An Introduction to Families, Deformations and Moduli

Contents......Page 9
Preface......Page 15
Introduction......Page 17
The Goals of this Book......Page 27
A Historical Note......Page 29
Overview of this Chapter......Page 33
Topological Coverings......Page 34
Branched and Unbranched Coverings of Riemann Surfaces......Page 36
Fundamental Groups and Coverings......Page 38
Uniformization of Riemann Surfaces......Page 40
The Fundamental Theorem......Page 41
Surfaces with Universal Covering the Complex Plane......Page 42
Surfaces with abelian Fundamental Group and Covering the Upper Half-plane......Page 43
Classification of Annuli up to Conformal Equivalence......Page 47
Set-theoretic Classification of Elliptic Curves......Page 49
Quotients, Projective Embeddings and Automorphic Functions......Page 51
The Riemann Surface Structure on U/PSL(2,Z)......Page 62
Overview of this Chapter......Page 65
The Definition of a Differentiable Family......Page 66
Examples of Differentiable Families......Page 68
Notions of Triviality and Operations on Differentiable Families......Page 69
Smooth Deformations of Complex Structure......Page 72
Complex Analytic Families......Page 77
The Definition of a Complex Analytic Family......Page 78
Notions of Triviality and Operations on Complex Analytic Families......Page 80
Remarks on Holomorphic Deformations of Complex Structure......Page 82
The Functor of Families......Page 83
The Complex Analytic Family B of Complex Tori......Page 85
The Complex Analytic Family C of Complex Tori......Page 86
Algebraizability and Analytic Deformations......Page 89
Algebraizability of Complex Tori......Page 90
Non-algebraic Deformations of Complex Algebraic Tori......Page 91
Discontinuous Variation of Complex Structure: Jump Phenomena......Page 92
Overview of this Chapter......Page 95
Infinitesimal Deformations for Differentiable Families......Page 98
Infinitesimal Kodaira-Spencer Maps for Differentiable Families......Page 100
The Fundamental Sequence of Vector Bundles for a Differentiable Family......Page 101
Reformulation of the Definition of Differentiable Family in Terms of Differentiable Fiber Bundles......Page 102
The Fundamental Sequence of Sheaves for a Differentiable Family......Page 103
The Global Kodaira-Spencer Map for a Differentiable Family......Page 104
Local Triviality and the Kodaira-Spencer Maps......Page 105
Infinitesimal Deformations and Kodaira-Spencer Maps for Complex Analytic Families......Page 108
Relationships of the Infinitesimal Kodaira-Spencer Maps for a Complex Analytic Family to those of the Underlying Differentiable Family......Page 111
The Global Kodaira-Spencer Map for a Complex Analytic Family......Page 114
The Relationship of the Global Kodaira-Spencer Map for a Complex Analytic Family to that of the Underlying Differentiable Family......Page 116
``Chain Rule'' for Derivative of Complex Structure......Page 121
Indispensability of the Assumption of Regularity......Page 123
Primary Obstructions to Infinitesimal Deformations......Page 124
Complete Families & the Theorem of Completeness......Page 128
Effective Families and the Number of Moduli......Page 130
Examples of Complete Effectively Parametrized Families: The Case of Complex Tori......Page 133
Obstructions to Infinitesimal Deformation of Complex Structure: A Reformulation......Page 137
The Theorem of Existence and Number of Moduli......Page 142
Differentiable and Complex Analytic Families of Complex Fiber Bundles......Page 144
Fundamental Sequences and Diagrams for Families of Complex Fiber Bundles......Page 146
The Global and Infinitesimal Kodaira-Spencer Maps for a Family of Complex Fiber Bundles......Page 148
Deformations of Complex Analytic Spaces......Page 151
Kuranishi's Theorem and Local Moduli Spaces......Page 153
Riemann's Formula: Local Moduli for Curves......Page 155
Local Moduli Spaces for Complex Tori......Page 156
Families of Vector Bundles and the Infinitesimal Deformation Maps......Page 157
Local Moduli for Simple Vector Bundles over a Compact Riemann Surface......Page 158
Deformation of Schemes and Geometric Vector Bundles over Schemes......Page 159
Algebraic Families of Schemes and Vector Bundles......Page 160
Infinitesimal Deformation Maps for Algebraic Families of Schemes......Page 162
Infinitesimal Deformation Maps for Algebraic Families of Algebraic Vector Bundles......Page 163
Overview of this Chapter......Page 165
Conditions on the Parameter Category C......Page 168
Remarks on the Above Definition......Page 169
The Functor of Equivalence Classes of Families......Page 170
Example: Problem of Moduli of Vector Bundles on a Compact Riemann Surface......Page 171
Example: Problem of Moduli of Elliptic Curves......Page 174
Yoneda's Lemma......Page 175
Examples of Representable Functors......Page 177
Moduli Problems......Page 180
Examples of Moduli Problems......Page 181
Fine Moduli Spaces......Page 182
Coarse Moduli Spaces......Page 184
The Picard Group of a Ringed Space......Page 188
The Chern Class of a Line Bundle......Page 189
The Degree of a Vector Bundle......Page 191
The Moduli Problem for Degree Zero Line Bundles......Page 193
Construction of the Jacobian......Page 194
Local Moduli for Degree Zero Line bundles and the Poincaré Family......Page 195
Fine Moduli for Line Bundles......Page 197
The Necessity of the Concept of a Coarse Moduli Space: The Example of Elliptic Curves......Page 200
Local Moduli for Elliptic Curves......Page 201
The Elliptic Modular Function Associated to a Family......Page 202
The Coarse Moduli Space for Elliptic Curves......Page 204
Local Obstructions to Existence of a Tautological Family......Page 209
Global Obstructions to Existence of a Fine Moduli Space......Page 213
Sheaves......Page 215
Pullbacks and Pushforwards of Sheaves of Modules......Page 218
Examples of Locally Ringed Spaces......Page 219
The Local Model for Complex Analytic Spaces......Page 220
Definition of Complex Analytic Space......Page 221
Integral Schemes......Page 222
Fiber Products......Page 223
Proper Morphisms and Projective Schemes......Page 224
Quasi-coherent and Coherent Algebraic Sheaves......Page 225
Proper Morphisms......Page 226
Sheaf Cohomology......Page 227
Cech Cohomology......Page 230
The Complex Analytic Space Associated to a Scheme of Finite Type......Page 231
The Coherent Analytic Sheaf Associated to a Coherent Algebraic Sheaf......Page 232
Properties of the Associated Complex Analytic Space......Page 233
The GAGA Correspondence......Page 234
References......Page 237