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.
Citation
Roberto Laface. "On Zariski Decomposition with and without Support." Taiwanese J. Math. 20 (4) 755 - 767, 2016. https://doi.org/10.11650/tjm.20.2016.6891
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