Adjoint asymptotic multiplier ideal sheaves associated to potential triples
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866917055734743040 |
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| author | Choi, Sung Rak Jang, Sungwook Kim, Donghyeon |
| author_facet | Choi, Sung Rak Jang, Sungwook Kim, Donghyeon |
| contents | In this paper, we explore the geometry of potential triples $(X,Δ,D)$, which by definition consists of a pair $(X,Δ)$ and an $\mathbb{R}$-Cartier pseudoeffective divisor $D$ on $X$. We define and study the asymptotic multiplier ideal sheaf $\mathcal{J}(X,Δ,\lVert D\rVert)$ associated to a potential triple $(X,Δ,D)$. As a first main result, when $D$ is big, we prove that the condition $\mathcal{J}(X,Δ,\lVert D\rVert)=\mathcal{O}_{X}$ is equivalent to the triple $(X,Δ,D)$ being potentially klt, which is a klt analog of the pair $(X,Δ)$. We also study the closed set defined by the ideal sheaf $\mathcal{J}(X,Δ,\lVert D\rVert)$ and prove a Nadel type cohomology vanishing theorem for $\mathcal{J}(X,Δ,\lVert D\rVert)$. As an application of the main result, we prove that we can run the $(K_X+Δ+D)$-MMP with scaling of an ample divisor for a pklt triple $(X,Δ,D)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_07441 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Adjoint asymptotic multiplier ideal sheaves associated to potential triples Choi, Sung Rak Jang, Sungwook Kim, Donghyeon Algebraic Geometry In this paper, we explore the geometry of potential triples $(X,Δ,D)$, which by definition consists of a pair $(X,Δ)$ and an $\mathbb{R}$-Cartier pseudoeffective divisor $D$ on $X$. We define and study the asymptotic multiplier ideal sheaf $\mathcal{J}(X,Δ,\lVert D\rVert)$ associated to a potential triple $(X,Δ,D)$. As a first main result, when $D$ is big, we prove that the condition $\mathcal{J}(X,Δ,\lVert D\rVert)=\mathcal{O}_{X}$ is equivalent to the triple $(X,Δ,D)$ being potentially klt, which is a klt analog of the pair $(X,Δ)$. We also study the closed set defined by the ideal sheaf $\mathcal{J}(X,Δ,\lVert D\rVert)$ and prove a Nadel type cohomology vanishing theorem for $\mathcal{J}(X,Δ,\lVert D\rVert)$. As an application of the main result, we prove that we can run the $(K_X+Δ+D)$-MMP with scaling of an ample divisor for a pklt triple $(X,Δ,D)$. |
| title | Adjoint asymptotic multiplier ideal sheaves associated to potential triples |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2311.07441 |