Equivalence of the sharp effectiveness results of strong openness property

TL;DR

This paper demonstrates the equivalence of two different approaches to sharp effectiveness results related to the strong openness property of multiplier ideal sheaves, connecting $\xi$-Bergman kernels and minimal $L^2$ integrals.

Contribution

It establishes the equivalence between the $\xi$-Bergman kernel approach and the minimal $L^2$ integrals approach for sharp effectiveness results in strong openness property.

Findings

Proves the equivalence of two methods for sharp effectiveness results

Bridges the gap between $\xi$-Bergman kernels and minimal $L^2$ integrals approaches

Enhances understanding of the strong openness property in complex analysis

Abstract

In this paper, we show the equivalence of the sharp effectiveness results of the strong openness property of multiplier ideal sheaves obtained in \cite{BG1} using $\xi-$Bergman kernels and in \cite{Guan19} using minimal $L^2$ integrals.