Classification of Higher Dimensional Algebraic Varieties

ISBN-13: 9783034602891 / Angielski / Miękka / 2010 / 220 str.

Christopher D. Hacon;Sandor Kovacs

Opis

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.

Recenzje

From the reviews:

"The present text presents the proofs of many results surrounding the minimal model program (MMP) for higher-dimensional varieties. ... This text treats the subject in the great generality which is required for getting the most recent results. Hence, it is laden with terminology, all necessary for the modern researcher. ... As such, the text will be invaluable for those currently living off of survey articles trying to grasp recent advances in higher-dimensional geometry." (Michael A. van Opstall, Mathematical Reviews, Issue 2011 f)

"The authors give a detailed account of these new results and the theory of compact moduli spaces of canonically polarised varieties. ... the book contains a considerable number of exercises as well as a chapter of hints to solve them. ... the authors have made quite an effort to write a text that is both an accessible introduction and a useful reference. ... I can only recommend it to researchers and advanced graduate students interested in this highly active field of mathematics." (Andreas Höring, Zentralblatt MATH, Vol. 1204, 2011)

Spis treści

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.

Blurb

This book focuses on recent advances in the classification of complex projective varieties. It is divided into two parts. The first part gives a detailed account of recent results in the minimal model program. In particular, it contains a complete proof of the theorems on the existence of flips, on the existence of minimal models for varieties of log general type and of the finite generation of the canonical ring. The second part is an introduction to the theory of moduli spaces. It includes topics such as representing and moduli functors, Hilbert schemes, the boundedness, local closedness and separatedness of moduli spaces and the boundedness for varieties of general type.

The book is aimed at advanced graduate students and researchers in algebraic geometry.