Preface vii
1 Elementary theory of several complex variables 1
1.1 Geometry of Cn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Holomorphic functions in several complex variables . . . . . . . . . 7
1.2.1 Definition of a holomorphic function . . . . . . . . . . . . . 7
1.2.2 Basic properties of holomorphic functions . . . . . . . . . . 10
1.2.3 Partially holomorphic functions and the Cauchy–Riemann
differential equations . . . . . . . . . . . . . . . . . . . . . . 13
1.3 The Cauchy Integral Formula . . . . . . . . . . . . . . . . . . . . . 17
1.4 O(U) as a topological space . . . . . . . . . . . . . . . . . . . . . . 19
1.4.1 Locally convex spaces . . . . . . . . . . . . . . . . . . . . . 20
1.4.2 The compact-open topology on C (U,E) . . . . . . . . . . . 23
1.4.3 The Theorems of Arzel`a–Ascoli and Montel . . . . . . . . . 28
1.5 Power series and Taylor series . . . . . . . . . . . . . . . . . . . . . 34
1.5.1 Summable families in Banach spaces . . . . . . . . . . . . . 34
1.5.2 Power series . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
1.5.3 Reinhardt domains and Laurent expansion . . . . . . . . . 38
2 Continuation on circular and polycircular domains 47
2.1 Holomorphic continuation . . . . . . . . . . . . . . . . . . . . . . . 47
2.2 Representation-theoretic interpretation of the Laurent series . . . . 54
2.3 Hartogs’ Kugelsatz, Special case . . . . . . . . . . . . . . . . . . . 56
3 Biholomorphic maps 59
3.1 The Inverse Function Theorem and Implicit Functions . . . . . . . 59
3.2 The Riemann Mapping Problem . . . . . . . . . . . . . . . . . . . 64
3.3 Cartan’s Uniqueness Theorem . . . . . . . . . . . . . . . . . . . . . 67
4 Analytic Sets 71
4.1 Elementary properties of analytic sets . . . . . . . . . . . . . . . . 71
4.2 The Riemann Removable Singularity Theorems . . . . . . . . . . . 75
vi Contents
5 Hartogs’ Kugelsatz 79
5.1 Holomorphic Differential Forms . . . . . . . . . . . . . . . . . . . . 79
5.1.1 Multilinear forms . . . . . . . . . . . . . . . . . . . . . . . . 79
5.1.2 Complex differential forms . . . . . . . . . . . . . . . . . . . 82
5.2 The inhomogenous Cauchy–Riemann Differential Equations . . . . 88
5.3 Dolbeaut’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
5.4 The Kugelsatz of Hartogs . . . . . . . . . . . . . . . . . . . . . . . 94
6 Continuation on Tubular Domains 97
6.1 Convex hulls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
6.2 Holomorphically convex hulls . . . . . . . . . . . . . . . . . . . . . 100
6.3 Bochner’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
7 Cartan–Thullen Theory 111
7.1 Holomorphically convex sets . . . . . . . . . . . . . . . . . . . . . . 111
7.2 Domains of Holomorphy . . . . . . . . . . . . . . . . . . . . . . . . 115
7.3 The Theorem of Cartan–Thullen . . . . . . . . . . . . . . . . . . . 118
7.4 Holomorphically convex Reinhardt domains . . . . . . . . . . . . . 121
8 Local Properties of holomorphic functions 125
8.1 Local representation of a holomorphic function . . . . . . . . . . . 125
8.1.1 Germ of a holomorphic function . . . . . . . . . . . . . . . 125
8.1.2 The algebras of formal and of convergent power series . . . 127
8.2 The Weierstrass Theorems . . . . . . . . . . . . . . . . . . . . . . . 135
8.2.1 The Weierstrass Division Formula . . . . . . . . . . . . . . 138
8.2.2 The Weierstrass Preparation Theorem . . . . . . . . . . . . 142
8.3 Algebraic properties of C{z1, . . . , zn} . . . . . . . . . . . . . . . . . 145
8.4 Hilbert’s Nullstellensatz . . . . . . . . . . . . . . . . . . . . . . . . 151
8.4.1 Germs of a set . . . . . . . . . . . . . . . . . . . . . . . . . 152
8.4.2 The radical of an ideal . . . . . . . . . . . . . . . . . . . . . 156
8.4.3 Hilbert’s Nullstellensatz for principal ideals . . . . . . . . . 160
Register of Symbols 165
Bibliography 167
Index 169