈𝜆 }𝜆∈𝛬 . ⊓ ⊔ Ð𝑁 Proposition 3.4.16 (Partition of unity). Let 𝛺 = 𝑗=1 𝑈 𝑗 (𝑁 ≤ ∞) be an open covering of an open set 𝛺 ⊂ C𝑛 which is at most countable and locally finite. Then there are continuous functions 𝜒 𝑗 ∈ 𝒞 0 (𝛺) such that: (i) 0 ≤ 𝜒 𝑗 ≤ 1, 1 ≤ 𝑗 ≤ 𝑁, (ii) Supp 𝜒 𝑗 ⊂ 𝑈 𝑗 , 1 ≤ 𝑗 ≤ 𝑁, Í (iii) 𝑁𝑗=1 𝜒 𝑗 (𝑥) ≡ 1, 𝑥 ∈ 𝛺. Proof. We take an open covering {𝑉 𝑗 } 𝑁𝑗=1 of 𝛺, so that the closures 𝑉¯ 𝑗 in 𝛺 are contained in 𝑈 𝑗 for all 𝑗. Then, since 𝑉¯ 𝑗 ∩ (𝛺 \𝑈 𝑗 ) = ∅ for each 𝑗, there is a function 𝜌 𝑗 ∈ 𝒞 0 (𝛺) (cf. Exercise 7 at the end of this chapter) such that: (i) 𝜌 𝑗 (𝑥) ≥ 0, 𝑥 ∈ 𝛺; (ii) 𝜌 𝑗 (𝑥) = 1, 𝑥 ∈ 𝑉 𝑗 ; (iii) Supp 𝜌 𝑗 ⊂ 𝑈 𝑗 . Í Since 𝑁𝑗=1 𝜌 𝑗 (𝑥) > 0 (∀𝑥 ∈ 𝛺), it su ces to put 𝜌 𝑗 (𝑥) 𝜒 𝑗 (𝑥) = Í 𝑁 , 𝑗=1 𝜌 𝑗 (𝑥)

𝑥 ∈ 𝛺.

⊓ ⊔

Definition 3.4.17. The above { 𝜒 𝑗 } is called a (continuous or 𝐶 0 -) partition of unity subordinated to the open covering {𝑈 𝑗 }. Definition 3.4.18 (Continuous Cousin data). Let {𝑈𝜆 }𝜆∈𝛬 be an open covering of 𝛺. A family 𝒞 = {(𝑈𝜆 , 𝑓𝜆 )}𝜆∈𝛬 of pairs of 𝑈𝜆 and 𝑓𝜆 ∈ 𝒞 0 (𝑈𝜆 ) is called Continuous Cousin data (or a continuous Cousin distribution) if for all 𝜆, 𝜇 ∈ 𝛬, 𝑓𝜆 − 𝑓 𝜇 ∈ 𝒪(𝑈𝜆 ∩ 𝑈 𝜇 )

on 𝑈𝜆 ∩ 𝑈 𝜇 .

Here, if 𝑈𝜆 ∩ 𝑈 𝜇 = ∅, we consider it to hold. Problem 3.4.19 (Continuous Cousin). Let 𝒞 = {(𝑈𝜆 , 𝑓𝜆 )}𝜆∈𝛬 be continuous Cousin data on 𝛺. Then, nd a continuous function 𝐹 on 𝛺, called a solution of 𝒞 such that

94

(3.4.20)

3 Domains of Holomorphy

𝐹 − 𝑓𝜆 ∈ 𝒪(𝑈𝜆 ),

∀𝜆 ∈ 𝛬.

Remark 3.4.21. (i) In the above Continuous Cousin Problem, according to Propo𝑁 of {𝑈𝜆 }𝜆∈𝛬 , which is at most sition 3.4.15 we take a re nement {𝑉𝜇 } 𝜇=1 countable and locally nite. Set 𝑔 𝜇 = 𝑓𝜆( 𝜇) | 𝑉𝜇 ∈ 𝒞 0 (𝑉𝜇 ),

1 ≤ 𝜇 ≤ 𝑁.

𝑁 Then 𝒞 ′ := {(𝑉𝜇 , 𝑔 𝜇 )} 𝜇=1 is continuous Cousin data, which is called the induced continuous Cousin data of 𝒞 by the re nement {𝑉𝜇 }. A solution 𝐺 ∈ 𝒞 0 (𝛺) of 𝒞 ′ is clearly a solution of 𝒞. (ii) Therefore any Continuous Cousin Problem is reduced to the one with respect to an open covering which is at most countable and locally nite; we assume this henceforth.

Theorem 3.4.22. If 𝛺 is a holomorphically convex domain (or equivalently a domain of holomorphy), then every Continuous Cousin data on 𝛺 has a solution. Proof. Suppose that continuous Cousin data 𝒞 = {(𝑈𝜆 , 𝑓𝜆 )}𝜆∈𝛬 is given on 𝛺. (1) Let P ⋐ 𝛺 be an analytic polyhedron. We rst obtain a solution of 𝒞 on P. In what follows, a solution is that of 𝒞. Let (3.4.23)

𝜑 : P −→ P𝛥 ⊂ C 𝑁

be the Oka map. We assume that 𝜑 is de ned on a slightly larger analytic polyhedron f ⋑ P𝛥: e P ⋑ P with the corresponding polydisk P𝛥 (3.4.24)

f 𝜑 :e P −→ P𝛥.

f We set 𝑆˜ = 𝜑(e P). By Riemann’s Mapping Theorem 1.4.9 we may assume that P𝛥 is an open cuboid with edges parallel to coordinate axes (it is the same below). We f containing 𝜑(P), and set take a closed cuboid 𝐸 0 in P𝛥, 𝑆 = 𝜑(e P) ∩ 𝐸 0 . We identify 𝜑 −1 𝑆(⊃ P) with 𝑆. In order to obtain a solution on 𝑆 we consider: Claim 3.4.25. For an arbitrary closed cuboid 𝐸 ⊂ 𝐸 0 there are a neighborhood 𝑊 (⊃ 𝐸 ∩ 𝑆) in 𝑆˜ and a continuous function 𝐺 on 𝑊 (cf. Fig. 3.2) such that 𝐺 − 𝑓𝜆 | 𝐸∩𝑆∩𝑈𝜆 ∈ 𝒪(𝐸 ∩ 𝑆 ∩ 𝑈𝜆 ),

∀𝜆 ∈ 𝛬;

that is, 𝐺 | 𝑊∩𝑈𝜆 − 𝑓𝜆 | 𝑊∩𝑈𝜆 ∈ 𝒪(𝑊 ∩ 𝑈𝜆 ) for every 𝜆 ∈ 𝛬. ∵ ) For the proof we modify the induction on the cuboid dimension dim 𝐸 used in 2.5. (a) The case of dim 𝐸 = 0. Let 𝐸 = {𝑎}. If 𝑎 ∉ 𝑆, then 𝐸 ∩ 𝑆 = ∅ and the claim holds. If 𝑎 ∈ 𝑆, there is a 𝑈𝜆 ∋ 𝑎, on which we set 𝐺 = 𝑓𝜆 .

3.4 Cousin Problem

95

˜ Fig. 3.2 𝐸 ∩ 𝑆 ⊂ 𝑊 ⊂ 𝑆.

(b) Suppose that the case of dim 𝐸 = 𝜈 − 1 (𝜈 ≥ 1) holds. Let dim 𝐸 = 𝜈. As in (2.5.3), 𝐸 may be assumed to be of the form: 𝐸 = 𝐹 × {𝑧 𝑁 : 0 ≤ ℜ𝑧 𝑁 ≤ 𝑇, |ℑ𝑧 𝑁 | ≤ 𝜃},

(3.4.26)

𝑇 > 0,

𝜃 ≥ 0.

For every 𝑡 ∈ [0,𝑇] 𝐸 𝑡 := 𝐹 × {𝑧 𝑁 : ℜ𝑧 𝑁 = 𝑡, |ℑ𝑧 𝑁 | ≤ 𝜃} is a closed cuboid of dimension 𝜈 − 1. By the induction hypothesis there is a solution on 𝐸 𝑡 ∩ 𝑆. By the Heine Borel Theorem there is a nite partition 0 = 𝑡0 < 𝑡1 < · · · < 𝑡 𝐿 = 𝑇

(3.4.27) such that with

𝐸 𝛼 = 𝐹 × {𝑧 𝑁 : 𝑡 𝛼−1 ≤ ℜ𝑧 𝑁 ≤ 𝑡 𝛼 , |ℑ𝑧 𝑁 | ≤ 𝜃},

1 ≤ 𝛼 ≤ 𝐿,

either 𝐸 𝛼 ∩ 𝑆 = ∅ or there is a solution 𝐺 𝛼 on 𝐸 𝛼 ∩ 𝑆 (≠ ∅). It is su cient to consider those 𝐸 𝛼 with 𝐸 𝛼 ∩ 𝑆 ≠ ∅. We say that the adjacent 𝐸 𝛼 and 𝐸 𝛼+1 is connected through 𝑆 if 𝐸 𝛼 ∩ 𝐸 𝛼+1 ∩ 𝑆 ≠ ∅. We take a maximal sequence of 𝐸 𝛼 mutually connected through 𝑆: 𝐸 ′ = 𝐸 𝛼0 ∪ 𝐸 𝛼0 +1 ∪ · · · ∪ 𝐸 𝛼1 . ˜ For the simplicity of indices We are going to construct a solution 𝐺 ′ on 𝐸 ′ ∩ 𝑆(⊂ 𝑆). we assume 𝛼0 = 1. By the choices of 𝐺 𝛼 (1 ≤ 𝛼 ≤ 𝛼1 ) we see that (3.4.28)

ℎ := 𝐺 1 | 𝐸1 ∩𝐸2 ∩𝑆 − 𝐺 2 | 𝐸1 ∩𝐸2 ∩𝑆 ∈ 𝒪(𝐸 1 ∩ 𝐸 2 ∩ 𝑆).

With small 𝛿 > 0 and a small open cuboid neighborhood 𝑉 ′ ⊂ C 𝑁 −1 of 𝐹 we have

96

3 Domains of Holomorphy

𝑉 :=𝑉 ′ × {𝑧 𝑁 : 𝑡 1 − 𝛿 < ℜ𝑧 𝑁 < 𝑡 1 + 𝛿, |ℑ𝑧 𝑁 | < 𝜃 + 𝛿}, ˜ ℎ ∈ 𝒪(𝑉 ∩ 𝑆). Note that 𝑉 is biholomorphic to a polydisk and 𝑉 ∩ 𝑆˜ is a submanifold of 𝑉. By the Oka Extension Theorem 2.5.14 there is a function 𝐻 ∈ 𝒪(𝑉) such that 𝐻| 𝑉∩𝑆˜ = ℎ (cf. Fig. 3.3). We consider the Cousin integral of 𝐻 along the oriented line segment   𝛿 𝛿 ℓ = 𝑧 𝑁 : ℜ𝑧 𝑁 = 𝑡1 , − 𝜃 − ≤ ℑ𝑧 𝑁 ≤ 𝜃 + 2 2 and obtain the Cousin decomposition of 𝐻: 𝐻 = 𝐻1 − 𝐻2

(on 𝐸 1 ∩ 𝐸 2 ),

𝐻1 ∈ 𝒪(𝐸 1 ), 𝐻2 ∈ 𝒪(𝐸 2 ).

This together with (3.4.28) implies (3.4.29)

(𝐺 1 − 𝐻1 | 𝐸1 ∩𝑆 )| 𝐸1 ∩𝐸2 ∩𝑆 = (𝐺 2 − 𝐻2 | 𝐸2 ∩𝑆 )| 𝐸1 ∩𝐸2 ∩𝑆 .

The left-hand side of the above equation is a solution on 𝐸 1 ∩ 𝑆, and the right-hand side is a solution on 𝐸 2 ∩ 𝑆; hence we obtain a solution 𝐺 2′ on (𝐸 1 ∪ 𝐸 2 ) ∩ 𝑆. We call 𝐺 2′ the merged solution of the solution 𝐺 1 (on 𝐸 1 ∩ 𝑆) and 𝐺 2 (on 𝐸 2 ∩ 𝑆). We then merge 𝐺 2′ and the solution 𝐺 3 on 𝐸 3 ∩ 𝑆 to obtain a solution 𝐺 3′ on (𝐸 1 ∪ 𝐸 2 ∪ 𝐸 3 ) ∩ 𝑆. Repeating this, we obtain a solution 𝐺 ′ := 𝐺 ′𝛼1 on (𝐸 1 ∪ · · · ∪ 𝐸 𝛼1 ) ∩ 𝑆 = 𝐸 ′ ∩ 𝑆. △ (2) We obtain a solution on 𝛺. It follows from Proposition 3.3.19 that there is an increasing sequence of analytic polyhedral domains P1 ⋐ P2 ⋐ · · · ⋐ P 𝜇 ⋐ · · · ,

∞ Ø

P 𝜇 = 𝛺.

𝜇=1

By the result of the above (1) we have a solution 𝐺 𝜇 on each P 𝜇 . Note that

Fig. 3.3 𝐸1 , . . . , 𝐸𝐿 .

3.4 Cousin Problem

97

(𝐺 𝜇+1 − 𝐺 𝜇 )| P 𝜇 ∈ 𝒪(P 𝜇 ). Set 𝐹1 = 𝐺 1 . Since (𝐺 2 − 𝐹1 )| P1 ∈ 𝒪(P1 ), the Oka Weil Approximation Theorem 3.3.12 implies the existence of a function ℎ2 ∈ 𝒪(𝛺) such that 1 . 2

∥𝐺 2 − 𝐹1 − ℎ2 ∥ P1
−𝛿}, 𝑈2 = {(𝑧, 𝑤) ∈ 𝛺 : ℑ𝑧 < 𝛿}, 𝛺 = 𝑈1 ∪ 𝑈2 , 𝑆 1 = 𝑆 ∩ 𝑈1 . Then 𝑆1 is a complex hypersurface of 𝛺. We take Cousin II data 𝒞𝐼 𝐼 with zero-set 𝑆1 de ned by 𝑓1 (𝑧, 𝑤) = 𝑓 (𝑧, 𝑤) = 𝑤 − 𝑧 + 1, (𝑧, 𝑤) ∈ 𝑈1 , 𝑓2 (𝑧, 𝑤) = 1, (𝑧, 𝑤) ∈ 𝑈2 . Claim 3.4.54. 𝒞𝐼 𝐼 has no solution.

104

3 Domains of Holomorphy

Fig. 3.4 Non-solvable Cousin II data 𝑆1 = {𝑤 = 𝑧 − 1} ∩𝑈1 ⊂ 𝛺.

∵ ) Suppose that 𝒞𝐼 𝐼 has an analytic solution 𝐹 (𝑧, 𝑤) ∈ 𝒪(𝛺). We take circles with anti-clockwise orientation in the 𝑧-plane and the 𝑤-plane as follows:     5 5 𝐶1 = |𝑧| = , 𝐶2 = |𝑤| = . 6 6 We then see that (cf. Fig. 3.4)  𝐶1 × 𝐶2 ⊂ 𝛺,

(𝐶1 × 𝐶2 ) ∩ 𝑆1 =

1 2 1 2 + 𝑖, − + 𝑖 2 3 2 3

 .

For 𝑧 ∈ 𝐶1 with ({𝑧} × 𝐶2 ) ∩ 𝑆1 = ∅ we set ∫ ∫ 1 𝜕𝑤 𝐹 (𝑧, 𝑤) 𝛩(𝑧) = 𝜕𝑤 arg 𝐹 (𝑧, 𝑤)𝑑𝑤 = 𝑑𝑤 ∈ 2𝜋Z, 𝑖 𝑤∈𝐶2 𝑤∈𝐶2 𝐹 (𝑧, 𝑤) where 𝜕𝑤 = 𝜕/𝜕𝑤. On 𝑈1 we can write 𝐹 (𝑧, 𝑤) = (𝑤 − 𝑧 + 1)𝜙(𝑧, 𝑤),

𝜙 ∈ 𝒪∗ (𝑈1 ).

Therefore on 𝑈1 we have ∫ ∫ 𝛩(𝑧) = 𝜕𝑤 arg(𝑤 − 𝑧 + 1)𝑑𝑤 + 𝜕𝑤 arg 𝜙(𝑧, 𝑤)𝑑𝑤, 𝑤∈𝐶2 𝑤∈𝐶2 ∫ 𝜏(𝑧) := 𝜕𝑤 arg 𝜙(𝑧, 𝑤)𝑑𝑤. 𝑤∈𝐶2

We rst consider 𝜏(𝑧). As 𝑧 moves on 𝐶1 ∩ {ℑ𝑧 ≥ 0} from 56 to − 56 , 𝜏(𝑧) moves continuously; the value is in 2𝜋Z. Hence, 𝜏(𝑧) = 𝜏0 must be a constant. On the other hand, we see that   ∫ 5 𝜕𝑤 arg 𝑤 − + 1 𝑑𝑤 = 2𝜋, 6 𝑤∈𝐶2

3.4 Cousin Problem

105



∫ 𝑤∈𝐶2

𝜕𝑤 arg 𝑤 +



5 + 1 𝑑𝑤 = 0. 6

It follows that     5 5 = 2𝜋 + 𝜏0 , 𝛩 − = 𝜏0 , 6 6     5 5 𝛩 − 𝛩 − = 2𝜋. 6 6 𝛩 (3.4.55)

Now, as 𝑧 moves on 𝐶1 ∩ {ℑ𝑧 ≤ 0} from 56 to − 56 , 𝐹 (𝑧, 𝑤) has no zero there. Similarly to the arguments on 𝜏(𝑧), 𝛩(𝑧) is a constant. Then     5 5 𝛩 −𝛩 − = 0 : 6 6 △

This contradicts (3.4.55)

3.4.7 Weierstrass’ Theorem (One Variable) In the present subsection we assume 𝑛 = 1. Let 𝛺 ⊂ C be a domain. Let 𝛤 ⊂ 𝛺 be a closed discrete subset; i.e., it is discrete and has no accumulation point in 𝛺. Let {𝑚 𝑎 ∈ Z \ {0} : 𝑎 ∈ 𝛤} be given arbitrarily. We consider a problem to construct a meromorphic function on 𝛺 such that it has a zero (in the case 𝑚 𝑎 > 0) or a pole (in the case 𝑚 𝑎 < 0) of order |𝑚 𝑎 | at every 𝑎 ∈ 𝛤 and has no other zero or pole. If 𝛤 is nite, it su ces to set Ö 𝑓 (𝑧) = (𝑧 − 𝑎) 𝑚𝑎 . 𝑎∈𝛤

The case of in nite 𝛤 is the problem. Theorem 3.4.56 (Weierstrass). Let 𝛺 ⊂ C be a domain. Let {𝑎 𝜈 }∞ 𝜈=1 ⊂ 𝛺 be a closed discrete subset with distinct 𝑎 𝜈 and let {𝑚 𝜈 }∞ 𝜈=1 be an arbitrary sequence in Z \ {0}. Then there is a meromorphic function on 𝛺 which has a zero (in the case 𝑚 𝜈 > 0) or a pole (in the case 𝑚 𝜈 < 0) of order |𝑚 𝜈 | at every 𝑎 𝜈 and has no zero nor pole at other points. Proof. Just for a convenience, we assume by a parallel transform (3.4.57)

0 ∈ 𝛺.

We take a sequence of subdomains P 𝜇 , 𝜇 = 1, 2, . . ., such that P 𝜇 ⋐ P 𝜇+1 and 𝛺 \ P¯ 𝜇 has no relatively compact connected component in 𝛺. Set

106

3 Domains of Holomorphy

𝑄 𝜇 (𝑧) =

Ö

(𝑧 − 𝑎 𝜈 ) 𝑚𝜈 ,

𝜇 = 1, 2, . . .

𝑎𝜈 ∈P 𝜇

We are going to modify 𝑄 𝜇 (𝑧) to obtain a topological solution of the Cousin II Problem with the given zeros and poles. 𝜇 = 1: We set 𝐹1 (𝑧) = 𝑄 1 (𝑧). 𝜇 = 2: We rst consider Ö (3.4.58) (𝑧 − 𝑎 𝜈 ) 𝑚𝜈 . 𝑎𝜈 ∈P2 \P1

For each 𝑎 𝜈 ∈ P2 \P1 we take the connected component 𝑊𝜈 of 𝛺 \P1 containing 𝑎 𝜈 . In ˆ we consider the boundary 𝜕𝑊𝜈 = (𝜕𝑊𝜈 ∩ 𝜕𝛺) ∪ (𝜕𝑊𝜈 ∩ 𝜕P1 ). the Riemann sphere C Note that (𝜕𝑊𝜈 ∩ 𝜕𝛺) ∩ (𝜕𝑊𝜈 ∩ 𝜕P1 ) = ∅. Since 𝑊𝜈 is not relatively compact in 𝛺, 𝜕𝑊𝜈 ∩ 𝜕𝛺 ≠ ∅. Therefore there is a point 𝑏 𝜈 ∈ 𝜕𝑊𝜈 ∩ 𝜕𝛺 such that there is a piecewise linear Jordan curve 𝐶𝜈 connecting 𝑎 𝜈 and 𝑏 𝜈 , and satisfying 𝐶𝜈 \ {𝑏 𝜈 } ⊂ 𝑊𝜈 (cf. Fig. 3.5). When 𝑏 𝜈 = ∞, a line segment connecting a point 𝑐 𝜈 of 𝑊𝜈 and ∞ is a half-line in ˆ \𝐶𝜈 is simply connected, C with one end at 𝑐 𝜈 . Note that 𝑏 𝜈 ≠ 0 by (3.4.57). Since C we may take a one-valued branch of (3.4.59)

log

𝑧 − 𝑎𝜈 , − 𝑏𝑧𝜈 + 1

ˆ \ 𝐶𝜈 , 𝑧∈C

where, in the case 𝑏 𝜈 = ∞, −𝑧/𝑏 𝜈 = 0. We then set

Fig. 3.5 Jordan curve 𝐶𝜈 .

3.4 Cousin Problem

(3.4.60)

107

Ö

𝐺 2 (𝑧) = 𝐹1 (𝑧)

𝑎𝜈 ∈P2 \P1

𝑧 − 𝑎𝜈 − 𝑏𝑧𝜈 + 1

! 𝑚𝜈 ∈ ℳ(𝛺).

The zeros and poles of 𝐺 2 (𝑧) in 𝛺 are the same as those of 𝑄 2 (𝑧) in 𝛺. Let 𝑈1 be a neighborhood of P1 su ciently close to P1 such that P1 ⋐ 𝑈1 ⋐ P2 ,

© Ø ª 𝑈1 ∩ ­ 𝐶𝜈 ® = ∅. «𝑎𝜈 ∈P2 \P1 ¬

On 𝑈1 we have by (3.4.59) that Í

(3.4.61)

𝑚𝜈 log

𝑧−𝑎 𝜈 𝑧

− +1 𝑏𝜈 𝐺 2 (𝑧) = 𝐹1 (𝑧)𝑒 𝜈 2 1 = 𝐹1 (𝑧)𝑒 ℎ2 (𝑧) , Õ 𝑧 − 𝑎𝜈 ℎ2 (𝑧) := 𝑚 𝜈 log 𝑧 ∈ 𝒪(𝑈1 ). − 𝑏𝜈 + 1 𝑎 ∈P \P

𝑎𝜈 ∈P2 \P1

Let 𝜆2 (𝑧) ∈ 𝒞 0 (𝛺) be such that 𝜆2 (𝑧) = 0 on P1 and 𝜆2 (𝑧) = 1 on 𝛺 \𝑈1 (cf. Exercise 7 at the end of this chapter). Set ( 𝐹1 (𝑧)𝑒 𝜆2 (𝑧) ℎ2 (𝑧) , 𝑧 ∈ 𝑈1 , 𝐹2 (𝑧) = 𝐺 2 (𝑧), 𝑧 ∈ P2 \ 𝑈1 . Then 𝐹2 /𝑄 2 ∈ 𝒞 ∗ (P2 ); that is, 𝐹2 (𝑧) is a topological solution of the Cousin II Problem on P2 . For 𝑎 𝜈 ∈ P3 \ P2 we take 𝑏 𝜈 ∈ 𝜕𝛺 in the same way as above, and set ! 𝑚𝜈 Ö 𝑧 − 𝑎𝜈 (3.4.62) 𝐺 3 (𝑧) = 𝐹2 (𝑧) . − 𝑏𝑧𝜈 + 1 𝑎𝜈 ∈P3 \P2

The zeros and poles of 𝐺 3 (𝑧) are the same as those of 𝑄 3 (𝑧). Let P2 ⋐ 𝑈2 ⋐ P3 be as in the case of 𝑈1 . On 𝑈2 we have Í

(3.4.63)

𝑚𝜈 log

𝑧−𝑎 𝜈 𝑧

− +1 𝑏𝜈 𝐺 3 (𝑧) = 𝐹2 (𝑧)𝑒 𝜈 3 2 = 𝐹2 (𝑧)𝑒 ℎ3 (𝑧) , Õ 𝑧 − 𝑎𝜈 ℎ3 (𝑧) := 𝑚 𝜈 log 𝑧 ∈ 𝒪(𝑈2 ). − 𝑏𝜈 + 1 𝑎 ∈P \P

𝑎𝜈 ∈P3 \P2

Let 𝜆3 (𝑧) ∈ 𝒞 0 (𝛺) be such that 𝜆3 (𝑧) = 0 on P2 and 𝜆3 (𝑧) = 1 on 𝛺 \ 𝑈2 . Set ( 𝐹2 (𝑧)𝑒 𝜆3 (𝑧) ℎ3 (𝑧) , 𝑧 ∈ 𝑈2 , 𝐹3 (𝑧) = 𝐺 3 (𝑧), 𝑧 ∈ P3 \ 𝑈2 .

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3 Domains of Holomorphy

Then 𝐹3 (𝑧) is a topological solution of the Cousin II Problem on P3 such that 𝐹3 | P2 = 𝐹2 on P2 . Inductively, we construct a topological solution 𝐹𝜇 of the Cousin II Problem on P 𝜇 (𝜇 = 1, 2, 3, . . .) such that 𝐹𝜇+1 | P 𝜇 = 𝐹𝜇 . Therefore we have a topological solution 𝐹 (𝑧) = lim 𝜇→∞ 𝐹𝜇 (𝑧) of the Cousin II Problem on 𝛺. Since 𝛺 is holomorphically convex, by Oka’s Theorem 3.4.51 there is a meromrophic function with given zeros and poles on 𝛺. ⊓ ⊔

3.4.8 𝝏¯ -Equation We brie y recall the notion of di erential forms on complex domains (cf., e.g., [39] 3.5.3 for more). Let 𝑧 = (𝑧 1 , . . . , 𝑧 𝑛 ) be the standard complex coordinate system of C𝑛 (for tensor calculus it is better and convenient to use upper indices). Let 𝑥 𝑗 , 𝑦 𝑗 be respectively 𝜕 𝜕 the real and imaginary parts of 𝑧 𝑗 , and let 𝜕𝑥 𝑗 and 𝜕𝑦 𝑗 be the constant vector elds de ned respectively by 𝑥 𝑗 , 𝑦 𝑗 (1 ≤ 𝑗 ≤ 𝑛) with their dual di erential forms 𝑑𝑥 𝑗 , 𝑑𝑦 𝑗 (1 ≤ 𝑗 ≤ 𝑛). We then de ne complex vector elds of type (0, 1) on C𝑛 and their dual di erential forms of type (0, 1) by   𝜕 1 𝜕 1 𝜕 (3.4.64) = − , 1 ≤ 𝑗 ≤ 𝑛, 𝜕 𝑧¯ 𝑗 2 𝜕𝑥 𝑗 𝑖 𝜕𝑦 𝑗 𝑑 𝑧¯ 𝑗 = 𝑑𝑥 𝑗 − 𝑖𝑑𝑦 𝑗 ,

1 ≤ 𝑗 ≤ 𝑛.

We consider a di erential form of type (0, 1) with 𝐶 ∞ function coe cients in an open set 𝑈 of C𝑛 (here, coe cient functions are always of 𝐶 ∞ -class): (3.4.65)

𝑓 = 𝑓1 𝑑 𝑧¯1 + · · · + 𝑓𝑛 𝑑 𝑧¯𝑛 ,

𝑓 𝑗 ∈ 𝒞 ∞ (𝑈).

¯ For a function 𝑢 ∈ 𝒞 ∞ (𝑈) we de ne the 𝜕-operator by ¯ = 𝜕𝑢 𝑑 𝑧¯1 + · · · + 𝜕𝑢 𝑑 𝑧¯𝑛 . 𝜕𝑢 𝜕 𝑧¯𝑛 𝜕 𝑧¯1 For a given di erential from 𝑓 of type (0, 1), a di erential equation with an unknown function 𝑢 is de ned by (3.4.66)

¯ =𝑓: 𝜕𝑢

¯ This is called the 𝜕-equation. The Cauchy Riemann condition (1.1.10) is equivalent to (3.4.67)

¯ = 0. 𝜕𝑢

3.4 Cousin Problem

109

If there is a solution 𝑢 ∈ 𝐶 ∞ (𝑈) of (3.4.66), then 𝜕 𝑓𝑗 𝜕2𝑢 = 𝑘, 𝑘 𝑗 𝜕 𝑧¯ 𝜕 𝑧¯ 𝜕 𝑧¯

𝜕𝑢 = 𝑓𝑗, 𝜕 𝑧¯ 𝑗 Since

𝜕2 𝑢 𝜕 𝑧¯ 𝑘 𝜕 𝑧¯ 𝑗

(3.4.68)

=

𝜕2 𝑢 , 𝜕 𝑧¯ 𝑗 𝜕 𝑧¯ 𝑘

1 ≤ 𝑗, 𝑘 ≤ 𝑛.

we have 𝜕 𝑓 𝑗 𝜕 𝑓𝑘 = , 𝜕 𝑧¯ 𝑘 𝜕 𝑧¯ 𝑗

1 ≤ 𝑗, 𝑘 ≤ 𝑛.

¯ This is a necessary condition for 𝜕-equation (3.4.66) to have a solution 𝑢, which is called the integrable condition of (3.4.66). A di erential form of type (0, 2) is de ned by Õ  𝜕 𝑓𝑘 𝜕 𝑓 𝑗  ¯ 𝜕𝑓 = − 𝑘 𝑑 𝑧¯ 𝑗 ∧ 𝑑 𝑧¯ 𝑘 . 𝑗 𝜕 𝑧 ¯ 𝜕 𝑧¯ 1≤ 𝑗 𝛺. Then there is a Proof. Let 𝐾 ⋐ 𝛺 be a compact subset. Suppose that 𝐾 b sequence of points 𝑎 𝜈 ∈ 𝐾𝛺 , 𝜈 = 1, 2, . . . with no accumulation point in 𝛺. By the assumption there is a function 𝐹 ∈ 𝒪(𝛺) such that 𝐹 (𝑎 𝜈 ) = 𝜈,

𝜈 = 1, 2, . . . .

On the other hand, |𝐹 (𝑎 𝜈 )| ≤ ∥𝐹 ∥ 𝐾 < ∞,

𝜈 = 1, 2, . . . . ⊓ ⊔

This is absurd.

A closed discrete subset 𝑇 of 𝛺 is a special case of complex submanifold (without connectedness) of dimension 0, and a map 𝑓 : 𝑎 𝜈 ∈ 𝑇 → 𝐴𝜈 ∈ C is regarded as a holomorphic function on 𝑇. From this viewpoint we prove a generalized interpolation by Oka’s Joku-Iko Principle: Theorem 3.5.4 (Generalized Interpolation). Let 𝛺 ⊂ C𝑛 be a holomorphically convex domain (or equivalently a domain of holomorphy). Let 𝑆 ⊂ 𝛺 be a complex submanifold. Then the restriction map 𝒪(𝛺) ∋ 𝐹 ↦→ 𝐹 | 𝑆 ∈ 𝒪(𝑆) is surjective; i.e., the following is exact: 𝒪(𝛺) ∋ 𝐹 ↦−→ 𝐹 | 𝑆 ∈ 𝒪(𝑆) −→ 0.

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3 Domains of Holomorphy

Proof. Take a function 𝑓 ∈ 𝒪(𝑆). (1) Let P (⋐ 𝛺) be an analytic polyhedral domain. We rst nd a function 𝐹 ∈ 𝒪(P) with 𝐹 | 𝑆∩P = 𝑓 | 𝑆∩P . Consider the Oka map 𝜑P : P −→ P𝛥. Then 𝜑P (𝑆 ∩ P) is a complex submanifold of P𝛥. By the Oka Extension Theorem 2.5.14 there is a function 𝐺 ∈ 𝒪(P𝛥) such that 𝐺 (𝜑P (𝑧)) = 𝑓 (𝑧) (𝑧 ∈ 𝑆 ∩ P). Then 𝐹 (𝑧) := 𝐺 (𝜑P (𝑧)) ∈ 𝒪(P), which is the required one. (2) By Proposition 3.3.19 there is an increasing sequence of analytic polyhedral domains of 𝛺, ∞ Ø P1 ⋐ P2 ⋐ · · · ⋐ P 𝜇 ⋐ · · · , P 𝜇 = 𝛺. 𝜇=1

For the Oka map of each P 𝜇 we write 𝜑 𝜇 : P 𝜇 → P𝛥 𝜇 . We may assume that every 𝜑 𝜇 is de ned in a slightly larger analytic polyhedral domain P˜ 𝜇 ⋑ P 𝜇 with an Oka map f 𝜇 (⋑ P𝛥 𝜇 ). 𝜑 𝜇 : P˜ 𝜇 → P𝛥 From the result of (1) above we obtain (3.5.5)

𝐹𝜇′ ∈ 𝒪( P˜ 𝜇 ),

𝐹𝜇′ | 𝑆∩P˜ 𝜇 = 𝑓 | 𝑆∩P˜ 𝜇 ,

𝜇 = 1, 2, . . . .

f 𝜇 , it follows from Since the image 𝜑 𝜇 (𝑆 ∩ P˜ 𝜇 ) is a complex submanifold of P𝛥 the Geometric Syzygy Lemma 2.5.11 (i) that there is a nite generator system of ℐ⟨ 𝜑 𝜇 (𝑆 ∩ P˜ 𝜇 )⟩ on every P𝛥 𝜇 , 𝜎𝜇ℎ ∈ 𝛤 (P𝛥 𝜇 , ℐ⟨ 𝜑 𝜇 (𝑆 ∩ P˜ 𝜇 )⟩),

1 ≤ ℎ ≤ 𝑙𝜇.

′ = 𝜑∗ 𝜎 The pull-backs 𝜎𝜇ℎ 𝜇 𝜇ℎ by 𝜑 𝜇 form a nite generator system of ℐ⟨ 𝑆⟩ on P 𝜇 : ′ 𝜎𝜇ℎ ∈ 𝛤 (P 𝜇 , ℐ⟨ 𝑆⟩),

1 ≤ ℎ ≤ 𝑙𝜇.

′ ∈ 𝒪(P ). In order to avoid the complication of symbols we identify them with 𝜎𝜇ℎ 𝜇

We inductively de ne 𝐹𝜇 ∈ 𝒪(P 𝜇 ). First, set 𝐹1 = 𝐹1′ . Suppose that 𝐹1 , . . . , 𝐹𝜇 are determined so that (3.5.6)

𝐹𝜈 | P𝜈 ∩𝑆 = 𝑓 | P𝜈 ∩𝑆 , ∥𝐹𝜈 − 𝐹𝜈−1 ∥ P𝜈−1
0 : P𝛥( 𝑝; 𝑠𝑟) ⊂ 𝔇} > 0

(resp. 𝛿B(𝑅) ( 𝑝, 𝜕𝔇) is de ned similarly). We call 𝛿P𝛥 ( 𝑝, 𝜕𝔇) (resp. 𝛿B(𝑅) ( 𝑝, 𝜕𝔇)) the boundary distance function of 𝔇 with respect to P𝛥 (resp. B(𝑅)). As in (3.2.2), (3.6.5)

|𝛿P𝛥 ( 𝑝 ′ , 𝜕𝔇) − 𝛿P𝛥 ( 𝑝 ′′ , 𝜕𝔇)| ≤ ∥𝜋( 𝑝 ′ ) − 𝜋( 𝑝 ′′ )∥ P𝛥 , 𝑝 ′ , 𝑝 ′′ ∈ P𝛥( 𝑝; 𝛿P𝛥 ( 𝑝, 𝜕𝔇)).

Therefore, 𝛿P𝛥 ( 𝑝, 𝜕𝔇) is continuous; similarly, so is 𝛿B(𝑅) ( 𝑝, 𝜕𝔇). (b) Holomorphic functions in domains over C𝑛 . Let 𝜋 : 𝔇 → C𝑛 be a domain. For 𝑝 ∈ 𝔇 the points 𝑞 of P𝛥( 𝑝; 𝛿P𝛥 ( 𝑝, 𝜕𝔇)) are uniquely parametrized by 𝜋(𝑞) = 𝑧 = (𝑧 𝑗 )1≤ 𝑗 ≤𝑛 ∈ P𝛥(𝜋( 𝑝); 𝛿P𝛥 ( 𝑝, 𝜕𝔇)𝑟), which we use for a local complex coordinate system. We write 𝑝(𝑧) for a point 𝑝 ∈ 𝔇 with the local complex coordinate system 𝜋( 𝑝) = 𝑧 = (𝑧1 , . . . , 𝑧 𝑛 ) ∈ C𝑛 . Let 𝑊 ⊂ 𝔇 be an open set. It makes sense to say that a function 𝑓 ( 𝑝) de ned in 𝑊 is holomorphic if 𝑓 ( 𝑝(𝑧)) is holomorphic locally in 𝑧. By de nition 𝜋 is a vectorvalued holomorphic function on 𝔇. We denote by 𝒪(𝑊) the set of all holomorphic functions in 𝑊. We de ne functions of 𝐶 𝑘 -class (0 ≤ 𝑘 ≤ ∞) in 𝑊 in the same way as above, and denote all of them by 𝒞 𝑘 (𝑊). All local properties of holomorphic functions in an open set of C𝑛 also hold for holomorphic functions in 𝑊. Let 𝑓 ∈ 𝒪(P𝛥( 𝑝(𝑎);𝑟)). Then 𝑓 ( 𝑝(𝑧)) is expanded as a function of 𝑧 to a power series with center at 𝑧 = 𝑎: Õ (3.6.6) 𝑓 ( 𝑝(𝑧)) = 𝑐 𝛼 (𝑧 − 𝑎) 𝛼 . 𝛼∈Z+𝑛

118

3 Domains of Holomorphy

We denote the power series above by 𝑓 : 𝑝

𝑓 =

(3.6.7)

𝑝

Õ 𝛼∈Z+𝑛

𝑐 𝛼 (𝑧 − 𝑎) 𝛼 .

We take a family ℱ(≠ ∅) ⊂ 𝒪(𝔇) and x it. Definition 3.6.8 (Holomorphic separation). A domain 𝜋 : 𝔇 → C𝑛 is said to be ℱseparable if for arbitrary two distinct points 𝑝, 𝑞 ∈ 𝔇 there is an element 𝑓 ∈ ℱ with 𝑓 ( 𝑝) ≠ 𝑓 (𝑞): In the case of ℱ = 𝒪(𝔇), 𝔇 is said to be holomorphically separable. Since 𝜋 is a vector-valued holomorphic function on 𝔇, the holomorphic separability condition makes sense for the case of 𝜋( 𝑝) = 𝜋(𝑞). Proposition 3.6.9. For a domain 𝜋 : 𝔇 → C𝑛 the following two conditions are equivalent. (i) 𝔇 is holomorphically separable. (ii) For arbitrary distinct points 𝑝, 𝑞 ∈ 𝔇 over 𝑎 ∈ C𝑛 , there is a function 𝑓 ∈ 𝒪(𝔇) with 𝑓 ≠ 𝑓 . 𝑝

𝑞

Proof. (i) ⇒ (ii): This is clear. (ii) ⇒ (i): As a holomorphic partial di erential operator 𝜕 𝛼 is de ned in (1.1.8), 𝛼 𝜕 is naturally de ned for 𝑓 ∈ 𝒪(𝔇) with respect to complex local coordinate 𝑧. It follows from the assumption that for distinct 𝑝, 𝑞 ∈ 𝔇 with 𝜋( 𝑝) = 𝜋(𝑞) there is an element 𝑓 ∈ 𝒪(𝔇) with 𝑓 ≠ 𝑓 ∈ 𝒪𝜋 ( 𝑝) ; that is, there is a holomorphic partial 𝑝 𝑞 derivation 𝜕 𝛼 such that 𝜕 𝛼 𝑓 ( 𝑝) ≠ 𝜕 𝛼 𝑓 (𝑞), and 𝜕 𝛼 𝑓 ∈ 𝒪(𝔇). ⊔ ⊓ Let 𝜌 : 𝔊 → C𝑚 be another domain over C𝑚 . A continuous map 𝜓 : 𝔇 → 𝔊 is called a holomorphic map if 𝜌 ◦ 𝜓 : 𝔇 → C𝑚 is a vector-valued holomorphic function. In that case, the pull-back by 𝜓 𝜓 ∗ : 𝑔 ∈ 𝒪(𝔊) → 𝑔 ◦ 𝜓 ∈ 𝒪(𝔇) is de ned, and 𝜓 ∗ is a (ring) homomorphism: that is, for 𝑔, ℎ ∈ 𝒪(𝔊), 𝜓 ∗ (𝑔 + ℎ) = 𝜓 ∗ 𝑔 + 𝜓 ∗ ℎ,

𝜓 ∗ (𝑔 · ℎ) = 𝜓 ∗ 𝑔 · 𝜓 ∗ ℎ.

In the case where 𝜓 ∗ is b ective, 𝜓 ∗ is called a (ring) isomorphism. If 𝑛 = 𝑚 and 𝜋 = 𝜌 ◦ 𝜓, 𝜓 : 𝔇 → 𝔊 is called a relative map over C𝑛 . A relative map over C𝑛 is holomorphic. If a relative map 𝜓 : 𝔇 → 𝔊 over C𝑛 is b ective, the inverse 𝜓 −1 is again a relative map over C𝑛 , and 𝜓 is called a relative isomorphism over C𝑛 ; also 𝔇 and 𝔊 are said to be relatively isomorphic over C𝑛 . (c) Analytic continuation and envelope of holomorphy. Let 𝜋 : 𝔇 → C𝑛 be a domain. Let 𝜓 : 𝔇 → 𝔊 be a relative map from 𝔇 into another domain 𝔊 over C𝑛 . If 𝑓 ∈ 𝒪(𝔇) and 𝑔 ∈ 𝒪(𝔊) satis es 𝜓 ∗ 𝑔 = 𝑓 , 𝑓 is said to be analytically continued to 𝑔. If 𝜓 ∗𝒪(𝔊) = 𝒪(𝔇), 𝔊 is called a holomorphic extension of 𝔇.

3.6 Unrami ed Domains over C𝑛

119

Let ℱ ⊂ 𝒪(𝔇) be a non-empty family. We would like to obtain a maximal domain, over which all functions of ℱ are analytically continued simultaneously; that is, if ℱ = 𝒪(𝔇), it is a maximal domain among all the holomorphic extensions of 𝔇. We x a point 𝑝 0 ∈ 𝔇, and take a curve 𝐶 in C𝑛 with the initial point 𝑎 = 𝜋( 𝑝 0 ). For a function 𝑓 ∈ ℱ we consider the analytic function 𝑓 ( 𝑝(𝑧)) of 𝑧 about 𝑎, and denote by 𝛤 the set of all curves 𝐶 such that 𝑓 ( 𝑝(𝑧)) is analytically continued along 𝐶. Let 𝐶 𝑏 ∈ 𝛤 be a curve with the terminal point 𝑏 ∈ C𝑛 . Then an analytic function 𝑓 ( 𝑝(𝑧)) ( 𝑓 ∈ ℱ) de ned about 𝑎 is analytically continued along 𝐶 𝑏 to an analytic function 𝑓𝐶 𝑏 (𝑧) about 𝑏. If 𝐶 𝑏 and 𝐶 ′𝑏 ∈ 𝛤 are mutually homotopic within 𝛤, 𝑓𝐶 𝑏 = 𝑓𝐶 ′𝑏 ; in this sense, {𝐶 𝑏 } denotes the homotopy class of 𝐶. We write 𝑏 𝑏 𝑓 {𝐶 𝑏 } := 𝑓𝐶 𝑏 . 𝑏

𝑏

Fix a polydisk P𝛥 with center at the origin in C𝑛 . Let 𝑠({𝐶 𝑏 }, 𝑓 ) be the supremum of 𝑠 > 0 such that the power expansion of 𝑓 {𝐶 𝑏 } (𝑧) at 𝑧 = 𝑏 converges in 𝑏 + 𝑠P𝛥 (𝑟 > 𝑏

0). We write 𝛤 † for the set of all {𝐶 𝑏 } with inf 𝑓 ∈ℱ 𝑠({𝐶 𝑏 }, 𝑓 ) > 0. ′ For two elements {𝐶 𝑏 }, {𝐶 ′𝑏 } of 𝛤 † we de ne an equivalence relation {𝐶 𝑏 } ∼ ′ ′𝑏 {𝐶 } by 𝑏 = 𝑏′ , 𝑓 {𝐶 𝑏 } = 𝑓 {𝐶 ′𝑏′ } , ∀ 𝑓 ∈ ℱ. 𝑏

𝑏′

[{𝐶 𝑏 }].

The equivalence class is denoted by The quotient set and the natural map are written as ˆ = 𝛤 † /∼, 𝜋ˆ : [{𝐶 𝑏 }] ∈ 𝔇 ˆ → 𝑏 ∈ C𝑛 . 𝔇 ˆ → C𝑛 is a domain. Since 𝔇 is arc-wise connected, 𝔇 ˆ is By the construction, 𝜋ˆ : 𝔇 ˆ over C𝑛 is independent of the choice of 𝑝 0 ∈ 𝔇, and the relative map 𝜂 : 𝔇 → 𝔇 naturally de ned. ˆ → C𝑛 is called the Definition 3.6.10. (i) The above-constructed domain 𝜋ˆ : 𝔇 ℱ-envelope of 𝔇. (ii) When ℱ = 𝒪(𝔇), the 𝒪(𝔇)-envelope of 𝔇 is called the envelope of holomorphy of 𝔇. (iii) When ℱ consists of one function 𝑓 ∈ 𝒪(𝔇), the { 𝑓 }-envelope of 𝔇 is called the domain of existence of 𝑓 . (iv) A domain 𝔇/C𝑛 is a domain of holomorphy, if 𝔇 is the envelope of holomorphy of itself. Remark 3.6.11. (i) The domains de ned in the above (i) (iv) are all holomorphically separable. ˆ (ℱ-envelope) over C𝑛 is injective if and only if (ii) The relative map 𝜂 : 𝔇 → 𝔇 ˆ 𝔇 is ℱ-separable; in this case, 𝔇 ⊂ 𝔇. Therefore we deduce: Theorem 3.6.12. Let 𝜋 : 𝔇 → C𝑛 be a holomorphically separable domain. ˆ → C𝑛 containing 𝔇 as a subdo(i) 𝔇 carries the envelope of holomorphy 𝜋ˆ : 𝔇 main with 𝜋| ˆ 𝔇 = 𝜋. In particular, a univalent domain 𝛺 (⊂ C𝑛 ) has necessarily the envelope of holomorphy of 𝛺.

