论文标题
边界点，最小$ l^{2} $积分和凹陷属性
Boundary points, Minimal $L^{2}$ integrals and Concavity property
论文作者
论文摘要
为了证明乘数理想带的强烈的开放性猜想，Jonsson-Mustaţ构成了增强的猜想，并证明了二维情况，其中说：lebesguger衡量了集合$ \ big \ big \ big \ {c_o^f（c_o^f（ψ）ψ-\ f | f | f | f | f | f | f | $ r $，用于plurisubharmonic函数$ψ$和holomorphic函数$ f $附近的原点$ o $。乔恩·穆斯塔（Jonsson-Mustaţ）的猜想是由关 - 周证明，这取决于强烈的开放性猜想的真相。但是，仍然有一个问题，是否可以在不使用强大的开放性属性的情况下证明Jonsson-Mustaţ的猜想，并为此猜想获得急剧的有效性。 在本文中，我们将$ l^2 $方法与权重函数$ψ-\ log | f | $一起使用，并首先在plurisubharmonic函数的Sublevel集合的边界点上考虑一个模块。通过研究最小$ l^{2} $积分在边界点相对于模块的plurisubharmonic函数的超级集合，我们建立了最小$ l^{2} $积分的凹陷属性。作为应用程序，我们获得了与Jonsson-Mustaţ的猜想有关的敏锐有效性结果，该结果完成了从猜想到强大的开放性属性的方法。我们还获得了模块的强大开放性和相对于模块的较低半持续性。
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.
