Zariski-decomposition and abundance.

*(English)*Zbl 1061.14018
MSJ Memoirs 14. Tokyo: Mathematical Society of Japan (ISBN 4-931469-31-0/pbk). xiii, 277 p. (2004).

The Zariski decomposition of an effective (or more generally, pseudo-effective) divisor on a smooth projective surface was introduced and studied systematically by 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 \(X\) such that \(D-K_X\) is ample then \(D\) admits a Zariski decomposition.

Then an affirmative answer to this conjecture implies the existence of flips, and conversely this conjecture follows from the existence and the termination of flips. Many results treated in the present monograph are new. The monograph is very useful to the people interested in classification theory of higher dimensional varieties.

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 \(X\) such that \(D-K_X\) is ample then \(D\) admits a Zariski decomposition.

Then an affirmative answer to this conjecture implies the existence of flips, and conversely this conjecture follows from the existence and the termination of flips. Many results treated in the present monograph are new. The monograph is very useful to the people interested in classification theory of higher dimensional varieties.

Reviewer: Lucian Bădescu (Genova)