Tasty Bits of Several Complex Variables A whirlwind tour of the subject Jiří Lebl December 23, 2020 (version 3.4) 2 Typeset in LATEX. Copyright ©2014–2020 Jiří Lebl License: This work is dual licensed under the Creative Commons Attribution-Noncommercial-Share Alike 4.0 International License and the Creative Commons Attribution-Share Alike 4.0 International License. To view a copy of these licenses, visit https://creativecommons.org/licenses/ by-nc-sa/4.0/ or https://creativecommons.org/licenses/by-sa/4.0/ or send a letter to Creative Commons PO Box 1866, Mountain View, CA 94042, USA. You can use, print, duplicate, share this book as much as you want. You can base your own notes on it and reuse parts if you keep the license the same. You can assume the license is either the CC-BY-NC-SA or CC-BY-SA, whichever is compatible with what you wish to do, your derivative works must use at least one of the licenses. Acknowledgments: I would like to thank Debraj Chakrabarti, Anirban Dawn, Alekzander Malcom, John Treuer, Jianou Zhang, Liz Vivas, Trevor Fancher, Nicholas Lawson McLean, Alan Sola, Achinta Nandi, Sivaguru Ravisankar, Richard Lärkäng, Elizabeth Wulcan, Tomas Rodriguez, and students in my classes for pointing out typos/errors and helpful suggestions. During the writing of this book, the author was in part supported by NSF grant DMS-1362337. More information: See https://www.jirka.org/scv/ for more information (including contact information). The LATEX source for the book is available for possible modification and customization at github: https://github.com/jirilebl/scv Contents Introduction 5 0.1 Motivation, single variable, and Cauchy’s formula ..................5 1 Holomorphic functions in several variables 11 1.1 Onto several variables ................................. 11 1.2 Power series representation .............................. 16 1.3 Derivatives ...................................... 24 1.4 Inequivalence of ball and polydisc .......................... 28 1.5 Cartan’s uniqueness theorem ............................. 32 1.6 Riemann extension theorem, zero sets, and injective maps .............. 35 2 Convexity and pseudoconvexity 40 2.1 Domains of holomorphy and holomorphic extensions ................ 40 2.2 Tangent vectors, the Hessian, and convexity ..................... 44 2.3 Holomorphic vectors, the Levi form, and pseudoconvexity ............. 51 2.4 Harmonic, subharmonic, and plurisubharmonic functions .............. 63 2.5 Hartogs pseudoconvexity ............................... 73 2.6 Holomorphic convexity ................................ 80 3 CR functions 86 3.1 Real-analytic functions and complexification ..................... 86 3.2 CR functions ..................................... 92 3.3 Approximation of CR functions ........................... 98 3.4 Extension of CR functions .............................. 105 ¯ 4 The m-problem 109 4.1 The generalized Cauchy integral formula ....................... 109 ¯ 4.2 Simple case of the m-problem ............................. 111 4.3 The general Hartogs phenomenon .......................... 113 5 Integral kernels 116 5.1 The Bochner–Martinelli kernel ............................ 116 5.2 The Bergman kernel ................................. 119 5.3 The Szegö kernel ................................... 123 4 CONTENTS 6 Complex analytic varieties 125 6.1 The ring of germs ................................... 125 6.2 Weierstrass preparation and division theorems .................... 127 6.3 The dependence of zeros on parameters ....................... 132 6.4 Properties of the ring of germs ............................ 135 6.5 Varieties ........................................ 137 6.6 Hypervarieties ..................................... 140 6.7 Irreducibility, local parametrization, and Puiseux theorem .............. 143 6.8 Segre varieties and CR geometry ........................... 146 A Basic notation and terminology 151 B Results from one complex variable 152 C Differential forms and Stokes’ theorem 162 D Basic terminology and results from algebra 170 Further reading 173 Index 174 List of Notation 179 Introduction This book is a polished version of my course notes for Math 6283, Several Complex Variables, given in Spring 2014, Spring 2016, and Spring 2019 semesters at Oklahoma State University. It is meant for a semester-long course. Quite a few exercises of various difficulty are sprinkled throughout the text, and I hope a reader is at least attempting all of them. Many are required later in the text. The reader should attempt exercises in sequence as earlier exercises can help or even be required to solve later exercises. The prerequisites are a decent knowledge of vector calculus, basic real analysis, and a working knowledge of complex analysis in one variable. Measure theory (Lebesgue