On Skoda's theorem for Nadel-Lebesgue multiplier ideal sheaves on singular complex spaces and regularity of weak K\"ahler-Einstein metrics

TL;DR

This paper characterizes regular points on singular complex spaces using vanishing, Ricci curvature positivity, and $ar{ ext{d}}$-equation solvability, and shows that weak K"ahler-Einstein metrics on such spaces are generally not smooth.

Contribution

It extends Skoda's theorem for Nadel-Lebesgue multiplier ideal sheaves to singular spaces and analyzes the regularity of weak K"ahler-Einstein metrics on singular varieties.

Findings

Weak K"ahler-Einstein metrics on singular varieties are generally not smooth.

Regular points characterized by local vanishing and Ricci positivity.

No singular normal K"ahler space admits a K"ahler-Einstein metric on the regular locus.

Abstract

In this article, we will characterize regular points respectively by the local vanishing, positivity of the Ricci curvature and $L^2$-solvability of the $\overline\partial$-equation together with Skoda's theorem for Nadel-Lebesgue multiplier ideal sheaves associated to plurisubharmonic (psh) functions on any (reduced) complex space of pure dimension. As a by-product, we show that any weak K\"ahler-Einstein metric on \emph{singular} $\mathbb{Q}$-Fano/Calabi-Yau/general type varieties cannot be smooth, and that in general there exists no \emph{singular} normal K\"ahler complex space such that the K\"ahler metric is K\"ahler-Einstein on the regular locus.