Lectures on Several Complex Variables

This monograph provides a concise, accessible snapshot of key topics in several complex variables, including the Cauchy Integral Formula, sequences of holomorphic functions, plurisubharmonic functions, the Dirichlet problem, and meromorphic functions. Based on a course given at Université de Montréal, this brief introduction covers areas of contemporary importance that are not mentioned in most treatments of the subject, such as modular forms, which are essential for Wiles' theorem and the unification of quantum theory and general relativity. Also covered is the Riemann manifold of a function, which generalizes the Riemann surface of a function of a single complex variable and is a topic that is well-known in one complex variable, but rarely treated in several variables. Many details, which are intentionally left out, as well as many theorems are stated as problems, providing students with carefully structured instructive exercises.Prerequisites for use of this book are functions of one complex variable, functions of several real variables, and topology, all at the undergraduate level. __Lectures on Several Complex Variables__ will be of interest to advanced undergraduate and beginning undergraduate students, as well as mathematical researchers and professors. ​​​This monograph provides a concise, accessible snapshot of key topics in several complex variables, including the Cauchy Integral Formula, sequences of holomorphic functions, plurisubharmonic functions, the Dirichlet problem, and meromorphic functions. Based on a course given at Université de Montréal, this brief introduction covers areas of contemporary importance that are not mentioned in most treatments of the subject, such as modular forms, which are essential for Wiles' theorem and the unification of quantum theory and general relativity. Also covered is the Riemann manifold of a function, which generalizes the Riemann surface of a function of a single complex variable and is a topic that is well-known in one complex variable, but rarely treated in several variables. Many details, which are intentionally left out, as well as many theorems are stated as problems, providing students with carefully structured instructive exercises. Prerequisites for use of this book are functions of one complex variable, functions of several real variables, and topology, all at the undergraduate level. Lectures on Several Complex Variables will be of interest to advanced undergraduate and beginning undergraduate students, as well as mathematical researchers and professors. Preface 6 Contents 8 1 Introduction 10 2 Basics 12 3 Cauchy Integral Formula 17 4 Sequences of Holomorphic Functions 23 5 Series 26 6 Zero Sets of Holomorphic Functions 32 7 Holomorphic Mappings 37 8 Plurisubharmonic Functions 44 9 The Dirichlet Problem 56 10 Uniform Approximation 58 11 Complex Manifolds 60 12 Examples of Manifolds 63 12.1 Domains 63 12.2 Submanifolds 63 12.3 Product Manifolds and Matrices 66 12.4 Lie Groups 67 12.5 Projective Space 68 12.6 Grassmann Manifolds 70 12.7 Tori 72 12.8 The Quotient Manifold with Respect to an Automorphism Group 74 12.9 Spread Manifolds 78 13 Holomorphic Continuation 80 13.1 Direct Holomorphic Continuation and Domains of Holomorphy 80 13.2 Indirect Holomorphic Continuation and Riemann Domains 83 14 The Tangent Space 87 14.1 Hermitian Manifolds 91 14.2 Symplectic Manifolds 97 14.3 Almost Complex Manifolds 99 15 Meromorphic Functions and Subvarieties 102 15.1 Meromorphic Functions 102 15.2 Subvarieties 105 References 107 Index 109