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3 Domains of Holomorphy

ˆ (ii) 𝒪(𝔇) = { 𝑓 | 𝔇 : 𝑓 ∈ 𝒪( 𝔇)}. ′ ′ 𝑛 (iii) Let 𝜋 : 𝔇 → C be a holomorphically separable domain which contains 𝜋 : 𝔇 → C𝑛 and 𝜋 ′ | 𝔇 = 𝜋. Assume that 𝒪(𝔇) = { 𝑓 | 𝔇 : 𝑓 ∈ 𝒪(𝔇′ )}. Then there ˆ (𝜋 ′ = 𝜋ˆ ◦ 𝜄) over C𝑛 . is a relative injection 𝜄 : 𝔇′ → 𝔇 ˆ Proof. (i), (ii): These are immediate by the construction of 𝔇. (iii): It follows from the arc-wise connectedness of a domain over C𝑛 and the ˆ construction of the envelope of holomorphy 𝔇. ⊓ ⊔ Example 3.6.13. We give an example such that 𝜂 in Remark 3.6.11 (ii) is not injective (cf. [39] Example 7.5.4). In the unit polydisk P𝛥 = 𝛥(1) 2 (⊂ C2 ) we set (describe the gure by oneself)  𝐾 = (𝑧1 , 𝑧 2 ) ∈ P𝛥 : 14 ≤ |𝑧 𝑗 | ≤ 34 , 𝑗 = 1, 2 , 𝛺 = P𝛥 \ 𝐾,      𝛺 H = 𝛥(1) × 𝛥 14 ∪ { 34 < |𝑧 1 | < 1} × 𝛥(1) ,  2     𝜔 = 𝛥 14 , 𝑈 = 𝛥(1) × 𝛥 14 , 𝑉 = 𝛥 14 × 𝛥(1). Then 𝑈 and 𝑉 are subdomains of 𝛺, and 𝑈 ∩𝑉 = 𝜔. We distinguish 𝜔 as a part of 𝑈 and that as a part of 𝑉, so that we obtain a 2-sheeted domain 𝜋 : 𝔇 → 𝛺 ⊂ C2 . Note that 𝛺 H (⊂ 𝔇) is a Hartogs domain, of which the envelope of holomorphy is P𝛥. ˆ = P𝛥, and hence 𝜂 : 𝔇 → P𝛥 is Therefore, the envelope of holomorphy of 𝔇 is 𝔇 not injective. There is no domain of holomorphy which contains 𝔇 as a subdomain. Remark 3.6.14. As seen in Example 3.6.1, if 𝑛 ≥ 2, there is a univalent domain in C𝑛 such that all holomorphic functions are analytically continued over an in nitely sheeted unrami ed domain over C𝑛 . Therefore it is theoretically necessary to deal with at least unrami ed domains over C𝑛 , multivalent in general. It is incomplete as a theory to deal only with univalent domains of C𝑛 : This is an important viewpoint of analytic function theory of several variables. (d) 𝜎-compact. Let 𝜋 : 𝔇 → C𝑛 be a domain. We take the euclidean metric on

C𝑛 ,

𝑑𝑠2 =

𝑛  Õ 𝑗=1

 𝑑𝑥 2𝑗 + 𝑑𝑦 2𝑗 ,

𝑧 𝑗 = 𝑥 𝑗 + 𝑖𝑦 𝑗 .

Through the local biholomorphism 𝜋 we lift it to a Riemann metric on 𝔇, denoted by 𝜋 ∗ 𝑑𝑠2 , which is called the euclidean metric on 𝔇. For a curve 𝐶 (𝜙 : [𝑡 0 , 𝑡 1 ] → 𝔇) of piecewise 𝐶 1 -class in 𝔇 we de ne the length with respect to 𝜋 ∗ 𝑑𝑠2 by v tÕ ∫ ∫ 𝑡1 u 𝑛   𝐿 (𝐶) = 𝜋 ∗ 𝑑𝑠 = (𝜙′𝑗1 (𝑡)) 2 + (𝜙′𝑗2 (𝑡)) 2 𝑑𝑡, 𝐶

𝑡0

𝑗=1

𝜙 𝑗 (𝑡) = 𝜙 𝑗1 (𝑡) + 𝑖𝜙 𝑗2 (𝑡)

(local expression).

Fix a point 𝑝 0 ∈ 𝔇 and a polydisk P𝛥 = P𝛥(𝑟) with polyradius 𝑟. With the boundary distance function 𝛿P𝛥 ( 𝑝, 𝜕𝔇) of 𝔇 we de ne

3.6 Unrami ed Domains over C𝑛

121

𝔇𝜌 = {𝑝 ∈ 𝔇 : 𝛿P𝛥 ( 𝑝, 𝜕𝔇) > 𝜌} for 𝜌 > 0. We set 𝑈 𝜎 ( 𝑝) = P𝛥( 𝑝; 𝜎𝑟),

0 < 𝜎 ≤ 𝛿P𝛥 ( 𝑝, 𝜕𝔇).

Choose 𝜌 > 0 so that 𝔇𝜌 ∋ 𝑝 0 . It is noticed that every point 𝑝 of the connected component of 𝔇𝜌 containing 𝑝 0 is connected by a piecewise 𝐶 1 curve 𝐶 ( 𝑝) in 𝔇. Set (3.6.15)

𝑑 𝜌 ( 𝑝) =

inf

𝐶 ( 𝑝) ⊂𝔇𝜌

𝐿(𝐶 ( 𝑝)).

As easily checked, the function 𝑑 𝜌 ( 𝑝) satis es the Lipschitz continuity condition: (3.6.16)

|𝑑 𝜌 ( 𝑝 ′ ) − 𝑑 𝜌 ( 𝑝 ′′ )| ≤ ∥𝜋( 𝑝 ′ ) − 𝜋( 𝑝 ′′ )∥ = ∥ 𝑝 ′ − 𝑝 ′′ ∥, 𝑝 ′ , 𝑝 ′′ ∈ 𝑈 𝛿P𝛥 ( 𝑝,𝜕𝔇) ( 𝑝).

Here we identify points 𝑝 ′ , 𝑝 ′′ contained in a univalent P𝛥( 𝑝; 𝛿P𝛥 ( 𝑝, 𝜕𝔇)𝑟) with 𝜋( 𝑝 ′ ), 𝜋( 𝑝 ′′ ), respectively. In this way, unless confusion occurs, we write a point of a univalent domain of 𝔇 as a point of C𝑛 for the sake of notational simplicity. Lemma 3.6.17. We have {𝑝 ∈ 𝔇𝜌 : 𝑑 𝜌 ( 𝑝) < 𝑏} ⋐ 𝔇 for every 𝑏 > 0. Proof. Without loss of generality we may assume B ⊂ P𝛥. For 𝑏 = 𝜌 we have {𝑝 ∈ 𝔇𝜌 : 𝑑 𝜌 ( 𝑝) < 𝜌} ⋐ 𝑈 𝛿P𝛥 ( 𝑝0 ,𝜕𝔇) ( 𝑝 0 ). Therefore, {𝑝 ∈ 𝔇𝜌 : 𝑑 𝜌 ( 𝑝) < 𝜌} ⋐ 𝔇, and so the assertion holds for 𝑏 = 𝜌. ¯𝜌: Now, suppose that the assertion holds for a number 𝑏 ≥ 𝜌; i.e., 𝐾 := {𝑝 ∈ 𝔇 ¯ 𝑑 𝜌 ( 𝑝) ≤ 𝑏} is compact. For every 𝑝 ∈ 𝐾, we get 𝑈𝜌/2 ( 𝑝) ⋐ 𝔇 and see that Ø 𝐾′ = 𝑈¯ 𝜌/2 ( 𝑝) 𝑝∈𝐾

is compact. For, if we take a sequence of points 𝑞 𝜈 ∈ 𝐾 ′ , 𝜈 ∈ N, then there are points 𝑝 𝜈 ∈ 𝐾 and 𝑤𝜈 ∈ 𝜌2 P𝛥 such that 𝑞 𝜈 = 𝑝 𝜈 + 𝑤𝜈 ,

𝜈 ∈ N.

Since 𝐾 is compact, after taking a subsequence we deduce that lim𝜈→∞ 𝑝 𝜈 = 𝑥0 ∈ 𝐾 and lim𝜈→∞ 𝑤𝜈 = 𝑤0 with 𝑤0 ∈ 𝜌2 P𝛥. It follows that lim 𝑞 𝜈 = 𝑥 0 + 𝑤0 ∈ 𝐾 ′ .

𝜈→∞

Since {𝑝 ∈ 𝔇𝜌 : 𝑑 𝜌 ( 𝑝) < 𝑏 + 𝜌/2} ⊂ 𝐾 ′ , the assertion holds for 𝑏 + 𝜌/2. Inductively, it holds for 𝜌 + 𝜈𝜌/2, 𝜈 = 1, 2, . . .; hence it holds for all 𝑏 > 0. ⊓ ⊔ The following property is said to be 𝜎-compact.

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3 Domains of Holomorphy

Proposition 3.6.18. For a domain 𝜋 : 𝔇 → C𝑛 , there is an open covering {𝔇𝜈 }∞ 𝜈=1 by an increasing sequence of subdomains 𝔇𝜈 of 𝔇 such that 𝔇𝜈 ⋐ 𝔇𝜈+1 ,

𝔇=

∞ Ø

𝔇𝜈 .

𝜈=1

Proof. We use the notation in the proof of Lemma 3.6.17. For a number 𝑠 > 0 we denote by 𝔇𝜌.𝑠 the set of all points 𝑝 ∈ 𝔇𝜌 which is connected with 𝑝 0 by a curve 𝐶 of piecewise 𝐶 1 -class in 𝔇𝜌 with length 𝐿(𝐶) < 𝑠. Then 𝔇𝜌,𝑠 is naturally connected, and 𝔇𝜌,𝑠 ⋐ 𝔇 by Lemma 3.6.17. Taking sequences 𝜌 𝜈 ↘ 0 and 𝑠 𝜈 ↗ ∞ (𝜈 = 1, 2, . . .), we obtain ∞ Ø 𝔇𝜈 := 𝔇𝜌𝜈 ,𝑠𝜈 ⋐ 𝔇𝜈+1 , 𝔇𝜈 = 𝔇. ⊓ ⊔ 𝜈=1

3.7 Stein Domains over C 𝒏 K. Oka dealt with the Approximation Problem and the Cousin Problem as well on unrami ed domains over C𝑛 in his unpublished ve papers of 1943 which historically rst solved a rmatively the Pseudoconvexity Problem in general dimension,6 and in the published paper Oka IX (1953) which was a revised version of the unpublished papers above, by making use of a part of the coherence theorems obtained in Oka VII and VIII. In the present section we see how the main results on univalent domains of C𝑛 extend to unrami ed domains over C𝑛 . Let 𝜋 : 𝔇 → C𝑛 be a domain. The notions of a holomorphically convex hull and holomorphic convexity are de ned in the same way as in the case of univalent domains ( 3.1). For example, 𝔇 is holomorphically convex if for every compact subset 𝐾 ⋐ 𝔇, b𝔇 := {𝑝 ∈ 𝔇 : | 𝑓 ( 𝑝)| ≤ ∥ 𝑓 ∥ 𝐾 , ∀ 𝑓 ∈ 𝒪(𝔇)} ⋐ 𝔇. 𝐾 Definition 3.7.1 (Stein domain). A domain 𝔇/C𝑛 is called a Stein domain if: (i) 𝔇 is holomorphically separable, and (ii) 𝔇 is holomorphically convex. An open set 𝛺 of 𝔇 is said to be Stein if every connected component of 𝛺 is a Stein domain. The results shown in 3.2 are stated for a domain 𝔇 over C𝑛 as follows: The proofs are the same. For example: 3.7.2.

The Cartan–Thullen Lemma 3.2.7 holds for 𝔇.

6 The case of univalent 2-dimensional domains had been a rmatively solved by Oka [47] (1941) and Oka VI (1942).

3.7 Stein Domains over C𝑛

123

By the proof of Theorem 3.2.11 we have: Theorem 3.7.3 (Cartan Thullen). A Stein domain is a domain of holomorphy. In the same way as we obtained Corollary 3.2.12 we have: Corollary 3.7.4. Let 𝔇/C𝑛 be a Ñ domain. Let {𝛺 𝛾 } 𝛾 ∈𝛤 be a family of Stein open subsets of 𝔇. Then the interior of 𝛾 ∈𝛤 𝛺 𝛾 is Stein. Remark 3.7.5. The converse in Theorem 3.7.3 holds in general, but we have to wait for the solution of the Pseudoconvexity Problem by Oka (see Theorem 5.3.2). The ow of the proof is as follows: 𝔇/C𝑛 , domain of holomorphy ⇒ 𝔇/C𝑛 , pseudoconvex ⇒ 𝔇/C𝑛 , Stein. Here is an essential di erence between the univalent case and the multivalent one. An analytic polyhedron of 𝔇 is de ned as in De nition 3.3.1 with 𝛺 = 𝔇. In the de nition of Oka map (3.3.3), however, with an analytic polyhedron P ⋐ 𝔇 de ned by 𝜑 𝑗 ∈ 𝒪(𝔇) (1 ≤ 𝑘 ≤ 𝑙) we add a condition such that
(3.7.6) 𝛷P : 𝑝(𝑧) ∈ P → 𝑧, 𝜑1 ( 𝑝(𝑧)), . . . , 𝜑𝑙 ( 𝑝(𝑧)) ∈ P𝛥, P𝛥 := P𝛥((𝑟 𝑗 )) × P𝛥((𝜌 𝑗 )) (⊂ C𝑛 × C𝑙 ), is injective. This is possible if 𝔇 is holomorphically separable (in particular, a domain of holomorphy: Cf. Remark 3.6.11). Then, 𝛷P (P) is a submanifold (not necessarily connected) of P𝛥. Theorem 3.7.7 (Oka Weil Approximation). Let 𝔇/C𝑛 be a holomorphically sepab𝔇 ⋐ 𝔇 be a holomorphically convex set. Then a holomorphic rable domain. Let 𝐾 = 𝐾 function in a neighborhood of 𝐾 can be uniformly approximated on 𝐾 by elements of 𝒪(𝔇). The proof is the same as that of Theorem 3.3.12. The notion of a Runge pair (𝔇′ , 𝔇) for a domain 𝔇/C𝑛 and a subdomain 𝔇′ ⊂ 𝔇 is de ned as in De nition 3.3.16. Theorem 3.7.8. Let 𝔇/C𝑛 be a domain. Ð Let 𝔇𝜈 (𝜈 ∈ N) be an increasing sequence of subdomains 𝔇𝜈 ⊂ 𝔇𝜈+1 with 𝔇 = ∞ 𝜈=1 𝔇 𝜈 . If for every 𝜈 ∈ N, 𝔇 𝜈 is Stein and (𝔇𝜈 , 𝔇𝜈+1 ) is a Runge pair, then 𝔇 is Stein. The proof is left to Exercise 13 at the end of this chapter. The following main results hold for domains over C𝑛 : Theorem 3.7.9 (Oka). Let 𝔇/C𝑛 be a Stein domain. (i) The Continuous Cousin Problem on 𝔇 is always solvable (Theorem 3.4.22). (ii) The Cousin I Problem on 𝔇 is always solvable (Theorem 3.4.31). (iii) The Cousin II Problem on 𝔇 is analytically solvable, if and only if it is topologically solvable (Theorem 3.4.51).

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3 Domains of Holomorphy

¯ ¯ = 𝑓 , for a function 𝑢 on 𝔇 with a 𝐶 ∞ differential form 𝑓 (iv) The 𝜕-equation, 𝜕𝑢 ¯ of type (0, 1) has a solution if 𝑓 is 𝜕-closed (Oka Dolbeault Theorem 3.4.75). (v) The general Interpolation Problem on 𝔇 is always solvable (Theorem 3.5.4). Similarly to Proposition 3.5.3 we have: Proposition 3.7.10. A domain 𝔇/C𝑛 is Stein if and only if an arbitrary Interpolation Problem on 𝔇 is solvable. Remark 3.7.11. It is noted that up to the present argument, Theorem 3.7.9 or Proposition 3.7.10 is not yet proved for domains of holomorphy over C𝑛 (cf. Remark 3.7.5). The notion of domains of holomorphy is natural from the viewpoint of analytic continuation, but di cult to deal with. On the other hand, the notion of holomorphic separability or holomorphic convexity is characterized only by the interior of the domain, so that it is easy for the abstraction. In fact the notion of Stein manifolds (cf. [39] De nition 4.5.7), consisting of holomorphic separability and holomorphic convexity, was introduced for abstract complex manifolds by K. Stein; the term “Stein domain” comes conversely from “Stein manifold”. It is not the case that the term “Stein domain” is used in Oka’s papers. Now, a “Stein neighborhood” or “Stein ∗ ∗” is a commonly used convenient wording, and so we follow it.

3.8 Supplement: Ideal Boundary Let 𝜋 : 𝔇 → C𝑛 be a domain. As 𝔇 is de ned an abstract Hausdor topological space, 𝔇 is the total space by itself, and the boundary may not be considered. But, if we intuitively think of it as a space spreading over C𝑛 , we may consider its boundary. Here we will formulate it mathematically. The content of this section is not absolutely necessary to read the present book, but is a part of the theoretical background. Also for those who have a concern about the boundary of 𝔇 it may be helpful for a better understanding. We would like to de ne the “boundary” of 𝔇 by the information from the inside and 𝜋. A family 𝛶˜ = {𝛶𝜈 }∞ 𝜈=1 7 of subsets of 𝔇 is a filter base if: (i) every 𝛶𝜈 ≠ ∅; (ii) for every 𝛶𝜈1 and 𝛶𝜈2 of 𝛶˜ there exists an element 𝛶𝜈3 ∈ 𝛶˜ such that 𝛶𝜈3 ⊂ 𝛶𝜈1 ∩𝛶𝜈2 . Example 3.8.1. A countable fundamental neighborhood system of a point of a topological space is a lter base. Two lter bases 𝛶˜ and 𝛶˜′ are de ned to be mutually equivalent (𝛶˜ ∼ 𝛶˜′ ) if for every element 𝛶𝜈 of 𝛶˜ (resp. 𝛶𝜈′ of 𝛶˜′ ) there is an element of 𝛶˜′ (resp. 𝛶˜ ) contained in 𝛶𝜈 (resp. 𝛶𝜈′ ); this de nes, in fact, an equivalence relation, and the equivalence class is denoted by [𝛶˜ ]. 7 The index set may be uncountable in general, but here it is su cient to assume it countable.

3.8 Supplement: Ideal Boundary

125

Definition 3.8.2. An equivalence class [𝛶˜ ] of a lter base with 𝛶˜ = {𝛶𝜈 } 𝜈 ∈N is an ideal boundary point of 𝔇 (more precisely, the triple (𝔇, 𝜋, C𝑛 )) if the following conditions are satis ed: (i) All 𝛶𝜈 are connected. (ii) 𝛶˜ = {𝛶𝜈 }∞ 𝜈=1 has no accumulation point in 𝔇; that is, for any point 𝑝 ∈ 𝔇 there is a neighborhood 𝑈 ( 𝑝) such that 𝛶𝜈 ∩ 𝑈 ( 𝑝) = ∅ except for nitely many 𝛶𝜈 . (iii) (a) 𝜋(𝛶𝜈 ) (𝜈 = 1, 2, . . .) has a unique accumulation point 𝑧 0 ∈ C𝑛 , or (b) 𝜋(𝛶𝜈 ) (𝜈 = 1, 2, . . .) diverges to in nity; i.e., for every 𝑅 > 0, 𝜋(𝛶𝜈 ) ∩ B(𝑅) = ∅ except for nitely many 𝛶𝜈 . (iv) In the case of (a) above, there is a fundamental neighborhood system {𝑉𝜈 }∞ 𝜈=1 of 𝑧0 with connected 𝑉𝜈 such that 𝛶𝜈 is a connected component of 𝜋 −1𝑉𝜈 . In the case of (iii) (a) above, the equivalence class [𝛶˜ ] is called a relative or ideal boundary point of 𝔇 (or 𝜋 : 𝔇 → C𝑛 ) over 𝑧0 , and 𝑧 0 is called the base point of it. The set 𝜕 ∗ 𝔇 of all relative (ideal) boundary points of 𝔇 is called the relative or ideal boundary of 𝔇 (over C𝑛 ). Remark 3.8.3. Even in the case of 𝔇 ⊂ C𝑛 , as 𝔇 is regarded as a domain over C𝑛 with the inclusion map 𝜄 : 𝔇 → C𝑛 , the ideal boundary 𝜕 ∗ 𝔇 does not coincide with the boundary 𝜕𝔇 as a subset of C𝑛 in general. For example, let 𝔇 = C \ {𝑥 ∈ R : 𝑥 ≥ 0}. Then, 𝜕𝔇 = {𝑥 ∈ R : 𝑥 ≥ 0}. But 𝜕 ∗ 𝔇 has two ideal boundary points over 𝑥 ∈ R, 𝑥 > 0; one is 𝑞 +𝑥 as a boundary point of the upper-half plane, and the other is 𝑞 −𝑥 as a boundary point of the lower-half plane (cf. Fig. 3.6): 𝜕 ∗ 𝔇 = {0} ∪ {𝑞 +𝑥 , 𝑞 −𝑥 : 𝑥 > 0}.

Fig. 3.6 Relative boundary.

Note 1. The works of Cartan Thullen in 3.2 were obtained in [11]; they are very important and provided the starting foundation of Oka’s study. Historically, Corollary 3.3.13 was announced in 1932 and proved in 1935 by A. Weil. Theorem 3.3.12 was proved by K. Oka in 1936/’37 (Oka I/II). The Oka Principle (Theorem 3.4.51) says that the existence of an analytic solution is completely characterized by a topological condition; it was then accepted with a surprise, and thereafter such a characterization was termed as the Oka Principle. It has advanced so that there are now “Grauert’s Oka Principle”, “Gromov’s Oka

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3 Domains of Holomorphy

Principle”, etc.; it has served as a model case of developments of the theory of complex analysis and geometry (cf. F. Forstneri [17]). ¯ Dolbeault’s Lemma 3.4.73 is a special case of the so-called 𝜕-Poincaré Lemma for general ( 𝑝, 𝑞)-forms, which is also referred to as Dolbeault’s Lemma or the Dolbeault Grothendieck Lemma. By Theorem 3.4.75 and Proposition 3.4.77 we see that the solvability of the ¯ Continuous Cousin Problem is equivalent to that of 𝜕-equation with integrable condition. In the present book (Oka theory) we will solve the Pseudoconvexity Problem by solving the Cousin I Problem, but in Hörmander [30] the approach is the ¯ opposite so that the solvability of 𝜕-equations on a pseudoconvex domain is shown rst, and then later the Cousin Problem is solved; the signi cant di erence is that the order of the solutions is reversed. The intention is clearly written in the “Summary” of [30] Chap. 4: In this chapter we abandon the classical methods for solving the Cousin problems (that is, solving the Cauchy Riemann equations), of which an example was given in section 2.7. Instead, we develop a technique for studying the Cauchy Riemann equations where the main point is an 𝐿 2 estimate proved in sections .....

Note 2 (Weil condition). Let 𝛺 ⊂ C𝑛 be a domain of holomorphy, and let 𝜑 ∈ 𝒪(𝛺). As an immediate consequence of Exercise 10 at the end of the chapter, there are holomorphic functions 𝐴 𝑗 (𝑧, 𝑤) ∈ 𝒪(𝛺 × 𝛺) (1 ≤ 𝑗 ≤ 𝑛) satisfying (3.8.4)

𝜑(𝑧) − 𝜑(𝑤) =

𝑛 Õ

𝐴 𝑗 (𝑧, 𝑤) (𝑧 𝑗 − 𝑤 𝑗 ).

𝑗=1

There is an interesting history to this equation. A. Weil (C.R. 1932) dealt with (3.8.4) for polynomial 𝜑(𝑧), and obtained an integral formula (so-called Weil’s integral formula) extending the Cauchy integral formula8 and an approximation theorem of Runge type for polynomially convex subsets. He probably tried to prove it for holomorphic functions in domains of holomorphy, but (3.8.4) was left to be a condition (Weil condition) in his full paper, Math. Ann. 111 (1935). Chronologically, H. Hefer took the problem and proved (3.8.4) in his Dissertation at Münster 1940; He died in the War in 1941, and the result was later published in Math. Ann. 122 (1950); his proof relied on the solution of the Cousin I Problem due to Oka I (1936) and II (1937). Without communications between Japan and Europe then, K. Oka V (1941) proved a weakened form (3.8.5)

(𝜑(𝑧) − 𝜑(𝑤))𝑅(𝑧, 𝑤) =

𝑛 Õ

𝐴 𝑗 (𝑧, 𝑤)(𝑧 𝑗 − 𝑤 𝑗 ),

𝑗=1

8 In his essay “Souvenirs d’apprentissage, Birkhäuser 1991”, A. Weil writes that he felt so much happiness by nding the formula to send a telegram on the nding from Aligarh to his friend in Calcutta, India.

3.8 Supplement: Ideal Boundary

127

where 𝑅(𝑧, 𝑤) is a holomorphic function in 𝛺 × 𝛺 with 𝑅(𝑧, 𝑧) = 1, independent of 𝜑(𝑧), and it is yet su cient to deduce Weil’s integral formula, which in 𝑛 = 2 is stated roughly as ∫ 1 Õ (3.8.6) 𝑓 (𝑧) = (2𝜋𝑖) 2 𝑗≠𝑘 { | 𝜑 𝑗 (𝑤) |=1}∩{ | 𝜑𝑘 (𝑤) |=1} ( 𝐴 𝑗1 (𝑤) 𝐴 𝑘2 (𝑤) − 𝐴 𝑘1 (𝑤) 𝐴 𝑗2 (𝑤)) 𝑓 (𝑤) 𝑑𝑤1 𝑑𝑤2 , (𝜑 𝑗 (𝑤) − 𝜑 𝑗 (𝑧)) (𝜑 𝑘 (𝑤) − 𝜑 𝑘 (𝑧)) 𝑧 = (𝑧1 , 𝑧 2 ) ∈ P, where P ⋐ 𝛺 is an analytic polyhedron de ned by |𝜑 𝑗 | < 1 (1 ≤ 𝑗 ≤ 𝑙) with 𝜑 𝑗 ∈ 𝒪(𝛺), (𝜑 𝑗 (𝑧) − 𝜑 𝑗 (𝑤))𝑅(𝑧.𝑤) = 𝐴 𝑗1 (𝑧1 − 𝑤1 ) + 𝐴 𝑗2 (𝑧2 − 𝑤2 ),

1 ≤ 𝑗 ≤ 𝑙,

¯ Oka VI (1942) used the formula to solve and 𝑓 (𝑧) is a holomrophic function on P. the Pseudoconvexity Problem (Levi’s Problem) for univalent domains in 𝑛 = 2: At the end of the paper he wrote, as mentioned in the Preface, L’auteur pense que cette conclusion sera aussi indépendante des nombres de variables complexes. (The author thinks that this conclusion is true too, independently of the number of complex variables.)

S. Hitotsumatsu ([28], 1949) generalized Oka’s result for univalent domains of general dimension 𝑛 ≥ 2 by making use of 𝑛 variable versions of (3.8.5) and (3.8.6). H.J. Bremermann ([8], 1954) extended Oka’s result for univalent domains of general dimension 𝑛 ≥ 2 by making use of Weil’s integral formula, referring to Hefer’s result (3.8.4) with 𝑅 ≡ 1, and independently F. Norguet ([45], 1954) for univalent domains of general dimension 𝑛 ≥ 2 by making use of Weil’s integral formula with Weil condtion (3.8.4), referring to the coherence theorem mentioned at the beginning. It is noted that Weil’s integral formula is not yet established for multivalent domains over C𝑛 (to the best of the author’s knowledge). In the forthcoming Chap. 5 of the present book we will solve the Pseuoconvexity Problem for unrami ed multivalent domains over C𝑛 by Oka’s Joku-Iko Principle based on his First First Coherence Theorem 2.3.16, and the Cauchy integral (or Cousin integral) formula. It is the same throughout these works to use the Fredholm integral equation of the second kind (type) formulated by Oka [47] (1941).

128

3 Domains of Holomorphy

Exercises 1. Prove each item of Example 3.1.12. 2. Show that 𝛿P𝛥 (𝑧) in (3.2.1) satis es the axioms of a norm of the complex vector space C𝑛 : For 𝑧, 𝑤 ∈ C𝑛 and 𝛼 ∈ C, ∥𝑧 + 𝑤∥ P𝛥 ≤ ∥𝑧∥ P𝛥 + ∥𝑤∥ P𝛥 ,

∥𝛼𝑧∥ P𝛥 = |𝛼| · ∥𝑧∥ P𝛥 ;

∥𝑧∥ P𝛥 ⇐⇒ 𝑧 = 0. 3. Let 𝛺 ⊂ C𝑛 be a domain, and let 𝐾 ⋐ 𝛺 be a compact subset. Assume that there is no connected component of 𝛺 \ 𝐾 which is relatively compact in 𝛺. Prove that 𝛺 \ 𝐾 consists of nitely many connected components. 4. Prove that the relation de ned by (3.4.3) is, in fact, an equivalence relation. 5. Let 𝛺 ⋐ C be a bounded domain of which boundary consists of nitely many piecewise 𝐶 1 closed Jordan curves 𝐶 𝑗 , (0 ≤ 𝑗 ≤ 𝑙). We suppose that 𝐶0 is the outer boundary, i.e., 𝐶0 is the boundary of the unbounded connected component of C \ 𝛺¯ and 𝐶 𝑗 (1 ≤ 𝑗 ≤ 𝑙) are the boundaries of the bounded connected components 𝜔 𝑗 of C \ 𝛺¯ (𝜕𝜔 𝑗 = 𝐶 𝑗 (1 ≤ 𝑗 ≤ 𝑙), the inner boundaries). Take a point 𝑝 𝑗 ∈ 𝜔 𝑗 for every 1 ≤ 𝑗 ≤ 𝑙. Show that a holomorphic function on 𝛺¯ can be approximated uniformly on compact subsets of 𝛺 by rational functions with poles only at 𝑝 𝑗 , 1 ≤ 𝑗 ≤ 𝑙. Í ∫ ) Hint: Use the Cauchy integral formula 𝑓 (𝑧) = 2 1𝜋𝑖 𝑙𝑗=0 𝑓𝜁 (𝜁 −𝑧 𝑑𝜁 and the proof of Lemma 3.3.20. Note that the integral is Riemann’s integral (cf. [38] 7.1). 6. Prove that (i) and (ii) of De nition 3.4.13 are equivalent. 7. Let 𝐴 ⊂ C𝑛 be a non-empty subset. Let 𝑑 (𝑧, 𝐴) be the distance from a point 𝑧 ∈ C𝑛 to 𝐴 de ned by (3.1.3). Prove the following: a. |𝑑 (𝑧, 𝐴) − 𝑑 (𝑧 ′ , 𝐴)| ≤ ∥𝑧 − 𝑧 ′ ∥ (𝑧, 𝑧 ′ ∈ C𝑛 ). In particular, 𝑑 (𝑧, 𝐴) is Lipschitz continuous. b. Let 𝛺 ⊂ C𝑛 be an open subset, and let 𝐸 ⊂ 𝛺 be a closed subset. Prove that for 𝑧 ∈ 𝛺, 𝑑 (𝑧, 𝐸) = 0 if and only if 𝑧 ∈ 𝐸. c. Let 𝐸, 𝐹 ⊂ 𝛺 be two closed subsets with 𝐸 ∩ 𝐹 = ∅. Set 𝜌(𝑧) =

𝑑 (𝑧, 𝐹) , 𝑑 (𝑧, 𝐸) + 𝑑 (𝑧, 𝐹)

𝑧 ∈ 𝛺.

Prove that 𝜌 ∈ 𝒞 0 (𝛺), 𝜌(𝑧) = 1 on 𝐸, and 𝜌(𝑧) = 0 on 𝐹. 8. Solve the interpolation problem of one variable by making use of the Weierstrass and Mittag-Le er Theorems. Hint: Cf. (3.5.2). 9. Let 𝛺 ⊂ C𝑛 be a holomorphically convex domain, and let 𝑓 𝑗 ∈ 𝒪(𝛺), 1 ≤ 𝑗 ≤ 𝑞, be nitely many holomorphic functions such that for every 𝑧 ∈ 𝛺 there is some 𝑓 𝑗 (𝑧) ≠ 0. Prove that there are holomorphic functions 𝑎 𝑗 ∈ 𝒪(𝛺), 1 ≤ 𝑗 ≤ 𝑞, satisfying

Exercises

129

𝑎 1 (𝑧) 𝑓1 (𝑧) + · · · + 𝑎 𝑞 (𝑧) 𝑓𝑞 (𝑧) = 1,

𝑧 ∈ 𝛺.

(This is due to H. Cartan [9] for compact cylinder domains.) Hint: Use Exercise 6 and 7 of the previous chapter. By the Oka map 𝜑 : P¯ → P𝛥 we embed an analytic polyhedron P ⋐ 𝛺 into a higher dimensional polydisk P𝛥, and extend 𝑓 𝑗 (1 ≤ 𝑗 ≤ 𝑞) to 𝑓˜𝑗 ∈ 𝒪(P𝛥). Moreover we add nitely many ¯ and have no common 𝑓˜𝑗 ∈ 𝒪(P𝛥), 𝑞 +1 ≤ 𝑗 ≤ 𝑞, ˜ which take the value 0 on 𝜑( P) ˜ zero on the common zero set of 𝑓 𝑗 (1 ≤ 𝑗 ≤ 𝑞), and then obtain a nite system of holomorphic functions 𝑓˜𝑗 , 1 ≤ 𝑗 ≤ 𝑞˜ without common zeros on P𝛥. Follow the proof of Theorem 3.5.4 for the rest. 10. Let 𝛺 ⊂ C𝑛 be a domain of holomorphy with the natural coordinate system (𝑧 1 , . . . , 𝑧 𝑛 ), and let 𝑆 = 𝛺 ∩ {𝑧 1 = 𝑧2 = · · · = 𝑧 𝑝 = 0} with 1 ≤ 𝑝 ≤ 𝑛. Show that if 𝑓 ∈ 𝒪(𝛺) satis es 𝑓 | 𝑆 ≡ 0,Íthen there are holomorphic functions 𝑓 𝑗 ∈ 𝒪(𝛺), 1 ≤ 𝑗 ≤ 𝑝, such that 𝑓 (𝑧) = 𝑝𝑗=1 𝑓 𝑗 (𝑧) · 𝑧 𝑗 (𝑧 ∈ 𝛺). 11. Let 𝛺 ⊂ C𝑛 be a domain and let ℬ𝒪(𝛺) denote the set of all bounded holomorphic functions on 𝛺. For a compact subset 𝐾 ⋐ 𝛺 the ℬ𝒪(𝛺)-hull is de ned by   bℬ𝒪 (𝛺 ) = 𝑧 ∈ 𝛺 : | 𝑓 (𝑧)| ≤ max | 𝑓 |, ∀ 𝑓 ∈ ℬ𝒪(𝛺) . 𝐾 𝐾

bℬ𝒪 (𝛺 ) . Prove that a holomorphic function in a neighborhood Assume 𝐾 = 𝐾 of 𝐾 is uniformly approximated on 𝐾 by elements of ℬ𝒪(𝛺). 12. Assume that a domain 𝜋 : 𝔇 → C𝑛 is nitely sheeted. Let P𝛥 be a polydisk with center at the origin of C𝑛 . For 𝑅, 𝑐 > 0 we set 𝐾 = {𝑝 ∈ 𝔇 : ∥𝜋( 𝑝) ∥ ≤ 𝑅, 𝛿P𝛥 ( 𝑝, 𝜕𝔇) ≥ 𝑐},

13. 14.

15. 16.

where ∥ · ∥ is the euclidean norm. Show that 𝐾 is a compact subset of 𝔇. Hint: Show that an arbitrary sequence of points
of 𝐾 has a convergent subseÐ quence. Use B¯ := {∥𝑧∥ ≤ 𝑅} ⊂ 𝑧 ∈ B¯ 𝑧 + 𝑐2 · P𝛥 (an open covering). Prove Theorem 3.7.8. ˆ 𝑛 be the envelope of holomorphy of 𝔇. Let 𝔇/C𝑛 be a domain and let 𝔇/C a. If 𝑓 ∈ 𝒪(𝔇) does not take the value 𝛼 ∈ C, then prove that the analytic ˆ of 𝑓 does not take the value 𝛼, either. continuation 𝑓ˆ ∈ 𝒪( 𝔇) b. With the same notation as above, if | 𝑓 | < 𝑀 (or | 𝑓 | ≤ 𝑀), show that | 𝑓ˆ| < 𝑀 (or | 𝑓ˆ| ≤ 𝑀). 𝑛. Prove Proposition 3.4.15 for a general multivalent unrami ed domain 𝔇/C  Let 𝐺 := {𝑥 + 𝑖𝑦 : 0 < 𝑥 < 1, 0 < 𝑦 < 1} \ 𝜈1 + 𝑖𝑦 : 0 < 𝑦 < 12 , 𝜈 = 1, 2, . . . ⊂ C. Describe the relative boundary 𝜕 ∗ 𝐺 with respect to the inclusion map 𝜄 : 𝐺 → C. Show that the set of base points of 𝜕 ∗ 𝐺 is not closed in C.

Chapter 4

Pseudoconvex Domains I — Problem and Reduction

In the present and the next chapter we solve the last Pseudoconvexity Problem of the Three Big Problems on unramified domains over C𝑛 . 1 We fully employ the results obtained previously, but the path is yet long. Here, introducing the notion of plurisubharmonic (or pseudoconvex) functions, we formulate the Pseudoconvexity Problems and discuss their relations.

4.1 Plurisubharmonic Functions 4.1.1 Subharmonic Functions (One Variable) With respect to the complex coordinate 𝑧 = 𝑥 + 𝑖𝑦 ∈ C the holomorphic partial dif𝜕 ferential operator 𝜕𝑧 and the anti-holomorphic partial di erential operator 𝜕𝜕𝑧¯ are de ned by (1.1.6). Repeating them, we have   𝜕2 𝜕 𝜕 1 𝜕2 𝜕2 1 (4.1.1) = = + = Δ. 𝜕𝑧𝜕 𝑧¯ 𝜕𝑧 𝜕 𝑧¯ 4 𝜕𝑥 2 𝜕𝑦 2 4 Here, Δ is well-known as the Laplacian. With respect to the polar coordinate 𝑧 = 𝑟𝑒 𝑖 𝜃 (𝑟 > 0) we have (4.1.2)

Δ=

𝜕2 1 𝜕 1 𝜕2 + + 𝜕𝑟 2 𝑟 𝜕𝑟 𝑟 2 𝜕𝜃 2

(cf. Exercise 2 at the end of this chapter). Let 𝜑(𝑧) be a function on 𝛥(𝑎; 𝑅) (𝑅 > 0). We de ne the mean integration of 𝜑(𝑧) on the circle {|𝑧 − 𝑎| = 𝑟 } by

1 For those studying the present subject for the rst time, it is recommended to read this chapter and the next, assuming that the domains are univalent.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 J. Noguchi, Basic Oka Theory in Several Complex Variables, Universitext, https://doi.org/10.1007/978-981-97-2056-9_4

131

132

(4.1.3)

4 Pseudoconvex Domains I

𝑀 𝜑 (𝑎;𝑟) =

1 2𝜋

∫

2𝜋

𝜑(𝑎 + 𝑟𝑒 𝑖 𝜃 )𝑑𝜃,

Problem and Reduction

0 ≤ 𝑟 < 𝑅,

0

provided that it exists. Lemma 4.1.4 (Jensen’s formula). For 𝜑 ∈ 𝐶 2 ( 𝛥(𝑎; 𝑅)) we have ∫ 𝑟 ∫ 1 𝑑𝑡 𝑀 𝜑 (𝑎;𝑟) = 𝜑(𝑎) + Δ 𝜑 𝑑𝜆, 0 ≤ 𝑟 < 𝑅. 2𝜋 0 𝑡 𝛥(𝑎;𝑡 ) Here, 𝑑𝜆 = 𝑑𝑥𝑑𝑦 with 𝑧 = 𝑥 + 𝑖𝑦 denotes the standard surface measure on C. Proof. By a parallel transform we may assume 𝑎 = 0. We set 𝑀 (𝑟) = 𝑀 𝜑 (0;𝑟). It follows from (4.1.2) that  ∫ 2𝜋 ∫ 2𝜋  2 𝜕 1𝜕 1 𝜕2 𝑖𝜃 (4.1.5) + + 𝜑(𝑟𝑒 𝑖 𝜃 )𝑑𝜃. Δ 𝜑(𝑟𝑒 )𝑑𝜃 = 𝜕𝑟 2 𝑟 𝜕𝑟 𝑟 2 𝜕𝜃 2 0 0 For each 𝑟, 𝜑(𝑎 + 𝑟𝑒 𝑖 𝜃 ) is a periodical function in 𝜃 with period 2𝜋, and of 𝐶 2 𝜕2 𝑖 𝜃 ) has a primitive function 𝜕 𝜑(𝑎 + 𝑟𝑒 𝑖 𝜃 ), which is class. The function 𝜕𝜃 2 𝜑(𝑎 + 𝑟𝑒 𝜕𝜃 periodic. Therefore we see that ∫

2𝜋

𝜕2 𝜑(𝑟𝑒 𝑖 𝜃 )𝑑𝜃 = 0, 2 𝜕𝜃 0  2  ∫ 2𝜋 𝑑 1𝑑 1 + 𝑀 (𝑟) = Δ 𝜑(𝑟𝑒 𝑖 𝜃 )𝑑𝜃. 2𝜋 0 𝑑𝑟 2 𝑟 𝑑𝑟   𝑑 𝑑 The left-hand side above is written as 𝑟1 𝑑𝑟 𝑟 𝑑𝑟 𝑀 (𝑟) , and so   ∫ 2𝜋 𝑑 𝑑 𝑟 𝑟 𝑀 (𝑟) = Δ 𝜑(𝑟𝑒 𝑖 𝜃 )𝑑𝜃. 𝑑𝑟 𝑑𝑟 2𝜋 0

(4.1.6) On the other hand,

𝑑 1 𝑟 𝑀 (𝑟) = 𝑑𝑟 2𝜋

∫ 0

2𝜋



 𝜕𝜑 𝜕𝜑 𝑟 cos 𝜃 + 𝑟 sin 𝜃 𝑑𝜃 → 0 𝜕𝑥 𝜕𝑦

(𝑟 → +0).

Thus, by integrating (4.1.6) in 𝑟, we get 𝑑 𝑟 𝑀 (𝑟) = 𝑑𝑟 𝑑 𝑀 (𝑟) = 𝑑𝑟

∫ 𝑟 ∫ 2𝜋 ∫ 1 1 𝑖𝜃 𝑠𝑑𝑠 Δ 𝜑(𝑠𝑒 )𝑑𝜃 = Δ 𝜑 𝑑𝜆; 2𝜋 0 2𝜋 𝛥(𝑟 ) 0 ∫ 1 Δ 𝜑 𝑑𝜆. 2𝜋𝑟 𝛥(𝑟 )

Since 𝑀 (0) = 𝜑(0), we integrate the above equation again to deduce

4.1 Plurisubharmonic Functions

133

𝑀 (𝑟) − 𝜑(0) =

1 2𝜋

∫ 0

𝑟

𝑑𝑡 𝑡

∫ 𝛥(𝑡 )

Δ 𝜑 𝑑𝜆.

⊓ ⊔

Let 𝑈 ⊂ C be an open set, and consider a function 𝜑 : 𝑈 → [−∞, ∞), which is allowed to take value −∞. Definition 4.1.7. The function 𝜑 is subharmonic if the following two conditions are satis ed: (i) (Upper semi-continuity) 𝜑 is upper semi-continuous; that is, for every 𝑐 ∈ R, {𝑧 ∈ 𝑈 : 𝜑(𝑧) < 𝑐} is an open set; this is equivalent to lim 𝜑(𝑧) ≤ 𝜑(𝑎),

𝑧→𝑎

∀𝑎 ∈ 𝑈.

(Cf. Exercise 3 at the end of this chapter.) (ii) (Submean property) For every disk 𝛥(𝑎;𝑟) ⋐ 𝑈, 𝜑(𝑎) ≤ 𝑀 𝜑 (𝑎;𝑟). By de nition, 𝜑 ≡ −∞ is a subharmonic function. If 𝜑 𝑗 ( 𝑗 = 1, 2) are subharmonic and 𝑐 𝑗 > 0 ( 𝑗 = 1, 2) are constants, then 𝑐 1 𝜑1 + 𝑐 2 𝜑2 is subharmonic. Theorem 4.1.8. Let 𝜑 ∈ 𝐶 2 (𝑈). (i) 𝜑 is subharmonic if and only if Δ 𝜑 ≥ 0. (ii) If 𝜑 is subharmonic and 𝛥(𝑎; 𝑅) ⊂ 𝑈, then 𝑀 𝜑 (𝑎;𝑟) is monotone increasing and continuous in 0 ≤ 𝑟 < 𝑅, and of 𝐶 1 -class in 0 < 𝑟 < 𝑅. Proof. (i) Assume that 𝜑 is subharmonic. For a point 𝑎 ∈ 𝑈, the Taylor expansion of 𝜑(𝑎 + 𝜁) up to order 2 in su ciently small 𝜁 ∈ C (i.e., small |𝜁 |) is written as (4.1.9)

𝜕𝜑 𝜕𝜑 (𝑎)𝜁 + (𝑎) 𝜁¯ 𝜕𝑧 𝜕 𝑧¯ 1 𝜕2 𝜑 1 𝜕2 𝜑 𝜕2 𝜑 + (𝑎)𝜁 2 + (𝑎) 𝜁¯2 + (𝑎)|𝜁 | 2 + 𝑜(|𝜁 | 2 ). 2 2 2 𝜕𝑧 2 𝜕 𝑧¯ 𝜕𝑧𝜕 𝑧¯

𝜑(𝑎 + 𝜁) = 𝜑(𝑎) +

With 𝜁 = 𝑟𝑒 𝑖 𝜃 we take the integration on 0 ≤ 𝜃 ≤ 2𝜋: 0≤

1 2𝜋𝑟 2

∫

2𝜋

(𝜑(𝑎 + 𝑟𝑒 𝑖 𝜃 ) − 𝜑(𝑎))𝑑𝜃 =

0

𝜕2 𝜑 (𝑎) + 𝑜(1). 𝜕𝑧𝜕 𝑧¯

As 𝑟 ↘ 0, we deduce (4.1.10)

Δ 𝜑(𝑎) = 4

𝜕2 𝜑 (𝑎) ≥ 0, 𝜕𝑧𝜕 𝑧¯

𝑎 ∈ 𝑈.

