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 \(\big \{c_o^F(\psi )\psi -\log |F|<\log r\big \}\) $\{c_o^F(\psi )\psi -log|F|<logr\}$ divided by \(r^2\) $r^2$ has a uniform positive lower bound independent of r, for a plurisubharmonic function \(\psi \) $\psi$ and a holomorphic function F near the origin o. After proving the strong openness conjecture, Guan–Zhou proved Jonsson–Mustaţă’s conjecture based on the truth of the strong openness conjecture. In this article, we use an \(L^2\) $L^2$ method with the weight functions \(\psi -\log |F|\) $\psi -log|F|$ and first consider a module at a boundary point of the sublevel sets of a plurisubharmonic function. By studying the minimal \(L^{2}\) $L^2$ 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 \(L^{2}\) $L^2$ integrals. As applications, we obtain a sharp effectiveness result related to Jonsson–Mustaţă’s conjecture independent of the truth of the strong openness conjecture, which completes the approach from Jonsson–Mustaţă’s conjecture to the strong openness conjecture. We also obtain a strong openness property of the module and a lower semi-continuity property with respect to the module.