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Description of  An introduction to families, deformations and moduli

Table of contents : 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 Poincare 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