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

Abstract

In this article, we consider minimal L 2 integrals on sublevel sets of a plurisubharmonic function with respect to a module at a boundary point of the sublevel sets, and establish a concavity property of the minimal L 2 integrals. As applications, we obtain a sharp effectiveness result related to a conjecture posed by Jonsson-Mustat ¸ȃ, 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.

Modules at boundary points, fiberwise Bergman kernels, and log-subharmonicity

“…Let f be a holomorphic function on D. Recall the definition of the minimal L 2 integral related to J ( [3,30])…”

“…Using the strong openness property, Guan-Zhou [40] proved Conjecture J-M. Independent of the strong openness property, Bao-Guan-Yuan [3] (see also [30]) considered minimal L 2 integrals with respect to a module at a boundary point of the sublevel sets, and established a concavity property of the minimal L 2 integrals, which deduced a sharp effectiveness result related to Conjecture J-M, and completed the approach from Conjecture J-M to the strong openness property.…”

Modules at boundary points, fiberwise Bergman kernels, and log-subharmonicity II -- on Stein manifolds

“…Can one complete the approach from Conjecture J-M to the strong openness property? Bao-Guan-Yuan [4] (see also [31]) gave an affirmative answer to the above question by establishing a concavity property of the minimal L 2 integrals with respect to a module at a boundary point of the sublevel sets.…”

Fiberwise Bergman kernels, vector bundles, and log-subharmonicity

“…Guan-Zhou [24] proved Conjecture J-M by using the strong openness property. Bao-Guan-Yuan [3] (see also [19] by Guan-Mi-Yuan) gave an approach to Conjecture J-M independent of the strong openness property by establishing a concavity property of the minimal L 2 integrals with respect to a module at a boundary point of the sub-level sets, and obtained a sharp effectiveness result of Conjecture J-M meanwhile.…”