Adjoint asymptotic multiplier ideal sheaves associated to potential triples

Abstract

In this paper, we explore the geometry of potential triples $(X,\Delta,D)$, which by definition consists of a pair $(X,\Delta)$ and an $\mathbb{R}$-Cartier pseudoeffective divisor $D$ on $X$. We define and study the asymptotic multiplier ideal sheaf $\mathcal{J}(X,\Delta,\lVert D\rVert)$ associated to a potential triple $(X,\Delta,D)$. As a first main result, when $D$ is big, we prove that the condition $\mathcal{J}(X,\Delta,\lVert D\rVert)=\mathcal{O}_{X}$ is equivalent to the triple $(X,\Delta,D)$ being potentially klt, which is a klt analog of the pair $(X,\Delta)$. We also study the closed set defined by the ideal sheaf $\mathcal{J}(X,\Delta,\lVert D\rVert)$ and prove a Nadel type cohomology vanishing theorem for $\mathcal{J}(X,\Delta,\lVert D\rVert)$. As an application of the main result, we prove that we can run the $(K_X+\Delta+D)$-MMP with scaling of an ample divisor for a pklt triple $(X,\Delta,D)$.