Mathematical Society of Japan Memoirs

Zariski-decomposition and Abundance

Noboru Nakayama

Book information

Noboru Nakayama

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

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

Permanent link to this book

Digital Object Identifier:


Zentralblatt MATH:

Mathematical Reviews (MathSciNet):

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]

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

Copyright © 2004, The Mathematical Society of Japan

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

Select/deselect all
  • Export citations
Select/deselect all
  • Export citations


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.