Classification of Higher Dimensional Algebraic Varieties

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Springer Science & Business Media, May 27, 2010 - Mathematics - 220 pages
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This book grew out of the Oberwolfach-SeminarHigherDimensionalAlgebraicGeo- tryorganizedbythetwoauthorsinOctober2008. Theaimoftheseminarwas tointroduce advanced PhD students and young researchers to recent advances and research topics in higher dimensional algebraic geometry. The main emphasis was on the minimal model program and on the theory of moduli spaces. The authors would like to thank the Mathematishes Forshunginstitut Oberwolfach for its hospitality and for making the above mentioned seminar possible, the participants to the seminar for their useful comments, and Alex Kuronya, Max Lieblich, and Karl Schwede for valuable suggestions and conversations. The ?rst named author was partially supported by the National Science Foundation under grant number DMS-0757897 and would like to thank Aleksandra, Stefan, Ana, Sasha, Kristina and Daniela Jovanovic-Haconfor their love and continuos support. The second named author was partially supported by the National Science Foun- tion under grant numbers DMS-0554697 and DMS-0856185, and the Craig McKibben and Sarah Merner Endowed Professorship in Mathematics at the University of Wa- ington. He would also like to thank Timea Tihanyi for her enduring love and support throughout and beyond this project and his other co-authors for their patience and und- standing. Contents I Basics 1 1INTRODUCTION 3 1. A. CLASSIFICATION 3 2PRELIMINARIES 17 2. A. NOTATION 17 2. B. DIVISORS 18 2. C. REFLEXIVE SHEAVES 20 2. D. CYCLIC COVERS 21 2. E. R-DIVISORS IN THE RELATIVE SETTING 22 2. F. FAMILIES AND BASE CHANGE 24 2. G. PARAMETER SPACES AND DEFORMATIONS OF FAMILIES 25 3SINGULARITIES 27 3. A."
 

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Contents

CHAPTER 1 Introduction
3
CHAPTER 2 Preliminaries
17
CHAPTER 3 Singularities
27
Part II Recent advances in the minimal model program
46
CHAPTER 4 Introduction
49
CHAPTER 5 The main result
51
CHAPTER 6 Multiplier ideal sheaves
67
CHAPTER 7 Finite generation of the restricted algebra
79
Part III Compact moduli spaces of canonically polarized varieties
103
CHAPTER 11 Moduli problems
105
CHAPTER 12 Hilbert schemes
111
CHAPTER 13 The construction of the moduli space
116
CHAPTER 14 Families and moduli functors
133
CHAPTER 15 Singularities of stable varieties
141
CHAPTER 16 Subvarieties of moduli spaces
148
Part IV Solutions and hints to some of the exercises
172

CHAPTER 8 Log terminal models
82
CHAPTER 9 Nonvanishing
89
CHAPTER 10 Finiteness of log terminal models
98
Bibliography
185
Index
203
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