Conversely, if Δ 𝜑 ≥ 0, Lemma 4.1.4 implies that 𝑀 𝜑 (𝑎;𝑟) ≥ 𝜑(0), and hence 𝜑 is subharmonic. (ii) It immediately follows from (i) above and Lemma 4.1.4. ⊓ ⊔

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4.1.2 Plurisubharmonic Functions Let 𝑛 ≥ 2 and let 𝑈 ⊂ C𝑛 be an open set. As in the previous subsection we consider a function 𝜑 : 𝑈 → [−∞, ∞). Definition 4.1.11. The function 𝜑 is said to be plurisubharmonic or pseudoconvex2 if the following conditions are satis ed: (i) 𝜑 is upper semi-continuous. (ii) For every point 𝑧 ∈ 𝑈 and vector 𝑣 ∈ C𝑛 \ {0} the function (4.1.12)

𝜁 ∈ C → 𝜑(𝑧 + 𝜁 𝑣) ∈ [−∞, ∞)

satis es the submean property (subharmonic) in 𝜁 ∈ C where it is de ned. We denote by 𝒫(𝑈) the set of all plurisubharmonic function on 𝑈, and set 𝒫 𝑘 (𝑈) = 𝒫(𝑈) ∩ 𝒞 𝑘 (𝑈) (0 ≤ 𝑘 ≤ ∞). Suppose that 𝜑(𝑧) (𝑧 = (𝑧 1 , . . . , 𝑧 𝑛 ) ∈ 𝑈) is of 𝐶 2 -class. In (ii) above, we have with 𝑣 = (𝑣1 , . . . , 𝑣𝑛 ) Õ 𝜕2 𝜑 𝜕 2 𝜑(𝑧 + 𝜁 𝑣) = (𝑧)𝑣 𝑗 𝑣¯ 𝑘 , 𝜕𝑧 𝑗 𝜕 𝑧¯ 𝑘 𝜕𝜁 𝜕 𝜁¯ 𝜁 =0 𝑗,𝑘

(4.1.13)

𝐿 [𝜑] (𝑣) = 𝐿 [𝜑] (𝑧; 𝑣) :=

Õ 𝜕2 𝜑 (𝑧)𝑣 𝑗 𝑣¯ 𝑘 . 𝜕𝑧 𝑗 𝜕 𝑧¯ 𝑘 𝑗,𝑘

We call 𝐿 [𝜑] (𝑣) (𝐿 [𝜑] (𝑧; 𝑣)) the Levi form of 𝜑. The Levi form 𝐿 [𝜑] (𝑣) is (resp. semi-) positive de nite if 𝐿 [𝜑] (𝑣) > 0 (resp. ≥ 0),

∀ 𝑣 ∈ C𝑛 \ {0}.

We write 𝐿 [𝜑] > 0 (resp. 𝐿 [𝜑] ≥ 0) for it. It follows from Theorem 4.1.8 (i): Theorem 4.1.14. For 𝜑 ∈ 𝐶 2 (𝑈), 𝜑 ∈ 𝒫(𝑈) if and only if 𝐿 [𝜑] ≥ 0. Definition 4.1.15. We say that a function 𝜑 ∈ 𝐶 2 (𝑈) is strongly plurisubharmonic or strongly pseudoconvex if 𝐿 [𝜑] > 0. By de nition, 𝜑 ≡ −∞ is plurisubharmonic. In general the following properties hold: Theorem 4.1.16. (i) If 𝜑 𝑗 ( 𝑗 = 1, 2) are plurisubharmonic and 𝑐 𝑗 ( 𝑗 = 1, 2) are positive constants, then 𝑐 1 𝜑1 + 𝑐 2 𝜑2 is plurisubharmonic. 2 This notion of functions was rst de ned by K. Oka who termed it “fonction pseudoconvexe” (Oka VI, 1942). Around that time, P. Lelong also introduced the same notion, calling it “fonction plurisousharmonique”; nowadays, the latter is used more frequently.

4.1 Plurisubharmonic Functions

135

(ii) Let 𝜑 ∈ 𝒫(𝑈) such that 𝜑(𝑎) > −∞ at some point 𝑎 ∈ 𝑈. Then 𝜑 is locally integrable in a connected component of 𝑈 containing 𝑎 with respect to the Lebesgue measure of C𝑛  R2𝑛 . (iii) (Maximum Principle) Let 𝜑 ∈ 𝒫(𝑈). If 𝜑 attains the maximum at 𝑎 ∈ 𝑈, then 𝜑 is a constant function in a connected component of 𝑈 containing 𝑎. (iv) Let 𝜑 ∈ 𝒫(𝑈) and let 𝜓 be a monotone increasing convex function defined on [inf 𝜑, sup 𝜑). Then 𝜓 ◦ 𝜑 ∈ 𝒫(𝑈); here, if inf 𝜑 = −∞, we set 𝜓(−∞) = lim𝑡→−∞ 𝜓(𝑡). (v) If 𝜑 𝜈 ∈ 𝒫(𝑈), 𝜈 = 1, 2, . . ., is a decreasing sequence of plurisubharmonic functions, then the limit 𝜑(𝑧) = lim𝜈→∞ 𝜑 𝜈 (𝑧) is a plurisubharmonic function. (vi) Let {𝜑𝜆 }𝜆∈𝛬 be a family of 𝒫(𝑈). If 𝜑(𝑧) := sup𝜆∈𝛬 𝜑𝜆 (𝑧) is upper semicontinuous, then 𝜑(𝑧) ∈ 𝒫(𝑈). In particular, for finite 𝛬, 𝜑(𝑧) is upper semicontinuous, and so plurisubharmonic. Proof. (i) It is immediate from the de nition. (ii) Suppose that 𝑈 is connected and 𝜑(𝑎) > −∞ at some 𝑎 ∈ 𝑈. Let 𝑑𝜆 denote the Lebesgue measure of C𝑛 . Take an open ball B(𝑎;𝑟) = 𝑎 + B(𝑟) ⋐ 𝑈. Since 𝜑 is upper semi-continuous, 𝜑 has a upper bound 𝐶 ∈ R on B(𝑎;𝑟). For 𝑤 ∈ B(𝑟) we get −∞ < 𝜑(𝑎) ≤ With 𝑉 (𝑟) := (4.1.17)

∫ B(𝑟 )

𝑑𝜆 =

𝜋𝑛 𝑛!

1 2𝜋

∫

2𝜋

𝜑(𝑎 + 𝑒 𝑖 𝜃 𝑤)𝑑𝜃 ≤ 𝐶.

0

𝑟 2𝑛 we obtain

1 −∞ < 𝜑(𝑎) ≤ 2𝜋𝑉 (𝑟)

∫ 0

2𝜋 ∫ B(𝑟 )

𝑑𝜆(𝑤)𝜑(𝑎 + 𝑒 𝑖 𝜃 𝑤)𝑑𝜃 ≤ 𝐶.

It follows from the change of order of the integrations and the rotation invariance of the Lebesgue measure, 𝑑𝜆(𝑤) = 𝑑𝜆(𝑒 𝑖 𝜃 𝑤), that ∫ 1 (4.1.18) −∞ < 𝜑(𝑎) ≤ 𝜑(𝑎 + 𝑤)𝑑𝜆(𝑤) ≤ 𝐶. 𝑉 (𝑟) B(𝑟 ) Therefore 𝜑 is integrable over B(𝑎;𝑟) (⋐ 𝑈), so that 𝜑(𝑏) > −∞ for almost all 𝑏 ∈ B(𝑎;𝑟) with respect to 𝑑𝜆. Let 𝑈 ′ be the set of all 𝑎 ∈ 𝑈 such that 𝜑 is integrable over every B(𝑎;𝑟) ⋐ 𝑈. From (4.1.18) we easily deduce that 𝑈 ′ is non-empty, open and closed. Hence, 𝑈 ′ = 𝑈, and 𝜑 is locally integrable on 𝑈. (iii) Suppose that 𝑈 is connected, and that 𝜑(𝑎) is the maximum for a point 𝑎 ∈ 𝑈. It follows from (4.1.18) that ∫ (𝜑(𝑎 + 𝑤) − 𝜑(𝑎)) = 0. B(𝑟 )

If there were a point 𝑤0 ∈ B(𝑟) with 𝑐 := 𝜑(𝑎 + 𝑤0 ) − 𝜑(𝑎) < 0, we deduce from the upper semi-continuity that {𝜑(𝑎 + 𝑤) − 𝜑(𝑎) < 𝑐/2} is a non-empty open set, and

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since 𝜑(𝑎 + 𝑤) − 𝜑(𝑎) ≤ 0 everywhere in B(𝑟), we get ∫ (𝜑(𝑎 + 𝑤) − 𝜑(𝑎)) < 0. B(𝑟 )

This contradicts (4.1.18). Therefore, 𝜑| B(𝑎;𝑟 ) ≡ 𝜑(𝑎). Set 𝑉 = {𝑏 ∈ 𝑈 : 𝜑(𝑏) = 𝜑(𝑎)}. It follows from the above arguments that 𝑉 is a non-empty open subset. By the upper semi-continuity of 𝜑(𝑧), 𝑉 is closed; hence, 𝑉 = 𝑈. (iv) It is rst noted that 𝜓 is continuous by the condition. Therefore 𝜓 ◦ 𝜑 is upper semi-continuous. Let B(𝑎; 𝑅) ⋐ 𝑈 be any open ball, and let 𝑣 ∈ C𝑛 be a vector with ∥𝑣∥ = 1. It su ces to show the submean property of 𝜓 ◦ 𝜑(𝑎 + 𝜁 𝑣) at 𝜁 = 0. It follows from the convexity of 𝜓 that  ∫ 2𝜋  ∫ 2𝜋 1 1 𝑖𝜃 𝑖𝜃 𝜓(𝜑(𝑎 + 𝑟𝑒 𝑣))𝑑𝜃 ≥ 𝜓 𝜑(𝑎 + 𝑟𝑒 𝑣)𝑑𝜃 , 0 < 𝑟 ≤ 𝑅. 2𝜋 0 2𝜋 0 Because of the submean property of 𝜑  ∫ 2𝜋  1 𝑖𝜃 𝜓 𝜑(𝑎 + 𝑟𝑒 𝑣)𝑑𝜃 ≥ 𝜓(𝜑(𝑎)). 2𝜋 0 Therefore, the submean property of 𝜓 ◦ 𝜑(𝑎 + 𝜁 𝑣) follows, and hence 𝜓 ◦ 𝜑 is subharmonic. (v) The limit function of a decreasing sequence of upper semi-continuous functions is upper semi-continuous. The submean property follows from Lebesgue’s monotone convergence theorem.3 (vi) It is immediate from the de nition. ⊓ ⊔ Example 4.1.19. (i) Let 𝑓 ∈ 𝒪(𝑈). By the computation of the Levi form we see log(𝑐 + | 𝑓 | 2 ) ∈ 𝒫(𝑈) (𝑐 > 0). As 𝑐 ↘ 0, log | 𝑓 | ∈ 𝒫(𝑈). Therefore, | 𝑓 | 𝜌 ∈ 𝒫0 (𝑈) for all 𝜌 > 0. (ii) It follows from the computations of the Levi forms that ∥𝑧∥ 2 and log(𝑐 + ∥𝑧∥ 2 ) with 𝑐 > are strongly plurisubharmonic in 𝑧 ∈ C𝑛 . As 𝑐 ↘ 0, log ∥𝑧∥ ∈ 𝒫(C𝑛 ); 𝜌 0 𝑛 ) (𝜌 > 0). hence, ∥𝑧∥ Í𝑛 ∈ 𝒫 (C Í 2 (iii) 𝜑(𝑧) = 𝑗=1 (ℜ𝑧 𝑗 ) (or 𝑛𝑗=1 (ℑ𝑧 𝑗 ) 2 ) is strongly plurisubharmonic.

4.1.3 Smoothing We denote the Lebesgue measure of C𝑛  R2𝑛 (∋ (𝑥1 , 𝑦 1 , 𝑥 2 , 𝑦 2 , . . . , 𝑥 𝑛 , 𝑦 𝑛 )) by 𝑑𝜆 = 𝑑𝑥1 𝑑𝑦 1 𝑑𝑥 2 𝑑𝑦 2 · · · 𝑑𝑥 𝑛 𝑑𝑦 𝑛 . 3 Those who are not familiar with the Lebesgue integration theory may assume that the functions are continuous and the convergence is locally uniform; they do not cause trouble in understanding Oka theory itself.

4.1 Plurisubharmonic Functions

For 𝜀 > 0 we put

137

𝑈 𝜀 = {𝑧 ∈ 𝑈 : 𝑑 (𝑧, 𝜕𝑈) > 𝜀}

(cf. (3.1.3)). We take 𝜒(𝑧) = 𝜒(|𝑧 1 |, . . . , |𝑧 𝑛 |) ∈ 𝐶0∞ (C𝑛 ) such that ∫ 𝜒(𝑧) ≥ 0, Supp 𝜒 ⊂ B(= B(1)), 𝜒(𝑧)𝑑𝜆 = 1, and put

𝜒 𝜀 (𝑧) = 𝜒(𝜀 −1 𝑧)𝜀 −2𝑛 ,

𝜀 > 0.

For a locally integrable function 𝜑 we de ne the smoothing by ∫ (4.1.20) 𝜑 𝜀 (𝑧) = 𝜑 ∗ 𝜒 𝜀 (𝑧) = 𝜑(𝑤) 𝜒 𝜀 (𝑤 − 𝑧)𝑑𝜆(𝑤) C𝑛 ∫ = 𝜑(𝑧 + 𝑤) 𝜒 𝜀 (𝑤)𝑑𝜆(𝑤) 𝑛 ∫C = 𝜑(𝑧 + 𝜀𝑤) 𝜒(𝑤)𝑑𝜆(𝑤), 𝑧 ∈ 𝑈 𝜀 . B

Remark 4.1.21. While we used a unit open ball B, it is also possible to use a polydisk P𝛥 with center at the origin 0; with 𝑈 𝜀 = {𝑧 ∈ 𝑈 : 𝛿P𝛥 (𝑧, 𝜕𝑈) > 𝜀} we may de ne similarly 𝜒 𝜀 , and the arguments below work similarly. Proposition 4.1.22. (i) 𝜑 𝜀 (𝑧) is of 𝐶 ∞ -class in 𝑈 𝜀 . (ii) If 𝜑 is continuous, the convergence lim 𝜀→+0 𝜑 𝜀 (𝑧) = 𝜑(𝑧) is locally uniform. Proof. (i) It follows from the change of the integration and a partial di erentiation in (4.1.20). (ii) It is immediate from the last line of (4.1.20). ⊓ ⊔ Theorem 4.1.23. Let 𝜑 ∈ 𝒫(𝑈) such that 𝜑 . −∞ on each connected component of 𝑈. (i) The smoothing 𝜑 𝜀 (𝑧) is a 𝐶 ∞ plurisubharmonic function in 𝑈 𝜀 . (ii) As 𝜀 ↘ 0, 𝜑 𝜀 (𝑧) converges monotone decreasingly to 𝜑(𝑧). Proof. (i) It has been already proved that 𝜑 𝜀 is of 𝐶 ∞ -class. For a vector 𝑣 ∈ C𝑛 \ {0} and 𝜁 ∈ C, we show the submean property of 𝜑 𝜀 (𝑧 + 𝜁 𝑣) in 𝜁 where it is de ned. By a parallel transform it su ces to show it at 𝜁 = 0: 𝜑 𝜀 (𝑧) ≤

1 2𝜋

∫ 0

2𝜋

𝜑 𝜀 (𝑧 + 𝑟𝑒 𝑖 𝜃 𝑣)𝑑𝜃.

This follows from the interchange of the integrations. (ii) With an arbitrary open set 𝑉 ⋐ 𝑈 we consider 𝑧 ∈ 𝑉,

0 < 𝜀 < 𝜀 0 :=

1 inf{∥𝑧 ′ − 𝑧 ′′ ∥ : 𝑧 ′ ∈ 𝑉, 𝑧 ′′ ∈ 𝜕𝑈}. 2

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By the rotational symmetries 𝑑𝜆(𝑤) = 𝑑𝜆(𝑒 𝑖 𝜃 𝑤), and 𝜒(𝑤) = 𝜒(𝑒 𝑖 𝜃 𝑤) (0 ≤ 𝜃 ≤ 2𝜋) we have ∫ 𝜑 𝜀 (𝑧) = 𝜑(𝑧 + 𝜀𝑤) 𝜒(𝑤)𝑑𝜆(𝑤) C𝑛

∫

(4.1.24)

∫ 2𝜋 1 𝑑𝜆(𝑤) 𝑑𝜃 𝜑(𝑧 + 𝜀𝑒 𝑖 𝜃 𝑤) 𝜒(𝑤) 2𝜋 0 C𝑛 ∫ ≥ 𝜑(𝑧) 𝜒(𝑤)𝑑𝜆(𝑤) = 𝜑(𝑧).

=

C𝑛

Note that 𝜑 has a upper bound 𝐶0 ∈ R on 𝑉 + B(𝜀 0 ) (⋐ 𝑈). It follows that (4.1.25)

𝜑(𝑧) ≤ 𝜑 𝜀 (𝑧) ≤ 𝐶0 ,

𝑧 ∈ 𝑉.

We rst prove the point-wise convergence (4.1.26)

lim 𝜑 𝜀 (𝑧) = 𝜑(𝑧),

𝜀→+0

where the case of 𝜑(𝑧) = −∞ is included. For, in the case of 𝜑(𝑧) > −∞, the upper semi-continuity implies that {𝑧 ′ ∈ 𝑉 : 𝜑(𝑧 ′ ) < 𝜑(𝑧) + 𝑐} is an open set containing 𝑧 for 𝑐 > 0. Therefore, for an arbitrary su ciently small 𝜀 > 0, 𝜑(𝑧) ≤ 𝜑 𝜀 (𝑧) ≤ 𝜑(𝑧) + 𝑐 ;

lim 𝜑 𝜀 (𝑧) = 𝜑(𝑧).

𝜀→+0

In the case of 𝜑(𝑧) = −∞, the above 𝑐 may be chosen arbitrarily small (𝑐 < 0, large |𝑐|); hence 𝜑 𝜀 (𝑧) ≤ 𝑐 ; lim 𝜑 𝜀 (𝑧) = −∞. 𝜀→+0

With a su ciently small 𝛿 > 0 we apply Theorem 4.1.8 (ii) for 𝜑 𝛿 (𝑧 + 𝜁 𝑤) (|𝜁 | = 𝑟) to obtain ∫ ∫ 2𝜋 1 (4.1.27) 𝑑𝜆(𝑤) 𝑑𝜃𝜑 𝛿 (𝑧 + 𝜀𝑒 𝑖 𝜃 𝑤) 2𝜋 0 C𝑛 ∫ ∫ 2𝜋 1 ≤ 𝑑𝜆(𝑤) 𝑑𝜃𝜑 𝛿 (𝑧 + 𝜀 ′ 𝑒 𝑖 𝜃 𝑤), 0 < 𝜀 < 𝜀 ′ < 𝜀 0 . 𝑛 2𝜋 C 0 Here, note that as 𝛿 → +0, 𝜑 𝛿 (𝑧) (𝑧 ∈ 𝑉) converges to 𝜑(𝑧) point-wise, and (4.1.25) holds for 𝜑 𝛿 . By Lebesgue’s convergence theorem4 we deduce from (4.1.27) by letting 𝛿 → 0 that (4.1.28)

𝜑 𝜀 (𝑧) ≤ 𝜑 𝜀 ′ (𝑧),

0 < 𝜀 < 𝜀 ′ < 𝜀0 .

This together with (4.1.26) implies the monotone convergence.

⊓ ⊔

4 If 𝜑 is continuous, the convergence follows directly from 4.1.22 (ii). In fact, it is such a case when we use it later.

4.2 Hartogs’ Separate Analyticity

139

Theorem 4.1.29. (i) The plurisubharmonicity is a local property. (ii) Let 𝑈 ⊂ C𝑚 and 𝑉 ⊂ C𝑛 be open sets, and let 𝑓 : 𝑉 → 𝑈 be a holomorphic map. For a plurisubharmonic function 𝜑 in 𝑈, the pull-back 𝑓 ∗ 𝜑 = 𝜑 ◦ 𝑓 is plurisubharmonic in 𝑉. Proof. By Theorem 4.1.23 it is su cient to show both for 𝐶 ∞ plurisubharmonic functions. (i) Let 𝜑(𝑧) ∈ 𝐶 ∞ (𝑈). The plurisubharmonicity of 𝜑 is characterized by the semi-positivity of the Levi form 𝐿 [𝜑] (𝑣), and so it is a local property. (ii) Suppose 𝑓 : 𝑉 → 𝑈 is given by (𝑧 𝑗 ) = ( 𝑓 𝑗 (𝜁1 , . . . , 𝜁 𝑚 )). Then the Levi form is transformed to ! 𝑚 𝑚 Õ Õ 𝜕 𝑓1 𝜕 𝑓𝑛 ∗ 𝐿 [ 𝑓 𝜑] (𝑣1 , . . . , 𝑣𝑚 ) = 𝐿 [𝜑] 𝑣𝑘 , . . . , 𝑣𝑘 . 𝜕𝜁 𝑘 𝜕𝜁 𝑘 𝑘=1 𝑘=1 Therefore, if 𝜑 is plurisubharmonic, 𝐿 [ 𝑓 ∗ 𝜑] is semi-positive de nite, so that 𝑓 ∗ 𝜑 is plurisubharmonic. ⊓ ⊔ Because of Theorem 4.1.29 we may de ne plurisubharmonic functions on a domain 𝔇/C𝑛 as follows. Definition 4.1.30. Let 𝔇/C𝑛 be an unrami ed domain over C𝑛 . A function 𝜑 : 𝔇 → [−∞, ∞) is plurisubharmonic or pseudoconvex if for every point 𝑝 ∈ 𝔇 there is a polydisk neighborhood P𝛥( 𝑝) of 𝑝 such that, regarded as P𝛥( 𝑝) ⊂ C𝑛 , 𝜑(𝑧) is plurisubharmonic in 𝑧 ∈ P𝛥( 𝑝). We denote the set of all plurisubharmonic functions on 𝔇 by 𝒫(𝔇), and set 𝒫 𝑘 (𝔇) = 𝒫(𝔇) ∩ 𝒞 𝑘 (𝔇) (0 ≤ 𝑘 ≤ ∞).

4.2 Hartogs’ Separate Analyticity In this section we give a proof of the separate analyticity (cf. De nition 1.1.9 (ii)) due to F. Hartogs, proved in 1906 (see Theorem 4.2.9). The statement is very simple but the proof is not so simple. Those readers who know the Baire Category Theorem may skip the next subsection.

4.2.1 Baire Category Theorem For a moment we deal with a general topological space, but in fact, we will soon assume it to be a metric space. Definition 4.2.1. A topological space 𝑋 is called a Baire space, if every countable Ð union 𝜈 ∈N 𝐹𝜈 of closed subsets 𝐹𝜈 (⊂ 𝑋) (𝜈 ∈ N) without interior points contains no interior point.

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4 Pseudoconvex Domains I

Problem and Reduction

Remark 4.2.2. (i) As an example of non-Baire space, the set Q (⊂ R) endowed with the induced topology from the metric topology of R is not a Baire space. For a singleton {𝑎} (⊂ Q) is a closed set, and Q is a countable set; however, {𝑎} contains no interior point. (ii) In De nition 4.2.1, if 𝐹𝜈 are only nitely many closed subsets, then the nite Ð union 𝐹𝜈 contains no interior point (see Exercise 1 at the end of the chapter). Theorem 4.2.3 (Baire). (i) (Baire Category Theorem) A complete metric space is Baire. (ii) The euclidean 𝑛-space R𝑛 with the standard metric is Baire. (iii) Any closed subset of R𝑛 is Baire. (iv) Any open subset of R𝑛 is Baire. Proof. (i) Let 𝑋 be a space with a complete metric 𝑑 (𝑥, Ð 𝑦). Let 𝐴 𝜈 ⊂ 𝑋, 𝜈 = 1, 2, . . . be closed subsets without interior points, and set 𝐴 = ∞ 𝜈=1 𝐴 𝜈 . For 𝑎 ∈ 𝑋 and 𝑟 > 0 we put 𝑈 (𝑎;𝑟) = {𝑥 ∈ 𝑋 : 𝑑 (𝑎, 𝑥) < 𝑟 }. It su ces to show that 𝑈 (𝑎;𝑟) ⊄ 𝐴 for every 𝑈 (𝑎;𝑟). Since 𝑈 (𝑎;𝑟) \ 𝐴1 ≠ ∅, there is a point 𝑥1 ∈ 𝑈 (𝑎;𝑟) \ 𝐴1 . Because 𝐴1 is closed, there is a number 𝑟 1 such that 0 < 𝑟 1 < 1/2 and 𝑈 (𝑥1 ;𝑟 1 ) = {𝑥 ∈ 𝑋 : 𝑑 (𝑥, 𝑥 1 ) ≤ 𝑟 1 } ⊂ 𝑈 (𝑎;𝑟) \ 𝐴1 . Similarly, since 𝑈 (𝑥 1 ;𝑟 1 ) \ 𝐴2 ≠ ∅, there are 𝑥2 ∈ 𝑈 (𝑥 1 ;𝑟 1 ) \ 𝐴2 and 0 < 𝑟 2 < 1/22 such that 𝑈 (𝑥2 ;𝑟 2 ) ⊂ 𝑈 (𝑥1 ;𝑟 1 ) \ 𝐴2 . Inductively, we have 𝑈 (𝑥 𝜈 ;𝑟 𝜈 ) ⊂ 𝑈 (𝑥 𝜈−1 ;𝑟 𝜈−1 ) \

𝜈 Ø

𝐴𝜇 ,

0 < 𝑟𝜈
𝛽 > 0, and assume that 𝑢 𝜈 (𝑧) ≤ 𝛼,

(4.2.5) (4.2.6)

lim 𝑢 𝜈 (𝑧) ≤ 𝛽,

𝜈→∞

𝑧 ∈ P𝛥(𝑟), 𝜈 = 1, 2, . . . , 𝑧 ∈ 𝛤.

Then, for every compact subset 𝐾 ⋐ P𝛥(𝑟) and an arbitrary 𝜀 > 0, there is a number 𝑁 ∈ N such that 𝑢 𝜈 (𝑧) ≤ 𝛽 + 𝜀, 𝑧 ∈ 𝐾, 𝜈 ≥ 𝑁. Proof. For 𝜈 ∈ N we set 𝐹𝜈 = {𝑧 ∈ 𝛤 : 𝑢 𝜇 (𝑧) < 𝛽 + 𝜀, 𝜇 ≥ 𝜈}. Note that 𝐹𝜈 are Borel sets and 𝐹𝜈 ⊂ 𝐹𝜈+1 ⊂ · · · ,

∞ Ø

𝐹𝜈 = 𝛤.

𝜈=1

Let 𝑚(∗) denote the Lebesgue measure of the arguments 𝜃 𝑗 = arg 𝑧 𝑗 of (𝑧 𝑗 ) ∈ 𝛤. Then we have (4.2.7)

lim 𝑚(𝛤 \ 𝐹𝜈 ) = 0.

𝜈→∞

With the Poisson kernel 𝑃(𝑟, 𝜌, 𝜃, 𝜗) of 𝑛 variables we have (4.2.8)

𝑢 𝜇 (𝜌1 𝑒 𝑖 𝜗1 , . . . , 𝜌 𝑛 𝑒 𝑖 𝜗𝑛 ) ∫ 1 ≤ 𝑃(𝑟, 𝜌, 𝜃, 𝜗)𝑢 𝜇 (𝑟 1 𝑒 𝑖 𝜃1 , . . . , 𝑟 𝑛 𝑒 𝑖 𝜃𝑛 )𝑑𝜃 1 · · · 𝑑𝜃 𝑛 (2𝜋) 𝑛 𝛤 ∫  ∫ 1 = + 𝑃(𝑟, 𝜌, 𝜃, 𝜗)𝑢 𝜇 (𝑟 1 𝑒 𝑖 𝜃1 , . . . , 𝑟 𝑛 𝑒 𝑖 𝜃𝑛 )𝑑𝜃 1 · · · 𝑑𝜃 𝑛 , (2𝜋) 𝑛 𝐹𝜈 𝛤\𝐹𝜈

where 𝜌 = (𝜌 𝑗 ) with 0 < 𝜌 𝑗 < 𝑟 𝑗 (1 ≤ 𝑗 ≤ 𝑛). We take a polydisk P𝛥(𝑠) of polyradius 𝑠 = (𝑠 𝑗 ) such that 𝐾 ⋐ P𝛥(𝑠) ⋐ P𝛥(𝑟). It follows that for 𝜇 ≥ 𝜈 ∈ N and 𝑧 ∈ 𝐾

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4 Pseudoconvex Domains I

𝑢 𝜇 (𝑧) ≤ 𝛽 + 𝜀 + 𝛼𝑚(𝛤 \ 𝐹𝜈 )

𝑛 Ö 𝑟𝑗 +𝑠𝑗 𝑗=1

𝑟𝑗 −𝑠𝑗

Problem and Reduction

.

By (4.2.7) there is a number 𝑁 ∈ N such that 𝛼𝑚(𝛤 \ 𝐹𝜈 )

𝑛 Ö 𝑟𝑗 +𝑠𝑗 𝑗=1

𝑟𝑗 −𝑠𝑗

< 𝜀,

𝜈 ≥ 𝑁.

Therefore, 𝑢 𝜇 (𝑧) < 𝛽 + 2𝜀 for 𝑧 ∈ 𝐾 and 𝜇 ≥ 𝑁; since 𝜀 is arbitrary, the proof is nished. ⊓ ⊔ Theorem 4.2.9 (Hartogs Separate Analyticity). A separately holomorphic function 𝑓 (𝑧) on an open set 𝑈 of C𝑛 is holomorphic in 𝑈. Proof. The problem is local; so it is su cient to prove: Claim 4.2.10. For any point 𝑎 ∈ 𝑈 there is a polydisk neighborhood P𝛥(𝑎;𝑟), where 𝑓 (𝑧) is holomorphic. We may assume 𝑎 = 0 by a parallel translation, and take a polydisk P𝛥(𝑟) ⋐ 𝑈 of polyradius 𝑟 = (𝑟 𝑗 ). Then 𝑓 (𝑧) is de ned on the closed polydisk P𝛥(𝑟) such that: 4.2.11.

for every 𝑗 (1 ≤ 𝑗 ≤ 𝑛), the function 𝑓 (𝑧1 , . . . , 𝑧 𝑗 , . . . , 𝑧 𝑛 ) with arbitrarily ¯ 𝑘 ) (𝑘 ≠ 𝑗) is a holomorphic function on the closed disk 𝛥(𝑟 ¯ 𝑗 ). xed 𝑧 𝑘 ∈ 𝛥(𝑟

We are going to prove 𝑓 (𝑧) ∈ 𝒪(P𝛥) by induction on 𝑛. The case of 𝑛 = 1 is trivial. We assume the case of 𝑛 − 1 (𝑛 ≥ 2) to hold. We write ¯ 1 ) × P𝛥(𝑟 ′ ) with 𝑟 ′ = (𝑟 2 , . . . , 𝑟 𝑛 ). 𝑧 = (𝑧1 , 𝑧 ′ ) ∈ 𝛥(𝑟 (a) (Bounded 𝑓 (𝑧)) We assume that 𝑓 (𝑧) is bounded on P𝛥(𝑟); so, there is a constant 𝑀 > 0 such that (4.2.12)

| 𝑓 (𝑧)| ≤ 𝑀,

𝑧 ∈ P𝛥(𝑟).

With a xed 𝑧 ′ ∈ P𝛥(𝑟 ′ ) we expand (4.2.13)

𝑓 (𝑧 1 , 𝑧 ′ ) =

∞ Õ

𝑐 𝜈 (𝑧 ′ )𝑧 1𝜈 ,

¯ 1 ). 𝑧 1 ∈ 𝛥(𝑟

𝜈=0

By the coe cient estimate we have (4.2.14)

|𝑐 𝜈 (𝑧 ′ )| ≤

𝑀 , 𝑟 1𝜈

𝜈 = 0, 1, 2, . . . .

From this estimate we easily infer that the convergence of (4.2.13) is uniform on every compact subset of 𝛥(𝑟 1 ) × P𝛥(𝑟 ′ ). Therefore it is su cient to show the next claim in order to obtain 𝑓 (𝑧) ∈ 𝒪(P𝛥(𝑟)): Claim 4.2.15. 𝑐 𝜈 (𝑧 ′ ) ∈ 𝒪(P𝛥(𝑟 ′ )), 𝜈 = 0, 1, 2, . . ..

4.2 Hartogs’ Separate Analyticity

143

When 𝜈 = 0, 𝑐 0 ((𝑧 ′ )) = 𝑓 (0, 𝑧 ′ ), which is holomorphic in P𝛥(𝑟 ′ ) by the induction hypothesis. Suppose that 𝑐 𝜇 (𝑧 ′ ) ∈ P𝛥(𝑟 ′ ) for 𝜇 = 0, 1, . . . , 𝜈 − 1 (𝜈 ≥ 2). We consider (4.2.16)

′

𝑔 𝜈 (𝑧 1 , 𝑧 ) :=

𝑓 (𝑧1 , 𝑧 ′ ) −

Í𝜈−1

𝜇=0 𝑐 𝜇 (𝑧 𝜈 𝑧1

′ )𝑧 𝜇 1

= 𝑐 𝜈 (𝑧 ′ ) +

∞ Õ 𝜇=1

𝜇

𝑐 𝜈+𝜇 (𝑧 ′ )𝑧 1 .

For each xed 𝑧1 ≠ 0, 𝑔 𝜈 (𝑧1 , 𝑧 ′ ) is holomorphic in 𝑧 ′ ∈ P𝛥(𝑟 ′ ), and by (4.2.14) Õ Õ ∞ ∞ 𝑀 |𝑧1 | 𝜇 |𝑧1 |𝑀 𝑐 𝜈+𝜇 (𝑧 ′ )𝑧 𝜇 ≤ → 0 (|𝑧1 | → 0). 𝜈+𝜇 = 𝜈+1 1 𝑟 (1 − |𝑧1 |/𝑟 1 ) 𝑟 𝜇=1 𝜇=1 1 1 It follows from (4.2.16) that 𝑐 𝜈 (𝑧 ′ ) is a uniform limit of holomorphic functions in 𝑧 ′ ∈ P𝛥(𝑟 ′ ), and so holomorphic there. (b) In the general case, we are going to prove rst that there is a non-empty open subset 𝜔 in 𝛥(𝑟 1 ) such that 𝑓 (𝑧1 , 𝑧 ′ ) is bounded on 𝜔 × P𝛥(𝑠′ ) with any 𝑛 − 1 dimensional polydisk P𝛥(𝑠′ ) ⋐ P𝛥(𝑟 ′ ); then, 𝑓 (𝑧1 , 𝑧 ′ ) ∈ 𝒪(𝜔 × P𝛥(𝑠′ )) by (a) above, and hence 𝑓 (𝑧1 , 𝑧 ′ ) ∈ 𝒪(𝜔 × P𝛥(𝑟 ′ )). Set Ù ¯ 1 ) : | 𝑓 (𝑧1 , 𝑧 ′ )| ≤ 𝜈}, 𝜈 = 1, 2, . . . . 𝐸𝜈 = {𝑧1 ∈ 𝛥(𝑟 𝑧 ′ ∈P𝛥(𝑠 ′ )

Note that by the assumption (resp. the induction hypothesis), 𝑓 (𝑧1 , 𝑧 ′ ) is holomorphic ¯ 1 ) (resp. 𝑧 ′ ∈ P𝛥(𝑟 ′ )) with each xed 𝑧 ′ ∈ P𝛥(𝑟 ′ ) (resp. 𝑧1 ∈ 𝛥(𝑟 1 )). in 𝑧 1 ∈ 𝛥(𝑟 Therefore, 𝐸 𝜈 are clearly closed sets, and 𝐸 𝜈 ⊂ 𝐸 𝜈+1 ⊂ · · · ,

∞ Ø

¯ 1 ). 𝐸 𝜈 = 𝛥(𝑟

𝜈=1

It follows from Baire Theorem 4.2.3 that some 𝐸 𝜈 contains a non-empty open subset 𝜔 ⊂ 𝛥(𝑟 1 ). ¯ 1 ) → 𝛥(𝑟 ¯ 1 ) be a linear transform with 𝜙(0) = 𝑎 0 . Note Let 𝑎 0 ∈ 𝜔 and let 𝜙 : 𝛥(𝑟 ¯ 1 ) and 𝜙(𝜕 𝛥(𝑟 1 )) = 𝜕 𝛥(𝑟 1 ). that 𝜙(𝑧 1 ) is holomorphic in a neighborhood of 𝛥(𝑟 ′ ′ ′ ¯ We consider 𝑓 (𝜙(𝑧 1 ), 𝑧 ) in (𝑧1 , 𝑧 ) ∈ 𝛥(𝑟 1 ) × P𝛥(𝑟 ). Then 𝑓 (𝜙(𝑧1 ), 𝑧 ′ ) satis es the separate analyticity condition 4.2.11 on P𝛥(𝑟); moreover, there is a number 𝛿1 > 0 such that 𝑓 (𝜙(𝑧1 ), 𝑧 ′ ) is holomorphic in (𝑧1 , 𝑧 ′ ) ∈ 𝛥(𝛿1 ) × P𝛥(𝑟 ′ ). Thus, we may rewrite 𝑓 (𝑧 1 , 𝑧 ′ ) for 𝑓 (𝜙(𝑧 1 ), 𝑧 ′ ), and show 𝑓 (𝑧1 , 𝑧 ′ ) ∈ 𝒪( 𝛥(𝑟 1 ) × P𝛥(𝑟 ′ )). Taking smaller 𝛿1 > 0 and P𝛥(𝑟 ′ ) if necessary, we may assume: ¯ 1 ) × P𝛥(𝑟 ′ ), 4.2.17. (i) 𝑓 (𝑧1 , 𝑧 ′ ) is separately holomorphic on 𝛥(𝑟 ′ ′ ¯ 1 ) × P𝛥(𝑟 ). (ii) 𝑓 (𝑧1 , 𝑧 ) is holomorphic on 𝛥(𝛿 Set 𝑀 = ∥ 𝑓 (𝑧 1 , 𝑧 ′ ) ∥ 𝛥( ¯ 𝛿1 ) ×P𝛥(𝑟 ′ ) . As in (4.2.13) and (4.2.14) we have (4.2.18)

𝑓 (𝑧 1 , 𝑧 ′ ) =

∞ Õ 𝜈=0

𝑐 𝜈 (𝑧 ′ )𝑧 1𝜈 ,

|𝑧1 | ≤ 𝑟 1 , 𝑧 ′ ∈ P𝛥(𝑟 ′ ),

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4 Pseudoconvex Domains I

𝑐 𝜈 (𝑧 ′ ) =

(4.2.19) (4.2.20)

1 2𝜋𝑖

𝑀 |𝑐 𝜈 (𝑧 ′ )| ≤ 𝜈 , 𝛿1

∫ | 𝑧1 |= 𝛿1

Problem and Reduction

𝑓 (𝑧1 , 𝑧 ′ ) 𝑑𝑧 1 ∈ 𝒪(P𝛥(𝑟 ′ )), 𝜈 = 0, 1, 2, . . . , 𝑧 1𝜈+1

𝑧 ′ ∈ P𝛥(𝑟 ′ ), 𝜈 = 0, 1, 2, . . . .

Now we set 𝑢 𝜈 (𝑧 ′ ) =

1 log |𝑐 𝜈 (𝑧 ′ )|, 𝜈

𝑧 ′ ∈ P𝛥(𝑟 ′ ), 𝜈 = 1, 2, . . . .

Because of the convergence radius of (4.2.18) we get (4.2.21)

lim 𝑢 𝜈 (𝑧 ′ ) ≤ − log𝑟 1 ,

𝜈→∞

𝑧 ′ ∈ P𝛥(𝑟 ′ ).

By (4.2.20) we have a uniform bound (4.2.22)

𝑢 𝜈 (𝑧 ′ ) ≤ − log 𝛿1 + log 𝑀,

𝑧 ′ ∈ P𝛥(𝑟 ′ ), 𝜈 = 1, 2, . . . .

Let P𝛥(𝑠) ⋐ P𝛥(𝑟) be given arbitrarily. By Lemma 4.2.4 with (4.2.21) and (4.2.22) there is a number 𝑁 ∈ N such that 𝑢 𝜈 (𝑧 ′ ) ≤ − log 𝑠1 ,

𝑧 ′ ∈ P𝛥(𝑠′ ), 𝜈 ≥ 𝑁.

It follows that |𝑐 𝜈 (𝑧 ′ )|𝑠1𝜈 ≤ 1 (𝜈 ≥ 𝑁); hence, the series (4.2.18) converges absolutely and uniformly on every compact subset of 𝛥(𝑠1 ) × P𝛥(𝑠′ ). Since every term 𝑐 𝜈 (𝑧 ′ )𝑧1𝜈 is holomorphic in P𝛥(𝑠) by (4.2.19), so is the limit 𝑓 (𝑧1 , 𝑧 ′ ). Since P𝛥(𝑠) ⋐ P𝛥(𝑟) is arbitrary, we see that 𝑓 (𝑧) is holomorphic in P𝛥(𝑟). ⊓ ⊔ Remark 4.2.23. (i) As seen in Remark 1.1.17 (ii), for real analytic functions, the separate analyticity even with boundedness condition does not imply the analyticity in all variables. Together with Hartogs’ phenomenon, Hartogs’ Theorem 4.2.9 represents a special feature of complex analytic functions of 𝑛 (≥ 2) variables. (ii) In the proof above, a little idea is to reduce 𝑎 0 ∈ 𝜔 to 𝑎 0 = 0 by making use of the linear transform 𝜙(𝑧1 ), so that it is possible to claim the analyticity of 𝑓 on the whole 𝛥(𝑟 1 ) × P𝛥(𝑟 ′ ). Cf., e.g., the proofs of Hörmander [30], Chap. 2, and Nishino [37], Chap. 1.

4.3 Pseudoconvexity 4.3.1 Pseudoconvexity Problem Let 𝜋 : 𝔇 → C𝑛 be an unrami ed domain over C𝑛 . We take and x a polydisk P𝛥 with center at the origin, and consider the boundary distance function 𝛿P𝛥 ( 𝑝, 𝜕𝔇).

4.3 Pseudoconvexity

145

As a fundamental basis we prove Oka’s Theorem of Boundary Distance Functions, which serves as a key towards the solution of the Pseudoconvexity Problem: Theorem 4.3.1 (Oka). If 𝔇 is a domain of holomorphy, then − log 𝛿P𝛥 ( 𝑝, 𝜕𝔇) ∈ 𝒫0 (𝔇). Proof. It follows from (3.6.5) that − log 𝛿P𝛥 ( 𝑝, 𝜕𝔇) is continuous. Since the plurisubharmonicity is a local property, it su ces to deal with it in a neighborhood 𝑈 of 𝑎 ∈ 𝔇. Since 𝜋 : 𝔇 → C𝑛 is locally biholomorphic, we may regard 𝑈 ⊂ C𝑛 . Take a complex line 𝐿 ⊂ C𝑛 (𝐿  C) passing through 𝑎. We then prove that the restriction − log 𝛿P𝛥 ( 𝑝, 𝜕𝔇)| 𝐿∩𝑈 satis es the submean property on a circle, of which the center may be assumed to be 𝑎 by a translation. We take a closed disk and the circle in 𝐿 ∩ 𝔇 with center at 𝑎: 𝐸 = {𝑎 + 𝜁 𝑣 : |𝜁 | ≤ 𝑅} ⊂ 𝐿 ∩ 𝔇,

𝐾 = {𝑎 + 𝜁 𝑣 : |𝜁 | = 𝑅},

where 𝑣 ∈ C𝑛 \ {0} is a directional vector of 𝐿. Noting that − log 𝛿P𝛥 (𝑎 + 𝜁 𝑣, 𝜕𝔇) is a continuous function on |𝜁 | ≤ 𝑅, we take the Poisson integral 𝑢(𝜁) of − log 𝛿P𝛥 ( 𝑝, 𝜕𝔇) on the closed disk |𝜁 | ≤ 𝑅 (cf. [38] Chap. 3 6). Then 𝑢(𝜁) is continuous on |𝜁 | ≤ 𝑅, harmonic (Δ 𝑢 = 0) in the interior |𝜁 | < 𝑅, and is equal to − log 𝛿P𝛥 (𝑎 + 𝜁 𝑣, 𝜕𝔇) on the circle |𝜁 | = 𝑅. We get (4.3.2)

𝑢(0) =

1 2𝜋

∫

2𝜋

− log 𝛿P𝛥 (𝑎 + 𝑅𝑒 𝑖 𝜃 𝑣, 𝜕𝔇)𝑑𝜃.

0

For an arbitrarily small 𝜀 > 0, − log 𝛿P𝛥 (𝑎 + 𝑅𝑒 𝑖 𝜃 𝑣, 𝜕𝔇) < 𝑢(𝑟𝑒 𝑖 𝜃 ) + 𝜀, where 𝑟 < 𝑅 is su ciently close to 𝑅. With the adjoint harmonic function 𝑢 ∗ (𝜁) of 𝑢(𝜁) in |𝜁 | < 𝑅, we obtain a holomorphic function 𝑔(𝜁) = 𝑢(𝜁) + 𝑖𝑢 ∗ (𝜁). Let (4.3.3)

𝑔(𝜁) =

∞ Õ

𝑐𝜈 𝜁 𝜈 ,

|𝜁 | < 𝑅

𝜈=0

be the power expansion. Since it converges uniformly on |𝜁 | = 𝑟, the partial sum Í 𝑁 series 𝑃(𝜁) = 𝜈=0 𝑐 𝜈 𝜁 𝜈 with su ciently large 𝑁 ∈ N satis es |𝑔(𝜁) − 𝑃(𝜁)| < 𝜀 (|𝜁 | = 𝑟). Therefore, 𝑟  (4.3.4) − log 𝛿P𝛥 (𝑎 + 𝜁 𝑣, 𝜕𝔇) < ℜ𝑃 𝜁 + 2𝜀, |𝜁 | = 𝑅. 𝑅 𝑛 𝑛 ˆ Since 𝐿 is an a ne
linear subspace of C , there is a polynomial 𝑃(𝑧) on C with 𝑟 ˆ 𝑃(𝑎 + 𝜁 𝑣) = 𝑃 𝑅 𝜁 . It follows from (4.3.4) that ˆ 𝛿P𝛥 (𝑧, 𝜕𝔇) > 𝑒 − 𝑃 (𝑧) −2𝜀 , 𝑧 ∈ 𝐾.

