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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.
Table des matières
Basics.- Preliminaries.- Singularities.- Recent advances in the minimal model program.- The main result.- Multiplier ideal sheaves.- Finite generation of the restricted algebra.- Log terminal models.- Non-vanishing.- Finiteness of log terminal models.- Compact moduli spaces of canonically polarized varieties.- Moduli problems.- Hilbert schemes.- The construction of the moduli space.- Families and moduli functors.- Singularities of stable varieties.- Subvarieties of moduli spaces.
Commentaire
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)
