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Demailly's approximation of general weights

Shijie Bao, Qi'an Guan

TL;DR

The paper addresses whether Demailly's approximation $V_m=(1/(2m))\log K_{mV}$ converges for general, non-psh weights. It introduces the psh envelope $\widetilde{V}=\sup\{\psi\in \mathrm{Psh}(\Omega): \psi\le V\}$ and the regularization concept $V^\star$, establishing that if $V$ is weakly upper semi-continuous on a bounded pseudoconvex domain, then $V_m\to \widetilde{V}$ pointwise on $\Omega$; continuity implies $V^\star=V$ and the same convergence. The proof leverages Bergman-kernel comparisons, mean-value inequalities, and the Ohsawa–Takegoshi $L^2$ extension theorem to obtain matching limsup and liminf bounds, yielding convergence. This extends Demailly's approximation to a broader class of weights and connects to analytic-singularity constructions in complex geometry.

Abstract

In this note, we demonstrate the convergence of the Demailly approximation of a general (weakly) upper semi-continuous weight.

Demailly's approximation of general weights

TL;DR

The paper addresses whether Demailly's approximation converges for general, non-psh weights. It introduces the psh envelope and the regularization concept , establishing that if is weakly upper semi-continuous on a bounded pseudoconvex domain, then pointwise on ; continuity implies and the same convergence. The proof leverages Bergman-kernel comparisons, mean-value inequalities, and the Ohsawa–Takegoshi extension theorem to obtain matching limsup and liminf bounds, yielding convergence. This extends Demailly's approximation to a broader class of weights and connects to analytic-singularity constructions in complex geometry.

Abstract

In this note, we demonstrate the convergence of the Demailly approximation of a general (weakly) upper semi-continuous weight.

Paper Structure

This paper contains 3 sections, 2 theorems, 16 equations.

Table of Contents

  1. Introduction
  2. Demailly's approximation of upper semi-continuous weights
  3. Proof of the Main Theorem

Key Result

Theorem 1.1

Let $V\in\mathrm{Psh}(\Omega)$. Then there are constants $C_1, C_2>0$ independent of $m$ such that for every $z\in\Omega$ and $r<\mathop{\mathrm{dist}}\nolimits(z, \partial\Omega)$. In particular, $V_m$ converges to $V$ pointwise and in $L_{\mathrm{loc}}^1$ topology on $\Omega$ when $m\to +\infty$.

Theorems & Definitions (4)

  • Theorem 1.1: Demailly's approximation theorem Dem92
  • Theorem 1.3: Main Theorem
  • Remark 1.4
  • proof : Proof of Theorem \ref{['thm-main.thm']}