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

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.