Classification of higher dimensional algebraic varieties
著者
書誌事項
Classification of higher dimensional algebraic varieties
（Oberwolfach seminars, 41）
Birkhäuser, c2010
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注記
Bibliography: p. [185]-202
Includes index
内容説明・目次
内容説明
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.
目次
I Basics.- 1 Introduction.- 1.A. Classification.- 2 Preliminaries.- 2.A. Notation.- 2.B. Divisors.- 2.C. Reflexive sheaves.- 2.D. Cyclic covers.- 2.E. R-divisors in the relative setting.- 2.F. Vanishing theorems.- 2.G. Families and base change.- 2.H. Parameter spaces and deformations of families.- 3 Singularities.- 3.A. Canonical singularities.- 3.B. Cones.- 3.C. Log canonical singularities.- 3.D. Normal crossings.- 3.E. Pinch points.- 3.F. Semi-log canonical singularities.- 3.G. Pairs.- 3.H. Rational and du Bois singularities.- II Recent advances in the MMP.- 4 Introduction.- 5 The main result.- 5.A. The cone and base point free theorems.- 5.B. Flips and divisorial contractions.- 5.C. The minimal model program for surfaces.- 5.D. The main theorem and sketch of proof.- 5.E. The minimal model program with scaling.- 5.F. PL-flips.- 5.G. Corollaries.- 6 Multiplier ideal sheaves.- 6.A. Asymptotic multiplier ideal sheaves.- 6.B. Extending pluricanonical forms.- 7 Finite generation of the restricted algebra.- 7.A. Rationality of the restricted algebra.- 7.B. Proof of (5.69).- 8 Log terminal models.- 8.A. Special termination.- 8.B. Existence of log terminal models.- 9 Non-vanishing.- 9.A. Nakayama-Zariski decomposition.- 9.B. Non-vanishing.- 10 Finiteness of log terminal models.- III Compact moduli spaces.- 11 Moduli problems.- 11.A. Representing functors.- 11.B. Moduli functors.- 11.C. Coarse moduli spaces.- 12 Hilbert schemes.- 12.A. The Grassmannian functor.- 12.B. The Hilbert functor.- 13 The construction of the moduli space.- 13.A. Boundedness.- 13.B. Constructing the moduli space.- 13.C. Local closedness.- 13.D. Separatedness.- 14 Families and moduli functors.- 14.A. An important example.- 14.B. Q-Gorenstein families.- 14.C. Projective moduli schemes.- 14.D. Moduli of pairs and other generalizations.- 15 Singularities of stable varieties.- 15.A. Singularity criteria.- 15.B. Applications to moduli spaces and vanishing theorems.- 15.C. Deformations of DB singularities.- 16 Subvarieties of moduli spaces.- 16.A. Shafarevich's conjecture.- 16.B. The Parshin-Arakelov reformulation.- 16.C. Shafarevich's conjecture for number fields.- 16.D. From Shafarevich to Mordell: Parshin's trick.- 16.E. Hyperbolicity and boundedness.- 16.F. Higher dimensional fibers.- 16.G. Higher dimensional bases.- 16.H. Uniform and effective bounds.- 16.I. Techniques.- 16.J. Allowing more general fibers.- 16.K. Iterated Kodaira-Spencer maps and strong non-isotriviality.- IV Solutions and hints to some of the exercises.
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