146

4 Pseudoconvex Domains I

Problem and Reduction

b𝔇 ⊃ 𝐸, and then by Lemma 3.2.7 The Maximum Principle implies 𝐾 ˆ 𝛿P𝛥 (𝑧, 𝜕𝔇) ≥ 𝑒 − 𝑃 (𝑧) −2𝜀 , 𝑧 ∈ 𝐸 . In particular, at 𝑧 = 𝑎 we get ˆ 𝛿P𝛥 (𝑎, 𝜕𝔇) ≥ 𝑒 − 𝑃 (𝑎) −2𝜀 = 𝑒 −𝑢(0) −2𝜀 . This together with (4.3.2) implies − log 𝛿P𝛥 (𝑎, 𝜕𝔇) ≤

1 2𝜋

∫

2𝜋

− log 𝛿P𝛥 (𝑎 + 𝑅𝑒 𝑖 𝜃 𝑣, 𝜕𝔇)𝑑𝜃 + 2𝜀.

0

Now as 𝜀 > 0 is arbitrary, − log 𝛿P𝛥 (𝑎, 𝜕𝔇) ≤

1 2𝜋

∫

2𝜋

− log 𝛿P𝛥 (𝑎 + 𝑅𝑒 𝑖 𝜃 𝑣, 𝜕𝔇)𝑑𝜃.

0

Thus − log 𝛿P𝛥 (𝑎 + 𝜁 𝑣, 𝜕𝔇) satis es the submean property.

⊓ ⊔

Together with Theorem 3.7.3 we immediately have: Corollary 4.3.5. If 𝔇 is holomorphically convex, then − log 𝛿P𝛥 ( 𝑝, 𝜕𝔇) ∈ 𝒫0 (𝔇). 4.3.6. (Pseudoconvexity Problem I) Does the converse of Theorem 4.3.1 hold? Furthermore, is 𝔇 Stein? Definition 4.3.7. (i) A real-valued function 𝜓 : 𝔇 → [−∞, ∞) with −∞ allowed as a value is an exhaustion function if 𝔇𝑐 := {𝑝 ∈ 𝔇 : 𝜓( 𝑝) < 𝑐} ⋐ 𝔇 for every 𝑐 ∈ R. The set 𝔇𝑐 is called a sublevel set. (ii) A domain 𝔇/C𝑛 is pseudoconvex if there exists a pseudoconvex exhaustion function 𝜓 on 𝔇.5 In the case when 𝜓 is chosen to be R-valued and of 𝐶 𝑘 -class (0 ≤ 𝑘 ≤ ∞), 𝔇 is said to be 𝐶 𝑘 -pseudoconvex. Remark 4.3.8. (i) In many references, the de nition of an exhaustion function includes the continuity. Also “𝐶 0 -pseudoconvex” or “𝐶 ∞ -pseudoconvex” is simply termed “pseudoconvex”. The reader should confer other books. (ii) Since a pseudoconvex exhaustion function 𝜓 : 𝔇 → [−∞, ∞) is only upper semi-continuous, we do not have necessarily 𝔇𝑐 ⋐ 𝔇𝑐′ for real numbers 𝑐 < 𝑐 ′ . If 𝑐 ′ is chosen su ciently large, naturally 𝔇𝑐 ⋐ 𝔇𝑐′ . Therefore 1/(𝑐 − 𝜓( 𝑝)) does not serve as an exhaustion function on the sublevel set 𝔇𝑐 ; it is not known immediately if 𝔇𝑐 is pseudoconvex or not. (With the continuity assumption it is immediate.) Remark 4.3.9. The notions de ned for a domain 𝔇/C𝑛 are listed as follows: 5 Note that 𝜓 is not assumed to be continuous and that the holomorphic separation is not assumed for 𝔇.

4.3 Pseudoconvexity

147

(i) 𝔇 is a domain of holomorphy (with the holomorphic separation). (ii) 𝔇 is holomorphically convex. (iii) 𝔇 is Stein (holomorphically convex + holomorphically separable). (iv) − log 𝛿P𝛥 ( 𝑝, 𝜕𝔇) ∈ 𝒫0 (𝔇). (v) 𝔇 is 𝐶 0 -pseudoconvex. (vi) 𝔇 is pseudoconvex. (i) is the most natural notion from the viewpoint of the domain of existence of analytic functions, and also historically it is the oldest. But it is de ned by the analytic continuation which requires the “outside” of the given domain; the existence of the “outside” is implicitly assumed (extrinsic). In comparison, (ii) is completely characterized only by the interior information of the domain (intrinsic). In the case of univalent domains, the two are equivalent (Theorem 3.2.11), but it is not yet proved here for multivalent domains in general at this moment (cf. Remark 3.7.5 and Theorem 5.3.2). The holomorphic separation is also a fundamental property of domains; added with it, one gets (iii). (v)⇒(vi) is clear. It is “Pseudoconvexity Problem” to ask if (i) (vi) are mutually equivalent: In particular, (vi)⇒(iii) is crucial. 4.3.10. (Pseudoconvexity Problem II) Is a pseudoconvex domain over C𝑛 Stein? Pseudoconvexity Problems I 4.3.6, and II 4.3.10 together with their relations were a rmatively solved nally by K. Oka; the aim of this chapter and the next is to give the proofs. Before going into the proofs, we investigate the mutual relations among the notions listed in Remark 4.3.9. Theorem 4.3.11 (Oka). If 𝔇/C𝑛 is a pseudoconvex domain, then − log 𝛿P𝛥 ( 𝑝, 𝜕𝔇) ∈ 𝒫0 (𝔇). Proof. We follow the proof of Theorem 4.3.1. Take a point 𝑝 0 ∈ 𝔇 and its polydisk neighborhood P𝛥 ⋐ 𝔇. We regard P𝛥 ⊂ C𝑛 . With a vector 𝑣 ∈ C𝑛 \ {0} we consider a closed disk 𝛥¯ = {𝑝 0 + 𝜁 𝑣 : |𝜁 | ≤ 𝑅} ⊂ P𝛥 ⊂ 𝔇. Let 𝑢(𝜁) be the Poisson integral of − log 𝛿( 𝑝 0 + 𝜁 𝑣, 𝜕𝔇) on the circle |𝜁 | = 𝑅. Take the holomorphic function 𝑔(𝜁) in (4.3.3) so that (4.3.12)

ℑ𝑔(0) = 0,

𝑔(0) = 𝑢(0).

Note that ℜ𝑔 = 𝑢. For every small 𝜀 > 0 we take 0 < 𝑡 < 1, su ciently close to 1, so that 𝑔(𝑡𝜁) is holomorphic in a neighborhood of |𝜁 | ≤ 𝑅 and |ℜ𝑔(𝑡𝜁) − 𝑢(𝜁)| < 𝜀, Thus we have

|𝜁 | = 𝑅.

148

(4.3.13)

4 Pseudoconvex Domains I

𝛿P𝛥 ( 𝑝 0 + 𝜁 𝑣, 𝜕𝔇) ≥ |𝑒 −𝑔 (𝑡 𝜁 ) − 𝜀 |,

Problem and Reduction

|𝜁 | = 𝑅.

We want to show that this inequality holds also at 𝜁 = 0. With an arbitrarily chosen vector 𝑤 ∈ P𝛥 we consider a holomorphic map 𝛹 : (𝜁, 𝜉) ↦→ 𝑝 0 + 𝜁 𝑣 + 𝜉𝑤𝑒 −𝑔 (𝑡 𝜁 ) − 𝜀 for 𝜉 ∈ C with |𝜉 | ≤ 1. If 𝜉 = 0, then𝛹 ({|𝜁 | ≤ 𝑅}, 0) = 𝛥¯ ⊂ 𝔇 by de nition. If |𝜁 | = 𝑅, then it follows from (4.3.13) that 𝛹 ({𝜁 | = 𝑅} × {|𝜉 | ≤ 1}) ⊂ 𝔇. Set (4.3.14)

𝐾 = ({|𝜁 | ≤ 𝑅} × {0}) ∪ ({|𝜁 | = 𝑅} × {|𝜉 | ≤ 1}) (⊂ C2 ).

Note that 𝛹 (𝐾) ⋐ 𝔇. We take an exhaustion function 𝜙 ∈ 𝒫(𝔇), and a sublevel set 𝔇𝑐 := {𝜙 < 𝑐} ⋑ 𝛹 (𝐾). Since the map𝛹 : 𝛺 H → 𝔇𝑐 is holomorphic in a neighborhood 𝛺 H (a Hartogs domain) of 𝐾 ⊂ C2 , we see that 𝜙 ◦𝛹 ∈ 𝒫(𝛺 H ). Let 𝑆 denote the set of all 𝑠 with 0 ≤ 𝑠 ≤ 1, satisfying the condition 𝛹 ({|𝜁 | ≤ 𝑅} × {|𝜉 | ≤ 𝑠}) ⊂ 𝔇. Since 𝑆 ∋ 0, 𝑆 ≠ ∅. By the condition, 𝑆 is an open set. Also, if the condition is satis ed, the Maximal Principle implies that 𝛹 ({|𝜁 | ≤ 𝑅} × {|𝜉 | ≤ 𝑠}) ⊂ 𝔇𝑐 . For an accumulation point 𝑠′ of 𝑠 ∈ 𝑆 ¯ 𝑐. 𝛹 ({|𝜁 | ≤ 𝑅} × {|𝜉 | ≤ 𝑠′ }) ⊂ 𝔇 Therefore, 𝑆 is closed, and so 𝑆 = [0, 1]. In particular, with 𝜁 = 0 and 𝜉 = 1 one obtains 𝑝 0 + 𝑤𝑒 −𝑔 (0) − 𝜀 ∈ 𝔇. Since 𝑤 ∈ P𝛥 is arbitrary, it follows together with (4.3.12) that 𝑝 0 + 𝑒 −𝑢(0) − 𝜀 · P𝛥 ⊂ 𝔇 ; 𝛿P𝛥 ( 𝑝 0 , 𝜕𝔇) ≥ 𝑒 −𝑢(0) − 𝜀 ; ∫ 2𝜋 1 − log 𝛿P𝛥 ( 𝑝 0 , 𝜕𝔇) ≤ − log 𝛿P𝛥 ( 𝑝 0 + 𝑅𝑒 𝑖 𝜃 𝑣)𝑑𝜃 + 𝜀 ; 2𝜋 0 ∫ 2𝜋 1 − log 𝛿P𝛥 ( 𝑝 0 , 𝜕𝔇) ≤ − log 𝛿P𝛥 ( 𝑝 0 + 𝑅𝑒 𝑖 𝜃 𝑣)𝑑𝜃. 2𝜋 0 Thus, − log 𝛿P𝛥 ( 𝑝, 𝜕𝔇) ∈ 𝒫0 (𝔇). We prove the converse of the above theorem.

⊓ ⊔

4.3 Pseudoconvexity

149

Theorem 4.3.15 (Oka). Let 𝜋 : 𝔇 → C𝑛 be a domain. If − log 𝛿P𝛥 ( 𝑝, 𝜕𝔇) is plurisubharmonic, then 𝔇 is 𝐶 0 -pseudoconvex. Proof. We construct a continuous pseudoconvex exhaustion function on 𝔇. (1) If 𝜋 : 𝔇 → C𝑛 is nitely sheeted, then the function de ned by 𝜆( 𝑝) = max{− log 𝛿P𝛥 ( 𝑝, 𝜕𝔇), ∥𝜋( 𝑝) ∥} is a continuous pseudoconvex exhaustion function on 𝔇. (2) The proof in the case when 𝔇/C𝑛 is in nitely sheeted is elementary, but a little bit long, so that we divide it into several steps. (a) Take a point 𝑝 0 ∈ 𝔇 and x it. In what follows we consider a number 𝜌 > 0 with 𝜌 < 𝛿P𝛥 ( 𝑝 0 , 𝜕𝔇). Set 𝔇𝜌 = the connected component of {𝑝 ∈ 𝔇 : 𝛿P𝛥 ( 𝑝, 𝜕𝔇) > 𝜌} containing 𝑝 0 . Ð For 0 < 𝜌 ′ < 𝜌, 𝔇𝜌 ⊂ 𝔇𝜌′ , and 𝜌>0 𝔇𝜌 = 𝔇. As stated in Remark 4.1.21, 𝑑𝜆 denotes the Lebesgue measure of C𝑛 . We choose a 𝐶 ∞ -function 𝜒(𝑧) ≥ 0 such that ∫ (4.3.16) Supp 𝜒 ⊂ P𝛥, 𝜒(𝑤)𝑑𝜆(𝑤) = 1. 𝑤∈C𝑛

For 𝜀 > 0 we put 𝜒 𝜀 (𝑤) = 𝜒(𝑤/𝜀)𝜀 −2𝑛 . Then ∫ Supp 𝜒 𝜀 ⊂ 𝜀P𝛥, 𝜒 𝜀 (𝑤)𝑑𝜆(𝑤) = 1. C𝑛

Let 𝑑 𝜌 ( 𝑝) ( 𝑝 ∈ 𝔇𝜌 ) be the continuous function de ned by (3.6.15). As far as a point in a univalent subdomain of 𝔇 is concerned, we write it as a point of C𝑛 , unless confusion occurs. For 0 < 𝜀 ≤ 𝜌 we obtain the smoothing of 𝑑 𝜌 , ∫ (𝑑 𝜌 ) 𝜀 ( 𝑝) = (𝑑 𝜌 ) ∗ 𝜒 𝜀 ( 𝑝) = 𝑑 𝜌 ( 𝑝 + 𝑤) 𝜒 𝜀 (𝑤)𝑑𝜆(𝑤), 𝑝 ∈ 𝔇𝜌 . 𝑤∈ 𝜀P𝛥

Let (𝑟 0 𝑗 ) be the polyradius of P𝛥, and set 𝐶0 = de nition and (3.6.16) that |(𝑑 𝜌 ) 𝜀 ( 𝑝) − 𝑑 𝜌 ( 𝑝)| ≤ 𝜀𝐶0 ,

qÍ

2 𝑗 𝑟0 𝑗 .

It follows from the

𝑝 ∈ 𝔇𝜌 .

By Lemma 3.6.17 we see that (4.3.17)

{𝑝 ∈ 𝔇𝜌 : (𝑑 𝜌 ) 𝜀 ( 𝑝) < 𝑏} ⋐ 𝔇,

∀ 𝑏 > 0.

(b) For 𝑝 ∈ 𝔇 we write the local complex coordinate system 𝜋( 𝑝) = (𝑧 𝑗 ) = (𝑥 𝑗 + 𝑖𝑦 𝑗 ). Let 𝜉 (∥𝜉 ∥ = 1) be one of the directional vectors in the space C𝑛  R2𝑛 of 𝑥 𝑗 , 𝑦 𝑗 , 1 ≤ 𝑗 ≤ 𝑛, and write 𝜕𝜕𝜉 for the directional derivative. With ℎ ∈ R we have

150

4 Pseudoconvex Domains I

Problem and Reduction

(𝑑 𝜌 ) 𝜀 ( 𝑝 + ℎ𝜉) − (𝑑 𝜌 ) 𝜀 ( 𝑝) 𝜕 (𝑑 𝜌 ) 𝜀 = ( 𝑝). ℎ→0 ℎ 𝜕𝜉 lim

On the other hand, we deduce from (3.6.16) the following estimate: (𝑑 𝜌 ) 𝜀 ( 𝑝 + ℎ𝜉) − (𝑑 𝜌 ) 𝜀 ( 𝑝) ℎ ∫ 1 = {𝑑 𝜌 ( 𝑝 + ℎ𝜉 + 𝑤) − 𝑑 𝜌 ( 𝑝 + 𝑤)} 𝜒 𝜀 (𝑤)𝑑𝜆(𝑤) ℎ 𝑤 ∫ 1 ≤ |𝑑 𝜌 ( 𝑝 + ℎ𝜉 + 𝑤) − 𝑑 𝜌 ( 𝑝 + 𝑤)| 𝜒 𝜀 (𝑤)𝑑𝜆(𝑤) |ℎ| 𝑤 1 ≤ |ℎ| · ∥𝜉 ∥ = 1. |ℎ| Therefore we have (4.3.18)

𝜕 (𝑑 𝜌 ) 𝜀 𝜕𝜉 ( 𝑝) ≤ 1,

𝑝 ∈ 𝔇𝜌 , 0 < 𝜀 ≤ 𝜌.

With 0 < 2𝜀 ≤ 𝜌 we consider 𝑑˜𝜌, 𝜀 ( 𝑝) = (𝑑 𝜌 ) 𝜀




𝜀 ( 𝑝),

𝑝 ∈ 𝔇𝜌 .

It follows that ∫ 𝜕 𝑑˜𝜌, 𝜀 𝜕 (𝑑 𝜌 ) 𝜀 ( 𝑝) = ( 𝑝 + 𝑤) 𝜒 𝜀 (𝑤)𝑑𝜆(𝑤) 𝜕𝜉 𝜕𝜉 𝑤 ∫ 𝑤− 𝑝 1 𝜕 (𝑑 𝜌 ) 𝜀 = (𝑤) 𝜒 𝑑𝜆(𝑤). 𝜕𝜉 𝜀 𝜀 2𝑛 𝑤 Similarly to 𝜕𝜕𝜉 , we take a directional derivative 𝑗 ≤ 𝑛, and then have 𝜕 2 𝑑˜𝜌, 𝜀 ( 𝑝) = 𝜕𝜂𝜕𝜉

∫ 𝑤

𝜕 𝜕𝜂

with respect to one of 𝑥 𝑗 , 𝑦 𝑗 , 1 ≤

𝜕 (𝑑 𝜌 ) 𝜀 𝜕 𝜒  𝑤 − 𝑝  −1 (𝑤) 𝑑𝜆(𝑤). 𝜕𝜉 𝜕𝜂 𝜀 𝜀 2𝑛+1

Together with (4.3.18) we see that ∫ 𝜕 2 𝑑˜ 𝜕 (𝑑 𝜌 ) 𝜀 𝜕 𝜒  𝑤 − 𝑝  1 𝜌, 𝜀 (4.3.19) ( 𝑝) ≤ (𝑤) · 𝑑𝜆(𝑤) 𝜕𝜂𝜕𝜉 𝜕𝜉 𝜕𝜂 𝜀 𝜀 2𝑛+1 𝑤 ∫ 1 𝜕 𝜒 𝐶1 ≤ (𝑤) 𝑑𝜆(𝑤) = . 𝜀 𝑤 𝜕𝜂 𝜀 Here, 𝐶1 is a positive constant independent from 𝜀 and 𝜌. We set

4.3 Pseudoconvexity

151

𝑑ˆ𝜌 ( 𝑝) = 𝑑˜𝜌, 𝜌 ( 𝑝), 2

𝑝 ∈ 𝔇𝜌 .

By (4.3.17) we have for 𝑑ˆ𝜌 ( 𝑝) that {𝑝 ∈ 𝔇𝜌 : 𝑑ˆ𝜌 ( 𝑝) < 𝑏} ⋐ 𝔇,

(4.3.20) With 𝐶2 ≫

2𝐶1 𝜌

(4.3.21)

∀ 𝑏 > 0.

we set 𝜑𝜌 ( 𝑝) = 𝑑ˆ𝜌 ( 𝑝) + 𝐶2 ∥𝜋( 𝑝) ∥ 2 .

Then from (4.3.19) we obtain Õ 𝜕 2 𝜑𝜌 𝜉 𝑗 𝜉¯𝑘 ≥ ∥ (𝜉 𝑗 )∥ 2 . 𝜕𝑧 𝑗 𝜕 𝑧¯ 𝑘 𝑗,𝑘 Summarizing the above, we get: Lemma 4.3.22. There is a strongly plurisubharmonic function 𝜑𝜌 ( 𝑝) > 0 of 𝐶 ∞ class on 𝔇𝜌 such that {𝑝 ∈ 𝔇𝜌 : 𝜑𝜌 ( 𝑝) < 𝑏} ⋐ 𝔇,

∀ 𝑏 > 0.

(c) Here we use the property, “− log 𝛿P𝛥 ( 𝑝, 𝜕𝔇) ∈ 𝒫0 (𝔇)”. Choose 𝑎 1 > 0 so that 𝛿P𝛥 ( 𝑝 0 ) > 𝑒 −𝑎1 , and take a divergent monotone increasing sequence, 𝑎 1 < 𝑎 2 < · · · < 𝑎 𝑗 ↗ ∞. We set 𝔇 𝑗 = 𝔇𝑒 −𝑎 𝑗 ( 𝑗 = 1, 2, . . .), which satisfy 𝔇 𝑗 ⊂ 𝔇 𝑗+1 and Ð 𝔇= ∞ 𝑗=1 𝔇 𝑗 . Applying Lemma 4.3.22 for each 𝔇 𝑗 (𝜌 = 𝑒 −𝑎 𝑗 ), we obtain a strongly plurisubharmonic function 𝜑 𝑗 ( 𝑝) of 𝐶 ∞ -class such that for every 𝑏 > 0 {𝑝 ∈ 𝔇 𝑗 : 𝜑 𝑗 ( 𝑝) < 𝑏} ⋐ 𝔇 𝑗+1 . We choose a monotone increasing sequence, 𝑏 1 < 𝑏 2 < · · · ↗ ∞ as follows. With an arbitrarily chosen 𝑏 1 > 0 we set 𝛥1 = {𝑝 ∈ 𝔇1 : 𝜑4 ( 𝑝) < 𝑏 1 }. It follows that (4.3.23)

𝜕 𝛥1 ⊂ {− log 𝛿P𝛥 ( 𝑝) = 𝑎 1 } ∪ {𝜑4 ( 𝑝) = 𝑏 1 }.

Since 𝛥1 ⋐ 𝔇2 , there is 𝑏 2 ≫ max{2, 𝑏 1 } such that 𝛥2 = {𝑝 ∈ 𝔇2 : 𝜑5 ( 𝑝) < 𝑏 2 } ⋑ 𝛥1 . Inductively, we choose 𝑏 𝑗 > max{ 𝑗, 𝑏 𝑗 −1 } so that (4.3.24)

𝛥 𝑗 = {𝑝 ∈ 𝔇 𝑗 : 𝜑 𝑗+3 ( 𝑝) < 𝑏 𝑗 } ⋑ 𝛥 𝑗 −1 .

152

4 Pseudoconvex Domains I

Problem and Reduction

It follows that (4.3.25)

𝔇=

∞ Ø

𝛥𝑗.

𝑗=1

We set 𝛷1 ( 𝑝) = 𝜑4 ( 𝑝) + 1 (> 1) for 𝑝 ∈ 𝛥4 . For 𝑗 ≥ 1, we assume that 𝛷 ℎ ( 𝑝), 1 ≤ ℎ ≤ 𝑗, are de ned so that the following conditions are satis ed: 4.3.26. (i) 𝛷 ℎ ( 𝑝) ∈ 𝒫0 ( 𝛥ℎ+3 ). (ii) 𝛷 ℎ ( 𝑝) > ℎ, ∀ 𝑝 ∈ 𝛥ℎ+2 \ 𝛥ℎ+1 , 1 ≤ ℎ ≤ 𝑗. (iii) 𝛷 ℎ ( 𝑝) = 𝛷 ℎ−1 ( 𝑝), ∀ 𝑝 ∈ 𝛥ℎ , 2 ≤ ℎ ≤ 𝑗. We de ne a continuous plurisubharmonic function on 𝛥 𝑗+4 by 𝜓 𝑗+1 ( 𝑝) = max{− log 𝛿P𝛥 ( 𝑝, 𝜕𝔇) − 𝑎 𝑗+1 , 𝜑 𝑗+4 ( 𝑝) − 𝑏 𝑗+1 },

𝑝 ∈ 𝛥 𝑗+4 .

Then 𝜓 𝑗+1 ( 𝑝) ≤ 0,

(4.3.27)

𝑝 ∈ 𝛥¯ 𝑗+1 ,

min 𝜓 𝑗+1 ( 𝑝) > 0.

𝛥¯ 𝑗+3 \𝛥 𝑗+2

From this it follows that for a su ciently large 𝑘 𝑗+1 > 0 ( (4.3.28)

)

min 𝑘 𝑗+1 𝜓 𝑗+1 ( 𝑝) > max 𝑗 + 1, max 𝛷 𝑗 ( 𝑝) .

𝛥¯ 𝑗+3 \𝛥 𝑗+2

𝛥¯ 𝑗+2

We de ne ( 𝛷 𝑗+1 ( 𝑝) =

max{𝛷 𝑗 ( 𝑝), 𝑘 𝑗+1 𝜓 𝑗+1 ( 𝑝)}, 𝑘 𝑗+1 𝜓 𝑗+1 ( 𝑝),

𝑝 ∈ 𝛥 𝑗+2 , 𝑝 ∈ 𝛥 𝑗+4 \ 𝛥 𝑗+2

(cf. Fig. 4.1). We see by (4.3.28) that in a neighborhood of 𝜕 𝛥 𝑗+2

Fig. 4.1 The de ning graph of 𝛷 𝑗+1 .

4.3 Pseudoconvexity

153

𝛷 𝑗+1 ( 𝑝) = 𝑘 𝑗+1 𝜓 𝑗+1 ( 𝑝), and so that 𝛷 𝑗+1 ( 𝑝) is a continuous plurisubharmonic function on 𝛥 𝑗+4 . It follows from (4.3.27) and (4.3.28) that 𝛷 𝑗+1 ( 𝑝) = 𝛷 𝑗 ( 𝑝), 𝛷 𝑗+1 ( 𝑝) > 𝑗 + 1,

𝑝 ∈ 𝛥 𝑗+1 , 𝑝 ∈ 𝛥 𝑗+4 \ 𝛥 𝑗+2 .

Therefore, inductively we obtain 𝛷 𝑗 ( 𝑝), 𝑗 = 1, 2, . . ., satisfying 4.3.26. Now we de ne 𝛷( 𝑝) = lim 𝛷 𝑗 ( 𝑝), 𝑝 ∈ 𝔇. 𝑗→∞

Then, 𝛷( 𝑝) is a continuous plurisubharmonic function on 𝔇. By 4.3.26 (ii) 𝛷( 𝑝) > 𝑗,

𝑝 ∈ 𝔇 \ 𝛥 𝑗+1 ,

𝑗 = 1, 2, . . . ,

so that 𝛷( 𝑝) is an exhaustion function. Corollary 4.3.29.

⊓ ⊔

If a domain 𝔇(/C𝑛 ) is pseudoconvex, then 𝔇 is 𝐶 0 -pseudoconvex.

Proof. This readily follows from Theorems 4.3.11 and 4.3.15.

⊓ ⊔

Remark 4.3.30. (i) By virtue of Oka’s Theorem 4.3.15, we see that Pseudoconvexity Problem I 4.3.6 follows from II 4.3.10. (ii) In terms of the items stated in Remark 4.3.9, we have shown the following implications: • • • • • • •

(iii) ⇒ (i) (Theorem 3.7.3); (iii) ⇒ (ii) (Obvious); (i) ⇒ (iv) (Theorem 4.3.1); (ii) ⇒ (iv) (Corollary 4.3.5); (vi) ⇒ (iv) (Theorem 4.3.11); (iv) ⇒ (v) (Theorem 4.3.15); (v) ⇒ (vi) (Trivial).

The nal di culty is the implication of (vi) ⇒ (iii) (Pseudoconvexity Problem II 4.3.10), which we will solve in the next chapter (cf. Afterword for historical comments). Theorem 4.3.31. Let 𝔇(/C𝑛 ) be a domain, and let 𝛺 𝛼 (𝛼 ∈ 𝛤) be a family of subdomains of Ñ 𝔇. If all 𝛺 𝛼 are pseudoconvex, then every connected component of the interior ( 𝛼∈𝛤 𝛺 𝛼 ) ◦ of the intersection is pseudoconvex. Proof. It follows from Theorem 4.3.11 that − log 𝛿P𝛥 (𝑧, 𝜕𝛺 𝛼 ) is plurisubharmonic Ñ in 𝛺 𝛼 . For any connected component 𝜔 of ( 𝛼∈𝛤 𝛺 𝛼 ) ◦ , − log 𝛿P𝛥 (𝑧, 𝜕𝜔) = sup − log 𝛿P𝛥 (𝑧, 𝜕𝛺 𝛼 ), 𝛼∈𝛤

𝑧 ∈ 𝜔,

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and − log 𝛿P𝛥 (𝑧, 𝜕𝜔) is continuous. By Theorem 4.1.16 (vi), we see that − log 𝛿P𝛥 (𝑧, 𝜕𝜔) (𝑧 ∈ 𝜔) is plurisubharmonic. Then Theorem 4.3.15 implies the (𝐶 0 -) pseudoconvexity of 𝜔. ⊓ ⊔

4.3.2 Bochner’s Tube Theorem We have introduced several notions of “convexity”. There is a class of domains for which they coincide. We would like to discuss it here. The topic is a little a way from the Three Big Problems, but has an interest of its own and as an application of Oka’s Theorem 4.3.1 of Boundary Distance Functions. Let 𝜑 : 𝑅 → R𝑛 be a real unrami ed domain; i.e., 𝑅 is a Hausdor topological space, and 𝜑 is a local homeomorphism. We consider a domain over C𝑛 of the following type: (4.3.32)

𝜋 : 𝑇𝑅 = 𝑅 × R𝑛  𝑅 + 𝑖R𝑛 ∋ 𝑥 + 𝑖𝑦 → 𝜑(𝑥) + 𝑖𝑦 ∈ C𝑛 ,

which is called a tube domain, or simply a tube. We call 𝑅 the base of the tube domain 𝑇𝑅 . In many references, the case of univalent 𝑅 ↩→ R𝑛 is dealt with, but here we consider the generalized case as a special unrami ed domain over C𝑛 . The univalent domains of this type are of fundamental importance in the theories of bounded symmetric domains and Laplace integral transforms in partial di erential equations. As a model case of tube domains, we consider the convergent domain 𝛺( 𝑓 ) of a power series in 1.1.4; using the same notation as there, we set 𝑇 := {(𝑧 1 , . . . , 𝑧 𝑛 ) ∈ C𝑛 : (𝑒 𝑧1 , . . . , 𝑒 𝑧𝑛 ) ∈ 𝛺( 𝑓 )} = log 𝛺 ∗ ( 𝑓 ) + 𝑖R𝑛 . We see by Theorem 1.1.25 that 𝑇 is a convex tube domain (univalent) with real convex domain log 𝛺 ∗ ( 𝑓 ) as the base. The function 𝑓 (𝑒 𝑧1 , . . . , 𝑒 𝑧𝑛 ) is a holomorphic function with period 2𝜋𝑖Z. An analytic function 𝑓 given in any neighborhood of an arbitrary shape of the origin has the convergent domain which is necessarily analytically continued over the convex hull of the given domain in the expression of 𝑓 (𝑒 𝑧1 , . . . , 𝑒 𝑧𝑛 ). In the univalent case, 𝑇𝑅 is convex if and only if 𝑅 = ch(𝑅), where ch(𝑅) stands for the convex hull of 𝑅 (see 1.1.1). In general, let 𝜋 : 𝛺 → C𝑛 be a domain. For distinct two points 𝑝, 𝑞 ∈ 𝛺, we take a line segment 𝐿 [𝜋( 𝑝), 𝜋(𝑞)] (⊂ C𝑛 ) connecting 𝜋( 𝑝) and 𝜋(𝑞). Let 𝐿 𝑝 be the connected component of 𝜋 −1 𝐿 [𝜋( 𝑝), 𝜋(𝑞)] containing 𝑝. In the case 𝑞 ∈ 𝐿 𝑝 , we write 𝐿 𝑝 = 𝐿 [ 𝑝, 𝑞] for 𝐿 𝑝 (note that it does not necessarily exist). If 𝐿 [ 𝑝, 𝑞] exists, it carries the natural parameterization, (4.3.33)

𝑡 ∈ [0, 1] → 𝜋| −1 𝐿 [ 𝑝,𝑞 ] ((1 − 𝑡)𝜋( 𝑝) + 𝑡𝜋(𝑞)) ∈ 𝐿 [ 𝑝, 𝑞].

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155

Lemma 4.3.34. 𝛺 is univalent and convex if and only if for arbitrary distinct two points 𝑝, 𝑞 ∈ 𝛺, there exists the line segment 𝐿 [ 𝑝, 𝑞] connecting 𝑝 and 𝑞. Proof. The necessity is obvious. We show the su ciency. For 𝐿 [ 𝑝, 𝑞] ⊂ 𝛺, the restriction 𝜋| 𝐿 [ 𝑝,𝑞 ] : 𝐿 [ 𝑝, 𝑞] → 𝐿 [𝜋( 𝑝), 𝜋(𝑞)] is a b ection, so that 𝜋( 𝑝) ≠ 𝜋(𝑞); this means the injectivity of 𝜋. The convexity is the de nition itself. ⊓ ⊔ Definition 4.3.35. A real-valued function 𝜓 : 𝛺 → R on 𝛺 is said to be convex if for an arbitrary 𝐿 [ 𝑝, 𝑞] ⊂ 𝛺 ( 𝑝 ≠ 𝑞) above, the restricted function 𝜓| 𝐿 [ 𝑝,𝑞 ] : 𝐿 [ 𝑝, 𝑞] → R is convex with respect to the parameter 𝑡 of (4.3.33). We consider the boundary distance 𝛿P𝛥 (𝑧, 𝜕𝑇𝑅 ) of a tube domain 𝑇𝑅 over C𝑛 . It follows that 𝛿P𝛥 ( 𝑝 + 𝑖𝑦, 𝜕𝑇𝑅 ) = 𝛿P𝛥 ( 𝑝, 𝜕𝑇𝑅 ),

(4.3.36)

∀ 𝑦 ∈ R𝑛 .

That is, with 𝑝 = 𝑥 + 𝑖𝑦 (𝑥 ∈ 𝑅, 𝑦 ∈ R𝑛 ), 𝛿P𝛥 ( 𝑝, 𝜕𝑇𝑅 ) is a function only of the real part 𝑥. Lemma 4.3.37. If − log 𝛿P𝛥 ( 𝑝, 𝜕𝑇𝑅 ) is plurisubharmonic, then it is convex. Proof. We rst assume that 𝛿P𝛥 ( 𝑝, 𝜕𝑇𝑅 ) is of 𝐶 2 -class. Set 𝜑( 𝑝) = − log 𝛿P𝛥 ( 𝑝, 𝜕𝑇). With 𝜋( 𝑝) = (𝑧 𝑗 ) = (𝑥 𝑗 + 𝑖𝑦 𝑗 ), the plurisubharmonicity with respect to the local coordinates (𝑧 𝑗 ) leads to 𝐿 [𝜑] ( 𝑝; 𝑣) =

(4.3.38)

Õ 𝜕2 𝜑 ( 𝑝)𝑣 𝑗 𝑣¯ 𝑘 ≥ 0, 𝜕𝑧 𝑗 𝜕 𝑧¯ 𝑘 𝑗,𝑘

∀ (𝑣 𝑗 ) ∈ C𝑛 .

By (4.3.36) we have 𝐿 [𝜑] ( 𝑝; 𝑣) =

1 Õ 𝜕2 𝜑 ( 𝑝)𝑣 𝑗 𝑣¯ 𝑘 ≥ 0. 4 𝑗,𝑘 𝜕𝑥 𝑗 𝜕𝑥 𝑘

Therefore, 𝑑𝑑𝑡𝜑2 (𝑡) ≥ 0 with respect to 𝑡 of (4.3.33), so that 𝜑(𝑡) is convex. In general, as 𝜑 is not necessarily of 𝐶 2 -class, we rst take 𝐿 [ 𝑝, 𝑞] arbitrarily. Since 𝐿 [ 𝑝, 𝑞] ⋐ 𝑇𝑅 , the smoothing 𝜑 𝜀 ( 𝑝) with su ciently small 𝜀 > 0 is de ned in a neighborhood of 𝐿 [ 𝑝, 𝑞], and is 𝐶 ∞ plurisubharmonic there. Since 𝜑 𝜀 ( 𝑝) satis es (4.3.36) too, 𝜑 𝜀 | 𝐿 [ 𝑝,𝑞 ] is a convex function as shown above. The convergence 𝜑 𝜀 ↘ 𝜑 ⊓ ⊔ (as 𝜀 ↘ 0) is uniform on 𝐿 [ 𝑝, 𝑞], and hence 𝜑| 𝐿 [ 𝑝,𝑞 ] is convex. 2

Theorem 4.3.39. For a tube domain 𝑇𝑅 (𝜑 : 𝑅 → R𝑛 ) over C𝑛 , the following conditions are equivalent: (i) 𝑇𝑅 is a domain of holomorphy. (ii) 𝑇𝑅 is holomorphically convex. (iii) 𝑇𝑅 is pseudoconvex. (iv) − log 𝛿P𝛥 (𝑧, 𝜕𝑇𝑅 ) is (continuous) plurisubharmonic. (v) 𝑇𝑅 is univalent and convex; that is, the base 𝑅 is a convex domain of R𝑛 .

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Proof. The implications, “(i), (ii), (iii) ⇒ (iv)”, and “(v) ⇒ (i), (ii), (iii)”, are the special cases of what have been proved. (iv) ⇒ (v): Assume that there is a line segment 𝐿 [ 𝑝, 𝑞] ⊂ 𝑅 connecting the two points 𝑝, 𝑞 ∈ 𝑅. In the case 𝑝 = 𝑞, we consider 𝐿 [ 𝑝, 𝑞] = {𝑝} to be a degenerate line segment. It follows from Lemma 4.3.37 that − log 𝛿P𝛥 ( 𝑝, 𝜕𝑇𝑅 ) is a continuous convex function, so that max − log 𝛿P𝛥 (𝑧, 𝜕𝑇𝑅 ) = max − log 𝛿P𝛥 (𝑧, 𝜕𝑇𝑅 ), 𝑝,𝑞

𝑧 ∈ 𝐿 [ 𝑝,𝑞 ]

(4.3.40)

min 𝛿P𝛥 (𝑧, 𝜕𝑇𝑅 ) = min 𝛿P𝛥 (𝑧, 𝜕𝑇𝑅 ). 𝑝,𝑞

𝑧 ∈ 𝐿 [ 𝑝,𝑞 ]

Set 𝑆 = {( 𝑝, 𝑞) ∈ 𝑅 2 : ∃ 𝐿 [ 𝑝, 𝑞] ⊂ 𝑅} ⊂ 𝑅 2 . It su ces to to prove 𝑆 = 𝑅 2 ; then, by Lemma 4.3.34 the proof is nished. Obviously, 𝑆 is a non-empty open set. We are going to show that 𝑆 is closed in 𝑅 2 . Let ( 𝑝, 𝑞) ∈ 𝑅 2 be an accumulation point of 𝑆. Then there is a sequence of points ( 𝑝 𝜈 , 𝑞 𝜈 ) ∈ 𝑆 (𝜈 = 1, 2, . . .) such that (4.3.41)

lim 𝑝 𝜈 = 𝑝,

𝜈→∞

lim 𝑞 𝜈 = 𝑞,

𝜈→∞

𝐿 [ 𝑝 𝜈 , 𝑞 𝜈 ] ⊂ 𝑅.

Let 0 < 𝜌 < min 𝑝,𝑞 𝛿P𝛥 (𝑧, 𝜕𝑇𝑅 ). Then we deduce from (4.3.41) and (4.3.40) that Ø (𝑧 + 𝜌P𝛥) ⊂ 𝑅. (4.3.42) 𝑈𝜈 := 𝑧 ∈ 𝐿 [ 𝑝𝜈 ,𝑞𝜈 ]

With a su ciently large 𝜈0 ∈ N, 𝑈𝜈 ⊃ 𝐿 [ 𝑝, 𝑞] for 𝜈 ≥ 𝜈0 (see Fig. 4.2). Therefore, 𝐿 [ 𝑝, 𝑞] ⊂ 𝑅, and hence ( 𝑝, 𝑞) ∈ 𝑆. ⊓ ⊔ The next is known as Bochner’s Tube Theorem. Theorem 4.3.43 (S. Bochner, K. Stein (𝑛 = 2)). The envelope of holomorphy of a univalent tube 𝑇𝑅 (⊂ C𝑛 ) is the convex hull 𝑇ch(𝑅) .

Fig. 4.2 Line segment 𝐿 [ 𝑝𝜈 , 𝑞𝜈 ].

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157

Proof. By Theorem 3.6.12, there exists the envelope of holomorphy 𝜋 : 𝑇ˆ → C𝑛 of 𝑇𝑅 , which is possibly multivalent, and contains 𝑇𝑅 as a subdomain. By the de nitions of tube and envelope of holomorphy, we have that 𝑇ˆ = 𝑇ˆ + 𝑖𝑦 (∀ 𝑦 ∈ R𝑛 ), and hence 𝑅ˆ = 𝜋 −1 R𝑛 (here, R𝑛 is the real part of C𝑛 = R𝑛 + 𝑖R𝑛 ). Setting 𝜛 = 𝜋| 𝑅ˆ , we have a real unrami ed domain 𝜛 : 𝑅ˆ −→ ch(𝑅) ⊂ R𝑛 such that 𝑇ˆ = 𝑅ˆ + 𝑖R𝑛 : 𝑇ˆ is a tube domain over C𝑛 . By Theorem 4.3.39, 𝑇ˆ has to be univalent and convex. Therefore it follows that 𝑇ˆ = ch(𝑅) + 𝑖R𝑛 . ⊓ ⊔

4.3.3 Pseudoconvex Boundary In order to consider the boundary of a domain 𝔇/C𝑛 , we assume the following condition through this subsection, unless otherwise mentioned: Condition 4.3.44. 𝔇 is a univalent domain, or a relatively compact subdomain of a larger domain 𝔊/C𝑛 ; in the latter case, 𝔇 is called a bounded domain of 𝔊. Under this condition, 𝜕𝔇 is de ned as a closed set. Some properties which we describe hold for more general domains, but the discussions are left to the readers. Theorem 4.3.45. For a domain 𝔇, the following are equivalent: (i) 𝔇 is pseudoconvex. (ii) For every boundary point 𝑎 ∈ 𝜕𝔇, there is a neighborhood 𝑈 of 𝑎 such that 𝑈 ∩ 𝔇 is pseudoconvex. Proof. Assume that 𝔇 is pseudoconvex. Let 𝜑 ∈ 𝒫(𝔇) be a pseudoconvex exhaustion function on 𝔇. For a boundary point 𝑎 ∈ 𝜕𝔇 we take a Stein neighborhood 𝑈 ∋ 𝑎. Then, − log 𝛿P𝛥 ( 𝑝, 𝜕𝑈) ∈ 𝒫0 (𝑈). Set 𝜓( 𝑝) := max{𝜑( 𝑝), − log 𝛿P𝛥 ( 𝑝, 𝜕𝑈)} ( 𝑝 ∈ 𝑈 ∩ 𝔇). It follows that 𝜓 ∈ 𝒫(𝑈 ∩ 𝔇) and 𝜓 is an exhaustion function on 𝑈 ∩ 𝔇. We show the converse. Assume that every point 𝑎 ∈ 𝜕𝔇 has a neighborhood 𝑈 with pseudoconvex 𝑈 ∩ 𝔇. By Theorem 4.3.11, − log 𝛿P𝛥 ( 𝑝, 𝜕 (𝑈 ∩ 𝔇)) ∈ 𝒫0 (𝑈 ∩ 𝔇). Taking a su ciently small neighborhood 𝑉 ⋐ 𝑈 of 𝑎, we have − log 𝛿P𝛥 ( 𝑝, 𝜕 (𝑈 ∩ 𝔇)) = − log 𝛿P𝛥 ( 𝑝, 𝜕𝔇),

𝑝 ∈ 𝑉 ∩ 𝔇.

Therefore we see that − log 𝛿P𝛥 ( 𝑝, 𝜕𝔇) ∈ 𝒫0 (𝑉 ∩ 𝔇). Covering 𝜕𝔇 by such 𝑉, we get a neighborhood 𝑊 of 𝜕𝔇 such that − log 𝛿P𝛥 ( 𝑝, 𝜕𝔇) ∈ 𝒫0 (𝑊 ∩ 𝔇). When 𝔇 is a bounded domain of 𝔊, there is a small 𝑐 > 0 such that {𝑝 ∈ 𝔇 : 𝛿P𝛥 ( 𝑝, 𝜕𝔇) < 𝑐} ⊂ 𝑊 ∩ 𝔇. With 𝜑( 𝑝) := max{− log 𝛿P𝛥 ( 𝑝, 𝜕𝔇), − log 𝑐}, 𝜑 ∈ 𝒫0 (𝔇) and 𝜑 is an exhaustion function.

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In the case of 𝔇 ⊂ C𝑛 , we set 𝐹 = 𝔇 \ 𝑊, which is closed in 𝔇. Let 𝜒(𝑡) (𝑡 ≥ 0) be a monotone increasing convex function in 𝑡 such that (4.3.46)

− log 𝛿P𝛥 (𝑧, 𝜕𝔇) < 𝜒(∥𝑧∥),

𝑧 ∈ 𝐹.

¯ For example, we set 𝐹 (𝑟) = 𝐹 ∩ B(𝑟) and take 𝑐 1 > max − log 𝛿P𝛥 (𝑧, 𝜕𝔇). 𝐹 (1)

Set 𝜒(𝑡) = 𝑐 1 + 𝑘 2 (𝑡 − 1) + (0 ≤ 𝑡 ≤ 2, 𝑘 2 > 1) with (𝑡 − 1) + = max{𝑡 − 1, 0}. Taking a large 𝑘 2 > 1, we have − log 𝛿P𝛥 (𝑧, 𝜕𝔇) < 𝜒(∥𝑧∥),

𝑧 ∈ 𝐹 (2).

With setting 𝜒(𝑡) := 𝑐 1 + 𝑘 2 (𝑡 − 1) + + 𝑘 3 (𝑡 − 2) + (0 ≤ 𝑡 ≤ 3) for a suitable 𝑘 3 > 𝑘 2 , we have − log 𝛿P𝛥 (𝑧, 𝜕𝔇) < 𝜒(∥𝑧∥), 𝑧 ∈ 𝐹 (3). Repeating this procedure, we get 𝜒(𝑡) with 𝜒(∥𝑧∥) ∈ 𝒫0 (C𝑛 ). It follows from (4.3.46) that 𝜓(𝑧) := max{− log 𝛿P𝛥 (𝑧, 𝜕𝔇), 𝜒(∥𝑧∥)}, is a continuous pseudoconvex exhaustion function on 𝔇.

