Boundary points, minimal L2 integrals and concavity property

Abstract

For the purpose of proving the strong openness conjecture of multiplier ideal sheaves, Jonsson-Mustat.a posed an enhanced conjecture and proved the two-dimensional case, which says that: the Lebesgue measure of the set { - log | F| < log r} divided by r 2 has a uniform positive lower bound independent of r, for a plurisubharmonic function. and a holomorphic function F near the origin o. After proving the strong openness conjecture, Guan-Zhou proved Jonsson-Mustat.a's conjecture based on the truth of the strong openness conjecture. In this article, we use an L2 method with the weight functions - log | F| and first consider a module at a boundary point of the sublevel sets of a plurisubharmonic function. By studying the minimal L2 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 L2 integrals. As applications, we obtain a sharp effectiveness result related to Jonsson-Mustata's conjecture independent of the truth of the strong openness conjecture, which completes the approach from Jonsson-Mustata'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.