Effectiveness of Demailly's strong openness conjecture and related problems

TL;DR

This paper investigates the effectiveness of Demailly's strong openness conjecture and related problems, providing explicit conditions, establishing effectiveness results, and demonstrating optimal bounds for related conjectures in complex analysis.

Contribution

It offers new explicit effectiveness results for Demailly's strong openness conjecture and related conjectures, extending previous solutions and identifying key properties like lower semicontinuity.

Findings

Established explicit effectiveness conditions for the conjecture.

Proved optimal effectiveness for Demailly-Kollár and Jonsson-Mustat conjectures.

Identified a lower semicontinuity property of plurisubharmonic functions with multipliers.

Abstract

In this article, stimulated by the effectiveness in Berndtsson's solution of the openness conjecture and continuing our solution of Demailly's strong openness conjecture, we discuss conditions to guarantee the effectiveness of the conjecture and establish such an effectiveness result. We explicitly point out a lower semicontinuity property of plurisubharmonic functions with a multiplier, which is implicitly contained in our paper arXiv:1401.7158. We also obtain optimal effectiveness of the conjectures of Demailly-Koll\'{a}r and Jonsson-Mustat\u{a} respectively.