𝑧∈𝔇 ⊓ ⊔

N.B. The property of Theorem 4.3.45 (ii) is called the local pseudoconvexity of 𝜕𝔇. We saw the equivalence of the (global) pseudoconvexity of 𝔇 and the local pseudoconvexity of 𝜕𝔇. After all, the Pseudoconvexity Problem is to ask if the statement remains valid after replacing “pseudoconvex” with “Stein”. Now we assume that a domain 𝔇/C𝑛 , multivalent in general, is 𝐶 0 -pseudoconvex, and that 𝜑 is a 𝐶 0 -pseudoconvex exhaustion function on 𝔇. Then the sublevel set 𝔇𝑐 = {𝜑 < 𝑐} with 1/(𝑐 − 𝜑( 𝑝)) is 𝐶 0 -pseudoconvex (in the sense that every connected component of it is 𝐶 0 -pseudoconvex). Here, 𝔇𝑐 satis es a better pseudoconvexity than that of 𝔇 because the de ning function 𝜑 is de ned over the outside of 𝜕𝔇𝑐 as a plurisubharmonic function. In fact, in such a case as 𝑎 ∈ 𝜕𝔇𝑐 , Theorem 4.3.74 proved later implies that 𝑈 ∩ 𝔇𝑐 is Stein for a Stein neighborhood 𝑈 (⊂ 𝔇) of 𝑎. By making use of this local Steinness on the boundary 𝜕𝔇𝑐 we deduce that 𝔇𝑐 is Stein, and then as the limit of the increasing Stein domains we conclude the Steinness of 𝔇: This is the ow of the argument to solve the problem. Definition 4.3.47. Let 𝔊/C𝑛 be a domain, possibly in nitely sheeted. Let 𝔇/C𝑛 be a subdomain of 𝔊, not necessarily bounded but with non-empty boundary 𝜕𝔇 in 𝔊. (i) A point 𝑎 ∈ 𝜕𝔇 is called a pseudoconvex boundary point if there are a neighborhood 𝑈 (⊂ 𝔊) of 𝑎 and 𝜑 ∈ 𝒫0 (𝑈) with 𝑈 ∩ 𝔇 = {𝜑 < 0}. If every point 𝑎 ∈ 𝜕𝔇 is a pseudoconvex boundary point, 𝜕𝔇 is said to be a pseudoconvex boundary.

4.3 Pseudoconvexity

159

(ii) In (i) above, if 𝜑 is chosen to be strongly plurisubharmonic, we call 𝑎 a strongly pseudoconvex boundary point. If every 𝑎 ∈ 𝜕𝔇 is a strongly pseudoconvex boundary point, 𝜕𝔇 is called a strongly pseudoconvex boundary. (Up to here, the boundedness of 𝔇 is not required.) (iii) If 𝔇 is bounded and has a strongly pseudoconvex boundary 𝜕𝔇, 𝔇 is called a strongly pseudoconvex domain.6 The function 𝜑 used above is called the defining function of 𝔇 about 𝑎 ∈ 𝜕𝔇. Let 𝔇 be 𝐶 0 -pseudoconvex, and let 𝜑 : 𝔇 → [−∞, ∞) be a 𝐶 0 -pseudoconvex exhaustion function on 𝔇. Then the boundary 𝜕𝔇′𝑐 of any connected component of 𝔇𝑐 is a pseudoconvex boundary. Proposition 4.3.48. Let 𝔇 be a domain. If 𝜕𝔇 is a pseudoconvex boundary, then 𝔇 is pseudoconvex. In particular, a strongly pseudoconvex domain is pseudoconvex. Proof. By Theorem 4.3.45, it su ces to show that for every point 𝑎 ∈ 𝜕𝔇, B(𝑎;𝑟) ∩ 𝔇 with an open ball neighborhood B(𝑎;𝑟) is pseudoconvex. (Here we identify a neighborhood of 𝑎 with that of the base point of 𝑎 in C𝑛 .) Taking B(𝑎;𝑟) smaller if necessary, we take the de ning function 𝜑 ∈ 𝒫0 (B(𝑎;𝑟)) of B(𝑎;𝑟) ∩ 𝜕𝔇. Set 𝜓(𝑧) = max{1/(𝑟 − ∥𝑧 − 𝑎∥), −1/𝜑(𝑧)},

𝑧 ∈ B(𝑎;𝑟) ∩ 𝔇.

Then, 𝜓(𝑧) is a pseudoconvex exhaustion function on B(𝑎;𝑟) ∩ 𝔇.

⊓ ⊔

The next lemma on di erentiable functions is frequently used. Lemma 4.3.49. For a pair of open sets 𝑉 ⋐ 𝑈 ⊂ 𝔇, there is a function 𝜌 ∈ 𝐶 ∞ (𝔇) satisfying (4.3.50)

𝜌 ≥ 0,

Supp 𝜌 ⋐ 𝑈,

𝜌( 𝑝) = 1 ( 𝑝 ∈ 𝑉).

Proof. Let 𝑊 be an open subset such that 𝑉 ⋐ 𝑊 ⋐ 𝑈, and de ne the set function of 𝑊 by ( 1, 𝑝 ∈ 𝑊, 𝜏( 𝑝) = 0, 𝑝 ∈ 𝔇 \ 𝑊 . There is a local biholomorphic map 𝜋 : 𝔇 → C𝑛 , and 𝑊 is compact. Therefore, for a su ciently small 𝜀 > 0, the smoothing 𝜏𝜀 ( 𝑝) of 𝜏( 𝑝) is de ned at least for 𝑝 ∈ 𝑊. With su ciently small 𝜀, Supp 𝜏𝜀 ⋐ 𝑈; the other properties follow immediately. ⊓ ⊔ Í In what follows, ⟨ (𝑣 𝑗 ), (𝑤 𝑗 )⟩ = 𝑗 𝑣 𝑗 𝑤¯ 𝑗 denotes the standard hermitian inner product of C𝑛 . Proposition 4.3.51. For a bounded domain 𝔇 ⋐ 𝔊 the following are equivalent: (i) (𝜕𝔇 local) 𝔇 is strongly pseudoconvex. 6 There are references where domains with the strong Levi pseudoconvex boundary condition (see De nition 4.3.58) are called strongly pseudoconvex domains; however, the equivalence will be shown.

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4 Pseudoconvex Domains I

Problem and Reduction

(ii) (𝜕𝔇 global) There are a neighborhood 𝑈 of 𝜕𝔇 and a function 𝜓 ∈ 𝒫0 (𝑈 ∪𝔇) such that 𝔇 = {𝜓 < 0} and 𝜓 is strongly plurisubharmonic in 𝑈. Proof. (ii)⇒(i) is trivial. (i)⇒(ii). By the assumption, there are a nite open covering {𝑈𝜈 }𝑙𝜈=1 of 𝜕𝔇, and strongly plurisubharmonic functions 𝜑 𝜈 on 𝑈𝜈 such that 𝑈𝜈 ∩ 𝔇 = {𝜑 𝜈 < 0},

1 ≤ 𝜈 ≤ 𝑙.

We take an open covering {𝑉𝜈 }𝑙𝜈=1 of 𝜕𝔇 with 𝑉𝜈 ⋐ 𝑈𝜈 , and real-valued functions 𝜒𝜈 ∈ 𝐶 ∞ (𝔊) such that 𝜒 ≥ 0,

𝜒𝜈 | 𝑉𝜈 ≡ 1,

Supp 𝜒𝜈 ⋐ 𝑈𝜈 .

With 𝑐 > 0 we set 𝜑(𝑧) =

𝑙  Õ 𝜈=1

 𝜒𝜈2 (𝑧)𝜑 𝜈 (𝑧) + 𝑐 𝜒𝜈 (𝑧)𝜑2𝜈 (𝑧) .

About an arbitrary point 𝑎 ∈ 𝜕𝔇 we compute the Levi from 𝐿 [𝜑] (𝑣) = 𝐿 [𝜑] (𝑎; 𝑣) (𝑣 ∈ C𝑛 \ {0}). Assume ∥𝑣∥ = 1. We identify a small neighborhood of 𝑎 with that of 𝜋(𝑎) in C𝑛 , and use the complex coordinates (𝑧1 , . . . , 𝑧 𝑛 ). For notation we write:   𝜕𝜑 𝜈 𝜕𝜑 𝜈 𝜕𝜑 𝜈 = 𝜕𝜑 𝜈 (𝑎) = (𝑎), . . . , (𝑎) , 𝜕𝑧 1 𝜕𝑧 𝑛   𝜕𝜑 𝜕𝜑 𝜈 𝜈 ¯ 𝜈 = 𝜕𝜑 ¯ 𝜈 (𝑎) = 𝜕𝜑 (𝑎), . . . , (𝑎) , 𝜕 𝑧¯1 𝜕 𝑧¯𝑛 𝑛 Õ 𝜕𝜑 𝜈 ¯ 𝜈 ⟩ = ⟨ 𝑣, 𝜕𝜑 ¯ 𝜈 (𝑎)⟩ = ⟨ 𝑣, 𝜕𝜑 𝑣𝑗 (𝑎), 𝑣 = (𝑣1 , . . . .𝑣𝑛 ). 𝜕𝑧 𝑗 𝑗=1 We have 𝜕𝜑( 𝑝) =

(4.3.52)

𝑙 Õ 𝜈=1

𝜒𝜈2 ( 𝑝)𝜕𝜑 𝜈 ( 𝑝),

𝑝 ∈ 𝜕𝔇.

By computations we further get: 𝐿 [𝜑] (𝑣) =

𝑙  Õ 𝜈=1

≥

𝑙  Õ 𝜈=1

=

   ¯ 𝜈 ⟩| 2 + 4𝜒𝜈 ℜ ⟨ 𝑣, 𝜕𝜑 ¯ 𝜈 ⟩ · ⟨ 𝑣, 𝜕¯ 𝜒𝜈 ⟩ + 𝜒𝜈2 𝐿 [𝜑 𝜈 ] (𝑣) 2𝑐 𝜒𝜈 |⟨ 𝑣, 𝜕𝜑

𝑙 Õ 𝜈=1

¯ 𝜈 ⟩| 2 − 4𝜒𝜈 |⟨ 𝑣, 𝜕𝜑 ¯ 𝜈 ⟩| · |⟨ 𝑣, 𝜕¯ 𝜒𝜈 ⟩| + 𝜒𝜈2 𝐿 [𝜑 𝜈 ] (𝑣) 2𝑐 𝜒𝜈 |⟨ 𝑣, 𝜕𝜑

¯ 𝜈 ⟩| 2 − 2𝑐 𝜒𝜈 |⟨ 𝑣, 𝜕𝜑

𝑙 Õ 𝜈=1

¯ 𝜈 ⟩| · |⟨ 𝑣, 𝜕¯ 𝜒𝜈 ⟩| + 4𝜒𝜈 |⟨ 𝑣, 𝜕𝜑

𝑙 Õ 𝜈=1



𝜒𝜈2 𝐿 [𝜑 𝜈 ] (𝑣).

4.3 Pseudoconvexity

161

Since the last term is positive, with a su ciently small 𝜂 > 0, the terms with ¯ 𝜈 ⟩| ≤ 𝜂 satis es |⟨ 𝑣, 𝜕𝜑 Õ

−

¯ 𝜈 ⟩| · |⟨ 𝑣, 𝜕¯ 𝜒𝜈 ⟩| + 4𝜒𝜈 |⟨ 𝑣, 𝜕𝜑

𝑙 Õ 𝜈=1

¯ 𝜈⟩|≤𝜂 𝜈:| ⟨ 𝑣, 𝜕𝜑

𝜒𝜈2 𝐿 [𝜑 𝜈 ] (𝑣) > 0.

¯ 𝜈 ⟩| > 𝜂, with a su ciently large 𝑐 > 0 we obtain On the other hand, for terms |⟨ 𝑣, 𝜕𝜑   Õ ¯ 𝜈 ⟩| 2 − 4𝜒𝜈 |⟨ 𝑣, 𝜕𝜑 ¯ 𝜈 ⟩| · |⟨ 𝑣, 𝜕¯ 𝜒𝜈 ⟩| 2𝑐 𝜒𝜈 |⟨ 𝑣, 𝜕𝜑 ¯ 𝜈 ⟩ |> 𝜂 𝜈:| ⟨ 𝑣, 𝜕𝜑

=

Õ

¯ 𝜈 ⟩| 𝑐|⟨ 𝑣, 𝜕𝜑 ¯ 𝜈 ⟩| − 2|⟨ 𝑣, 𝜕¯ 𝜒𝜈 ⟩| 2𝜒𝜈 |⟨ 𝑣, 𝜕𝜑




¯ 𝜈 ⟩ |> 𝜂 𝜈:| ⟨ 𝑣, 𝜕𝜑

≥

Õ

2𝜒𝜈 𝜂(𝑐𝜂 − 2|⟨ 𝑣, 𝜕¯ 𝜒𝜈 ⟩|) ≥ 0.

¯ 𝜈 ⟩ |> 𝜂 𝜈:| ⟨ 𝑣, 𝜕𝜑

Thus, 𝐿 [𝜑] is positive de nite at 𝑎 ∈ 𝜕𝔇, and so 𝐿 [𝜑] is positive de nite in a neighborhood of 𝑎. Since 𝜕𝔇 is compact, with su ciently large 𝑐 > 0, 𝐿 [𝜑] is positive de nite in a neighborhood 𝑊 of 𝜕𝔇. Let 𝛿 > 0 be su ciently small such that 𝑈 := {𝑝 ∈ 𝑊 : |𝜑( 𝑝)| < 𝛿} ⋐ 𝑊. Now we set ( −𝛿, 𝑧 ∈ 𝔇 ∪ {𝑝 ∈ 𝑊 : 𝜑( 𝑝) ≤ −𝛿}, 𝜓(𝑧) = 𝜑(𝑧), 𝑧 ∈ {𝑝 ∈ 𝑊 : 𝜑( 𝑝) > −𝛿}. Then 𝑈 and 𝜓 satisfy the required properties.

⊓ ⊔

The above proof is due to H. Grauert [23]. The notion of a strongly pseudoconvex boundary (point) plays an important role in a number of places. Proposition 4.3.53. Let 𝔇/C𝑛 be a pseudoconvex domain. Then there is an increasing sequence of strongly pseudoconvex domains 𝔇 𝑗 ⋐ 𝔇 𝑗+1 ⋐ 𝔇 ( 𝑗 = 1, 2, . . .) with Ð 𝔇= ∞ 𝑗=1 𝔇 𝑗 . Proof. Ð By Theorem 4.3.11, 𝔇 satis es the assumption of Theorem 4.3.15. Let 𝔇= ∞ 𝑗=1 𝛥 𝑗 be the covering obtained by (4.3.25). By (4.3.24) every point of 𝜕 𝛥 𝑗 is a strongly pseudoconvex boundary point. We x a point 𝑝 0 ∈ 𝛥1 and denote by 𝔇 𝑗 the connected component of 𝛥 𝑗 containing 𝑝 0 for 𝑗 ≥ 1. For every point 𝑝 ∈ 𝔇 there is a curve 𝐶 connecting 𝑝 and Ð 𝑝 0 . Since 𝐶 is compact, there is a 𝛥 𝑗 with 𝛥 𝑗 ⊃ 𝐶; thus, 𝔇 𝑗 ∋ 𝑝. We have 𝔇 = 𝑗 𝔇 𝑗 with strongly pseudoconvex domains 𝔇 𝑗 ⋐ 𝔇 𝑗+1 . ⊓ ⊔ Because of Proposition 4.3.53 the following provides an important step in the solution of the Pseudoconvexity Problem. 4.3.54. (Pseudoconvexity Problem III) Is a strongly pseudoconvex domain Stein?

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4 Pseudoconvex Domains I

Problem and Reduction

4.3.4 Levi Pseudoconvexity We keep Condition 4.3.44. The content of this subsection is not absolutely necessary for the development of Oka theory. It is, however, useful to understand the notion of a (strongly) pseudoconvex boundary of De nition 4.3.47 and the historical advances. De nition 4.3.47 of a (strongly) pseudoconvex boundary point is dependent on the choice of 𝜒 ∈ 𝒫0 (𝑈), and not determined only by 𝜕𝔇. Assuming the regularity of 𝜕𝔇 (i.e., no singularities), we study here the dependence (or the independence) from the viewpoint of the shape (geometry) of 𝜕𝔇 itself. For a moment we assume 𝜒 ∈ 𝐶 1 (𝑈) (𝑈 ∋ 𝑎) to satisfy the following about 𝑎 ∈ 𝜕𝔇:

(4.3.55)

𝑈 ∩ 𝔇 = { 𝜒 < 0}, grad 𝜒(𝑎 ′ ) ≠ 0,  𝜕𝜒 ′ 𝜕𝜒 ′ 𝜕 𝜒(𝑎 ′ ) = (𝑎 ), . . . , (𝑎 ) ≠ 0, 𝜕𝑧 1 𝜕𝑧 𝑛

𝑈 ∩ 𝜕𝔇 = { 𝜒 = 0}, ∀ 𝑎 ′ ∈ 𝑈 ∩ 𝜕𝔇, ∀ 𝑎 ′ ∈ 𝑈 ∩ 𝜕𝔇.

The conditions of the second and the third lines are equivalent. By the implicit function Theorem 1.4.6, 𝑈 ∩ 𝜕𝔇 is a real (2𝑛 − 1)-dimensional submanifold (real hypersurface) of 𝐶 1 -class. For 𝑎 ∈ 𝑈 ∩ 𝜕𝔇 we set (4.3.56)

𝑛   Õ     𝜕𝜒 T𝑎 (𝜕𝔇) = (𝑣 𝑗 ) ∈ C𝑛 : 𝑣𝑗 (𝑎) = 0 ,   𝜕𝑧 𝑗 𝑗=1  

which is called the holomorphic tangent space of 𝜕𝔇 at 𝑎. We consider the restriction 𝐿 [ 𝜒]| T𝑎 (𝜕𝔇) of the Levi form 𝐿 [ 𝜒] (𝑣) of 𝜒 to the holomorphic tangent space T𝑎 (𝜕𝔇). Proposition 4.3.57. The holomorphic tangent space T𝑎 (𝜕𝔇) is independent from the choice of 𝜒 ∈ 𝒞 1 (𝑈). Proof. Let 𝜒1 ∈ 𝒞 1 (𝑈) be a function satisfying (4.3.55). By the implicit function Theorem 1.4.6, 𝜏(𝑧) := 𝜒(𝑧)/𝜒1 (𝑧) ∈ 𝒞 0 (𝑈), and 𝜒(𝑧) = 𝜏(𝑧) 𝜒1 (𝑧),

𝜏(𝑧) > 0,

𝑧 ∈ 𝑈.

Since 𝜒(𝑎) = 𝜒1 (𝑎) = 0, with ℎ ∈ R we have 𝜒(𝑎 1 , . . . , 𝑎 𝑗 + ℎ, . . . , 𝑎 𝑛 ) 𝜕𝜒 (𝑎) = lim ℎ→0 𝜕𝑥 𝑗 ℎ 𝜒1 (𝑎 1 , . . . , 𝑎 𝑗 + ℎ, . . . 𝑎 𝑛 ) = lim 𝜏(𝑎 1 , . . . , 𝑎 𝑗 + ℎ, . . . , 𝑎 𝑛 ) ℎ→0 ℎ 𝜕 𝜒1 = 𝜏(𝑎) (𝑎). 𝜕𝑥 𝑗

4.3 Pseudoconvexity

Similarly,

𝜕𝜒 𝜕𝑦 𝑗

163

1 (𝑎) = 𝜏(𝑎) 𝜕𝜒 𝜕𝑦 𝑗 (𝑎). Therefore,

𝜕𝜒 𝜕 𝜒1 (𝑎) = 𝜏(𝑎) (𝑎). 𝜕𝑧 𝑗 𝜕𝑧 𝑗

𝜕 𝜒(𝑎) = 𝜏(𝑎)𝜕 𝜒1 (𝑎).

Thus, T𝑎 (𝜕𝔇) coincides with that de ned by 𝜒1 .

⊓ ⊔

Definition 4.3.58. Suppose that 𝜒 ∈ 𝒞 2 (𝑈) satis es (4.3.55). Then a point 𝑎 ∈ 𝑈 ∩ 𝜕𝔇 is called a Levi pseudoconvex point if 𝐿 [ 𝜒] (𝑎; 𝑣) ≥ 0,

(4.3.59)

𝑣 ∈ T𝑎 (𝜕𝔇) \ {0};

if it is positive de nite, 𝑎 is called a strongly Levi pseudoconvex point. If every 𝑏 ∈ 𝑈 ∩ 𝜕𝔇 is a (resp. strongly) Levi pseudoconvex point, 𝑈 ∩ 𝜕𝔇 is called (resp. strongly) Levi pseudoconvex. Condition (4.3.59) is called the Levi condition or the Levi–Krzoska condition; if it is positive de nite, it is called the strong Levi condition or the strong Levi–Krzoska condition. Lemma 4.3.60. The property of 𝑎 ∈ 𝜕𝔇 being a (strongly) Levi convex point does not depend on the choice of the defining function 𝜒 of 𝜕𝔇 about 𝑎. Proof. Let 𝜒, 𝜒1 ∈ 𝒞 2 (𝑈) be the de ning functions of 𝔇 about 𝑎 ∈ 𝜕𝔇, satisfying (4.3.55). Assume that 𝑎 is a strongly pseudoconvex point with respect to 𝜒. Set 𝜒1 (𝑧) = 𝜏(𝑧) 𝜒(𝑧),

𝜏(𝑧) > 0,

𝑧 ∈ 𝑈.

By the implicit function Theorem 1.4.6 and a computation, we have that 𝜏 ∈ 𝐶 1 (𝑈). We compute the Levi form 𝜕 2 𝐿 [ 𝜒1 ] (𝑎; 𝑣) = 𝜒1 (𝑎 + 𝑡𝑣) 𝜕𝑡𝜕 𝑡¯ 𝑡=0 for a vector 𝑣 ∈ T𝑎 (𝜕𝔇) \ {0}. We have Õ 𝜕𝜒 Õ 𝜕𝜏 𝜕 𝜒1 (𝑎 + 𝑡𝑣) = 𝜏(𝑎 + 𝑡𝑣) (𝑎 + 𝑡𝑣) 𝑣¯ 𝑗 + 𝜒(𝑎 + 𝑡𝑣) (𝑎 + 𝑡𝑣) 𝑣¯ 𝑗 . 𝜕 𝑡¯ 𝜕 𝑧¯ 𝑗 𝜕 𝑧¯ 𝑗 𝑗=1 𝑗=1 𝑛

𝑛

Next we compute   𝜕 2 1 𝜕 𝜒1 𝜕 𝜒1 𝜒 (𝑎 + 𝑡𝑣) = lim (𝑎 + 𝑡𝑣) − (𝑎) . 1 𝑡→0 𝑡 𝜕𝑡𝜕 𝑡¯ 𝑡=0 𝜕 𝑡¯ 𝜕 𝑡¯ Noting that 𝜒(𝑎) =

Í𝑛

𝜕𝜒 𝑗=1 𝜕 𝑧¯ 𝑗

(𝑎) 𝑣¯ 𝑗 =

Í𝑛

𝜕𝜒 𝑗=1 𝜕𝑧 𝑗

(𝑎)𝑣 𝑗 = 0, we obtain

𝐿 [ 𝜒1 ] (𝑎; 𝑣) = 𝜏(𝑎)𝐿 [ 𝜒] (𝑎; 𝑣),

𝑣 ∈ T𝑎 (𝜕𝔇) \ {0}.

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⊓ ⊔

Then, the assertion follows immediately.

Lemma 4.3.61. For a point 𝑎 ∈ 𝜕𝔇 with (4.3.55) the following conditions are equivalent: (i) 𝑎 is a strongly pseudoconvex point. (ii) 𝑎 is a strongly Levi pseudoconvex point. Proof. (i)⇒(ii) is trivial. Assume (ii). Let 𝜒 ∈ 𝒞 2 (𝑈) be a de ning function of 𝔇 about 𝑎 ∈ 𝜕𝔇, and let 𝑐 > 0 be a constant determined later. We set 𝜑(𝑧) = 𝑒 𝑐𝜒 (𝑧) − 1,

𝑧 ∈ 𝑈.

Then, 𝜑 is a de ning function of 𝔇 about 𝑎 ∈ 𝜕𝔇. For 𝑣 ∈ C𝑛 \ {0}, we have   𝐿 [𝜑] (𝑎; 𝑣) = 𝑐𝑒 𝑐𝜒 𝐿 [ 𝜒] (𝑎; 𝑣) + 𝑐|⟨𝑣, 𝜕¯ 𝜒(𝑎)⟩| 2 . Since T𝑎 (𝜕𝔇) is the orthogonal complement space of 𝜕¯ 𝜒(𝑎), we decompose 𝑣 into the orthogonal factors: 𝑣 = 𝑤 ⊕ 𝜁 𝜕¯ 𝜒(𝑎),

𝑤 ∈ T𝑎 (𝜕𝔇), 𝜁 ∈ C.

It follows that (4.3.62)

* !+ Õ 𝜕2 𝜒 © 𝐿 [𝜑] (𝑎; 𝑣) = 𝑐 ­ 𝐿 [ 𝜒] (𝑎; 𝑤) + 2ℜ𝜁 𝜕¯ 𝜒(𝑎), (𝑎)𝑤 𝑘 𝜕𝑧 𝑘 𝜕 𝑧¯ 𝑗 𝑘 𝑗 « ! + |𝜁 | 2 𝐿 [ 𝜒] (𝑎; 𝜕¯ 𝜒(𝑎)) + 𝑐|𝜁 | 2 ∥ 𝜕¯ 𝜒(𝑎)∥ 4 .

By the assumption of 𝐿 [𝜑]| T(𝜕𝔇) ≫ 0, there are positive constants 𝐶1 and 𝐶2 such that 𝐿 [ 𝜒] (𝑎; 𝑤) ≥ 𝐶1 ∥𝑤∥ 2 , * ! + Õ 𝜕2 𝜒 𝜕¯ 𝜒(𝑎), ≤ 𝐶2 ∥𝑤∥. (𝑎)𝑤 𝑘 𝜕𝑧 𝜕 𝑧 ¯ 𝑘 𝑗 𝑘 𝑗 Then, it follows from (4.3.62) that 𝐿 [𝜑] (𝑎; 𝑣) ≥ 𝑐 𝐶1 ∥𝑤∥ 2 − 2𝐶2 ∥𝑤∥|𝜁 | + (𝑐∥ 𝜕¯ 𝜒(𝑎) ∥ 4 − |𝐿 [ 𝜒] (𝑎; 𝜕¯ 𝜒(𝑎))|)|𝜁 | 2
= 𝑐 𝐶1 ∥𝑤∥ 2 − 2𝐶2 ∥𝑤∥|𝜁 | + 𝐶3 |𝜁 | 2 ,




where 𝐶3 := 𝑐∥ 𝜕¯ 𝜒(𝑎)∥ 4 − |𝐿 [ 𝜒] (𝑎; 𝜕¯ 𝜒(𝑎))| > 0 with a large 𝑐 > 0. Note that 𝐶3 may be chosen arbitrarily large. Now, we have

4.3 Pseudoconvexity

𝐿 [𝜑] (𝑎; 𝑣) ≥

165


𝑐
𝑐 𝐶1 ∥𝑤∥ 2 + 𝐶3 |𝜁 | 2 + 𝐶1 ∥𝑤∥ 2 − 4𝐶2 ∥𝑤∥|𝜁 | + 𝐶3 |𝜁 | 2 . 2 2

With a large 𝐶3 such that 𝐶1 𝐶3 ≥ 4𝐶22 , the second quadric term of the right-hand side above is non-negative. It follows that 𝐿 [𝜑] (𝑎; 𝑣) ≥


𝑐 𝐶1 ∥𝑤∥ 2 + 𝐶3 |𝜁 | 2 > 0. 2

Hence we deduce that 𝜑 is strongly pseudoconvex in a neighborhood about 𝑎.

⊓ ⊔

Remark 4.3.63. It is known that the above lemma remains valid under the semipositive de nite condition, but the proof is more involved, not simply computations, but needs arguments similar to the proof of Theorem 4.3.45 (cf. [30] Theorem 2.6.12). Here we omit it because it is not particularly necessary. Proposition 4.3.64. Let 𝔇 be a bounded domain of 𝔊/C𝑛 . If every point 𝑎 ∈ 𝜕𝔇 is a strongly Levi pseudoconvex boundary point, then there is a strongly pseudoconvex function 𝜑 in a neighborhood 𝑈 (⊂ 𝔊) of 𝜕𝔇 such that 𝜕𝜑 ≠ 0 on 𝜕𝔇, and 𝑈 ∩ 𝔇 = {𝜑 < 0}. In particular, 𝜑 satisfies a strong Levi condition at every point of 𝜕𝔇. Proof. It follows from Lemma 4.3.61 and Proposition 4.3.51. Here the condition, 𝜕𝜑 ≠ 0 on 𝜕𝔇, follows from (4.3.52). ⊓ ⊔ Historically, the Pseudoconvexity Problem was posed in the following form: 4.3.65 (Levi’s Problem, [4] Kap. IV). Assume that the boundary of a bounded domain 𝔇 (⋐ 𝔊) is strongly Levi pseudoconvex. Then, is 𝔇 a domain of holomorphy? Note. Thereafter a number of notions of pseudoconvexity were proposed, and their relations were investigated; it might be evidence that the notion of “pseudoconvex” was ambiguous (this does not mean each de nition of “pseudoconvex” was ambiguous). Oka went back to Hartogs’ phenomenon, and then obtained the notion of pseudoconvex functions (equivalently, plurisubharmonic functions), with which he solved the Pseudoconvexity Problem. Because of the procedure, he termed the problem Hartogs’ Inverse Problem, formulating Pseudoconvexity Problems I 4.3.6, II 4.3.10, and III 4.3.54. As observed already, Levi’s Problem above is implied by Pseudoconvexity Problem III 4.3.54. In the next chapter we will give the solution of Levi’s Problem, where we will not use Levi’s condition; but, we solve Pseudoconvexity Problem III 4.3.54 stated in terms of strongly pseudoconvex (plurisubharmonic) functions. This circumstance is the same in the other proofs by Grauert’s method and also by the 𝐿 2 -𝜕¯ method of Hörmander.

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4.3.5 Strongly Pseudoconvex Boundary Points and Stein Domains Let 𝜋 : 𝔊 → C𝑛 be a domain. We consider a bounded domain 𝔇 ⋐ 𝔊 and its boundary 𝜕𝔇. Suppose that there is a real continuous function 𝜆 de ned in a neighborhood 𝑈 of 𝜕𝔇, and that 𝔇 = {𝜆 < 0}, 𝜕𝔇 = {𝜆 = 0}. Now we assume that 𝜆 is strongly plurisubharmonic in a neighborhood 𝑉 of a boundary point 𝑝 0 ∈ 𝜕𝔇. For a while we consider only in a neighborhood of 𝑝 0 , so that we regard 𝑉 ⊂ C𝑛 , and by translation, we assume 𝑝 0 = 0 ∈ C𝑛 . Then 𝜆(𝑧) is developed to a Taylor expansion with center at 𝑧 = 0: (4.3.66)

𝑛   Õ 𝜕 2𝜆   Õ   𝜕𝜆 𝜆(𝑧) = ℜ 2 (0)𝑧 𝑗 + (0)𝑧 𝑗 𝑧 𝑘   𝜕𝑧 𝜕𝑧 𝑗 𝜕𝑧 𝑘 𝑗,𝑘  𝑗=1 𝑗  Õ 𝜕 2𝜆 + (0)𝑧 𝑗 𝑧¯ 𝑘 + 𝑜(∥𝑧∥ 2 ). 𝜕𝑧 𝜕 𝑧 ¯ 𝑗 𝑘 𝑗,𝑘

With a small 𝑟 0 > 0, Õ 𝜕 2𝜆 1 Õ 𝜕 2𝜆 (0)𝑧 𝑗 𝑧¯ 𝑘 + 𝑜(∥𝑧∥ 2 ) ≥ (0)𝑧 𝑗 𝑧¯ 𝑘 > 0, 𝜕𝑧 𝑗 𝜕 𝑧¯ 𝑘 2 𝑗,𝑘 𝜕𝑧 𝑗 𝜕 𝑧¯ 𝑘 𝑗,𝑘

0 < ∥𝑧∥ < 𝑟 0 .

Introducing a new parameter 𝑡 ∈ C, we de ne holomorphic functions in 𝑧 and (𝑡, 𝑧) by (4.3.67)

𝑃(𝑧) = 2

𝑛 Õ Õ 𝜕 2𝜆 𝜕𝜆 (0)𝑧 𝑗 + (0)𝑧 𝑗 𝑧 𝑘 , 𝜕𝑧 𝑗 𝜕𝑧 𝑗 𝜕𝑧 𝑘 𝑗=1 𝑗,𝑘

𝜎(𝑡, 𝑧) = 𝑡 − 𝑃(𝑧). Note that 𝑃(𝑧) is a non-zero quadric. The function 𝑃(𝑧) (resp. 𝜎(𝑡, 𝑧)) is caled the Levi (resp. Levi–Oka) polynomial of 𝜆 (or the boundary 𝜕𝔇) at 𝑝 0 = 0.7 We consider a family of complex hypersurfaces de ned by 𝛴𝑡 = {𝑧 ∈ C𝑛 : 𝜎(𝑡, 𝑧) = 0} ⊂ C𝑛 ,

𝑡∈C

which are called the Oka hypersurfaces of 𝜆 (or the boundary 𝜕𝔇) at 𝑝 0 = 0 (cf. Fig. 4.3).8 7 For 𝑃 (𝑧) the term is due to Range [53] Chap. II, 2.8, and Lieb [33] 12; the introduction of 𝜎 (𝑡 , 𝑧) and 𝛴𝑡 is inspired by Oka IX of 1943 (unpublished) in [50], and Oka IX (1953) in [48], [49]; cf. Remark 4.3.73 and Note at the end of this chapter. 8 The term is due to Oka VI (1942), IX (1953) in [48], [49] and IX of 1943 (unpublished) in [50]; cf. ibid.

4.3 Pseudoconvexity

167

Fig. 4.3 Oka hypersurface 𝛴𝑡 .

In a neighborhood of (0, 0), we restrict 𝑡 to the real axis; if 𝑡 > 0, 𝜎(𝑡, 𝑧) = 0 implies 𝜆(𝑧) > 𝑡 > 0, so that ¯ ∩ {∥𝑧∥ < 𝑟 0 } = ∅, 𝛴𝑡 ∩ 𝔇 and if 𝑡 = 0, then

¯ ∩ {𝜎(0, 𝑧) = 0} ∩ {∥𝑧∥ < 𝑟 0 } = {0}. 𝔇

For small 𝛿0 > 0 we have !

Ø

(4.3.68)

¯ = ∅. 𝛴𝑡 ∩ {∥𝑧∥ = 𝑟 0 } ∩ 𝔇

0≤𝑡 ≤ 𝛿0

¯ su ciently close to 𝔇 ¯ (cf. Fig. 4.3), and a Taking a neighborhood 𝑈 ′ ⊂ 𝑈 of 𝔇 su ciently small neighborhood 𝑉 of the closed internal [0, 𝛿0 ] in complex plane C, we have {(𝑡, 𝑧) ∈ 𝑉 × 𝑈 ′ : 𝜎(𝑡, 𝑧) = 0} ⋐ 𝑉 × {∥𝑧∥ < 𝑟 0 }. Set 𝑊0 = 𝑉 × (𝑈 ′ ∩ {∥𝑧∥ < 𝑟 0 }). Then, the complex hypersurface {𝜎(𝑡, 𝑧) = 0} ∩ 𝑊 of 𝑊 := 𝑉 × 𝑈 ′ is that of 𝑊0 . With 𝑊1 := (𝑉 × 𝑈 ′ ) \ {𝜎(𝑡, 𝑧) = 0}, we have an open covering 𝑊 = 𝑊0 ∪ 𝑊1 . Set (4.3.69)

𝑔0 =

1 𝜎(𝑡, 𝑧)


(𝑡, 𝑧) ∈ 𝑊0 ;

Then {(𝑊 𝑗 , 𝑔 𝑗 )} 𝑗=0,1 is Cousin I data on 𝑊. Summarizing the above, we obtain:

𝑔1 = 0 (on 𝑊1 ).

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Lemma 4.3.70. Let the notation be as above. Assume that 𝜆( 𝑝) is strongly plurisubharmonic in a neighborhood of a boundary point 𝑝 0 ∈ 𝜕𝔇 (𝜋( 𝑝 0 ) = 𝑧0 ). Then we have: (i) There are a small neighborhood 𝑈0 of 𝑝 0 and a small 𝛿0 > 0 with the identification of 𝑝 ∈ 𝑈0 and 𝜋( 𝑝) = 𝑧 such that the Oka hypersurfaces 𝛴𝑡 of 𝜆 at 𝑝 0 = 𝑧 0 satisfy the following:

(4.3.71)

𝑎. 𝑧 0 ∈ 𝛴0 . 𝑏. 𝛴𝑡 ∩ 𝛴𝑡 ′ = !∅, 𝑡 ≠ 𝑡 ′ ∈ [0, 𝛿0 ]. Ø ¯ ∩ 𝑈0 = {𝑧 0 }. 𝑐. 𝛴𝑡 ∩ 𝔇 0≤𝑡 ≤ 𝛿0 ! Ø ¯ = ∅. 𝑑. 𝛴𝑡 ∩ 𝜕𝑈0 ∩ 𝔇 0≤𝑡 ≤ 𝛿0

Here we may take 𝑈0 = {𝑧 : ∥𝑧 − 𝑧0 ∥ < 𝑟 0 } with an arbitrarily small 𝑟 0 > 0. ¯ and 𝜔 (⊂ C) of [0, 𝛿0 ] such that on 𝑊 := (ii) There are neighborhoods 𝑈 ′ of 𝔇 𝜔 × 𝑈 ′ there is Cousin I data with polar set {𝜎(𝑡, 𝑧) = 0} ∩ 𝑊 defined by (4.3.69), where 𝜎(𝑡, 𝑧) is the Levi–Oka polynomial of 𝜆 at 𝑝 0 = 𝑧0 . ¯ has a Stein neighborhood, then with Stein 𝑈 ′ above, the above Cousin I (iii) If 𝔇 data has a solution 𝐺 (𝑡, 𝑝) ∈ ℳ(𝑊) on 𝑊 = 𝜔 × 𝑈 ′ . In particular, we have: a. 𝐺 (0, 𝑝) has poles on the Oka hypersurface 𝛴0 touching 𝜕𝔇 only at 𝑝 0 from the outside, ¯ \ {𝑝 0 }, b. 𝐺 (0, 𝑝) is holomorphic on 𝔇 c. lim 𝑝→ 𝑝0 ,𝑧 ∈ 𝔇\{ |𝐺 (0, 𝑝)| = ∞. ¯ 𝑝0 } Example 4.3.72. We consider the unit open ball B(1) = {∥𝑧∥ < 1} (⋐ C𝑛 ), a typical example of strongly pseudoconvex domains. Let 𝑧 = (𝑧1 , . . . , 𝑧 𝑛 ) ∈ C𝑛 be the coordinate system and let 𝑝 0 = ( 𝑝 0 𝑗 ) = (1, 0, . . . , 0) ∈ 𝜕B(1). We compute the Oka hypersurfaces 𝛴𝑡 in Lemma 4.3.70 at 𝑝 0 . Note that 𝜆(𝑧) := ∥𝑧∥ 2 − 1 is strongly plurisubharmonic and B(1) = {𝜆(𝑧) < 0}, and that 𝛴𝑡 is given by 𝜎(𝑡, 𝑧) of (4.3.67). 𝜕𝜆 Since 𝜕𝑧 = 𝑧¯ 𝑗 , the Levi Oka polynomial and the Oka hypersurfaces of 𝜕B(1) at 𝑝 0 𝑗 are written as n 𝑡o 𝜎(𝑡, 𝑧) = 𝑡 − 2(𝑧 1 − 1), 𝛴𝑡 = 𝑧1 = 1 + . 2 In particular, 𝛴0 = {𝑧 1 = 1} and ! Ø 𝛴𝑡 ∩ B(1) = {𝑝 0 }. 𝑡 ≥0

Remark 4.3.73. The method of constructing the Oka hypersurfaces 𝛴𝑡 above will be used in a number of places henceforward. The idea to consider not only 𝛴0 but 𝛴𝑡 with a parameter 𝑡 and to apply the Cousin I Problem in the total space of 𝑧 and 𝑡 is due to the last one of Oka’s unpubished ve papers of 1943 (cf. [44], Part II, 8).

4.3 Pseudoconvexity

169

The purpose is to obtain a holomorphic function on 𝔇 which diverges to in nity at 𝑝 0 , and then deform it so that it is holomorphic at 𝑝 0 but has nite values with large moduli. Here we showed that it is possible to construct Cousin I data on 𝔇 with such a divergence property at a strongly pseudoconvex boundary point 𝑝 0 ∈ 𝜕𝔇. If 𝑛 = 1, it su ces to take a rational function 1/(𝑧 − 𝑝 0 ) for any boundary point 𝑝 0 ∈ 𝜕𝔇. In the case of 𝑛 ≥ 2, the function 1/𝑃(𝑧) with the Levi polynomial 𝑃(𝑧) which is constructed from the strongly plurisubharmonic function 𝜆( 𝑝) about 𝑝 0 , plays the role of 1/(𝑧 − 𝑝 0 ). For this purpose Oka invented in VI (1942) the notion of (strongly) pseudoconvex (plurisubharmonic) functions. However, while 1/(𝑧 − 𝑝 0 ) is already de ned on C, 1/𝑃(𝑧) is de ned only locally about 𝑝 0 ; the gap is considerably large. Theorem 4.3.74.9 Let 𝜋 : 𝔇 → C𝑛 be a Stein domain, and let 𝜆 : 𝔇 → [−∞, ∞) be a plurisubharmonic function. Then, 𝔇𝑐 = {𝑝 ∈ 𝔇 : 𝜆( 𝑝) < 𝑐} is 𝒪(𝔇)-convex, and Stein. Proof. (a) We rst assume that 𝜆 is strongly plurisubharmonic. Take any compact b := 𝐾 b𝔇 ⋐ 𝔇. We are going to show 𝐾 b ⊂ 𝔇𝑐 . Set subset 𝐾 ⋐ 𝔇𝑐 . Since 𝔇 is Stein, 𝐾 𝑐 0 = max 𝜆 = 𝜆(𝑧 0 ), b 𝐾

b 𝑧 0 ∈ 𝐾.

it su ces to prove 𝑐 0 < 𝑐. Suppose 𝑐 0 ≥ 𝑐. It is noted that 𝑧0 ∉ 𝐾. We take an 𝒪(𝔇)b Applying Lemma 4.3.70 to 𝔇 = 𝔇𝑐0 and 𝑝 0 = 𝑧 0 , we analytic polyhedron P ⋑ 𝐾. obtain the Levi Oka polynomial 𝜎(𝑡, 𝑧) of 𝜕𝔇𝑐0 at 𝑧0 , and the Oka hypersurfaces 𝛴𝑡 = {𝑧 : 𝜎(𝑡, 𝑧) = 0} satisfying (4.3.71). For 𝑈 ′ of Lemma 4.3.70 (ii) we can choose a su ciently small P such that 𝑈 ′ ⊃ P. With a connected neighborhood 𝜔 ⊃ [0, 𝛿0 ] (closed interval) in C, the open set 𝑊 := 𝜔 × P is Stein. We consider the Cousin I data {(𝑊 𝑗 , 𝑔 𝑗 )} 𝑗=0,1 on 𝑊 de ned by (4.3.69). By Theorem 3.4.31 there is a meromorphic function 𝐺 (𝑡, 𝑝) on 𝑊 such that 𝐺 (𝑡, 𝑧) −

(4.3.75)

1 ∈ 𝒪(𝑊0 ). 𝜎(𝑡, 𝑧)

Since 𝐺 (𝑡, 𝑝) is holomorphic on [0, 𝛿0 ] × 𝐾, 𝑀 := max |𝐺 (𝑡, 𝑝)| < ∞. [0, 𝛿0 ] ×𝐾

It follows from (4.3.75) that there is a point 𝑡 0 ∈ (0, 𝛿0 ) with |𝐺 (𝑡 0 , 𝑧 0 )| > 𝑀 + 1 and b The construction implies that 𝐺 (𝑡 0 , 𝑧) ∈ 𝒪( 𝐾). max |𝐺 (𝑡0 , 𝑝)| ≤ 𝑀 < 𝑀 + 1 < |𝐺 (𝑡 0 , 𝑧 0 )|, 𝑝∈𝐾

b 𝐾 ∪ {𝑧0 } ⊂ 𝐾.

9 This is a well-known fact, but it is new in the sense that the proof is given before the solution of the Pseudoconvexity Problem, or Levi’s Problem.

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By the Oka Weil Approximation Theorem 3.7.7 we approximate 𝐺 (𝑡0 , 𝑝) by eleb Then we see that 𝑧 0 ∉ 𝐾; b this is a contradiction. ments of 𝒪(𝔇), uniformly on 𝐾. (b) Let 𝜆 be a plurisubharmonic function in general. Taking a compact set 𝐾 ⋐ 𝔇𝑐 , b := 𝐾 b𝔇 ⋐ 𝔇. Take an 𝒪(𝔇)-analytic polyhedron P with 𝐾 b ⋐ P ⋐ 𝔇. Since P we set 𝐾 b b is 𝒪(𝔇)-convex, 𝐾P = 𝐾. Take 𝑐 ′ ∈ R such that max 𝜆 < 𝑐 ′ < 𝑐. 𝐾





With 𝜀 0 := min 𝛿B (𝑧, 𝜕𝔇) : 𝑧 ∈ P¯ , we consider the smoothing 𝜆 𝜀 of 𝜆, where 0 < 𝜀 < 𝜀 0 . We would like to show: Claim 4.3.76. For a sufficiently small 𝜀 > 0, 𝐾 ⊂ {𝜆 𝜀 ( 𝑝) < 𝑐 ′ } ∩ P. ∵ ) Take any point 𝑝 ∈ 𝐾, and 𝑐 ′′ with 𝜆( 𝑝) < 𝑐 ′′ < 𝑐 ′ . Since 𝜆 is upper semicontinuous, 𝜆 < 𝑐 ′′ in a neighborhood 𝑈 of 𝑝. For an arbitrarily small 𝜀 > 0 we have 𝜆 𝜀 ( 𝑝) ≤ 𝑐 ′′ < 𝑐 ′ . Therefore there is a neighborhood 𝑈 ′ (⊂ 𝑈) of 𝑝 such that 𝜆 𝜀 ( 𝑝 ′ ) < 𝑐 ′ (∀ 𝑝 ′ ∈ 𝑈 ′ ). Because of the compactness of 𝐾, there are nitely many such 𝑈 ′ covering 𝐾. Hence, for su ciently small 𝜀 > 0 we obtain that {𝜆 𝜀 < 𝑐 ′ } ∩ P ⊃ 𝐾. △ ¯ In a neighborhood of P we set ′ 𝜆˜ 𝜀𝜀 ( 𝑝) = 𝜆 𝜀 ( 𝑝) + 𝜀 ′ ∥𝜋( 𝑝)∥ 2 ,

𝜀 ′ > 0.

