Boundary points, Minimal $L^{2}$ integrals and Concavity property Article Swipe

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Shijie Bao , Qi’an Guan , Zheng Yuan · YOU? · · 2022 · Open Access · · DOI: https://doi.org/10.48550/arxiv.2203.01648

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.

Type

preprint

Language

en

Landing Page

http://arxiv.org/abs/2203.01648

PDF

https://arxiv.org/pdf/2203.01648

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green

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10

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https://openalex.org/W4221142556

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https://openalex.org/W4221142556

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DOI

https://doi.org/10.48550/arxiv.2203.01648

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Title

Boundary points, Minimal $L^{2}$ integrals and Concavity property

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Type

preprint

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Language

en

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Publication year

2022

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Publication date

2022-03-03

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Authors

Shijie Bao, Qi’an Guan, Zheng Yuan

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Landing page

https://arxiv.org/abs/2203.01648

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https://arxiv.org/pdf/2203.01648

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Open access

Yes

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OA status

green

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https://arxiv.org/pdf/2203.01648

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Property (philosophy), Boundary (topology), Mathematics, Mathematical analysis, Geometry, Applied mathematics, Philosophy, Epistemology

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0

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Related works (count)

10

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