On Zariski Decomposition with and without Support

Abstract

We study Zariski decomposition with support in a negative definite cycle, a variation of Zariski decomposition introduced by Miyaoka [4]: given a negative definite cycle $G$, any $\mathbb{Q}$-divisor $D$ decomposes into the sum of a $G$-nef and a rigid $\mathbb{Q}$-divisor. We prove that such a decomposition actually exists for an arbitrary $\mathbb{Q}$-divisor. Moreover, we show that, under the hypothesis that $D$ is pseudo-effective, we can drop the assumption of $G$ being negative definite, and obtain decompositions of $D$ with respect to arbitrary cycles. Our methods are inspired by a work of Bauer [1], in which he gives a simpler proof of Zariski's original result [5], and by adapting his proof to other cases, we are able to provide an alternative approach to this circle of ideas.