Since P¯ is compact, 𝑀 := max ∥𝜋( 𝑝)∥ 2 < ∞. P¯

𝜀′

¯ and The function 𝜆˜ 𝜀 ( 𝑝) is strongly plurisubharmonic in a neighborhood of P, satis es ′ ¯ 𝜆 𝜀 ( 𝑝) ≤ 𝜆˜ 𝜀𝜀 ( 𝑝) ≤ 𝜆 𝜀 ( 𝑝) + 𝜀 ′ 𝑀, 𝑝 ∈ P. With 𝜀 ′ > 0 such that 𝑐 ′ + 𝜀 ′ 𝑀 < 𝑐 the following implications hold: n ′ o 𝐾 ⋐ {𝜆 𝜀 < 𝑐 ′ } ∩ P ⊂ 𝜆˜ 𝜀𝜀 < 𝑐 ′ + 𝜀 ′ 𝑀 ∩ P ⊂ {𝜆 𝜀 < 𝑐 ′ + 𝜀 ′ 𝑀 } ∩ P ⊂ {𝜆 < 𝑐} ∩ P. Since P is Stein, it follows from (a) above that n o bP ⋐ 𝜆˜ 𝜀𝜀′ < 𝑐 ′ + 𝜀 ′ 𝑀 ∩ P ⊂ {𝜆 < 𝑐} ∩ P. 𝐾 b=𝐾 bP ⋐ 𝔇𝑐 . Therefore, 𝐾

⊓ ⊔

Corollary 4.3.77. Let 𝜆 : C𝑛 → [−∞, ∞) be a plurisubharmonic function. Then for every 𝑐 ∈ R, each connected component of {𝑧 ∈ C𝑛 : 𝜆(𝑧) < 𝑐} is a domain of holomorphy.

4.3 Pseudoconvexity

171

That is, in the case of a univalent domain, the Pseudoconvexity Problem is solved by Corollary 4.3.77, if the pseudoconvex function de ning the pseudoconvex domain is de ned on C𝑛 . Let 𝔇/C𝑛 be domain and let 𝐾 ⊂ 𝔇 be a subset. As the holomorphically convex b𝔇 is de ned in (3.1.10), we de ne the pseudoconvex hull of 𝐾 in 𝔇 by hull 𝐾   b (4.3.78) 𝐾𝒫 (𝔇) = 𝑝 ∈ 𝔇 : |𝜑( 𝑝)| ≤ sup 𝜑, ∀ 𝜑 ∈ 𝒫(𝔇) . 𝐾

Theorem 4.3.79. Let 𝔇/C𝑛 be a Stein domain. If 𝐾 ⋐ 𝔇 is a compact subset, then, b𝒫 (𝔇) = 𝐾 b𝔇 . 𝐾 b𝒫 (𝔇) ⊂ 𝐾 b𝔇 . Suppose that the equality does not hold. There Proof. By de nition, 𝐾 b is a point 𝑎 ∈ 𝐾𝔇 such that there is a function 𝜑 ∈ 𝒫(𝔇) with 𝜑(𝑎) > sup𝐾 𝜑. Take a number 𝑐 ∈ R so that 𝜑(𝑎) > 𝑐 > sup𝐾 𝜑. Then 𝐾 ⋐ 𝔇𝑐 = {𝜑 < 𝑐}. It follows from Theorem 4.3.74 that 𝔇𝑐 is 𝒪(𝔇)-convex, and then Theorem 3.3.18, which holds for b𝔇 = 𝐾 b𝔇𝑐 ∌ 𝑎; this is absurd. Stein domains over C𝑛 , implies 𝐾 ⊓ ⊔ We consider a univalent domain 𝛺 ⊂ C𝑛 . For a compact subset 𝐾 ⋐ 𝛺 we have the following relations for its hulls in general: (4.3.80)

★ b𝛺 ⊂ 𝐾 b𝒫 (𝛺 ) 𝐾 ⊂ 𝐾𝛺 ⊂𝐾

★ b𝛺 , 𝜕𝛺) ≥ 𝑑 ( 𝐾 b𝒫 (𝛺 ) , 𝜕𝛺). 𝑑 (𝐾, 𝜕𝛺) = 𝑑 (𝐾𝛺 , 𝜕𝛺) ≥ 𝑑 ( 𝐾

Cf. Proposition 3.1.7 for the equality above. From Theorem 4.3.79 we immediately obtain: Corollary 4.3.81. Let 𝐾 ⋐ 𝛺 (⊂ C𝑛 ) be as above. Assume that 𝛺 is a domain of holomorphy. Then, (4.3.82)

★ b𝛺 = 𝐾 b𝒫 (𝛺 ) 𝐾 ⊂ 𝐾𝛺 ⊂𝐾

★ b𝛺 , 𝜕𝛺) = 𝑑 ( 𝐾 b𝒫 (𝛺 ) , 𝜕𝛺). 𝑑 (𝐾, 𝜕𝛺) = 𝑑 (𝐾𝛺 , 𝜕𝛺) = 𝑑 ( 𝐾

Remark 4.3.83. Theorem 4.3.74 suggests a close relationship between holomorphic functions and plurisubharmonic functions. It is interesting to notice that Theorems 4.3.74 and 4.3.79 imply the solution of the Pseudoconvexity Problem if the domain is univalent and the given plurisubharmonic function is de ned on a domain of holomorphy, or equivalently, a holomorphically convex domain; i.e., they are proved before the solution of the Psudoconvexity Problem. In Hörmander [30] 4.3 the statements similar to those above are proved for a pseudoconvex domain of C𝑛 by ¯ 2 making use of the solution of the Pseudoconvexity Problem by means of the 𝜕-𝐿 method. Note. The pseudoconvexity properties (iv) (vi) stated in Remark 4.3.9 are described only in terms of real functions, where only local complex coordinates are used in the de nition of plurisubharmonicity. Moreover, as described after 4.3.3,

172

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the pseudoconvexity of 𝔇 is a local property of the boundary 𝜕𝔇. Thus, the properties appear to be far from the existence of the global holomorphic functions. These di culties might be the reason why the Pseudoconvexity Problem was rst believed to be unsolvable (R. Remmert, [49] Vorwort). We have summarized the Pseudoconvexity Problem as in the present chapter, but it should be reminded that such a notion as “plurisubharmonic functions” or “pseudoconvex functions” was not yet invented when the problem was proposed. K. Oka introduced the new notion of “pseudoconvex functions” in Oka VI (1942) in order to solve the problem; this was the critical di erence in the notions of the pseudoconvexity of Levi and Oka (see also IX (unpublished, 1943) in [50], and IX 13 (published, 1953) in [48], [49]): Thanks to the notion of Oka’s pseudoconvex functions, it is then possible to approximate a strongly pseudonvex domain by strongly pseudonvex domains from the outside that contain the closure of the original one. Thus, a bridge from the Cousin I Problem to the Pseudoconvexity Problem is provided; these are the reason why 𝜎(𝑡, 𝑧) and 𝛴𝑡 are termed the Levi-Oka polynomial and the Oka hypersurfaces, respectively. Around that time, P. Lelong introduced the same notion from the potential theoretic viewpoint. The present proof of Bochner’s Theorem 4.3.43 is due to the author.10 Equation (4.3.40) was inspired by an exercise in Fritzsche Grauert [18] p. 87. While the existence of an envelope of holomorphy is used in the present proof, in [18] that notion is introduced later, so that the points of consideration seem to be di erent. Theorem 4.3.39 is due to Makoto Abe [1]. It is nice to see that these are obtained as applications of Oka’s Theorem of Boundary Distance Functions 4.3.1. Cf. Exercises 8 and 9 below. S. Bochner proved Theorem 4.3.43 rst for bounded holomorphic functions in [6] (1937), and in [7] (1938) he removed the boundedness condition. The method of the proof was very technical, by means of Legendre polynomial developments. Around that time, K. Stein ([56] 1937) proved the result independently by making use of ellipses in the case of 𝑛 = 2. Since the theorem has a special importance in the theories of homogeneous spaces, hyperfunctions, and partial di erential equations, it has had a various proofs since then. Hörmander [30] 2.5 gives a proof by making use of ellipses, close to Stein’s proof in 𝑛-dimensional case; it is rather involved. In Hörmander [30] p. 43 Example, some relation with the theory of partial di erential equations is mentioned, and in [31] H. Komatsu proved a localized form of Bochner’s Tube Theorem obtained due to M. Kashiwara who used it in a fundamental part of M. Sato’s hyperfunctions. Furthermore, Hartogs’ phenomenon has an application to the Edge of the Wedge Theorem due to N. Bogoliubov in theoretical physics (cf. Rudin [55] and the references there).

10 J. Noguchi, A brief proof of Bochner’s tube theorem and a generalized tube, arXiv, 2020.

Exercises

173

Exercises 1. Show that an upper semi-continuous function on a compact set 𝐾 of C𝑛 attains the maximum on 𝐾. 2. Prove (4.1.2). 3. Prove that a function 𝜑 : 𝑈 → [−∞, ∞) on an open set 𝑈 ⊂ C𝑛 is upper semi-continuous if and only if lim 𝜑(𝑧) ≤ 𝜑(𝑎) at every point 𝑎 ∈ 𝑈. 𝑧→𝑎

4. Show that an upper semi-continuous function on an open set of C𝑛 is the limit of a decreasing sequence of continuous functions on 𝑈. 5. Let 𝜑 be a subharmonic function on a domain 𝑈 ⊂ C, and assume 𝜑 . −∞. Show that 𝜑 is integrable on the boundary circle 𝐶 (𝑎;𝑟) of any disk 𝛥(𝑎;𝑟) ⋐ 𝑈. 6. (Hadamard’s three circle theorem) Let 𝑅2 > 𝑅1 > 0 and let 𝜑(𝑧) be a subharmonic function in a neighborhood of the closed ring domain {𝑧 ∈ C : 𝑅1 ≤ |𝑧| ≤ 𝑅2 }. For 𝑅1 ≤ 𝑟 ≤ 𝑅2 we set 𝑀 𝜑 (𝑟) = max { | 𝑧 |=𝑟 } 𝜑(𝑧). Then, show that 𝑀 𝜑 (𝑟) ≤

(log 𝑅2 − log𝑟) 𝑀 𝜑 (𝑅1 ) + (log𝑟 − log 𝑅1 ) 𝑀 𝜑 (𝑅2 ) . log 𝑅2 − log 𝑅1

Therefore, 𝑀 𝜑 (𝑟) is a continuous convex function in log𝑟. Hint: 𝜓(𝑧) = 𝜑(𝑧) − 𝛼 log |𝑧| (𝛼 ∈ R) is subharmonic. Take 𝛼 ∈ R so that 𝑀 𝜓 (𝑅1 ) = 𝑀 𝜓 (𝑅2 ). By the Maximum Principle, 𝑀 𝜓 (𝑟) = 𝑀 𝜑 (𝑟) − 𝛼 log𝑟 ≤ 𝑀 𝜓 (𝑅1 ). 7. With real numbers 𝛼 < 𝛽 we de ne a 1-dimensional tube domain 𝑇 = {𝑧 + 𝑖𝑦 ∈ C : 𝛼 < 𝑥 < 𝛽, 𝑦 ∈ R}. Let 𝜑(𝑧) be a continuous function on 𝑇 such that 𝜑(𝑧) = 𝜑(𝑧 + 𝑖𝑦) (∀ 𝑦 ∈ 𝑅). Show that 𝜑(𝑧) is subharmonic if and only if 𝜑(𝑥) (𝛼 < 𝑥 < 𝛽) is a convex function in 𝑥. Hint: Use the smoothing. 8. Let 𝛺 ⊂ C𝑛 be a Reinhardt domain (cf. Chap. Exercise 5). We de ne the logarithmic image of it by log 𝛺 = {(log |𝑧 𝑗 |) ∈ R𝑛 : (𝑧 𝑗 ) ∈ 𝛺, ∀ 𝑧 𝑗 ≠ 0}. On the other hand, for a real domain 𝑅 ⊂ R𝑛 we de ne exp 𝑅 = {(𝑧 𝑗 ) ∈ C𝑛 : |𝑧 𝑗 | < 𝑒 𝜌 𝑗 , 1 ≤ 𝑗 ≤ 𝑛, ∃ (𝜌 𝑗 ). ∈ 𝑅}. Assume that 𝛺 contains the origin. Show that the envelope of holomorphy of 𝛺 is exp ch(log 𝛺). Hint: Use Theorem 4.3.43. 9. Let 𝑓 ∈ 𝒪(C𝑛 ) and set 𝑄 = {𝑥 = (𝑥 𝑗 ) ∈ R𝑛 : 𝑓 (𝑥 + 𝑖𝑦) ≠ 0, ∀ 𝑦 ∈ R𝑛 }. Prove that every connected component of 𝑄 is convex. Remark: This is non-trivial even if 𝑓 (𝑧) is polynomial. 10. With two real domains 𝑅, 𝑄 ⊂ R𝑛 (𝑛 ≥ 2) we consider a domain 𝛺 = 𝑅 + 𝑖𝑄 ⊂ C𝑛 of real imaginary product type. Show that 𝛺 is a domain of holomorphy if and only if both 𝑅 and 𝑄 are (a ne) convex.

174

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Hint: Use that in a small neighborhood of 𝜕𝛺 ⊃ (𝜕𝑅 + 𝑖𝑄) ∪ (𝑅 + 𝑖𝜕𝑄), − log 𝛿P𝛥 (𝑧, 𝜕𝛺) is plurisubharmonic. 11. Let 𝔇/C𝑛 be a domain, and let 𝑣 ∈ C𝑛 \ {0} be a directional vector. Set 𝑅𝑣 (𝑧) = sup{𝜌 > 0 : 𝑧 + 𝜁 𝑣 ∈ 𝔇, ∀ |𝜁 | < 𝜌} for 𝑧 ∈ 𝔇. Show that if 𝔇 is a domain of holomorphy, then − log 𝑅𝑣 (𝑧) is plurisubharmonic. The function 𝑅𝑣 (𝑧) is called the Hartogs radius with respect to 𝑣. Hint: After a linear change of coordinates we may assume 𝑣 = (1, 0, . . . , 0). With a small 𝑡 > 0 we take a polydisk P𝛥𝑡 = {(𝑧1 , 𝑧 2 , . . . , 𝑧 𝑛 ) : |𝑧1 | < 1, |𝑧 𝑗 | < 𝑡, 2 ≤ 𝑗 ≤ 𝑛}. As 𝑡 ↘ 0, 𝛿P𝛥𝑡 (𝑧, 𝜕𝔇) converges increasingly to 𝑅𝑣 (𝑧). 12. Let 𝑀 ∈ R and let 𝐷 ⊂ C be a domain. Let 𝑓 (𝑧, 𝑤) be a holomorphic function in 𝛺 = {(𝑧, 𝑤) ∈ C2 : 𝑧 ∈ 𝐷, ℜ𝑤 > 𝑀 }. For a point 𝑧0 ∈ 𝐷 we denote by 𝜎(𝑧0 ) the in mum of 𝑠 ∈ R such that 𝑓 (𝑧, 𝑤) is analytically continued over a neighborhood of {𝑧 = 𝑧0 , ℜ𝑤 > 𝑠}. Show that 𝜎(𝑧) is subharmonic in 𝑧 ∈ 𝐷. Hint: Set a family ℱ = { 𝑓 (𝑧, 𝑤 + 𝑖𝑏) : 𝑏 ∈ R}. Let 𝛺˜ be the ℱ-envelope of 𝛺. Then, 𝛺˜ = {(𝑧, 𝑤) : 𝑧 ∈ 𝐷, ℜ𝑤 > 𝜏(𝑧)} with some real-valued function 𝜏(𝑧), which is a domain of holomorphy. Let 𝑅𝑣 (𝑧, 𝑤) (= 𝑅𝑣 (𝑧, 𝑤 + 𝑖𝑏), 𝑏 ∈ R) be the Hartogs radius with respect to 𝑣 = (0, 1). Then, − log 𝑅𝑣 (𝑧, 𝑤) is plurisubharmonic. For 𝑇 > 𝑀, 𝜏(𝑧) = 𝑇 − 𝑅𝑣 (𝑧,𝑇), and log(𝑇 − 𝜏(𝑧)) −1 is subharmonic in 𝑧 ∈ 𝐷. The function log(1 − 𝜏(𝑧)/𝑇) −𝑇 is subharmonic in 𝑧, and monotone decreasing in 𝑇 > 𝑀; as 𝑇 ↗ ∞, the limit is 𝜏(𝑧).11 13. Let 𝔇/C𝑛 be a domain. Let 𝛺 H ⊂ C2 be any Hartogs domain (see 1.1 (b)). Let 𝛹 : 𝛺 H → 𝔇 be an arbitrary holomorphic map. If 𝛹 is always analytically continued to a holomorphic map 𝛹ˆ : 𝛺ˆ H → 𝔇 from the envelope holomorphy 𝛺ˆ H (a bi-disk) of 𝛺 H into 𝔇, 𝔇 is said to be Hartogs pseudoconvex. Prove that if 𝔇/C𝑛 is Hartogs pseudoconvex, then − log 𝛿P𝛥 ( 𝑝, 𝜕𝔇) ∈ 𝒫0 (𝔇). Hint: Follow the proof of Oka’s Theorem 4.3.11, in particular, the arguments after (4.3.14). 14. Let 𝔇/C𝑛 be a domain, and let 𝔇𝜈 ⊂ 𝔇 (𝜈 = 1, 2, . . .) be a sequence of subdomains of 𝔇. Assume that every 𝔇𝜈 is pseudoconvex, monotone increasing Ð 𝔇𝜈 ⊂ 𝔇𝜈+1 , and 𝔇 = ∞ 𝜈=1 𝔇 𝜈 . Then, show that 𝔇 is pseudoconvex. 15. Show that the domain de ned by |𝑧1 | < |𝑧2 | < · · · < |𝑧 𝑛 | in C𝑛 with coordinates (𝑧 𝑗 )1≤ 𝑗 ≤𝑛 is a domain of holomorphy. 16. In Example 4.3.72, write down the Oka hypersurfaces 𝛴𝑡 in terms of coordinates at the boundary point 𝑝 0 = ( 𝑝 01 , . . . , 𝑝 0𝑛 ) ∈ 𝜕B(1).

11 This is a bit di cult and due to Bochner Martin, Several Complex Variables, Princeton Univ. Press, 1948, Chap. 7 8.

Chapter 5

Pseudoconvex Domains II — Solution

In the previous chapter the Pseudoconvexity Problem is reduced to the problem of asking if a pseudoconvex domain is Stein. In this chapter we solve it affirmatively. It is the high point to prove that a bounded domain with strongly pseudoconvex boundary is Stein (Levi’s Problem). We shall give two proofs to it; the first is K. Oka’s original one due to an unpublished paper of 1943 by means of the Fredholm integral equation of the second kind type combined with the Joku-Iko Principle, and the second is due to H. Grauert (1958) through L. Schwartz’s Fredholm Theorem for compact operators and the bumping method. The comparison is interesting. Each proof has its own advantage.

5.1 The Oka Extension with Estimate 5.1.1 Preparation from Topological Vector Spaces In the present book all vector spaces are assumed to be de ned over complex numbers. Let 𝐸 be a vector space. In general, 𝐸 is called a topological vector space if 𝐸 is endowed with a topology such that the algebraic operations, including those with C, are continuous. There is a method to introduce such a topology on 𝐸 by a system of semi-norms on 𝐸 as follows. A semi-norm ∥𝑥∥ (𝑥 ∈ 𝐸) is a real-valued function satisfying the following conditions: (i) ∥𝑥∥ ≥ 0, 𝑥 ∈ 𝐸. (ii) ∥𝜆𝑥∥ = |𝜆| · ∥𝑥∥, 𝜆 ∈ C, 𝑥 ∈ 𝐸. (iii) ∥𝑥 + 𝑦∥ ≤ ∥𝑥∥ + ∥𝑦∥, 𝑥, 𝑦 ∈ 𝐸. If ∥𝑥∥ = 0 implies 𝑥 = 0, ∥𝑥∥ is called a norm, and de nes naturally a metric topology, with which 𝐸 is a topological vector space. Assume that 𝐸 is given a system of semi-norms ∥𝑥∥ 𝑗 ( 𝑗 = 1, 2, . . .). For a nite set 𝛤 ⊂ N and positive numbers 𝜀 𝑗 > 0 ( 𝑗 ∈ 𝛤) we consider a subset de ned by

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 J. Noguchi, Basic Oka Theory in Several Complex Variables, Universitext, https://doi.org/10.1007/978-981-97-2056-9_5

175

176

(5.1.1)

5 Pseudoconvex Domains II

Solution

𝑈 (𝛤, {𝜀 𝑗 } 𝑗 ∈𝛤 ) = {𝑥 ∈ 𝐸 : ∥𝑥∥ 𝑗 < 𝜀 𝑗 , ∀ 𝑗 ∈ 𝛤}.

Then it is easy to see that the family of all such 𝑈 (𝛤, {𝜀 𝑗 } 𝑗 ∈𝛤 ) forms a fundamental system of neighborhoods of 0. For a general point 𝑎 ∈ 𝐸, the family of 𝑎 + 𝑈 (𝛤, {𝜀 𝑗 } 𝑗 ∈𝛤 ) forms a fundamental system of neighborhoods of 𝑎, and so 𝐸 gives rise to a topological vector space. In this case, 𝐸 is a Hausdor space if and only if for every 𝑥 ∈ 𝐸 \ {0} there is a 𝑗 ∈ N with ∥𝑥∥ 𝑗 ≠ 0. We assume all through the present book that: 5.1.2. Topological vector spaces are Hausdor . For two points 𝑥, 𝑦 ∈ 𝐸 we set 𝑑 (𝑥, 𝑦) =

(5.1.3)

∞ Õ ∥𝑥 − 𝑦∥ 𝑗 1 · . 𝑗 1 + ∥𝑥 − 𝑦∥ 2 𝑗 𝑗=1

The function 𝑡/(1 + 𝑡) is increasing in 𝑡 ≥ 0, and by an easy computation 𝑡+𝑠 𝑡 𝑠 − − ≤ 0, 1+𝑡 +𝑠 1+𝑡 1+𝑠

𝑡, 𝑠 ≥ 0.

Thus, 𝑑 (𝑥, 𝑦) satis es the axioms of a metric (distance). Note that 𝑑 (𝑥, 𝑦) has the following invariant property: (5.1.4)

𝑑 (𝑥 + 𝑣, 𝑤 + 𝑣) = 𝑑 (𝑥, 𝑤),

𝑑 (−𝑥, 0) = 𝑑 (𝑥, 0).

Lemma 5.1.5. The topology of 𝐸 is homeomorphic to the metric topology by 𝑑 (𝑥, 𝑦). Proof. Since the two topologies are invariant with respect to translations of 𝐸, it is su cient to compare them at 0 ∈ 𝐸. For 𝑟 > 0 we set 𝑈 (𝑟) = {𝑥 ∈ 𝐸 : 𝑑 (𝑥, 0) < 𝑟 }. Noting that ∞ ∞ Õ Õ ∥𝑥 − 𝑦∥ 𝑗 1 1 1 · < = 𝑁 −1 , 𝑗 1 + ∥𝑥 − 𝑦∥ 𝑗 2 2 2 𝑗 𝑗=𝑁 𝑗=𝑁 we see the following: 5.1.6. (i) For any neighborhood 𝑈 (𝛤, {𝜀 𝑗 } 𝑗 ∈𝛤 ) of 0, there is some 𝑈 (𝑟) with 𝑈 (𝑟) ⊂ 𝑈 (𝛤, {𝜀 𝑗 } 𝑗 ∈𝛤 ). (ii) For any 𝑈 (𝑟), there is some 𝑈 (𝛤, {𝜀 𝑗 } 𝑗 ∈𝛤 ) such that 𝑈 (𝛤, {𝜀 𝑗 } 𝑗 ∈𝛤 ) ⊂ 𝑈 (𝑟). Therefore the two topologies are homeomorphic to each other.

⊓ ⊔

Definition 5.1.7. (i) When the above-de ned metric 𝑑 (𝑥, 𝑦) is complete, 𝐸 is called a Fréchet space. (ii) A topological vector space is called a Baire vector space if it is Baire as a topological space (cf. De nition 4.2.1). Proposition 5.1.8. A Fréchet space is Baire. Proof. By Baire Theorem 4.2.3.

⊓ ⊔

5.1 The Oka Extension with Estimate

177

Theorem 5.1.9 (Banach’s Open Map Theorem). Let 𝐸 be a Fréchet space and let 𝐹 be a Baire vector space. If 𝐴 : 𝐸 → 𝐹 is a continuous surjective homomorphism, then 𝐴 is an open map. Proof. By the assumption, 𝐸 carries a complete metric 𝑑 (𝑥, 𝑤) de ned by (5.1.3). Note that besides (5.1.4), 𝑑 (𝑥, 𝑤) has the following property: (5.1.10)

𝑑 (𝑥 + 𝑤, 0) ≤ 𝑑 (𝑥 + 𝑤, 𝑤) + 𝑑 (𝑤, 0) = 𝑑 (𝑥, 0) + 𝑑 (𝑤, 0).

We put 𝑈 (𝜀) = {𝑥 ∈ 𝐸 : 𝑑 (𝑥, 0) < 𝜀}, 𝜀 > 0. It su ces to show that for every 𝜀 > 0, 𝐴(𝑈 (𝜀)) contains 0 ∈ 𝐹 as an interior point. We rst show: Claim 5.1.11. The closure 𝐴(𝑈 (𝜀)) contains 0 ∈ 𝐸 as an interior point. ∵ ) From the continuity of the operation (𝑥, 𝑦) ∈ 𝐸 × 𝐸 → 𝑥 − 𝑦 ∈ 𝐸 Ð it follows that there is a neighborhood 𝑊 of 0 ∈ 𝐸 with 𝑊 − 𝑊 ⊂ 𝑈 (𝜀). Since 𝐸 = ∞ 𝜈=1 𝜈𝑊, Ð∞ 𝐹 = 𝜈=1 𝜈 𝐴(𝑊). Since 𝜈 𝐴(𝑊) is a closed subset, by the assumption there is some 𝜈0 ∈ N such that 𝜈0 𝐴(𝑊) contains an interior point. Therefore 𝐴(𝑊) contains an interior point 𝑥0 , and so 0 ∈ 𝐹 is an interior point of 𝐴(𝑊) − 𝑥 0 . Since 0 ∈ 𝐴(𝑊) − 𝑥0 ⊂ 𝐴(𝑊) − 𝐴(𝑊) = 𝐴(𝑊 − 𝑊) ⊂ 𝐴(𝑈 (𝜀)), the zero vector 0 ∈ 𝐹 is an interior point of 𝐴(𝑈 (𝜀)).

△

It follows from Claim 5.1.11  that there is a neighborhood 𝑉 of 0 ∈ 𝐹 with 𝑉 ⊂ 𝜀 𝐴(𝑈 (𝜀)). We set 𝑈𝜈 = 𝑈 2𝜈+1 , 𝜈 = 1, 2, . . .. For each 𝐴(𝑈𝜈 ) there is a neighborhood Ñ 𝑉𝜈 of 0 ∈ 𝐹 such that 𝑉𝜈 ⊂ 𝐴(𝑈𝜈 ), 𝑉𝜈 ⊃ 𝑉𝜈+1 and ∞ 𝜈=1 𝑉𝜈 = {0}. The following nishes the proof: Claim 5.1.12. 𝐴(𝑈 (𝜀)) ⊃ 𝑉1 . ∵ ) Take arbitrarily a point 𝑦 = 𝑦 1 ∈ 𝑉1 . Because of 𝑦 1 ∈ 𝐴(𝑈1 ), (𝑦 1 −𝑉2 ) ∩ 𝐴(𝑈1 ) ≠ ∅; therefore, there are 𝑦 2 ∈ 𝑉2 and 𝑥1 ∈ 𝑈1 with 𝑦 1 − 𝑦 2 = 𝐴(𝑥1 ). Since 𝑦 2 ∈ 𝐴(𝑈2 ), (𝑦 2 −𝑉3 ) ∩ 𝐴(𝑈2 ) ≠ ∅; similarly, there are 𝑦 3 ∈ 𝑉3 and 𝑥2 ∈ 𝑈2 with 𝑦 2 − 𝑦 3 = 𝐴(𝑥2 ). Thus inductively we take 𝑥 𝜈 ∈ 𝑈𝜈 and 𝑦 𝜈 ∈ 𝑉𝜈 so that 𝑦 𝜈 − 𝑦 𝜈+1 = 𝐴(𝑥 𝜈 ),

𝜈 = 1, 2, . . . .

Because of the choices of {𝑉𝜈 } 𝜈 and {𝑦 𝜈 } 𝜈 , lim𝜈→∞ 𝑦 𝜈 = 0, and (5.1.13)

𝑦 = 𝑦 1 = 𝐴(𝑥 1 ) + 𝑦 2 = 𝐴(𝑥1 ) + 𝐴(𝑥 2 ) + 𝑦 3

𝜈 ©Õ ª 𝐴(𝑥 𝑗 ) + 𝑦 𝜈+1 = 𝐴 ­ 𝑥 𝑗 ® + 𝑦 𝜈+1 . 𝑗=1 « 𝑗=1 ¬ Í 𝜀 We check the convergence of ∞ 𝜈=1 𝑥 𝜈 . Since 𝑥 𝜈 ∈ 𝑈 𝜈 = 𝑈 ( 2𝜈+1 ), a computation with (5.1.10) implies that for every 𝜈, 𝜇 ∈ N,

= ··· =

𝜈 Õ

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Solution

𝜈+𝜇 𝜈+𝜇 𝜈+𝜇 𝜈 ©Õ Õ ª © Õ ª Õ 𝑑 ­ 𝑥 𝑗, 𝑥 𝑗 ® = 𝑑 ­0, 𝑥 𝑗® ≤ 𝑑 (0, 𝑥 𝑗 ) 𝑗=1 𝑗=1 𝑗=𝜈+1 𝑗=𝜈+1 « ¬ « ¬ 𝜈+𝜇 Õ 𝜀 𝜀 < < 𝜈+1 . 𝑗+1 2 2 𝑗=𝜈+1

Í Therefore ∞ 𝑥 satis es the Cauchy condition, and hence converges. We set the Í∞𝜈=1 𝜈 limit, 𝑤 = 𝜈=1 𝑥 𝜈 ; by (5.1.13), 𝑦 = 𝐴(𝑤), and also 𝑑 (0, 𝑤) ≤

∞ Õ

𝑑 (0, 𝑥 𝜈 ) ≤

𝜈=1

∞ Õ 𝜀 1 = 𝜀 < 𝜀. 𝜈+1 2 2 𝜈=1

Thus, the inclusion 𝐴(𝑈 (𝜀)) ⊃ 𝑉1 follows.

⊓ ⊔ C𝑁 ,

Example 5.1.14. A nite-dimensional vector space, isomorphic to some with the natural topology is a Fréchet space. While there are a number of examples of Fréchet spaces (cf. Exercise 3 at the end of the chapter), the following is of importance in complex analysis. (1) Let 𝛺 ⊂ C𝑛 be a domain. We take an increasing open covering {𝛺 𝑗 }∞ 𝑗=1 such that (5.1.15)

∅ ≠ 𝛺 𝑗 ⋐ 𝛺 𝑗+1 ,

𝛺=

∞ Ø

𝛺𝑗.

𝑗=1

We de ne a system of semi-norms on 𝒪(𝛺) by ∥ 𝑓 ∥ 𝑗 := ∥ 𝑓 ∥ 𝛺 𝑗 = sup | 𝑓 | (= max | 𝑓 |), 𝛺𝑗

𝛺¯ 𝑗

𝑗 = 1, 2, . . . ,

𝑓 ∈ 𝒪(𝛺).

With the system of semi-norms ∥ 𝑓 ∥ 𝛺¯ 𝑗 ( 𝑗 ∈ N), 𝒪(𝛺) gives rise to a Fréchet space: The topology of 𝒪(𝛺) is independent of the choice of such an open covering, and equivalent to the topology of locally uniform convergence. (2) In the case of an unrami ed domain 𝜋 : 𝔇 → C𝑛 , we use 𝔇𝜈 of Proposition 3.6.18 to de ne a system of semi-norms ∥ 𝑓 ∥ 𝜈 := ∥ 𝑓 ∥ 𝔇𝜈 = sup | 𝑓 | (= max | 𝑓 |), 𝔇𝜈

¯𝜈 𝔇

𝜈 = 1, 2, . . . ,

𝑓 ∈ 𝒪(𝔇).

Then 𝒪(𝔇) is a Fréchet space.

5.1.2 The Oka Extension with Estimate We prove the extension of the Joku-Iko Principle with estimate, which will play an important role in the proof of the Pseudoconvexity Problem (Levi’s Problem):

5.2 Strongly Pseudoconvex Domains

179

Lemma 5.1.16 (Oka Extension with Estimate). Let P𝛥 be a polydisk, let 𝛴 ⊂ P𝛥 be a complex submanifold, and let 𝐿 ⋐ P𝛥 be a compact subset. Then for a bounded holomorphic function 𝑓 ∈ 𝒪(𝛴), there is some 𝐹 ∈ 𝒪(P𝛥) such that 𝐹 |𝛴 = 𝑓 ,

∥𝐹 ∥ 𝐿 ≤ 𝐶 ∥ 𝑓 ∥ 𝛴 .

Here, 𝐶 > 0 is a positive constant independent from 𝑓 . Proof. Note that 𝒪(P𝛥) and 𝒪(𝛴) are Fréchet spaces; in particular, 𝒪(𝛴) is Baire. By the Oka Extension Theorem 2.5.14 the restriction map 𝐴 : 𝐹 ∈ 𝒪(P𝛥) −→ 𝐹 | 𝛴 ∈ 𝒪(𝛴) is a continuous surjective homomorphism. Note that 𝑈 = {𝐹 ∈ 𝒪(P𝛥) : ∥𝐹 ∥ 𝐿 < 1} is a neighborhood of 0 in 𝒪(P𝛥). By Banach’s Open Map Theorem 5.1.9, 𝐴(𝑈) contains a neighborhood of 0 in 𝒪(𝛴). Therefore, there is a compact subset 𝐾 ⋐ 𝛴 and 𝜀 > 0 such that 𝐴(𝑈) ⊃ { 𝑓 ∈ 𝒪(𝛴) : ∥ 𝑓 ∥ 𝐾 < 𝜀}. It su ces to choose 𝐶 = 1/𝜀.

⊓ ⊔

5.2 Strongly Pseudoconvex Domains The aim of this section is to solve Pseudoconvexity Problem III 4.3.54. Let 𝜋 : 𝔊 → C𝑛 be a domain. Let 𝔇 ⋐ 𝔊 be a bounded subdomain. Lemma 5.2.1. If 𝔇 is strongly pseudoconvex, then 𝔇 is Stein. N.B. The original Levi Problem 4.3.65 is solved by this lemma. Now we rst give the proof of Lemma 5.2.1 based on Oka’s unpublished papers of 1943.

5.2.1 Oka’s Method (a) Oka’s Heftungslemma. For the sake of simplicity we say in general that the ¯ of a domain 𝔇 is Stein if 𝔇 ¯ satis es the following conditions: closure 𝔇 (i) 𝜕𝔇 is the boundary of the set of exterior points of 𝔇. ¯ has a fundamental system of Stein open neighborhoods. (ii) 𝔇 In this case, 𝔇 is itself Stein by Corollary 3.7.4. Let 𝔇 ⋐ 𝔊 be a strongly pseudoconvex domain. Then 𝜕𝔇 satis es (i) above, so that we have: ¯ is Stein, then 𝔇 is Stein. Lemma 5.2.2. If 𝔇

180

5 Pseudoconvex Domains II

Solution

This will be used implicitly and frequently. 𝜋 Suppose that the strongly pseudoconvex domain 𝔇 (⋐ 𝔊) → C𝑛 with the rst coordinate 𝑧 1 = 𝑥1 + 𝑖𝑦 1 spreads over an open interval containing 𝑥1 = 0. Taking 𝑎 2 < 0 < 𝑎 1 in the interval, we set (5.2.3)

𝔇1 = 𝔇 ∩ {𝑥1 < 𝑎 1 },

𝔇2 = 𝔇 ∩ {𝑥 1 > 𝑎 2 },

𝔇3 = 𝔇1 ∩ 𝔇2 ≠ ∅.

N.B. In general, 𝔇𝜈 (𝜈 = 1, 2, 3) may have several connected components. ¯ 𝜈 (𝜈 = 1, 2) are Stein, so Lemma 5.2.4 (Oka’s Heftungslemma). If the closures 𝔇 ¯ is 𝔇, too. ¯ 3 is Stein. Remark 5.2.5. By Corollary 3.7.4 (also by Theorem 4.3.74), 𝔇 ¯ in Lemma 5.2.1 by real The idea here is to take a su ciently ne partition of 𝔇 hyperplanes parallel to real and imaginary axes of the complex coordinates, so that ¯ is contained in a univalent Stein domain. Then we every piece of such partition of 𝔇 sew them together horizontally and vertically with keeping the Steinness, and hence ¯ nally get the Steinness of the whole 𝔇. ¯ in Lemma 5.2.4. We use the Cousin I Problem to show the Steinness of 𝔇 ¯ 𝜈 (𝜈 = 1, 2) are Stein. Then for 𝑓 ∈ 𝒪( 𝔇 ¯ 3 ) there are Lemma 5.2.6. Assume that 𝔇 ¯ 𝑓 𝜈 ∈ 𝒪( 𝔇𝜈 ) (𝜈 = 1, 2) such that 𝑓1 (𝑧) − 𝑓2 (𝑧) = 𝑓 (𝑧),

𝑧 ∈ 𝔇3 .

(b) Integral equation of Fredholm type. We prepare for the proof of Lemma 5.2.6. We keep the notation above. With su ciently large 𝑟 > 𝑟 0 > 0, we take the double neighborhoods of the closure of 𝜋(𝔇) as follows: 𝜋(𝔇) ⋐ P𝛥0 = {(𝑧 𝑗 ) : |𝑧 𝑗 | < 𝑟 0 } ⋐ P𝛥 = {(𝑧 𝑗 ) : |𝑧 𝑗 | < 𝑟 }. In the 𝑧 1 -plane we take a line segment, which will be used later for Cousin’s integral, ℓ0 = {𝑧 1 = 𝑖𝑡 : −𝑟 0 ≤ 𝑡 ≤ 𝑟 0 }, where the orientation is the one as 𝑡 increases. With 𝛿 > 0 such that 𝑎 2 < −𝛿 < 0 < 𝛿 < 𝑎 1 , we set P𝛥 𝛿 = P𝛥 ∩ {|𝑥1 | < 𝛿}. We choose 𝜑 𝑘 ∈ 𝒪(𝔇3 ), 1 ≤ 𝑘 ≤ 𝑚, satisfying the following (cf. Fig. 5.1). 5.2.7 (Heftungscondition). we have

(i) With 𝔇3′ := {𝑧 ∈ 𝔇3 : |𝜑 𝑘 (𝑧)| < 1, 1 ≤ 𝑘 ≤ 𝑚},

(𝜕𝔇3′ ) \ 𝜕𝔇3 ⋐ {𝑎 2 < 𝑥 1 < 𝑎 1 },

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181

Fig. 5.1 Oka Heftungs strongly pseudoconvex domains.

(𝜕𝔇3 ) \ 𝜕𝔇3′ ⋑ (𝜕𝔇3 ) ∩ {|𝑥 1 | ≤ 𝛿}. (ii) With 𝔇3′ 𝛿 := 𝔇3′ ∩ {|𝑥1 | < 𝛿} ⋐ 𝔇3 and 𝜑(𝑧) = (𝜑1 (𝑧), . . . , 𝜑 𝑚 (𝑧)) (𝑧 ∈ 𝔇3 ), the Oka map

(5.2.8)

f 𝛿 := P𝛥 𝛿 × 𝛥(1) 𝑚 𝔇3′ 𝛿 ∋ 𝑧 ↦−→ (𝑧, 𝜑(𝑧)) ∈ P𝛥

is a proper embedding. Lemma 5.2.9. For every compact subset 𝐾 ⋐ 𝔇 the above {𝜑 𝑘 } 𝑚 𝑘=1 ⊂ 𝒪(𝔇3 ) can be chosen so that 𝐾 ∩ 𝔇3 ⊂ 𝔇3′ and Heftungscondition 5.2.7 is satisfied. Proof. For each point 𝑞 ∈ (𝜕𝔇3 ) ∩ {|𝑥1 | ≤ 𝛿} we apply Lemma 4.3.70 (iii) to obtain ¯ 3 \ {𝑞} with polar locus on the Oka hypersurface a holomorphic function 𝜑( 𝑝) on 𝔇 𝛴0 (∋ 𝑞) of 𝜕𝔇3 at 𝑞 (hence, |𝜑(𝑞)| = ∞1). Multiplying a small constant (≠ 0) to 𝜑( 𝑝), we have ∥𝜑∥ 𝐾 < 1,

¯ 3 = ∅. {|𝜑| ≥ 1} ∩ {𝑥1 = 𝑎 1 , 𝑎 2 } ∩ 𝔇

1 This stands for lim 𝑝→𝑞 | 𝜑 ( 𝑝) | = ∞; the same in the sequel.

182

5 Pseudoconvex Domains II

Solution

Since 𝑞 ∈ {|𝜑| > 1} ∩ (𝜕𝔇3 ) ∩ {|𝑥 1 | ≤ 𝛿} and (𝜕𝔇3 ) ∩ {|𝑥1 | ≤ 𝛿} is compact, there are nitely many such 𝜑 𝑘 satisfying ! 𝑚 Ø {|𝜑 𝑘 | > 1} ∩ (𝜕𝔇3 ) ∩ {|𝑥 1 | ≤ 𝛿} ⊃ (𝜕𝔇3 ) ∩ {|𝑥1 | ≤ 𝛿}. 𝑘=1

¯ 3 is Stein (cf. Remark 5.2.5), we may increase the members of {𝜑 𝑘 } so that Since 𝔇 (5.2.8) is injective, and the Heftungscondition (ii) is satis ed. ⊓ ⊔ We take 𝜌0 , 𝜌1 (0 < 𝜌0 < 𝜌1 < 1), both close to 1, 𝑟 1 (𝑟 0 < 𝑟 1 < 𝑟) close to 𝑟, and 𝛿1 (0 < 𝛿1 < 𝛿) close to 𝛿; we x them. Set (5.2.10)

𝔇1(0) := (𝔇1 \ 𝔇3 ) ∪ {𝑧 ∈ 𝔇3 : 𝑥1 < 𝛿, |𝜑 𝑘 (𝑧)| < 𝜌0 , 1 ≤ 𝑘 ≤ 𝑚}, 𝔇2(0) := (𝔇2 \ 𝔇3 ) ∪ {𝑧 ∈ 𝔇3 : 𝑥1 > −𝛿, |𝜑 𝑘 (𝑧)| < 𝜌0 , 1 ≤ 𝑘 ≤ 𝑚}

(cf. Fig. 5.1). We rst show Lemma 5.2.6 for a little bit smaller 𝔇1(0) , 𝔇2(0) : Here is the main part of Oka’s method and the proof gets a bit long.   Lemma 5.2.11. With the notation above, there exist ℎ 𝜈 ∈ 𝒪 𝔇𝜈(0) (𝜈 = 1, 2) such that (5.2.12)

ℎ1 (𝑧) − ℎ2 (𝑧) = 𝑓 (𝑧),

𝑧 ∈ 𝔇1(0) ∩ 𝔇2(0) .

Proof. With 𝑤 = (𝑤 𝑘 )1≤ 𝑘 ≤𝑚 ∈ C𝑚 we set
f𝛿 . 𝐿 1 = {(𝑧, 𝑤) ∈ C𝑛 × C𝑚 : |𝑥 1 | ≤ 𝛿1 , ∀ |𝑧 𝑗 | ≤ 𝑟 1 , ∀ |𝑤 𝑘 | ≤ 𝜌1 } ⋐ P𝛥 We consider a holomorphic function 𝑔 ∈ 𝒪(𝔇3′ 𝛿 ) such that ∥𝑔∥ 𝔇3′ 𝛿 ≤ 𝑀 (< ∞). By virtue of the Oka map (5.2.8) and Lemma 5.1.16 (the Oka Extension with Estimate) f 𝛿 ) such that there exists 𝐺 (𝑧, 𝑤) ∈ 𝒪( P𝛥 (5.2.13)

𝐺 (𝑧, 𝜑(𝑧)) = 𝑔(𝑧) (𝑧 ∈ 𝔇3′ 𝛿 ),

∥𝐺 ∥ 𝐿1 ≤ 𝐶 ∥𝑔∥ 𝔇3′ 𝛿 ≤ 𝐶 𝑀,

where 𝐶 is a positive constant independent from 𝑔. We take Cousin’s integral (decomposition) of 𝐺 (𝑧, 𝑤) along ℓ0 : ∫ 1 𝐺 (𝑡, 𝑧 ′ , 𝑤) 𝐺 𝜈 (𝑧, 𝑤) = 𝑑𝑡, 𝜈 = 1, 2, 2𝜋𝑖 ℓ0 𝑡 − 𝑧 1 |𝑧1 | < 𝑟 0 , 𝑧 ′ = (𝑧2 , . . . , 𝑧 𝑛 ), |𝑧 𝑗 | < 𝑟, 2 ≤ 𝑗 ≤ 𝑛,

𝑤 = (𝑤 𝑘 ), |𝑤 𝑘 | < 1, 1 ≤ 𝑘 ≤ 𝑚. Here 𝐺 1 (resp. 𝐺 2 ) is the branch of the integral de ned in the left (resp. right)-hand side of ℓ0 , and analytically continued across ℓ0 up to {𝑥 1 < 𝛿} (resp. {𝑥 1 > −𝛿}). We then have

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183

𝐺 1 (𝑧, 𝑤) − 𝐺 2 (𝑧, 𝑤) = 𝐺 (𝑧, 𝑤), |𝑥 1 | < 𝛿, |𝑧 1 | < 𝑟 0 , |𝑧 𝑗 | < 𝑟, 2 ≤ 𝑗 ≤ 𝑛, |𝑤 𝑘 | < 1, 1 ≤ 𝑘 ≤ 𝑚. Now, substituting 𝑤 = 𝜑(𝑧) (𝑧 ∈ 𝔇3′ 𝛿 ), we set 𝑔 𝜈 (𝑧) = 𝐺 𝜈 (𝑧, 𝜑(𝑧)) ∈ 𝒪(𝔇3′ 𝛿 ) (𝜈 = 1, 2), which satisfy ∫ 1 𝐺 (𝑡, 𝑧 ′ , 𝜑(𝑧 1 , 𝑧 ′ )) ′ 𝑔 𝜈 (𝑧1 , 𝑧 ) = 𝑑𝑡, 𝜈 = 1, 2, 2𝜋𝑖 ℓ0 𝑡 − 𝑧1 𝑔1 (𝑧) − 𝑔2 (𝑧) = 𝑔(𝑧),

𝑧 ∈ 𝔇3′ 𝛿 .

