Mathematical Society of Japan Memoirs

Zariski-decomposition and Abundance

Noboru Nakayama

Book information

Author
Noboru Nakayama

Publication information
MSJ Memoirs, Volume 14
Tokyo, Japan: The Mathematical Society of Japan, 2004
277 pp.

Dates
Publication date: 2004
First available in Project Euclid: 17 January 2014

Permanent link to this book
https://projecteuclid.org/euclid.msjm/1389986103

Digital Object Identifier:
doi:10.2969/msjmemoirs/014010000

ISBN:
978-4-931469-31-0

Zentralblatt MATH:
1061.14018

Mathematical Reviews (MathSciNet):
MR2104208

Subjects
Primary: 14C20: Divisors, linear systems, invertible sheaves 14E30: Minimal model program (Mori theory, extremal rays) 14J10: Families, moduli, classification: algebraic theory 14E15: Global theory and resolution of singularities [See also 14B05, 32S20, 32S45]

Keywords
Zariski-decomposition toric varieties minimal models Kodaira dimension plurigenera abundance conjecture

Rights
Copyright © 2004, The Mathematical Society of Japan

Citation
Noboru Nakayama, Zariski-decomposition and Abundance (Tokyo: The Matematical Society of Japan, 2004)

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Abstract

Dr. Noboru Nakayama, the author of this book, studies the birational classification of algebraic varieties and of compact complex manifolds. This book is a collection of his works on the numerical aspects of divisors of algebraic varieties. The notion of Zariski-decomposition introduced by Oscar Zariski is a powerful tool in the study of open surfaces. In the higher dimensional generalization, we encounter interesting phenomena on the numerical aspects of divisors. The author treats the higher dimensional Zariski-decomposition systematically. The abundance conjecture predicts that the numerical Kodaira dimension of a minimal variety coincides with the usual Kodaira dimension. The Kodaira dimension is an invariant of the canonical divisor of a variety. The numerical analogue used to be defined only for nef divisors, but it is now extended to arbitrary divisors in this book. Explained in details are many important results on the numerical Kodaira dimension related to the abundance, to the addition theorem for fiber spaces, and to the deformation invariance.