Higher-dimensional algebraic geometry
type: Item
instance of: scholarly article
title: Higher-dimensional algebraic geometry
zbMATH Open document ID: 0978.14001
author: Olivier Debarre
published in: Universitext
publication date: 2001-08-16T00:00:00Z
review text: Higher-dimensional algebraic geometry, that is the study of algebraic varieties of dimension greater than two, is a branch of algebraic geometry which is largely still in progress. Opposed to the theories of algebraic curves and surfaces, which are basically classical and almost perfectly understood, in the meantime, the classification theory of higher-dimensional varieties is a comparatively young subdiscipline of algebraic geometry and, in its systematic framework, barely older than twenty-five years. However, an amazing quantity of knowledge in this new area has accumulated, in the course of the past twenty years, together with a vast spectrum of new, specific concepts, methods, techniques, and deep-going results. The whole subject has now reached a stage where it is rather hard for the beginner to enter this magic world built up by S. Mori, F. Campana, M. Reid, Y. Kawamata, J. Kollár, Y. Miyaoka, V. Shokurov, V. Iskovskikh, E. Viehweg, and many others. As a result, higher-dimensional algebraic geometry, in its present state of art, somehow makes the impression of a private club of ultimate experts in the field, and has remained much more confidential than it should have. Certainly, this is particularly due to the fact that there has been no beginner-friendly textbook on this topic, so far, guiding the non-expert through the specific terminology, the considerable amount of technicalities, and the systematic strategy underlying the classification theory of higher-dimensional algebraic varieties. The book under review represents the first attempt to provide an easily accessible introduction to this subject, and that in the form of a systematic textbook. Of course, there are already several excellent survey articles on topics in higher-dimensional algebraic geometry, and there are also a few special research monographs and high-level reference books, for example \textit{J. Kollár}'s book: ``Rational curves on algebraic varieties'' (1996; Zbl 0877.14012) but those ca…
reviewed by: Werner Kleinert
Mathematics Subject Classification ID: 14-01
Mathematics Subject Classification ID: 14Jxx
Mathematics Subject Classification ID: 14E30
Mathematics Subject Classification ID: 14C20
zbMATH DE Number: 1634463
zbMATH Keywords: higher-dimensional algebraic varieties
zbMATH Keywords: birational geometry
zbMATH Keywords: birational classification theory
zbMATH Keywords: minimal model program
zbMATH Keywords: Mori theory
zbMATH Keywords: cohomological vanishing theorems
zbMATH Keywords: cohomological nonvanishing theorems
zbMATH Keywords: Cartier divisors
zbMATH Keywords: morphisms from curves
zbMATH Keywords: varieties with many rational curves
zbMATH Keywords: rational quotient of a variety
zbMATH Keywords: cone theorem
zbMATH Keywords: contraction theorem
zbMATH Keywords: extremal rays
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1634463