Boundary points, Minimal $L^{2}$ integrals and Concavity property
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866914736630661120 |
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| author | Bao, Shijie Guan, Qi'an Yuan, Zheng |
| author_facet | Bao, Shijie Guan, Qi'an Yuan, Zheng |
| contents | For the purpose of proving the strong openness conjecture of multiplier ideal sheaves, Jonsson-Mustaţă posed an enhanced conjecture and proved the two-dimensional case, which says that: the Lebesgue measure of the set $\big\{c_o^F(ψ)ψ-\log|F|<\log r\big\}$ divided by $r^2$ has a uniform positive lower bound independent of $r$, for a plurisubharmonic function $ψ$ and a holomorphic function $F$ near the origin $o$. Jonsson-Mustaţă's conjecture was proved by Guan-Zhou depending on the truth of the strong openness conjecture. However, it is still a question whether one can prove Jonsson-Mustaţă's conjecture without using the strong openness property, and obtain a sharp effectiveness result for this conjecture.
In this article, we use an $L^2$ method with the weight functions $ψ-\log|F|$ and firstly consider a module at at a boundary point of the sublevel sets of a plurisubharmonic function. By studying the minimal $L^{2}$ integrals on the sublevel sets of a plurisubharmonic function with respect to the module at the boundary point, we establish a concavity property of the minimal $L^{2}$ integrals. As applications, we obtain a sharp effectiveness result related to Jonsson-Mustaţă's conjecture, which completes the approach from the conjecture to the strong openness property. We also obtain a strong openness property of the module and a lower semi-continuity property with respect to the module. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2203_01648 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Boundary points, Minimal $L^{2}$ integrals and Concavity property Bao, Shijie Guan, Qi'an Yuan, Zheng Complex Variables Algebraic Geometry For the purpose of proving the strong openness conjecture of multiplier ideal sheaves, Jonsson-Mustaţă posed an enhanced conjecture and proved the two-dimensional case, which says that: the Lebesgue measure of the set $\big\{c_o^F(ψ)ψ-\log|F|<\log r\big\}$ divided by $r^2$ has a uniform positive lower bound independent of $r$, for a plurisubharmonic function $ψ$ and a holomorphic function $F$ near the origin $o$. Jonsson-Mustaţă's conjecture was proved by Guan-Zhou depending on the truth of the strong openness conjecture. However, it is still a question whether one can prove Jonsson-Mustaţă's conjecture without using the strong openness property, and obtain a sharp effectiveness result for this conjecture. In this article, we use an $L^2$ method with the weight functions $ψ-\log|F|$ and firstly consider a module at at a boundary point of the sublevel sets of a plurisubharmonic function. By studying the minimal $L^{2}$ integrals on the sublevel sets of a plurisubharmonic function with respect to the module at the boundary point, we establish a concavity property of the minimal $L^{2}$ integrals. As applications, we obtain a sharp effectiveness result related to Jonsson-Mustaţă's conjecture, which completes the approach from the conjecture to the strong openness property. We also obtain a strong openness property of the module and a lower semi-continuity property with respect to the module. |
| title | Boundary points, Minimal $L^{2}$ integrals and Concavity property |
| topic | Complex Variables Algebraic Geometry |
| url | https://arxiv.org/abs/2203.01648 |