𝑔 to (0) .) (Roughly speaking, Î𝑚 we would like to extend the de ning domain of each 𝜈 𝔇𝜈 Setting 𝛤0 = 𝑘=1 {|𝑤 𝑘 | = 𝜌0 }, we have by Cauchy’s integral formula ∫ 1 𝐺 (𝑧, 𝑢 1 , . . . , 𝑢 𝑚 ) 𝐺 (𝑧, 𝑤) = 𝑑𝑢 1 · · · 𝑑𝑢 𝑚 , (2𝜋𝑖) 𝑚 𝛤0 (𝑢 1 − 𝑤1 ) · · · (𝑢 𝑚 − 𝑤𝑚 ) |𝑤 𝑘 | < 𝜌0 , 1 ≤ 𝑘 ≤ 𝑚. We write 𝑢 0 = 𝑡, 𝑢 = (𝑢 1 , . . . , 𝑢 𝑚 ), 𝑢˜ = (𝑢 0 , 𝑢), 𝑑 𝑢˜ = 𝑑𝑢 0 𝑑𝑢 1 · · · 𝑑𝑢 𝑚 . (Pay attention to the variable 𝑧1 in what follows.) For 𝜈 = 1, 2 we set ∫ 1 𝐺 (𝑢 0 , 𝑧 ′ , 𝑢) 𝐺 𝜈 (𝑧1 , 𝑧 ′ , 𝑤) = 𝑑 𝑢, ˜ 𝑚+1 (2𝜋𝑖) ℓ0 ×𝛤0 (𝑢 0 − 𝑧 1 )(𝑢 1 − 𝑤1 ) · · · (𝑢 𝑚 − 𝑤𝑚 ) 1 𝜒( 𝑢, ˜ 𝑧1 , 𝑧′ ) = , 𝑚+1 (2𝜋𝑖) (𝑢 0 − 𝑧1 )(𝑢 1 − 𝜑1 (𝑧 1 , 𝑧 ′ )) · · · (𝑢 𝑚 − 𝜑 𝑚 (𝑧 1 , 𝑧 ′ )) ∫ 𝑚 Ù 𝑔 𝜈 (𝑧 1 , 𝑧 ′ ) = 𝜒( 𝑢, ˜ 𝑧 1 , 𝑧 ′ )𝐺 (𝑢 0 , 𝑧 ′ , 𝑢)𝑑 𝑢, ˜ 𝑧 ∈ 𝔇3′ 𝛿 ∩ {|𝜑 𝑘 (𝑧)| < 𝜌0 }. ℓ0 ×𝛤0

𝑘=1

With a su ciently small cylinder neighborhood 𝑉 (Stein) of ℓ0 × 𝛤0 ⊂ C𝑚+1 , we are going to de ne Cousin I data on 𝑉 × 𝔇1 covered by 𝑉 × 𝔇3 and 𝑉 × (𝔇1 ∩ {𝑥1 < 𝑎 2′ }) with 𝑎 2′ > 𝑎 2 chosen su ciently close to 𝑎 2 below. We rstly take 𝜒( 𝑢, ˜ 𝑧 1 , 𝑧 ′ ) ∈ ℳ(𝑉 × 𝔇3 ). Note that its polar set is a closed subset of 𝑉 × 𝔇3 , and has no intersection with the real hyperplanes 𝑥1 = 𝑎 𝜈 (𝜈 = 1, 2), and hence with 𝑥1 = 𝑎 2′ > 𝑎 2 close to 𝑎 2 . We secondly take 0 on 𝑉 × (𝔇1 ∩ {𝑥1 < 𝑎 2′ }), so that we obtain Cousin I data on 𝑉 × 𝔇1 . Since 𝑉 × 𝔇1 is Stein, the Cousin I Problem for the data has a solution by Theorem 3.7.9: ∃ 𝜒1 ( 𝑢, ˜ 𝑧 1 , 𝑧 ′ ) ∈ ℳ(𝑉 × 𝔇1 ), 𝜒 − 𝜒1 | 𝑉 ×𝔇3 ∈ 𝒪(𝑉 × 𝔇3 ). Similarly, we have ∃ 𝜒2 ( 𝑢, ˜ 𝑧 1 , 𝑧 ′ ) ∈ ℳ(𝑉 × 𝔇2 ),

𝜒 − 𝜒2 | 𝑉 ×𝔇3 ∈ 𝒪(𝑉 × 𝔇3 ).

184

5 Pseudoconvex Domains II

Solution

It is noticed that the de ning domain of the variable 𝑧 of each 𝜒𝜈 ( 𝑢, ˜ 𝑧) (𝜈 = 1, 2) is extended from 𝔇3 to 𝔇𝜈 , keeping the same poles as those of 𝜒( 𝑢, ˜ 𝑧). We extended the de ning domain of 𝜒 by virtue of Cousin I, but the substitution of 𝜒 by 𝜒𝜈 causes necessarily an error; to make the error small, we modify 𝜒𝜈 . ¯ ′ ⋐ 𝑉 ×𝔇3 . Note that 𝑉 ×𝔇3 is 𝒪(𝑉 ×𝔇𝜈 )-convex (𝜈 = 1, 2) and that (ℓ0 × 𝛤0 ) × 𝔇 3𝛿 By the Oka Weil Approximation Theorem 3.7.7 we have for every 𝜀 > 0: ∃ 𝛾 𝜈 ∈ 𝒪(𝑉 × 𝔇𝜈 ),

3𝛿

𝐾 𝜈 := 𝜒 − 𝜒𝜈 − 𝛾 𝜈 ∈ 𝒪(𝑉 × 𝔇𝜈 ), Set

∫

′

𝐼 𝜈 (𝐺) (𝑧 1 , 𝑧 ) =

(5.2.14)

ℓ0 ×𝛤0

∫ =

𝜀 , 2

∥ 𝜒 − 𝜒𝜈 − 𝛾 𝜈 ∥ (ℓ0 ×𝛤0 ) ×𝔇¯ ′
0 so that 𝜃 := 𝜀𝐶1 𝐶 < 1. Then, (viii) ∥ 𝑓 𝜇 ∥ 𝔇3′ 𝛿 ≤ 𝜃 𝜇 𝑀.


f𝛿 , (ix) By Lemma 5.1.16 (Oka Extension with Estimate), 𝑓 𝜇 ⇝ 𝐹𝜇 ∈ 𝒪 P𝛥 𝐹𝜇 | 𝔇3′ 𝛿 = 𝑓 𝜇 , and ∥𝐹𝜇 ∥ 𝐿1 ≤ 𝜃 𝜇 𝐶 𝑀. (x) 𝑔 := 𝑓0 + 𝑓1 + 𝑓2 + · · · ∈ 𝒪(𝔇3′ 𝛿 ): This converges uniformly on 𝔇3′ 𝛿 by a majorant convergent series, and is bounded. (xi) 𝐺 := 𝐹0 + 𝐹1 + 𝐹2 + · · · ∈ 𝒪(𝐿 1◦ ) with the interior 𝐿 1◦ of 𝐿 1 converges uniformly on 𝐿 1 by a majorant convergent series.

′ ◦ ′ (xii) 𝐺 (𝑧, 𝜑(𝑧)) = 𝑔(𝑧), 𝑧 ∈ 𝔇3(1) 𝛿1 := {𝑧 ∈ 𝔇3 𝛿 : (𝑧, 𝜑(𝑧)) ∈ 𝐿 1 } = {𝑧 ∈ 𝔇3 𝛿 : |𝑥 1 | < ′ 𝛿1 , |𝜑 𝑘 (𝑧)| < 𝜌1 , 1 ≤ 𝑘 ≤ 𝑚} (⋐ 𝔇3 𝛿 ); that is, 𝐺 is an extension of 𝑔 through ◦ the Oka map 𝔇3(1) 𝛿1 ∋ 𝑧 ↦→ (𝑧, 𝜑(𝑧)) ∈ 𝐿 1 , and

𝑔 = 𝑓0 + 𝐾 (𝐹0 ) + 𝐾 (𝐹1 ) + · · · = 𝐾 (𝐺) + 𝑓0

(on 𝔇3(1) 𝛿1 ).

  Claim 5.2.16. With the preparation above, we define 𝐼 𝜈 (𝐺) (𝑧) ∈ 𝒪 𝔇𝜈(0) (𝜈 = 1, 2) by (5.2.14). Then, 𝐼1 (𝐺) (𝑧) − 𝐼2 (𝐺)(𝑧) = 𝑔(𝑧) − 𝐾 (𝐺)(𝑧) = 𝑓 (𝑧),

𝑧 ∈ 𝔇1(0) ∩ 𝔇2(0) ∩ 𝔇3(1) 𝛿1 .

In fact, with 𝑓 𝜇 and 𝐹𝜇 (𝜇 = 0, 1, 2, . . .) constructed above inductively, we get 𝐼1 (𝐹𝜇 ) − 𝐼2 (𝐹𝜇 ) = 𝑓 𝜇 − 𝐾 (𝐹𝜇 ),

𝜇 ≥ 0.

Summing up each side with respect to 𝜇 = 0, 1, . . . , 𝑁, we deduce from 𝑓0 = 𝑓 and 𝑓 𝜇+1 = 𝐾 (𝐹𝜇 ) that

186

5 Pseudoconvex Domains II

Solution

𝑁 𝑁 𝑁 𝑁 ©Õ ª ©Õ ª Õ ©Õ ª 𝐼1 ­ 𝐹𝜇 ® − 𝐼2 ­ 𝐹𝜇 ® = 𝑓 𝜇 − 𝐾 ­ 𝐹𝜇 ® = 𝑓 − 𝐾 (𝐹𝑁 ). « 𝜇=0 ¬ « 𝜇=0 ¬ 𝜇=0 « 𝜇=0 ¬

Since 𝐾 (𝐹𝑁 )(𝑧) → 0 (𝑧 ∈ 𝔇3(1) 𝛿1 ) as 𝑁 → ∞, it follows that (5.2.17) 𝐼1 (𝐺) (𝑧) − 𝐼2 (𝐺)(𝑧) = 𝑔(𝑧) − 𝐾 (𝐺)(𝑧) = 𝑓 (𝑧),

𝑧 ∈ 𝔇1(0) ∩ 𝔇2(0) ∩ 𝔇3(1) 𝛿1 .

Thus integral equation (5.2.15) is solved in 𝑧 ∈ 𝔇1(0) ∩ 𝔇2(0) ∩ 𝔇3(1) 𝛿1 .

Here, since 𝐼 𝜈 (𝐺) (𝜈 = 1, 2) are holomorphic in 𝔇𝜈(0) , and 𝛿1 can be arbitrarily close to 𝛿, it follows from the Identity Theorem 1.1.46 that (5.2.17) holds in 𝔇1(0) ∩ 𝔇2(0) . Hence we obtain (5.2.12) with ℎ 𝜈 = 𝐼 𝜈 (𝐺) (𝜈 = 1, 2). ⊓ ⊔ Proof of Lemma 5.2.6. By the assumption we take a little bit larger strongly ˜ ⋑ 𝔇, ¯ and de ne 𝔇 ˜ 𝜈 (𝜈 = 1, 2, 3) as in (5.2.3), so that the pseudoconvex domains 𝔇   ˜ ˜ 3 . We then apply the closures 𝔇𝜈 (𝜈 = 1, 2) are Stein. Take an element 𝑓 ∈ 𝒪 𝔇 ˜ 𝜈 (𝜈 = 1, 2, 3). By Lemma 5.2.9 we may assume above arguments for the domains 𝔇   (0) ˜ ˜ 𝜈(0) that 𝔇𝜈 ⊃ 𝔇𝜈 (𝜈 = 1, 2). Therefore, by restricting the obtained 𝐼 𝜈 (𝐺) ∈ 𝒪 𝔇 (𝜈 = 1, 2) to 𝔇𝜈 , we get the required solutions 𝑓 𝜈 .

□

Proof of Oka’s Heftungslemma 5.2.4. (i) Similarly to the above we take double strongly pseudoconvex domains as follows: ˜ =𝔇 ˜ 1 ∪𝔇 ˜2⋑𝔇 ˜′=𝔇 ˜ ′ ∪𝔇 ˜ ′ ⋑ 𝔇 = 𝔇1 ∪ 𝔇2 , 𝔇 1 2 ˜ 𝜈 and 𝔇 ˜ ′𝜈 (𝜈 = 1, 2) are Stein. We would like to show 𝔇 ˜ ′ to be Stein. We are where 𝔇 ˜ ′ , and holomorphic going to construct a meromorphic function with a pole at 𝑞 ∈ 𝜕 𝔇 ˜ ′ \ {𝑞}. Assume 𝑞 ∈ 𝜕 𝔇 ˜ ′ . By a parallel transition we may assume 𝑞 ∉ 𝔇 ˜ ′ . By on 𝔇 1 2 ′ ˜ Lemma 4.3.70 (iii) there is a meromorphic function 𝑓 ∈ ℳ( 𝔇1 ) with polar locus ˜ ′ at 𝑞 (| 𝑓 (𝑞)| = ∞) and holomorphic on on the Oka hypersurface 𝛴0 (∋ 𝑞) of 𝜕 𝔇 ˜ ′ \ {𝑞}. By Lemma 5.2.6 there are functions 𝑓 𝜈 ∈ 𝒪( 𝔇 ˜ ′𝜈 ) (𝜈 = 1, 2) satisfying 𝔇 1 𝑓2 (𝑧) − 𝑓1 (𝑧) = 𝑓 (𝑧),

˜ ′ ∩𝔇 ˜ ′. 𝑧∈𝔇 1 2

˜ ′ with a pole at 𝑞 ∈ 𝜕 𝔇 ˜′ Then 𝐹 = 𝑓 + 𝑓1 = 𝑓2 is a meromorphic function on 𝔇 ˜ ′ \ {𝑞}. Therefore 𝔇 ˜ ′ is holomorphically convex. (|𝐹 (𝑞)| = ∞) and holomorphic on 𝔇 𝜋 ˜′→ (ii) To show the holomorphic separation we assume that 𝔇 C𝑛 is not univalent. ′ ˜ be distinct two points with 𝑧0 := 𝜋( 𝑝 1 ) = 𝜋( 𝑝 2 ). We consider a half Let 𝑝 1 , 𝑝 2 ∈ 𝔇 line ℓ in C𝑛 with initial point 𝑧0 and with any direction. Let ℓ𝜈 (𝜈 = 1, 2) be the ˜ ′ is bounded, ℓ𝜈 connected component of 𝜋 −1 ℓ containing 𝑝 𝜈 , respectively. Since 𝔇 ′ ˜ . We consider a moving point 𝑃 on ℓ starting from must intersect the boundary 𝜕 𝔇 𝑧 0 . Correspondingly, a moving point 𝑄 𝜈 with 𝜋(𝑄 𝜈 ) = 𝑃 on ℓ𝜈 is obtained for each ˜ ′ on the same time or 𝜈. Whichever will be ne, we suppose that 𝑄 1 reaches to 𝜕 𝔇

5.2 Strongly Pseudoconvex Domains

187

˜ ′ , and the corresponding point on earlier than 𝑄 2 . We denote the point 𝑞 1 ∈ ℓ1 ∩ 𝜕 𝔇 ˜ ′ with 𝜋(𝑞 1 ) = 𝜋(𝑞 2 ). Then, 𝑞 1 ≠ 𝑞 2 . By the same arguments as ℓ2 by 𝑞 2 ∈ ℓ2 ∩ 𝔇 ˜′ above 𝐹 was obtained, we obtain a meromorphic function 𝑔 with a pole at 𝑞 1 ∈ 𝜕 𝔇 ′ ˜ \ {𝑞 1 }. By the uniqueness of analytic continuation, (|𝑔(𝑞 1 )| = ∞), holomorphic on 𝔇 we have as convergent power series locally in 𝑧 − 𝑧 0 that (5.2.18)

𝑔

𝑝1

≠𝑔 . 𝑝2

˜ ′ → C𝑛 is holomorphically sepaTherefore it follows from Proposition 3.6.9 that 𝔇 rable.2 ˜ ′ is Stein. Since 𝔇 ˜ ′ is a Stein neighborhood of 𝔇, ¯ arbitrarily close to 𝔇, ¯ Now, 𝔇 and 𝜕𝔇 is a strongly pseudoconvex boundary, the Steinness of 𝔇 is deduced from Corollary 3.7.4. □ ¯ (c) Proof of Lemma 5.2.1. (c1) We rst partition 𝔇 by real hyperplanes parallel to the real and imaginary axes to small closed domains, each piece of which is a closed Stein domain. Let 𝜆 be the de ning strongly pseudoconvex function of the boundary 𝜕𝔇. For a point 𝑝 0 ∈ 𝜕𝔇 we take a Stein neighborhood 𝑉 (∋ 𝑝) contained in the open set where 𝜆 is de ned, so that 𝑉 ∩ 𝔇 is Stein by Theorem 4.3.74. ¯ are Stein. Similarly, we have that 𝑉¯ and 𝑉¯ ∩ 𝔇 (c2) With complex coordinates 𝑧 𝑗 = 𝑥2 𝑗 −1 +𝑖𝑥 2 𝑗 (1 ≤ 𝑗 ≤ 𝑛) in real and imaginary parts and a su ciently large 𝑁 ∈ N we have ¯ ⋐ {(𝑥 𝑘 ) : −𝑁 < 𝑥 𝑘 < 𝑁, 1 ≤ 𝑘 ≤ 2𝑛}. 𝜋( 𝔇) We set a closed cuboid 𝐸 0 = {(𝑥 𝑘 ) : −𝑁 ≤ 𝑥 𝑘 ≤ 𝑁, 1 ≤ 𝑘 ≤ 2𝑛}. For axes 𝑥 𝑘 we take partitions (5.2.19)

−𝑁 = 𝑐 𝑘0 < 𝑐 𝑘1 < · · · < 𝑐 𝑘 𝐿 = 𝑁

(1 ≤ 𝑘 ≤ 2𝑛)

and set 𝐸 ℎ1 ℎ2 ...ℎ2𝑛 = {(𝑥 𝑘 ) : 𝑐 𝑘ℎ𝑘 −1 ≤ 𝑥 𝑘 ≤ 𝑐 𝑘ℎ𝑘 , 1 ≤ 𝑘 ≤ 2𝑛} (1 ≤ ℎ1 , . . . , ℎ2𝑛 ≤ 𝐿). Taking the partitions (5.2.19) su ciently ne, we see by the arguments of (c1) −1 ¯ (𝑙) ¯ that every non-empty connected component 𝔇 ℎ1 ℎ2 ...ℎ2𝑛 of (𝜋 𝐸 ℎ1 ℎ2 ...ℎ2𝑛 ) ∩ 𝔇 with nitely many 𝑙 is univalent and Stein. Firstly with xed ℎ2 , ℎ3 , . . . , ℎ2𝑛 arbitrarily, we ′ ¯ (𝑙) ¯ (𝑙 ) run ℎ1 from 1 to 𝐿 − 1. We consider two adjacent 𝔇 1ℎ2 ...ℎ2𝑛 and 𝔇2ℎ2 ...ℎ2𝑛 sharing the edge 𝑥 1 = 𝑐 11 . By the de nition there is a pseudoconvex neighborhood 𝑈 of the union ′ ¯ (𝑙) ¯ (𝑙 ) 𝔇 1ℎ2 ...ℎ2𝑛 ∪ 𝔇2ℎ2 ...ℎ2𝑛 . It follows from Proposition 4.3.53 that there is a neighborhood ′ ¯ (𝑙) ¯ (𝑙 ) 𝑉 with strongly pseudoconvex boundary such that 𝔇 ∪𝔇 ⋐ 𝑉 ⋐ 𝑈. 1ℎ2 ...ℎ2𝑛

2ℎ2 ...ℎ2𝑛

2 The present proof is taken from Oka’s unpublished paper XI of 1943. In the published paper Oka IX (1953) it is shown by another proof. In Gunning Rossi [26] the proof relies on the coherence of a direct image sheaf by a nite morphism, which is rather sophisticated. The present proof of Oka XI of 1943 is not found in the literature to the best of the author’s knowledge, and hence is original even now.

188

5 Pseudoconvex Domains II

Solution

We now apply Oka’s Heftungslemma 5.2.4 to deduce that 𝑉 is Stein, and hence ′ ′ ¯ (𝑙) ¯ (𝑙 ) ¯ (𝑙) ¯ (𝑙 ) 𝔇 1ℎ2 ...ℎ2𝑛 ∪ 𝔇2ℎ2 ...ℎ2𝑛 is Stein. Next, we consider 𝔇1ℎ2 ...ℎ2𝑛 ∪ 𝔇2ℎ2 ...ℎ2𝑛 and the ′′ ¯ (𝑙 ) adjacent 𝔇 sharing the edge 𝑥 1 = 𝑐 12 . In the same way as above by applying 3ℎ2 ...ℎ2𝑛

¯ (𝑙) Oka’s Heftungslemma 5.2.4, we deduce that the merged closed domain 𝔇 1ℎ2 ...ℎ2𝑛 ∪ ′ ′′ (𝑙 ) (𝑙 ) ¯ ¯ 𝔇 ∪𝔇 is Stein; we proceed with this up to ℎ1 = 𝐿 − 1, and for all 2ℎ2 ...ℎ2𝑛

3ℎ2 ...ℎ2𝑛

¯ (𝑙) connected components 𝔇 1ℎ2 ...ℎ2𝑛 . Denote the merged connected components by (𝑙) ¯ 𝔇 ℎ2 ...ℎ2𝑛 . (Note that this is no longer univalent in general.) We do the same for the ¯ is Stein. next axis 𝑥2 , and then up to 𝑥2𝑛 . Thus we deduce that 𝔇 □ Remark 5.2.20.

(i) In general, an integral equation of the type ∫ 𝑔(𝑧) = 𝐾 (𝑧, 𝜁)𝑔(𝜁)𝑑𝜁 + 𝑓 (𝑧)

is called a Fredholm integral equation of the second kind, where 𝐾 (𝑧, 𝑤) (socalled kernal function) and 𝑓 (𝑧) are given and 𝑔(𝑧) is unknown. In (5.2.15) the integral part is given by 𝐾 (𝐺) with the Oka extension 𝐺 (𝑧) of 𝑔(𝑧). So we call (5.2.15) a Fredholm integral equation of the second kind type. More generally, one may consider ∫ 𝑔(𝑧) = 𝜆 𝐾 (𝑧, 𝜁)𝑔(𝜁)𝑑𝜁 + 𝑓 (𝑧) with parameter 𝜆(∈ C) (spectrum); the associated Fredholm integral equation of the rst kind is given by ∫ 𝑔(𝑧) = 𝜆 𝐾 (𝑧, 𝜁)𝑔(𝜁)𝑑𝜁 . Therefore one may consider: 𝑔(𝑧) = 𝜆𝐾 (𝐺) (𝑧) + 𝑓 (𝑧), 𝑔(𝑧) = 𝜆𝐾 (𝐺) (𝑧). Here, we deal with only the case 𝜆 = 1. (ii) This idea was rst described in K. Oka, Proc. Imperial Acad. Tokyo (1941) and then in Oka VI, Tôhoku Math. J. 49 (1942), in which he solved the Pseudoconvexity Problem for 2-dimensional univalent domains by making use of Weil’s integral formula. This idea of thinking in the reverse direction is non-trivial. The way of writing the present part is due to Oka’s unpublished paper XI (1943); in the published Oka IX (1953) the contents shown there are the same, but integral equation (5.2.15) is not written explicitly as an equation, probably because it was the third time for him to write down the part.

5.2 Strongly Pseudoconvex Domains

189

N.B. With the preparation above we are now close to the proof of Oka’s pseudoconvexity Theorem 5.3.1. For those readers who are learning the present subject for the rst time, it is recommended to move to Theorem 5.3.1 and to nish its proof or to read 5.3, and then proceed to the next section after grasping the whole gure.

5.2.2 Grauert’s Method (1) Topological Vector Spaces. In Grauert’s method it is necessary to prepare one step more in topological linear spaces. Let 𝐸 be a topological vector space with semi-norms ∥ · ∥ 𝑗 ( 𝑗 ∈ N) as in 5.1.1. We begin with the following elementary property. Proposition 5.2.21. Let 𝐸 be Baire and let 𝐹 ⊂ 𝐸 be a closed vector subspace. Then the quotient space 𝐸/𝐹 is Baire. Proof. Let 𝑞 : 𝐸 → 𝐸/𝐹 be the quotient homomorphism. By de nition 𝑞 is an open map. Therefore, if 𝐺 ⊂ 𝐸/𝐹 contains no interior point, so does 𝑞 −1 𝐺. Therefore, if 𝐺 𝜈 ⊂ 𝐸/𝐹 (𝜈 ∈ N) are countably many closed subsets without interior point, the Ð union 𝜈 ∈N 𝐺 𝜈 contains no interior point. ⊓ ⊔ Proposition 5.2.22. A finite-dimensional vector subspace of 𝐸 is closed. Proof. Let 𝐹 be a nite-dimensional vector subspace of 𝐸 with basis {𝑣 𝑗 } 𝑞𝑗=1 . We set 𝑣0 = 0 and denote by 𝐹ℎ (⊂ 𝐹) the vector subspace spanned by 𝑣0 , 𝑣1 , . . . , 𝑣ℎ (0 ≤ ℎ ≤ 𝑞). We use the induction on ℎ. By the Hausdor assumption 𝐹0 is closed. Suppose that 𝐹ℎ−1 (1 ≤ ℎ ≤ 𝑞) is closed. Taking an arbitrary element 𝑥 ∈ 𝐹ℎ \ 𝐹ℎ−1 , we set 𝑥 = 𝑦 + 𝛼𝑣ℎ ,

𝑦 ∈ 𝐹ℎ−1 , 𝛼 ∈ C \ {0},

where 𝛼 is uniquely determined. Since 𝑣ℎ ∉ 𝐹ℎ−1 , it follows from the induction hypothesis that there are 𝑁 ∈ N and 𝛿 > 0 satisfying (𝑣ℎ + {𝑢 ∈ 𝐸 : ∥𝑢∥ 𝑗 < 𝛿, 1 ≤ 𝑗 ≤ 𝑁 }) ∩ 𝐹ℎ−1 = ∅. Since 𝑣ℎ − (1/𝛼)𝑥 = −(1/𝛼)𝑦 ∈ 𝐹ℎ−1 , the above implies (1/𝛼)𝑥 ∉ {𝑢 ∈ 𝐸 : ∥𝑢∥ 𝑗 < 𝛿, 1 ≤ 𝑗 ≤ 𝑁 }, max ∥(1/𝛼)𝑥∥ 𝑗 ≥ 𝛿,

1≤ 𝑗 ≤ 𝑁

(5.2.23)

|𝛼| ≤

1 max ∥𝑥∥ 𝑗 , 𝛿 1≤ 𝑗 ≤ 𝑁

𝑥 ∈ 𝐹ℎ .

Note that the last equation holds trivially for 𝛼 = 0. Suppose that a sequence {𝑥 𝜈 }∞ 𝜈=1 of points of 𝐹ℎ converges to 𝑥 0 ∈ 𝐸. Write

190

5 Pseudoconvex Domains II

𝑥 𝜈 = 𝑦 𝜈 + 𝛼𝜈 𝑣ℎ ,

𝑦 𝜈 ∈ 𝐹ℎ−1 ,

Solution

𝜈 = 1, 2, . . . .

For every 𝜀 > 0 there is a number 𝑀 ∈ N such that max ∥𝑥 𝜈 − 𝑥 𝜇 ∥ < 𝜀𝛿,

1≤ 𝑗 ≤ 𝑁

𝜈, 𝜇 ≥ 𝑀.

From this and (5.2.23) it follows that |𝛼𝜈 − 𝛼 𝜇 | < 𝜀,

𝜈, 𝜇 ≥ 𝑀.

Therefore {𝛼𝜈 } forms a Cauchy sequence, and hence converges. With lim 𝛼𝜈 = 𝛼0 we have that lim 𝑦 𝜈 = 𝑦 0 = 𝑥 0 − 𝛼0 𝑣ℎ ; since 𝐹ℎ−1 is closed, 𝑦 0 ∈ 𝐹ℎ−1 . Thus 𝑥0 = 𝑦 0 + 𝛼0 𝑣ℎ ∈ 𝐹ℎ , and so 𝐹ℎ is closed. ⊓ ⊔ Definition 5.2.24. A linear continuous map 𝜓 : 𝐸 → 𝐹 between topological vector spaces is said to be completely continuous or a compact operator if there exists a neighborhood 𝑈 of the origin of 𝐸 with 𝜓(𝑈) ⋐ 𝐹 (i.e., the closure 𝜓(𝑈) is compact in 𝐹). Now we prove L. Schwartz’s Fredholm Theorem:3 Theorem 5.2.25 (L. Schwartz). Let 𝐸 be a Fréchet space and let 𝐹 be a Baire vector space. Let 𝜙 : 𝐸 → 𝐹 be a continuous linear surjection, and let 𝜓 : 𝐸 → 𝐹 be a completely continuous linear map. Then the image (𝜙 + 𝜓) (𝐸) is closed and the co-kernel Coker(𝜙 + 𝜓) := 𝐹/(𝜙 + 𝜓)(𝐸) is finite dimensional. Proof. 4 (a) Because of the length of the proof (2.5 pp.) we present a rough plan of it. We will imitate the proof of “a locally compact topological vector space being finite dimensional”. First by the assumption we take a neighborhood 𝑈 = −𝑈 of 0 ∈ 𝐸 with compact 𝐾 := 𝜓(𝑈). By the Open Map Theorem 5.1.9 𝑉 = 𝜙(𝑈) is an open neighborhood of 0 ∈ 𝐹. Since 𝐾 is compact,  there are nitely many points 𝑏 𝑗 ∈ 𝐾, Ð𝑙 1 1 ≤ 𝑗 ≤ 𝑙, such that 𝐾 ⊂ 𝑗=1 𝑏 𝑗 + 2 𝑉 . Let 𝐺 ⊂ 𝐹 denote the linear subspace
spanned by {𝑏 𝑗 }1≤ 𝑗 ≤𝑙 . Set 𝐹 → 𝐻 := 𝐹/ (𝜙 + 𝜓) (𝐸) + 𝐺 (assuming the quotient is de ned). Then 𝑉 has an image 𝑊, an open neighborhood of 0 ∈ 𝐻. It follows   𝜈

from the construction that 𝑊 ⊂ 12 𝑊. Hence, 𝑊 ⊂ 12 𝑊 (∀ 𝜈 ∈ N), and so 𝑊 = {0}; 𝐹/(𝜙 + 𝜓)(𝐸)  𝐺/(𝐺 ∩ (𝜙 + 𝜓) (𝐸)) is nite dimensional. But the closedness of (𝜙 + 𝜓)(𝐸) + 𝐺 is not proved, and hence the argument is incomplete. In fact, we will prove them simultaneously.

3 This naming is due to A. Huckleberry, Jahresber. Dtsch. Math.Ver. 115 (2013), 21 45). According to him, it is due to A. Andreotti. Since this counters the Fredholm integral equation of the second kind type (5.2.15) which Oka used, it is a quite appropriate term (see p. ix footnote 3)). 4 The idea of the following proof, largely shortened and simpli ed, is due to J.-P. Demailly (cf. [39] 7.3.4). The former proofs are rather long, twenty thirty pages (cf., e.g., Bers [5], Grauert Remmert [24] VI), and so it was often left out or relied on other sources on a key point (e.g., Hitotsumatsu [29] Chap. 12 2, Gunning Rossi [26] Appendix B, Fritzsche Grauert [18] 5.3, etc.).

5.2 Strongly Pseudoconvex Domains

191

(b) Set 𝜙0 = 𝜙 + 𝜓 : 𝐸 → 𝐹. It su ces to show the existence of a nite dimensional subspace 𝑆 ⊂ 𝐹 such that the composite 𝜙ˇ0 of 𝜙0 and the quotient map 𝐹 → 𝐹/𝑆 is surjective. For with the direct decomposition 𝑆 = 𝑆 ′ ⊕ (𝑆 ∩ 𝜙0 (𝐸)) 𝐹 = 𝜙0 (𝐸) ⊕ 𝑆 ′ algebraically. Note that 𝑆 ′ is nite dimensional and hence Fréchet, and that 𝐸/ker 𝜙0 is Hausdor . We consider the following continuous linear surjection and injective surjection: 𝜙e0 : 𝑥 ⊕ 𝑦 ∈ 𝐸 ⊕ 𝑆 ′ → 𝜙0 (𝑥) + 𝑦 ∈ 𝐹, b := (𝐸/Ker 𝜙0 ) ⊕ 𝑆 ′ → 𝜙0 (𝑥) + 𝑦 ∈ 𝐹. 𝜙b0 : [𝑥] ⊕ 𝑦 ∈ 𝐸 By Theorem 5.1.9 𝜙˜0 is an open map, and so is 𝜙b0 . Therefore 𝜙b0 is a linear homeomorb is closed, so is 𝜙b0 ((𝐸/Ker 𝜙0 ) ⊕ {0}) = 𝜙0 (𝐸). phism. Since (𝐸/Ker 𝜙0 ) ⊕ {0} (⊂ 𝐸) Because Coker 𝜙0 = 𝐹/𝜙0 (𝐸)  𝑆 ′ , Coker 𝜙0 is nite dimensional. We show the existence of 𝑆 above: By the assumption there is a convex neighborhood 𝑈0 of 0 ∈ 𝐸 such that −𝑈0 = 𝑈0 and 𝐾 := 𝜓(𝑈0 ) is compact. Since 𝜙 is surjective, 𝑉0 := 𝜙(𝑈  0 ) is open by Theorem 5.1.9. We consider an open covering Ð 𝐾 ⊂ 𝑏∈𝐾 𝑏 + 12 𝑉0 . Since 𝐾 is compact, there are nitely many points 𝑏 𝑗 ∈ 𝐾,  Ð  1 ≤ 𝑗 ≤ 𝑙, such that 𝐾 ⊂ 𝑙𝑗=1 𝑏 𝑗 + 12 𝑉0 . Let 𝑆 = ⟨𝑏 1 , . . . , 𝑏 𝑙 ⟩ denote the vector subspace of nite dimension spanned by 𝑏 𝑗 , 1 ≤ 𝑗 ≤ 𝑙. Then 𝑆 is closed by Proposition 5.2.22, so that the quotient space 𝐹/𝑆 is Hausdor and Baire by Proposition 5.2.21. Let 𝜋 : 𝐹 → 𝐹/𝑆 be the quotient map, and set 𝑉˜0 = 𝜋(𝑉0 ). Note that 𝐾˜ := 𝜋(𝐾) is compact and 𝐾˜ ⊂ 12 𝑉˜0 . Replacing 𝐹 with 𝐹/𝑆 we may assume from the beginning that 1 𝐾 ⊂ 𝑉0 ; 2 then we are going to show the surjectivity of 𝜙0 . (c) Since 𝜙0 (𝐸) is a linear subspace of 𝐹, the following claim implies 𝜙0 (𝐸) = 𝐹. Claim 5.2.26.

𝜙0 (𝐸) ⊃ 𝑉0 .

∵ ) Take an arbitrary point 𝑦 0 ∈ 𝑉0 . There is a point 𝑥0 ∈ 𝑈0 with 𝜙(𝑥0 ) = 𝑦 0 . Since   1 1 𝑦 1 = 𝑦 0 − 𝜙0 (𝑥0 ) = −𝜓(𝑥0 ) ∈ 𝐾 ⊂ 𝑉0 = 𝜙 𝑈0 , 2 2 there is a point 𝑥1 ∈ 12 𝑈0 with 𝜙(𝑥1 ) = 𝑦 1 , and then 

 1 1 𝑦 2 := 𝑦 1 − 𝜙0 (𝑥1 ) = −𝜓(𝑥 1 ) ∈ 𝜓 𝑈0 = 𝜓(𝑈0 ) 2 2   1 1 1 ⊂ 𝐾 ⊂ 2 𝑉0 = 𝜙 2 𝑈0 . 2 2 2 Hence there is a point 𝑥 2 ∈ 212 𝑈0 with 𝑦 2 = 𝜙(𝑥2 ). Inductively we choose 𝑥 𝜈 ∈ 𝑦 𝜈 = 𝜙(𝑥 𝜈 ), 𝜈 = 1, 2, . . ., so that

1 2𝜈 𝑈0 ,

192

5 Pseudoconvex Domains II



𝑦 𝜈+1 = 𝑦 𝜈 − 𝜙0 (𝑥 𝜈 ) ∈

Solution



1 1 𝐾 ⊂ 𝜙 𝜈+1 𝑈0 . 𝜈 2 2

For any seminorm ∥ · ∥ 𝑗 of 𝐹 there is an 𝑀 𝑗 > 0 with ∥𝑦∥ 𝑗 ≤ 𝑀 𝑗 (∀ 𝑦 ∈ 𝐾). It follows that 𝑀𝑗 ∥𝑦 𝜈 ∥ ≤ 𝜈−1 → 0, 𝜈 → ∞. 2 Thus, lim𝜈→∞ 𝑦 𝜈 = 0 and (5.2.27)

𝑦 𝜈+1 = 𝑦 𝜈 − 𝜙0 (𝑥 𝜈 ) = 𝑦 𝜈−1 − 𝜙0 (𝑥 𝜈−1 ) − 𝜙0 (𝑥 𝜈 )

𝜈 ©Õ ª = · · · = 𝑦 0 − 𝜙0 ­ 𝑥 𝑗 ® . « 𝑗=0 ¬ Í We would like to re-choose them so that ∞ 𝑗=0 𝑥 𝑗 converges. Let 𝑑 𝐸 (·, ·) be the complete distance de ned by (5.1.3) on 𝐸. Set 𝑈 (𝑟) = {𝑥 ∈ 𝐸 : 𝑑 (𝑥, 0) < 𝑟 } (𝑟 > 0). We take a fundamental neighborhood system {𝑈 𝑝 }∞ 𝑝=0 of 0 ∈ 𝐸 as follows:

(i) 𝑈0 is already taken, and may be assumed to satisfy 𝑈0 ⊂ 𝑈 (1). Moreover, 𝑈 𝑝 ⊂ 𝑈 (2− 𝑝 ), 𝑝 = 1, 2, . . .. (ii) Every 𝑈 𝑝 is convex and symmetric; i.e., −𝑈 𝑝 = 𝑈 𝑝 . (iii) 𝑈 𝑝+1 ⊂ 12 𝑈 𝑝 , 𝑝 = 0, 1, . . .. We consider an open covering of 𝐾,   ∞ ∞
1 ©©Ø ª 1 ª Ø 𝐾 ⊂ 𝜙 ­­ 2 𝜇 𝑈 𝑝 ® ∩ 𝑈0 ® = 𝜙 2 𝜇 𝑈 𝑝 ∩ 𝑈0 . 2 2 «« 𝜇=1 ¬ ¬ 𝜇=1 Then there is a number 𝑁 ( 𝑝)(≥ 1) such that   1  (5.2.28) 𝐾 ⊂ 𝜙 2 𝑁 ( 𝑝) 𝑈 𝑝 ∩ 𝑈0 . 2 We may assume that 𝑁 ( 𝑝) < 𝑁 ( 𝑝 + 1) (𝑝 = 1, 2, . . .). For 0 ≤ 𝜈 ≤ 𝑁 (1) we take 𝑥 𝜈 chosen above, and set 𝑥˜0 = 𝑥0 + · · · + 𝑥 𝑁 (1) . For 𝑁 ( 𝑝) < 𝜈 ≤ 𝑁 ( 𝑝 + 1) (𝑝 = 1, 2, . . .) we have by (5.2.28)   1  1 𝑁 ( 𝑝) −𝜈+1 𝐾⊂𝜙 2 𝑈 𝑝 ∩ 𝜈 𝑈0 . 2 2𝜈−1
1 Since 𝑦 𝜈 ∈ 2𝜈−1 𝐾, we take 𝑥 𝜈 ∈ 2 𝑁 ( 𝑝) −𝜈+1𝑈 𝑝 ∩ 21𝜈 𝑈0 with 𝜙(𝑥 𝜈 ) = 𝑦 𝜈 . Then, (5.2.29)

  1 1 𝑥˜ 𝑝 := 𝑥 𝑁 ( 𝑝)+1 + · · · + 𝑥 𝑁 ( 𝑝+1) ∈ 1 + + · · · + 𝑁 ( 𝑝+1) − 𝑁 ( 𝑝) −1 𝑈 𝑝 2 2

5.2 Strongly Pseudoconvex Domains

193

 ⊂ 2𝑈 𝑝 ⊂ 𝑈 𝑝−1 ⊂ 𝑈 (5.2.30)

𝑑 𝐸 ( 𝑥˜ 𝑝 , 0)
𝑞 > 𝑞 0 we have by (5.1.10) and (5.2.30) ! 𝑝 𝑞 𝑝 𝑝 Õ Õ Õ Õ 𝑑𝐸 𝑥˜ 𝜈 , 𝑥˜ 𝜈 ≤ 𝑑 𝐸 ( 𝑥˜ 𝜈 , 0) < 𝜈=0

𝜈=0

1 𝜈−1 2 𝜈=𝑞+1

𝜈=𝑞+1


0 such that 𝔇 ∩ 𝑇 = {𝜑 < 0},

𝜕𝔇 = {𝜑 = 0} ⊂ {−𝑐 < 𝜑 < 𝑐} ⋐ 𝑇 .

We extend 𝜑 to be −𝑐 on 𝔇\ {−𝑐 < 𝜑 < 0} as a continuous plurisubharmonic function ˜ := 𝔇 ∪ 𝑇. We set 𝑇𝑐 = {𝜑 < 𝑐}. on 𝔇 Step 1. For each point 𝑎 ∈ 𝜕𝔇 we take a univalent open ball neighborhood 𝑈 = B(𝑎; 𝛿) ⋐ 𝑇𝑐 . By Theorem 4.3.74, 𝑈 ∩ 𝔇 is Stein. We take the double neighborhoods 𝑉 = B(𝑎; 𝛿/2) ⋐ 𝑈 of 𝑎; 𝑉 ∩ 𝔇 is also Stein. Since 𝜕𝔇 is compact, we can cover it with nitely many such 𝑉𝑖 ⋐ 𝑈𝑖 : 𝜕𝔇 ⊂

(5.2.53)

𝑙 Ø 𝑖=1

Since 𝔇 \ borhoods: (5.2.54)

Ð𝑙

𝑖=1 𝑉𝑖

𝑉𝑖 ⋐

𝑙 Ø

𝑈𝑖

(𝑙 < ∞).

𝑖=1

is compact, it is covered by nitely many double open ball neigh-

𝑉𝑖 = 𝐵(𝑎 𝑖 ; 𝛿𝑖 /2) ⋐ 𝑈𝑖 = 𝐵(𝑎 𝑖 ; 𝛿𝑖 ) ⋐ 𝔇,

𝑖 = 𝑙 + 1, . . . , 𝐿.

Since 𝒱 = {𝔇 ∩ 𝑉𝑖 } and 𝒰 = {𝔇 ∩ 𝑈𝑖 } are Stein coverings of 𝔇, it follows from Lemma 5.2.49 that (5.2.55)

𝐻 1 (𝔇,𝒪𝔇 )  𝐻 1 (𝒱,𝒪𝔇 )  𝐻 1 (𝒰,𝒪𝔇 ).

Step 2. Take a 𝐶 ∞ function 𝑐 1 (𝑧) ≥ 0 such that Supp 𝑐 1 ⊂ 𝑈1 ,

𝑐 1 | 𝑉1 = 1.

5.2 Strongly Pseudoconvex Domains

199

Fig. 5.2 Boundary bumping method 1.

Fig. 5.3 Boundary bumping method 2.

With a su ciently small 𝜀 > 0, 𝜑 𝜀 (𝑧) := 𝜑(𝑧) − 𝜀𝑐 1 (𝑧) is strongly plurisubharmonic in 𝑇𝑐 . We set 𝑊1 = 𝑈1 ∩ {𝜑 𝜀 < 0}

(5.2.56)

(cf. Fig. 5.2). By Theorem 4.3.74 𝑊1 is Stein. With 𝜀 > 0 smaller if necessary we have 𝑉1 ∩ 𝔇 ⋐ 𝑊1 . Moreover, for other 𝑈 𝑗 having non-empty intersection with 𝑈1 , {𝑧 ∈ 𝑈 𝑗 : 𝜑 𝜀 (𝑧) < 0} is Stein (cf. Fig. 5.3). We set 𝑈1(1) = 𝑊1 , 𝒰 (1)

𝑈 𝑗(1) = 𝑈 𝑗 ∩ 𝔇, 𝑗 ≥ 2, 𝐿 Ø = {𝑈 𝑗(1) } 𝐿𝑗=1 , 𝔇 (1) = 𝑈 𝑗(1) . 𝑗=1

It follows that 𝒰 (1) is a Stein covering of 𝔇 (1) , and that for 𝑈 𝑗0 ,𝑈 𝑗1 ( 𝑗0 ≠ 𝑗 1 ) and 𝑈 𝑗(1) ,𝑈 𝑗(1) with the same pair of indices ( 𝑗 0 , 𝑗 1 ), 𝑈 𝑗0 ∩ 𝑈 𝑗1 ∩ 𝔇 = 𝑈 𝑗(1) ∩ 𝑈 𝑗(1) . 0 1 0 1 Therefore we obtain the following equality and the surjection induced from the

200

5 Pseudoconvex Domains II

Solution

Fig. 5.4 Boundary bumping method 3.

restrictions: 𝑍 1 (𝒰,𝒪𝔇 ) = 𝑍 1 (𝒰 (1) ,𝒪𝔇 (1) ),

(5.2.57)

𝐻 1 (𝔇 (1) ,𝒪𝔇 (1) )  𝐻 1 (𝒰 (1) ,𝒪𝔇 (1) ) → 𝐻 1 (𝒰,𝒪𝔇 )  𝐻 1 (𝔇,𝒪𝔇 ) → 0. Step 3. We change the covering of 𝔇 (1) as follows. Now, 𝑊1 is already given in (5.2.56). Set 𝑊 𝑗 = 𝔇 (1) ∩ 𝑈 𝑗 ,

𝑗 ≥ 2,

𝒲 = {𝑊 𝑗 } 𝐿𝑗=1 . Since all 𝑊 𝑗 are Stein, we get (5.2.58)

𝐻 1 (𝔇 (1) ,𝒪𝔇 (1) )  𝐻 1 (𝒲,𝒪𝔇 (1) ).

