Adjoint asymptotic multiplier ideal sheaves associated to potential triples

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