Classification of Higher Dimensional Algebraic Varieties by Christopher D. HaconClassification of Higher Dimensional Algebraic Varieties by Christopher D. Hacon

Classification of Higher Dimensional Algebraic Varieties

byChristopher D. Hacon, Sándor Kov

Paperback | May 27, 2010

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Higher Dimensional Algebraic Geometry presents recent advances in the classification of complex projective varieties. Recent results in the minimal model program are discussed, and an introduction to the theory of moduli spaces is presented.

Title:Classification of Higher Dimensional Algebraic VarietiesFormat:PaperbackDimensions:220 pages, 0.22 × 0.16 × 0.02 inPublished:May 27, 2010Publisher:Springer-Verlag/Sci-Tech/TradeLanguage:English

The following ISBNs are associated with this title:

ISBN - 10:3034602898

ISBN - 13:9783034602891


Table of Contents

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.

Editorial Reviews

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)