Strong openness conjecture and related problems for plurisubharmonic functions

TL;DR

This paper proves the strong openness conjecture for multiplier ideal sheaves of plurisubharmonic functions, confirms related conjectures on volume growth and jumping numbers, and offers new proofs for existing conjectures without relying on the ACC conjecture.

Contribution

It solves the strong openness conjecture for multiplier ideal sheaves and confirms related conjectures on volume growth and jumping numbers, providing new proofs and applications.

Findings

Proved the strong openness conjecture for multiplier ideal sheaves.

Confirmed conjectures on volume growth of sublevel sets and jumping numbers.

Provided a new proof of the lower semicontinuity conjecture without the ACC conjecture.

Abstract

In this article, we solve the strong openness conjecture on the multiplier ideal sheaves for the plurisubharmonic functions posed by Demailly. We prove two conjectures about the growth of the volumes of the sublevel sets of plurisubharmonic functions related to the complex singularity exponents and quasi-plurisubharmonic functions related to the jumping numbers, which were posed by Demailly-Koll\'{a}r and Jonsson-Mustat\u{a} respectively. We give a new proof of a lower semicontinuity conjecture posed by Demailly-Koll\'{a}r without using the ACC conjecture. Other applications by combining with well-known results are also mentioned.