On Zariski Decomposition with and without support

TL;DR

This paper explores advanced forms of Zariski Decomposition, extending its applicability to any $Q$-divisor and support cycles, and introduces an iterative approach for pseudo-effective divisors.

Contribution

It extends Zariski Decomposition with support to all $Q$-divisors and arbitrary cycles, and proposes an iterative method for pseudo-effective divisors.

Findings

Extended Zariski Decomposition to all $Q$-divisors

Generalized support cycles beyond negative definite cycles

Introduced iterative approach for pseudo-effective divisors

Abstract

In this paper we study Zariski Decomposition with support in a negative definite cycle, a variation introduced by Y. Miyaoka. We provide two extensions of the original statement, which was originally meant for effective $\Q$-divisors: we can either state it for any $\Q$-divisor, or we can take the support to be in any cycle. Ultimately, we present a new approach to Zariski Decomposition of pseudo-effective $\Q$-divisors, which consists in iterating Zariski Decomposition with support.