Zariski-decomposition and abundance
type: Item
instance of: scholarly article
title: Zariski-decomposition and abundance
zbMATH Open document ID: 1061.14018
author: Noboru Nakayama
published in: MSJ Memoirs
publication date: 2004-12-09T00:00:00Z
full work available at URL: http://projecteuclid.org/euclid.msjm/1389986103
review text: The Zariski decomposition of an effective (or more generally, pseudo-effective) divisor on a smooth projective surface was introduced and studied systematically by \textit{O. Zariski} [Ann. Math. (2) 76, 560--615 (1962; Zbl 0124.37001)]. The Zariski decomposition of the canonical divisor on a smooth projective surface of non-negative Kodaira dimension is closely related with the problem of minimal models for surfaces. This theory represents a very important tool to study open algebraic surfaces as well. The generalization of Zariski decomposition of divisors on higher-dimensional varieties encounters many difficulties and very interesting phenomena. The aim of this book is to give a systematic account of the known results connected with this problem, with special emphasis to the most recent ones. This theory is closely related with a number of fundamental conjectures in the classification theory of complex projective manifolds, such as: i) The canonical ring of a complex projective manifold \(X\) is finitely generated (still open in dimension \(\geq 4\)), ii) Iitaka's conjecture \(C_{n,m}\): \(\kappa(X)\geq\kappa(X| Y)+\kappa(Y)\) for every algebraic fiber space, iii) Existence and termination of flips in the Mori's minimal model program (open in dimension \(\geq 4\)), iv) The abundance conjecture: the canonical class \(K_X\) is semi-ample if \(X\) is a minimal model, and v) Deformation invariance of plurigenera. The author discusses in detail the following topics: Zariski decomposition in the higher dimensional case, numerical \(D\)-dimension, addition theorems and invariance of plurigenera. As far as the Zariski decomposition is concerned, the author tried to prove the existence of the Zariski decomposition for arbitrary pseudo-effective \(\mathbb R\)-divisors and found counterexamples for big divisors. There many important aspects of this problem, e.g. the following Conjecture: If \(D\) is a pseudo-effective \(\mathbb R\)-divisor on a complex projective manifold …
Mathematics Subject Classification ID: 14E30
Mathematics Subject Classification ID: 14J10
Mathematics Subject Classification ID: 14C20
zbMATH DE Number: 2121550
zbMATH Keywords: divisors
zbMATH Keywords: Kodaira dimension
zbMATH Keywords: Mori minimal model program
reviewed by: Lucian Bădescu
MaRDI profile type: Publication
OpenAlex ID: W560578122
DOI: 10.2969/MSJMEMOIRS/014010000
description: scientific article; zbMATH DE number 2121550