Ð Step 4. For 𝔇 (1) = 𝑗 𝑊 𝑗 and 𝑊2 we practice the procedures of Steps 2 and 3. Repeating this procedure 𝑙-times, we enlarge outward all 𝑈𝑖 ∩ 𝜕𝔇, 𝑖 = 1, 2, . . . , 𝑙, and denote the resulting covering of 𝜕𝔇 by 𝑈˜ 1 , 𝑈˜ 2 , , . . . , 𝑈˜ 𝑙 (cf. Fig. 5.4), and after the (𝑙 + 1)th we put without change, 𝑈˜ 𝑖 = 𝑈𝑖 ,

𝑙 + 1 ≤ 𝑖 ≤ 𝐿.

Now we set 𝐿 𝒰˜ = {𝑈˜ 𝑖 }𝑖=1 ,

˜ = 𝔇

From the construction and (5.2.57) we infer that

𝐿 Ø 𝑖=1

𝑈˜ 𝑖 .

5.2 Strongly Pseudoconvex Domains

(5.2.59)

201

𝑉𝑖 ∩ 𝔇 ⋐ 𝑈˜ 𝑖 , 1 ≤ 𝑖 ≤ 𝐿, 1 ˜ 𝜌˜ : 𝐻 ( 𝒰,𝒪 ˜ ) → 𝐻 1 (𝒱,𝒪𝔇 ) → 0. 𝔇

Here, 𝜌˜ is the homomorphism naturally induced from the restrictions. Therefore we obtain the following surjective homomorphism: (5.2.60)

˜ ˜ ) ⊕ 𝐶 0 (𝒱,𝒪𝔇 ) → 𝜌(𝜉) + 𝛿𝜂 ∈ 𝑍 1 (𝒱,𝒪𝔇 ) → 0, 𝛹 : 𝜉 ⊕ 𝜂 ∈ 𝑍 1 ( 𝒰,𝒪 𝔇

˜ ˜ ) → 𝑍 1 (𝒱,𝒪𝔇 ) denotes the homomorphism induced from the where 𝜌 : 𝑍 1 ( 𝒰,𝒪 𝔇 restrictions from 𝑈˜ 𝛼 ∩ 𝑈˜ 𝛽 to 𝑉 𝛼 ∩𝑉𝛽 ∩ 𝔇, and by de nition 𝐻 1 (𝒱,𝒪𝔇˜ ) = 𝑍 1 (𝒱,𝒪𝔇˜ )/𝛿𝐶 0 (𝒱,𝒪𝔇˜ ). Since 𝑉 𝛼 ∩ 𝑉𝛽 ∩ 𝔇 ⋐ 𝑈˜ 𝛼 ∩ 𝑈˜ 𝛽 , 𝜌 is completely continuous (see Example 5.2.31). It follows from Schwartz’s Fredholm Theorem 5.2.25 that Coker(𝛹 − 𝜌) = 𝑍 1 (𝒱,𝒪𝔇 )/𝛿𝐶 0 (𝒱,𝒪𝔇 ) = 𝐻 1 (𝒱,𝒪𝔇 ) is nite dimensional. We see by (5.2.55) that dim 𝐻 1 (𝔇,𝒪𝔇 ) < ∞.

⊓ ⊔

The above method of the proof is called the bumping method of Grauert, and is e ective even for so-called complex spaces with singularities (cf. [39] Chap. 8). (4) Proof of Lemma 5.2.1. (a) (Holomorphic convexity) Let 𝜑 be a plurisubharmonic function taken in 5.2.52 such that 𝔇 = {𝜑 < 0} and 𝜑 is strongly plurisubharmonic in a neighborhood of 𝜕𝔇. For a boundary point 𝑝 0 ∈ 𝜕𝔇 we take the Levi polynomial 𝑃(𝑧) of 𝜑 at 𝑝 0 , de ned by (4.3.67). Restricting (4.3.69) to 𝑡 = 0, we have for a su ciently small 𝑟 > 0 Cousin I data with 𝑈 ′ = 𝔇𝑟 = {𝜑 < 𝑟 } for 1 𝑈 ′ in the argument of (4.3.67) and 𝑔 = 𝑃 (𝑧) in a neighborhood of 𝑝 0 such that 1 |𝑔( 𝑝 0 )| = ∞. With 𝑔 𝑘 = 𝑃 (𝑧) 𝑘 (𝑘 = 1, 2, . . .) in the neighborhood of 𝑝 0 and 0 on 𝔇𝑟 \ {𝑃 = 0} we obtain Cousin I data on 𝔇𝑟 , which induce 1st cohomology classes 𝑓 𝑘 ∈ 𝐻 1 (𝔇𝑟 ,𝒪𝔇𝑟 ) (𝑘 = 1, 2, . . .). Since the vector space 𝐻 1 (𝔇𝑟 ,𝒪𝔇𝑟 ) is nite dimensional by Grauert’s Theorem 5.2.50, there are nitely many 𝑐 1 , 𝑐 2 , . . . , 𝑐 𝑘 ∈ C (𝑐 𝑘 ≠ 0) such that (5.2.61)

𝑐 1 𝑓1 + 𝑐 2 𝑓2 + · · · + 𝑐 𝑘 𝑓 𝑘 = 0 ∈ 𝐻 1 (𝔇𝑟 ,𝒪𝔇𝑟 ).

This means that there is a meromorphic function 𝐹 (𝑧) in 𝔇𝑟 such that (5.2.62)

𝐹 (𝑧) −

𝑘 Õ 𝑐𝑗 𝑃(𝑧) 𝑗 𝑗=1

is holomorphic in a neighborhood of 𝑝 0 and 𝐹 (𝑧) has no other poles. Since 𝑐 𝑘 ≠ 0, ¯ \ {𝑝 0 }). There is such a function 𝐹 (𝑧) for every point of |𝐹 ( 𝑝 0 )| = ∞ and 𝐹 ∈ 𝒪( 𝔇 𝜕𝔇, and hence 𝔇 is holomorphically convex.

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(b) (Holomorphic separation) Let 𝑝 1 , 𝑝 2 ∈ 𝔇 be points such that 𝑝 1 ≠ 𝑝 2 and 𝜋( 𝑝 1 ) = 𝜋( 𝑝 2 ). We use arguments similar to the proof (b) of Oka’s Heftungslemma 5.2.4, and use the same notation as there. As in (5.2.61) and (5.2.62), we de ne Cousin I data such that in a neighborhood of 𝑞 1 ∈ 𝜕𝔇 poles are given by 𝑘 Õ 𝑐𝑗 , 𝑗 𝑃(𝑧) 𝑗=1

𝑐 𝑘 ≠ 0,

and in a neighborhood of another 𝑞 2 it has no pole (holomorphic). Solving the Cousin I Problem, we have a meromorphic function 𝑔 on 𝔇𝑟 . It follows that 𝑔 ∈ 𝒪(𝔇) and as in (5.2.18), 𝑔 ≠ 𝑔 by the uniqueness of analytic continuation; Proposition 3.6.9 𝑝1 𝑝2 implies that 𝔇 is holomorphically separated. Thus, 𝔇 is Stein. □

5.3 Oka’s Pseudoconvexity Theorem By making use of Lemma 5.2.1 proved by two methods due to Oka and Grauert we describe the solution of the nal goal of the Pseudoconvexity Problem. Theorem 5.3.1 (Oka 1943/’53). A pseudoconvex domain over C𝑛 is Stein. Proof. Let 𝜋 : 𝔇 → C𝑛 be a pseudoconvex domain with a pseudoconvex exhaustion function 𝜑 : 𝔇 → [−∞, ∞). We x a point 𝑝 0 ∈ 𝔇. Taking 𝜑( 𝑝 0 ) < 𝑐 1 < 𝑐 2 < 𝑐 3 (∈ R) arbitrarily, we denote by 𝔇𝜈 the connected components of {𝜑 < 𝑐 𝜈 } (⋐ 𝔇) containing 𝑝 0 (𝜈 = 1, 2, 3). With a su ciently small 𝜀 > 0 we de ne the smoothing 𝜑 𝜀 ( 𝑝) of 𝜑 ¯ 3 . Since ∥𝜋( 𝑝)∥ 2 is strongly plurisubharmonic, de ned on 𝔇 𝜑˜ 𝜀 ( 𝑝) = 𝜑 𝜀 ( 𝑝) + 𝜀∥𝜋( 𝑝)∥ 2 ,

¯3 𝑝∈𝔇

¯ 2 ⋐ 𝔇3 . is strongly plurisubharmonic. Here 𝑐 3 is chosen larger if necessary, so that 𝔇 For any neighborhood 𝑊 ⊃ 𝜕𝔇3 there is a su ciently small 𝜀 > 0 such that 𝔇2 ⋐ 𝔇3′ := {𝑝 ∈ 𝔇3 : 𝜑˜ 𝜀 ( 𝑝) < 𝑐 3 }. By Lemma 5.2.1, 𝔇3′ is Stein. Since 𝜑 is a plurisubharmonic function on 𝔇3′ and 𝔇𝜈 (𝜈 = 1, 2) are connected components of the sublevel sets, Theorem 4.3.74 implies that they are Stein and moreover 𝔇1 is 𝒪(𝔇2 )-convex. Therefore, (𝔇1 , 𝔇2 ) is a Runge pair. Taking a divergent sequence 𝑐 𝜈 ↗ ∞ (𝜈 = 1, 2, . . . , 𝑐 1 > 𝜑( 𝑝 0 )), we set 𝔇𝜈 to be Ð the connected components of {𝜑 < 𝑐 𝜈 } containing 𝑝 0 . Then 𝔇 = ∞ 𝜈=1 𝔇 𝜈 and it is inferred from Theorem 3.7.8 that 𝔇 is Stein. Thus the proof of Oka’s Pseudoconvexity Theorem is completed. ⊓ ⊔ We summarize the results obtained above:

5.3 Oka’s Pseudoconvexity Theorem

203

Theorem 5.3.2. Let 𝔇 (/C𝑛 ) be a domain and let 𝛿P𝛥 ( 𝑝, 𝜕𝔇) be the boundary distance function. Then the following seven conditions are equivalent: (i) 𝔇 is Stein. (ii) 𝔇 is holomorphically convex. (iii) 𝔇 is 𝐶 0 -pseudoconvex. (iv) 𝔇 is pseudoconvex. (v) − log 𝛿P𝛥 ( 𝑝, 𝜕𝔇) is plurisubharmonic. (vi) 𝔇 is Hartogs pseudoconvex.5 (vii) 𝔇 is a domain of holomorphy. The proof is a corollary of what has already been proved, but it is stated since the content has its own interest; the proof is left to the readers (see Exercise 8 at the end of the chapter). Remark 5.3.3. The plurisubharmonicity of − log 𝛿P𝛥 ( 𝑝, 𝜕𝔇) of condition (i) of Theorem 5.3.2 is su cient only “near the boundary of 𝔇”; e.g., it is su cient to assume that − log 𝛿P𝛥 ( 𝑝, 𝜕𝔇) is plurisubharmonic in {𝑝 ∈ 𝔇 : 𝛿P𝛥 ( 𝑝, 𝜕𝔇) < 𝑐} with a small 𝑐 > 0. Remark 5.3.4 (Singular spaces). It is an interesting and signi cant problem to ask what happens for domains with singularities. Because of the introductory nature of the present book we are not going into the details of the case of singular spaces, but just give an overview how the results obtained above are extended to singular spaces and leave the details to the references. A so-called complex space with allowing singularities is de ned (cf., e.g., Noguchi [39] 6.9, Chap. 8). It is possible to de ne plurisubharmonic functions and strongly plurisubharmonic functions on a complex space (cf., e.g., Vâjâitu [58] and the references there). There are the generalizations of Oka’s method to singular complex spaces by Nishino [36], [37] Chap. 9, and Andreotti Narasimhan [3]; as for Grauert’s method, cf. Narasimhan [34]. By V. Vâjâitu [58] Lemma 1 we have the following general result: Theorem 5.3.5. If a complex space 𝑋 satisfies the following two conditions, then 𝑋 is Stein. (i) 𝑋 is pseudoconvex. (ii) Every relatively open subset of 𝑋 admits a strongly plurisubharmonic function. Remark 5.3.6 (Generalizations of 𝜋 : 𝔇 → C𝑛 ). Docquire Grauert [14] in 1960 generalized Theorem 5.3.1 to an abstract unrami ed domain 𝜛 : 𝑋 → 𝑌 such that 𝑋 and 𝑌 are complex manifolds and 𝜛 is a locally biholomorphic map. They proved that if every point 𝑝 ∈ 𝑌 has a Stein neighborhood 𝑉 with the Stein inverse 𝜛 −1𝑉 (locally Stein) and if 𝑌 is Stein, then 𝑋 is Stein. The result was further generalized to the case of singular complex spaces by Vâjâitu [59] in 2008 (cf. also the references there). 5 See Chap. 4, Exercise 13.

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There is a generalization of Theorem 5.3.1 to another direction by R. Fujita [19] in 1963 and by A. Takeuchi [57] in 1964 for the compacti cation of C𝑛 by 𝑛-dimensional complex projective space P𝑛 (C): Theorem 5.3.7 (Fujita, Takeuchi). If 𝜛 : 𝑋 → P𝑛 (C) is a locally biholomorphic map and locally Stein (as above), then 𝑋 is Stein. Remark 5.3.8 (Rami ed domains over C𝑛 ). We say that a complex space 𝑋 is weakly holomorphically separable6 if the following condition is satis ed: For every point 𝑎 ∈ 𝑋 there are nitely many holomorphic functions 𝑓 𝑘 (1 ≤ 𝑘 ≤ 𝑙) on 𝑋 vanishing at 𝑎 such that 𝑎 is an isolated point of the analytic subset {𝑥 ∈ 𝑋 : 𝑓 𝑘 (𝑥) = 0, 1 ≤ 𝑘 ≤ 𝑙}. Andreotti Narasimhan [3] p. 356 proved that: Lemma 5.3.9. A weakly holomorphically separable complex space 𝑋 satisfies condition (ii) of Theorem 5.3.5. We consider the case where 𝜋 : 𝑋 → C𝑛 is a rami ed domain. (cf. Remark 3.6.3); naturally, such 𝑋 is weakly holomorphically separable. Therefore in this case, the Pseudoconvexity Problem is valid if there is a pseudoconvex exhaustion function on 𝑋 (cf. [3] Theorem 3 for 𝐶 0 -pseudoconvex case): Theorem 5.3.10. If there is a pseudoconvex exhaustion function on a ramified domain 𝜋 : 𝑋 → C𝑛 , then 𝑋 is Stein. Now the Pseudoconvexity Problem for rami ed domains is stated as follows: Problem 5.3.11 (Pseudoconvexity Problem for Rami ed Domains). Let 𝑋 be a complex space and let 𝜋 : 𝑋 → C𝑛 be a rami ed domain. If every point 𝑧 ∈ C𝑛 carries a neighborhood 𝑈 with the Stein inverse 𝜋 −1𝑈, then is 𝑋 Stein? The Pseudoconvexity Problem for Rami ed Domains 5.3.11 is, however, countered by an example of J.E. Fornæss [16] with a two-sheeted rami ed domain over C2 , so that the problem cannot be true in its original form (cf. Noguchi [41] for an a rmative result). Subsequently, the problem is reduced to asking: Problem 5.3.12. What separates the validity of Theorem 5.3.10 and the non-validity of Problem 5.3.11? Note. The Pseudoconvexity Problem is often referred to as Levi’s Problem in a large sense; e.g., cf. I. Lieb [33].7 As written in Oka IX (Jpn. J. Math. 23 (1953), p. 138; [48], p. 211) Oka IX was published as an intermediate report and was a modi ed version of of the unpublished papers VII XI of 1943 (in Japanese, [50] Posthumous Papers Vol. 1) which gave the rst complete solution of the Pseudoconvexity Problem for unrami ed domains over C𝑛 with 𝑛 ≥ 2 general: The papers 6 Cf. “Weak Sepration Axiom” in Grauert Remmert [24] Introduction. 7 This is a ne-detailed survey on Levi’s Problem, beginning with the origin of the problem, by which we understand the state of the art of the study in Europe then. It devotes much space in detail to the works of K. Oka; however, in the journal names of the referred papers, the use of the abbreviation “Jap.” should be avoided and replaced with “Jpn.” or the full-spelling “Japanese”.

5.3 Oka’s Pseudoconvexity Theorem

205

were modi ed by means of “coherence” or “idéal de domaines indéterminés”, which were invented in order to solve the problem for domains with ramifications (Oka VII written 1948, VIII 1951). He titled a section at p. 138 of Oka IX 1953 as B. Problème inverse de Hartogs.—Point de départ. The present book began with Hartogs’ phenomenon in Chap. 1. Theorem 5.3.2 (vi) is just a statement, returning there. In Levi’s Problem 4.3.65 a regularity (di erentiability) of the boundary is assumed, but there is no such assumption in Theorem 5.3.2: Here one sees that K. Oka investigated the problem by going back to the origin. While the Pseudoconvexity Problem was clearly grasped as a boundary problem by assuming the boundary regularity, it is interesting to learn that the problem was solved by introducing a new class of pseudoconvex or plurisubharmonic functions without regularity. Oka’s method directly proves the solvability of the Continuous Cousin Problem on a strongly pseudoconvex domain, that is, 𝐻 1 (𝔇,𝒪𝔇 ) = 0. The key Lemma 5.2.4 is called “Oka’s Heftungslemma” (German), which is due to Andreotti-Narasimhan [3]; this paper is in English and Oka’s in French. One wonders why a German term was used. Grauert’s Theorem 5.2.50 asserts the nite dimensionality, relaxing the claim. Although the claim is relaxed, it holds as far as the boundary is strongly pseudoconvex even in a complex space, and hence has a number of applications; for instance, Kodaira’s Embedding Theorem in a generalized form for singular complex spaces is deuced from it (cf., e.g, [39] Chap. 8). On the other hand, Oka’s method is to solve an integral equation for given Cousin I data by means of iterative approximations; it is direct and constructive. It is also possible to generalize the proof for singular complex spaces (cf. Nishino [36], [37] Chap. 9, Andreotti-Narasimhan [3]). H. Grauert wrote the following comments in his “Selected Papers” (Vol. I pp. 155 156, Springer 1994) together with C.L. Siegel’s comments: Oka’s methods are very complicated. At rst he proved (rather simply) that in any unbranched pseudoconvex domain 𝑋 there is a continuous strictly plurisubharmonic function 𝑝 ( 𝑥 ) which converges to +∞ as 𝑥 goes to the (ideal) boundary of 𝑋.8 Then he got the existence of holomorphic functions 𝑓 from this property. In [19]9 the existence of the 𝑓 comes from a theorem of L. Schwartz in functional analysis (topological vector spaces, see: H. Cartan, Séminaire E.N.S. 1953/54, Exposés XVI and XVII). The approach is much simpler, but my predecessor in Göttingen C.L. Siegel nevertheless did not like it: Oka’s method is constructive and this one is not!

If the proof of L. Schwartz’s Theorem 5.2.2510 is included in Grauert’s method, it might be not so convincing to say “much simpler”. It is rather interesting to see a 8 Theorem 4.3.15 in the present book. 9 [22] at the end of the present book. 10 Now it is not so long as seen in the present book, but it was originally more than twenty pages or so.

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strong similarity in the arguments of the two methods; they are the part that solves the Fredholm integral equation of the second kind type (5.2.15) in Oka’s method, and in Grauert’s, the part that obtains a convergent series in the proof of L. Schwartz’s Fredholm Theorem 5.2.25.11 The present author does not know how they are really related mathematically.

Exercises 1. Let 𝑋 be a Hausdor topological space, and let 𝐺 1 , 𝐺 2 (⊂ 𝑋) be closed sets without interior points. Show that 𝐺 1 ∪ 𝐺 2 does not contain an interior point. 2. Show 5.1.6 (i), (ii). 3. Let the notation be as in Example 5.1.14 (1). For 𝑓 ∈ 𝒞 0 (𝔇) we de ne semi-norms ∥ 𝑓 ∥ 𝔇¯ 𝑗 = max𝔇¯ 𝑗 | 𝑓 | ( 𝑗 ∈ N), and endow 𝒞 0 (𝔇) with the topology de ned by ∥ 𝑓 ∥ 𝑗 ( 𝑗 ∈ N). Show that 𝒞 0 (𝔇) is Fréchet. 4. Let 𝒰 = {𝑈 𝛼 } 𝛼∈𝛤 be an open covering of a domain 𝔇(/C𝑛 ). Show that for every compact subset 𝐾 ⋐ 𝔇 the cardinality of {𝛼 ∈ 𝛤 : 𝑈 𝛼 ∩ 𝐾 ≠ ∅} is nite if and only if 𝒰 is locally nite. 5. Let 𝛿 : ( 𝑓 𝛼 ) ∈ 𝐶 0 (𝒰,𝒪) → 𝛿( 𝑓 𝛼 ) ∈ 𝐶 1 (𝒰,𝒪) be de ned as (5.2.34). Show that 𝛿( 𝑓 𝛼 ) = 0 if and only if there exists an element 𝑓 ∈ 𝒪(𝔇) with 𝑓 𝛼 = 𝑓 |𝑈𝛼 (∀𝛼 ∈ 𝛤). 6. Prove (5.2.39). 7. Let 𝔇(/C𝑛 ) be a domain and let 𝜑 : 𝔇 → R be an upper semi-continuous exhaustion function. Fix a point 𝑝 0 ∈ 𝔇. With an arbitrarily given monotone increasing sequence, 𝜑( 𝑝 0 ) < 𝑐 1 < 𝑐 𝜈 ↗ ∞ (𝜈 = 2, 3, . . .), we denote by 𝔇𝜈 Ð the connected component of {𝜑 < 𝑐 𝜈 } containing 𝑝 0 . Prove 𝔇 = ∞ 𝔇 𝜈=1 𝜈 . 8. Give a proof of Theorem 5.3.2. 9. (Behnke Stein Theorem) Let 𝔇/C𝑛 be a domain and let 𝔇𝜈 ⊂ 𝔇 (𝜈 = 1, 2, . . .) be a sequence of subdomains such that all 𝔇𝜈 are Stein, 𝔇𝜈 ⊂ 𝔇𝜈+1 , and Ð 𝜈 𝔇 𝜈 = 𝔇. Then, prove that 𝔇 is Stein. Hint: Use Oka’s Theorem of Boundary Distance Functions. 10. Let 𝔇/C𝑛 be a Stein domain, and let ℱ ⊂ 𝒪(𝔇) be an in nite family. We de ne that 𝑧 ∈ 𝔇 belongs 𝔇(ℱ) if there are a neighborhood 𝑈 ∋ 𝑧 and 𝑀 > 0 such that | 𝑓 (𝑧)| ≤ 𝑀, ∀ 𝑧 ∈ 𝑈, ∀ 𝑓 ∈ ℱ. Suppose that 𝔇(ℱ) ≠ ∅. Then, prove that 𝔇(ℱ) is Stein as follows: a. For 𝑧 ∈ 𝔇 we set 𝜑(𝑧) = sup 𝑓 ∈ℱ | 𝑓 (𝑧)| ≤ ∞, and furthermore 𝜑(𝑧) ˜ = lim 𝜑( 𝑝), 𝑝→𝑧

11 For the wording, see p. ix, Footnote 3).

𝛺 = {𝑧 ∈ 𝔇 : 𝜑(𝑧) ˜ < ∞}.

Exercises

207

Show that 𝛺 = 𝔇(ℱ). b. Show that 𝜑˜ : 𝔇(ℱ) → R is a plurisubharmonic function. c. Let 𝜓 : 𝔇 → [−∞, ∞) be a pseudoconvex exhaustion function, and set 𝛷(𝑧) = max{ 𝜑(𝑧), ˜ 𝜓(𝑧)} ∈ R,

𝑧 ∈ 𝔇(ℱ).

Show that 𝛷(𝑧) is an exhaustion function on 𝔇(ℱ).12 11. A family ℱ of holomorphic functions (or maps into C𝑛 ) on a domain 𝛺 is called a normal family if every sequence of ℱ admits a locally uniformly convergent subsequence. 𝜈 z }| { 𝑛 𝑛 𝜈 Let 𝑓 : C → C be a holomorphic map, and let 𝑓 := 𝑓 ◦ · · · ◦ 𝑓 denote the 𝜈th iterates. Set ℱ = { 𝑓 𝜈 : 𝜈 ∈ N}. ℱ is called a complex dynamical system on C𝑛 . We denote by 𝐹 ( 𝑓 ), called a Fatou set, the set of all points 𝑧 ∈ C𝑛 having a neighborhood 𝑈 such that the restriction ℱ|𝑈 = { 𝑓 𝜈 |𝑈 : 𝜈 ∈ N} is a normal family. Then, prove that 𝐹 ( 𝑓 ) is Stein. 12. Extend Corollary 4.3.81 for unrami ed domains over C𝑛 .

12 In general, 𝜑˜ (𝑧) is upper semi-continuous, but is not assumed to be continuous, so that if the pseudoconvex exhaustion function is restricted to being continuous, 𝔇(ℱ) being Stein does not follow.

Afterword — Historical Comments

The Pseudoconvexity Problem has an origin in a study of the shape of singularities of holomorphic or meromorphic functions in domains of several variables. According to K. Oka’s lecture [51], K. Weierstrass thought that the shape of singularities is arbitrary, and hence the study in that direction had been delayed considerably. Later, however, into the 1900s, due to the studies of E. Fabry, W.G. Osgood, F. Hartogs, E.E. Levi, etc., it was found that the singularities have a speci c feature; it is the notion of “pseudoconvexity”. Behnke and Thullen published a monograph [4] in 1934 that summarized the current results and problems. K. Oka read it, was attracted to the Three Big Problems described there, and began to concentrate on the study of them, discontinuing his research on the composition problem of holomorphic functions and some works related to complex dynamics. These Three Big Problems were then not thought to be solvable. The chronicle of the a rmative solutions of the problems is roughly as follows (cf. [42]): (i) The 1st Problem (Approximation) and the 2nd Problem (Cousin I, II): The case of univalent domains of general dimension by Oka I (1936) III (1939) ([48], [49]). (ii) The 3rd Problem (Pseudoconvexity): The case of univalent domains of dimension 𝑛 = 2 by Oka [47] (1941) (announcement), Oka VI (1942) ([48], [49]). (iii) The 3rd Problem (Pseudoconvexity) together with the 1st and 2nd Problems: The case of unrami ed domains over C𝑛 of general dimension 𝑛 ≥ 2 by Oka’s unpublished papers VII XI of 1943 ([50], [44]). In VII (1943) Oka formulated and proved a kind of “primitive coherence” which is su cient for the proofs. (iv) The 3rd Problem (Pseudoconvexity): The case of univalent domains of general dimension 𝑛 ≥ 2 by S. Hitotsumatsu [28] (1949).1 (v) The 3rd Problem (Pseudoconvexity): The case of unrami ed domains over C𝑛 of general dimension 𝑛 ≥ 2 by Oka IX (1953) ([48], [49]). 1 Probably because the paper is written in Japanese, it is rarely referred to, unfortunately.

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 J. Noguchi, Basic Oka Theory in Several Complex Variables, Universitext, https://doi.org/10.1007/978-981-97-2056-9

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Afterword

Historical Comments

(vi) The 3rd Problem (Pseudoconvexity): The case of univalent domains of general dimension 𝑛 ≥ 2 by Bremermann [8] (1954) and by Norguet [45] (1954), independently. Their method relies on Weil’s integral formula, which was developed in (ii) above and is the same as Hitotsumatsu’s in (iv) above. (vii) The 3rd Problem (Pseudoconvexity): Another proof of (v) above by Grauert [22] (1958). Just after Oka wrote VII (1948/50), the cohomology theory due to Cartan Serre ¯ for coherent sheaves was initiated, and then after Oka IX (1953) the theory of 𝜕equation as a part of elliptic partial di erential equations due to Morrey, Kohn, Hörmander, ... , developed; new theories and new analytic methods were introduced (cf., e.g., Hörmander [30], Henkin Leiter [27], Range [53], Laurent-Thiébaut [32], Ohsawa [46], etc.). After solving a rmatively the Three Big Problems in (iii) above without publishing them, Oka continued the study in order to establish them for rami ed domains over C𝑛 which may have singularities, and invented the notion of “coherence” or “idéal de domaines indéterminés” in his words (Oka’s Three Coherence Theorems proved by Oka VII (1948/50) and VIII (1951); as for the 2nd Coherence Theorem, H. Cartan (1950) gave his own proof; cf. [39] Chap. 9). From the viewpoint of analytic continuation of analytic functions rami cation points are naturally introduced, and so there is a de nite necessity to deal with rami cations and singularities in domains. Oka’s solution of the Pseudoconvexity Problem in 1943 was published ten years later as Oka IX (1953), where the proofs were modi ed in a form to use the 1st and 2nd Coherence Theorems of Oka VII and VIII. Oka himself wrote that Oka IX is an intermediate report towards the nal goal. Oka’s Three Coherence Theorems developed into the theory of Oka Cartan Serre Grauert · · · , giving a broad in uence in mathematics, and changing the description style of complex analysis and related elds after the 1950s; the in uence has reached even to theoretical physics as mentioned in M. Gonokami 2 [21]. As mentioned after Problem 5.3.11, the Pseudoconvexity Problem for rami ed domains was countered by J.E. Fornæss [16]. This created uncertainty in how to formulate the problem (Oka’s Dream, Oka VII Introduction). While there are some a rmative results (cf. Narasimhan [35], Noguchi [41]), it is expected to be explored more.

2 He was the President of the University of Tokyo. His research subjects are quantum electronics and quantum photonics.

References

1. Makoto Abe, Tube domains over C𝑛 , Memoirs Fac. Sci., Kyushu Univ. Ser. A 39 (2) (1985), 253 259. 2. Y. Aihara and J. Noguchi, Complex Analysis in One and Several Variables (in Japanese), Shokabo, Tokyo, 2024. 3. A. Andreotti and R. Narasimhan, Oka’s Heftungslemma and the Levi problem, Trans. Amer. Math. Soc. 111 (1964), 345 366. 4. H. Behnke and P. Thullen, Theorie der Funktionen mehrerer komplexer Veränderlichen, Ergeb. Math. Grenzgeb. Bd. 3, Springer-Verlag, Heidelberg, 1934. 5. L. Bers, Introduction to Several Complex Variables, Lecture Notes, Courant Inst. Math. Sci., New York University, 1964. 6. S. Bochner, Bounded analytic functions in several variables and multiple Laplace integrals, Amer. J. Math. 59 (1937), 732 738. 7. , A theorem of analytic continuation of functions in several variables, Ann. Math. 39 no. 1 (1938), 14 19. 8. H.J. Bremermann, Über die Äquivalenz der pseudokonvexen Gebiete und der Holomorphiegebiete im Raum von 𝑛 komplexen Veränderlichen, Math. Ann. 128 (1954), 63 91. 9. H. Cartan, Sur les matrices holomorphes de 𝑛 variables complexes, J. Math. pure appl. 19 (1940), 1 26. 10. H. Cartan, Idéaux et modules de fonctions analytiques de variables complexes Bull. Soc. Math. France 78 (1950), 29 64. 11. H. Cartan und P. Thullen, Regularitäts- und Konvergenzbereiche, Math. Ann. 106 (1932), 617 647. 12. P. Cousin, Sur les fonctions de 𝑛 variables complexes, Acta Math. 19 (1895), 1 61. 13. A. Czarnecki, M. Kulczycki and W. Lubawski, On the connectedness of boundary and complement for domains, Ann. Polon. Math. 103 (2) (2012), 189 191. 14. F. Docquier und H. Grauert, Levischen problem und Rungescher Satz für Teilgebiete Steinscher Mannigfaltigkeiten, Math. Ann. 140 (1960), 94 123. 15. E. Fabry, Sur les rayons de convergence d’une série double, C. R. Acad. Sci., Paris 134 (1902), 1190 1192. 16. J.E. Fornæss, A counterexample for the Levi problem for branched Riemann domains over C𝑛 , Math. Ann. 234 (1978), 275 277. 17. F. Forstneri , Stein Manifold and Holomorphic Mappings, Ergeb. Math. Grenzgeb. 3, Vol. 56, Springer-Verlag, Berlin Heidelberg, 2011: 2nd Edition, 2017. 18. K. Fritzsche and H. Grauert, From Holomorphic Functions to Complex Manifolds, G.T.M. 213, Springer-Verlag, New York, 2002. 19. R. Fujita, Domaines sans point critique intérieur sur l’espace projectif complexe, J. Math. Soc. Jpn. 15 (1963), 443 473.

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Index

A absolute convergence, 7 additive group (commutative group, abelian group), 31 analytic continuation, 14, 118 analytic function, 4 analytic interpolation problem, 112 analytic local ring, 33 analytic polyhedral domain, 83 analytic polyhedron, 83, 123 analytic (sub)set, 24 analytic sheaf, 34 anti-holomorphic partial di erential operator, 3 Approximation Theorem, 18, 85, 89, 123 B Baire Category Theorem, 140 Baire space, 139 Baire vector space, 176 ball, 1 ball neighborhood, 117 Banach’s Open Map Theorem, 177 base of a tube domain, 154 base point, 116 Behnke Stein Theorem, 206 biholomorphic, 23 b ection, xv b ective, xv Bochner’s Tube Theorem, 156 Bogolyubov, Nikolay (1909 1992), 172 boundary distance function, 74, 78, 117 bounded domain, 157 bumping method of Grauert, 201

C Cartan, Henri (1904 2008), vii, 39, 53 Cartan’s matrix decomposition, 57 Cartan Thullen Theorem, 78, 81 Cauchy condition (uniform), 7 Cauchy integral transform, 109 Cauchy kernel, 5 Cauchy’s integral formula, 4 Cauchy Riemann equations, 4 ech cohomology, 195 𝐶 𝑘 -pseudoconvex, 146 closed domain, 2 closed polydisk, 2 closed rectangle, 57 coboundary, 194 coboundary operator, 194 cochain, 193 cocycle, 193 cocycle condition, 193 coherent analytic sheaf, 39 coherent sheaf, 39 cohomology, 195 cohomology class ( ech), 195 commutative ring, 32 compact operator, 190 completely continuous, 190 complex hypersurface, 24 complex Jacobi matrix, 21 complex Jacobian, 21 complex submanifold, 27 continuous, 3 continuous Cousin data, 93 Continuous Cousin Problem, 93 convergent majorant, 7 convergent power series, 8 convex (a ne), 2 convex function, 155

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 J. Noguchi, Basic Oka Theory in Several Complex Variables, Universitext, https://doi.org/10.1007/978-981-97-2056-9

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216

convex hull, 2 Cousin, Pierre (1867 1933), 28 Cousin decomposition, 21, 28 Cousin I data, 91 Cousin I Problem, 91 Cousin II data, 101 Cousin II Problem, 101 Cousin integral, 20 covering cohomology, 194 covering cohomology class, 194 cuboid, 57 cylinder, 2 cylinder domain, 2 D de ning function of 𝔇, 159 ¯ 𝜕-closed, 109 ¯ 𝜕-equation, 108 dimension of cuboid, 63 direct limit, 195 direct product sheaf, 34 disk, 1 Dolbeault, Pierre (1924 2015), 111 Dolbeault’s Lemma, 110 domain, 2 domain (over C𝑛 ), 116 domain of convergence, 9 domain of existence, 75, 119 domain of holomorphy, 75, 119 E Edge of the Wedge Theorem, 172 entire function, 11 envelope of holomorphy, 119 equivalence relation, xv euclidean norm, 1 exact sequence, 67 exhaustion function, 146 Extension Lemma, 68 F ℱ-convex domain, 75 ℱ-convex hull, 75 ℱ-convex set, 75 ℱ-envelope, 119 ℱ-separable, 118

Index

Fabry, Eugène (1856 1944), 9, 28, 209 lter base, 124 nite generator system, 36 nitely sheeted, 117 Fréchet space, 176 Fredholm integral equation of the second kind, vi, 188 Fredholm integral equation of the second kind type, vi, vii, 184, 188 Fujita, R., 204 G generalized interpolation, 113 geometric ideal sheaf, 38 geometric syzygy, 66 Grauert, Hans (1930 2011), 198 H Hadamard’s three circles theorem, 173 Hartogs, Friedlich (1874 1943), v, 14, 98,209 Hartogs domain, 16 Hartogs Extension, 98 Hartogs pseudoconvex, 174 Hartogs separate analyticity theorem, 142 Hartogs’ Inverse Problem, 165 Hartogs’ phenomenon, 14 Hartogs’ radius, 174 Hartogs’ Theorem, 98 Heftungslemma, 179 holomorphic, 4, 117 holomorphic extension, 118 holomorphic function on a submanifold, 28 holomorphic local coordinate system, 27 holomorphic map (or mapping), 22, 118 holomorphic partial di erential operator, 3 holomorphic tangent space, 162 holomorphically convex, 75, 122

Index

holomorphically convex domain, 75 holomorphically convex hull, 75 holomorphically convex set, 75 holomorphically isomorphic, 23 holomorphically separable, 118 homogeneous polynomial expansion, 11 homomorphism (ring), 118 I ideal, 32 ideal boundary, 125 ideal boundary point, 125 idéal de domaines indéterminés, 70 ideal sheaf of 𝐴, 37 ideal sheaves of analytic subsets, 38 ideal sheaves of complex submanifolds, 38 Identity Theorem, 12 implicit function theorem, 22 induced continuous Cousin data, 94 induction on cuboid dimension, 67, 68, 94 inductive limit, 195 in nitely sheeted, 117 injection, xv injective, xv interpolation problem, 112 inverse function theorem, 22 isomorphism (ring), 118 J Jensen’s formula, 132 Joku-Iko Principle, v, vi, vii, 63, 68 L Lelong, Pierre (1912 2011), 134, 172 Levi, Eugenio Elia (1883 1917), v, 162, 172, 209 Levi condition, 163 Levi form, 134 Levi polynomial, 166 Levi pseudoconvex, 163 Levi pseudoconvex point, 163 Levi’s Problem, 165 Levi Krzoska condition, 163

217

Levi Oka polynomial, 166 Liouville’s Theorem, 12 local extension, 28 local pseudoconvexity, 158 locally nite, 36, 92 locally nite generator system, 36 logarithmically convex, 10 M main part, 100 majorant, 7 Maximum Principle, 13, 135 merged system, 63 meromorphic function, 89 Mittag-Le er Theorem, 100 module, 32 monotone decreasing, xv monotone increasing, xv multi-sheeted, 117 multivalent, 117 N Nishino, Toshio (1932 2005), 203 non-singular point, 27 normal family, 207 O Oka, Kiyoshi (1901 1978), v, vii, 47, 172 Oka Extension, 68 Oka Extension with Estimate, 179, 182, 185 Oka hypersurface, 166 Oka map, 83 Oka Principle, 102, 125 Oka Syzygy, 63 Oka’s First Coherence Theorem, 39, 47 Oka’s Heftungslemma, 179, 180 Oka’s Joku-Iko Principle, 63, 68, Oka’s Pseudoconvexity Theorem, 202 Oka’s Second Coherence Theorem, 39 Oka’s Theorem of Boundary Distance Functions, 83, 145 Oka’s Third Coherence Theorem, 40

218

Oka Weil Approximation Theorem, 85, 123 open ball, 1 Open Map Theorem, 177 operator norm, 54 order of zero, 24 Osgood, William Fogg (1864 1943), 98, 209 P partial sum, 7 partition of unity, 93 plurisubharmonic, 134, 139 Poincaré, Henri (1854 1912), 23, 28 point over 𝑧, 116 polar set, 90, 91 pole, 90 polydisk, 2, 117 polydisk neighborhood, 117 polynomial polyhedral domain, 83 polynomial polyhedron, 83 polynomial ring, 33 polynomial-like element, 49 polynomial-like germ, 49 polynomial-like section, 49 polynomially convex domain, 75 polynomially convex set, 75 polyradius, 2 power series, 8 proper, xvi pseudoconvex, 134, 139, 146 pseudoconvex boundary, 158 pseudoconvex boundary point, 158 pseudoconvex hull, 171 Pseudoconvexity Problem, 165 Pseudoconvexity Problem I, 146 Pseudoconvexity Problem II, 147 Pseudoconvexity Problem III, 161 Pseudoconvexity Problem for rami ed domains, 204 R rami ed domain (over C𝑛 ), 116 re nement, 92 Reinhardt domain, 29, 173

Index

relation sheaf, 39 relative boundary, 125 relative boundary point, 125 relative isomorphism, 118 relative map, 118 relatively compact, xv relatively compact hull, 73 relatively isomorphic, 118 Remmert, Reinhold (1930 2016), 172 removability of totally real subspaces, 16 restriction of sheaves, 34 Riemann’s Extension Theorem, 26 Riemann’s Mapping Theorem, 23 ring, 32 Runge Approximation Theorem, 89 Runge pair, 86, 123 S schlicht domain, 116 Schwartz’s Fredholm Theorem, 190 section, 34, 35 section space, 35 semi-norm, 175 separate analyticity, 139 separately analytic, 4 separately holomorphic, 4 sequence of functions, 6 series of functions, 7 sheaf, 33 sheaf of ideals, 34 sheet number, 117 𝜎-compact, 121 singular point, 27 smooth point, 27 smoothing, 137 solvable (analytically), 101 solvable (topologically), 101 standard coordinate system, 26, 43 standard polydisk, 26, 43 Stein, Karl (1913 2000), 122, 156 Stein covering, 197 Stein domain, 122 strong Levi condition, 163

Index

strong Levi Krzoska condition, 163 strongly Levi pseudoconvex, 163 strongly Levi pseudoconvex point, 163 strongly plurisubharmonic, 134 strongly pseudoconvex, 134 strongly pseudoconvex boundary, 159 strongly pseudoconvex boundary point, 159 strongly pseudoconvex domain, 159 subharmonic, 133 sublevel set, 146 submodule, 32 sup-norm (supremum norm), 18 support, xv surjection, xv surjective, xv syzygy, 67 T Takeuchi, A., 203 Three Big Problems, vi topological solution, 101 topological vector space, 175 topologically solvable, 101 totally real subspace, 16 tube, 154

219

tube domain, 154 Tube Theorem, 156 U unit, 32 unit ball, 1 unit disk, 1 unit element, 32 unit polydisk, 2 univalent domain, 116 unrami ed domain, 116 unrami ed Riemann domain, 116 W weakly holomorphicaly separable, 204 Weierstrass decomposition, 47 Weierstrass polynomial, 46 Weierstrass Theorem, 105 Weierstrass’ Preparation Theorem, 43 Weil, André (1906 1998), 85 Weil condition, 126 Z zero set (of meromorphic functions), 90 zero sheaf, 34

Symbols

𝐴 ⋐ 𝐵, xv 𝐴ˆ ℱ , 75 𝐴ˆ 𝛺 , 75 𝐴ˆ poly , 75 |𝛼|, 3 𝛼!, 3 B(𝑎;𝑟), 1 B( 𝑝; 𝑅), 117 B(𝑟), 1 𝒞 ∗ (𝑈), 101 C[(𝑧 𝑗 )], 4 C[𝑧, 𝜑], 83 ch(𝐵), 2 ch(𝑅), 154 𝐶 𝑘 , xv 𝒞 𝑘 (𝑈), xv 𝒞 𝑘 (𝑊), 117 𝑑 ( 𝐴, 𝐵), 74 𝑑 (𝑧, 𝐴), 74 𝑑 (𝑧, 𝜕𝛺), 74 𝑑 𝑧¯ 𝑗 , 108 𝜕 ∗ 𝔇, 125 𝜕 𝛼, 4 ¯ 108 𝜕, 𝛥(𝑎;𝑟), 1 𝛥(𝑟), 1 ¯ 𝛥(𝑎;𝑟), 1 ¯ 𝛥(𝑟), 1 𝛿B(𝑅) ( 𝑝, 𝜕𝔇), 117

𝛿B(𝑅) (𝑧, 𝜕𝛺), 78 𝛿P𝛥 ( 𝐴, 𝜕𝛺), 79 𝛿P𝛥 ( 𝑝, 𝜕𝔇), 117 𝛿P𝛥 (𝑧, 𝜕𝛺), 78 𝜕𝑈, xv 𝑑𝑧 𝑗 , 108 𝑓 𝑎 , 33 𝑓 | 𝐸 , xv ℱ|𝑈 , 34 𝑓 , 118 𝑝

𝛤 (𝐸, ℱ), 35 𝛤 (𝑈, ℱ), 35 ℐ⟨𝐴⟩, 37 ℑ𝑧, xv |𝜑(𝑞)| = ∞, 181  , 23 bℱ , 75 𝐾 b𝒫 (𝔇) , 171 𝐾 ★ , 73 𝐾𝛺 𝐿 [𝜑] (𝑣), 134 𝐿 [𝜑] (𝑧; 𝑣), 134 𝑀 (𝑆,𝑇), 56 ℳ(𝛺), 89 𝑀 𝜑 (𝑎;𝑟), 132 𝑁 (𝑆,𝑇), 56 ∥ 𝐴∥, 54 ∥ 𝐴∥ 𝐸 , 54 ∥ 𝑓 ∥ 𝐴, 18 𝒪(𝛺), 4

𝒪∗ (𝑈), 101 𝒪𝑎 , 33 𝒪(𝐸), 35 𝛺( 𝑓 ), 9 𝛺 H , 16 𝒪𝛺 , 33 𝑞 𝒪𝛺 , 34 𝑜(𝑟 𝛼 ), xvi ord𝑎 𝑓 , 24 𝒪(𝑆), 28 𝒪(𝑊), 117 𝒫(𝔇), 139 𝒫(𝑈), 134 𝑝(𝑧), 117 P𝛥(𝑎; (𝑟 𝑗 )), 2 P𝛥( 𝑝;𝑟), 117 P𝛥((𝑟 𝑗 )), 2 𝒫 𝑘 (𝔇), 139 𝒫 𝑘 (𝑈), 134 𝑟 𝛼, 8 ℜ𝑧, xv R+ , xv 𝑅[𝑋1 , . . . , 𝑋 𝑁 ], 33 𝒰 ≺ 𝒱, 92 𝑍0 , 90 𝑧 𝛼, 8 𝑍∞ , 90 ∥𝑧∥ P𝛥 , 78 Z+ , xv

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 J. Noguchi, Basic Oka Theory in Several Complex Variables, Universitext, https://doi.org/10.1007/978-981-97-2056-9

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