integral and its con- vergence theorems) is useful, but it is not essential except in a couple of places later in the book. The first two chapters and most of the third is accessible to beginning graduate students after one semester of a standard single-variable complex analysis graduate course. From time to time (e.g. proof of Baouendi–Trèves in chapter 3, and most of chapter 4, and chapter 5), basic knowledge of differential forms is useful, and in chapter 6 we use some basic ring theory from algebra. By design, it can replace the second semester of complex analysis. This book is not meant as an exhaustive reference. It is simply a whirlwind tour of several complex variables with slightly more material than can be covered within a semester. See the end of the book for a list of books for reference and further reading. There are also appendices for a list of one-variable results, an overview of differential forms, and some basic algebra. See appendix B , appendix C, and appendix D . 0.1 Motivation, single variable, and Cauchy’s formula Let us start with some standard notation. We use C for the complexp numbers, R for real numbers, Z for integers, N = f1, 2, 3,...g for natural numbers, 8 = −1. Throughout this book, we use the standard terminology of domain to mean connected open set. We will try to avoid using it if connectedness is not needed, but sometimes we use it just for simplicity. p As complex analysis deals with the complexp numbers, perhaps we should begin with −1. We start with the real numbers, R, and we add −1 into our field. We call this square root 8, and we write the complex numbers, C, by identifying C with R2 using I = G ¸ 8H, where I 2 C, and ¹G, Hº 2 R2. A subtle philosophical issue is that there are two square roots of −1. Two chickens are running around in our yard, and because we like to know which is which, we catch one and write “8” on it. If we happened to have caught the other chicken, we would have gotten an exactly equivalent theory, which we could not tell apart from the original. 6 INTRODUCTION Given a complex number I, its “opposite” is the complex conjugate of I and is defined as def I¯ = G − 8H. The size of I is measured by the so-called modulus, which is just the Euclidean distance: def p q jIj = II¯ = G2 ¸ H2. If I = G ¸ 8H 2 C for G, H 2 R, then G is called the real part and H is called the imaginary part. We write I ¸ I¯ I − I¯ Re I = Re¹G ¸ 8Hº = = G, Im I = Im¹G ¸ 8Hº = = H. 2 28 A function 5 : * ⊂ R= ! C for an open set * is said to be continuously differentiable, or 퐶1 if the first (real) partial derivatives exist and are continuous. Similarly, it is 퐶: or 퐶: -smooth if the first : partial derivatives all exist and are differentiable. Finally, a function is said to be 퐶1 or ∗ : simply smooth if it is infinitely differentiable, or in other words, if it is 퐶 for all : 2 N. Complex analysis is the study of holomorphic (or complex-analytic) functions. Holomorphic functions are a generalization of polynomials, and to get there one leaves the land of algebra to arrive in the realm of analysis. One can do an awful lot with polynomials, but sometimes they are just not enough. For example, there is no nonzero polynomial function that solves the simplest of differential equations, 5 0 = 5 . We need the exponential function, which is holomorphic. Let us start with polynomials. In one variable, a polynomial in I is an expression of the form 3 ∑︁ 9 %¹Iº = 29 I , 9=0 where 29 2 C and 23 < 0. The number 3 is called the degree of the polynomial %. We can plug in some number I and simply compute %¹Iº, so we have a function % : C ! C. We try to write 1 ∑︁ 9 5 ¹Iº = 29 I 9=0 and all is very fine until we wish to know what 5 ¹Iº is for some number I 2 C. We usually mean 1 3 ∑︁ 9 ∑︁ 9 29 I = lim 29 I . 3!1 9=0 9=0 As long as the limit exists, we have a function. You know all this; it is your one-variable complex analysis. We usually start with the functions and prove that we can expand into series. Let * ⊂ C be open. A function 5 : * ! C is holomorphic (or complex-analytic) if it is complex-differentiable at every point, that is, if 5 ¹I ¸ bº − 5 ¹Iº 5 0¹Iº = lim exists for all I 2 *. b2C!0 b ∗While 퐶1 is the common definition of smooth, not everyone always means the same thing by the word smooth.I have seen it mean differentiable, 퐶1, piecewise-퐶1, 퐶1, holomorphic, . 0.1. MOTIVATION, SINGLE VARIABLE, AND CAUCHY’S FORMULA 7 Importantly, the limit is taken with respect to complex b. Another vantage point is to start with a ∗ continuously differentiable 5 , and say 5 = D ¸8 E is holomorphic if it satisfies the Cauchy–Riemann equations: mD mE mD mE = , = − . mG mH mH mG The so-called Wirtinger operators, m 1 m m m 1 m m def= − 8 , def= ¸ 8 , mI 2 mG mH mI¯ 2 mG mH provide an easier way to understand the Cauchy–Riemann equations.