Kodaira vanishing for singular varieties revisited

TL;DR

This paper revisits and corrects a Kodaira-type vanishing theorem for singular varieties, extending its applicability by relating cohomology vanishing to the depth and dimension of the singular locus.

Contribution

It corrects and slightly strengthens a previous vanishing theorem for singular varieties, establishing new conditions for cohomology vanishing based on singularity properties.

Findings

Corrected proof of Kodaira vanishing for singular varieties

Extended vanishing range depending on singular locus depth and dimension

Applicable to nef and big line bundles in characteristic zero

Abstract

We correct the proof and slightly strengthen a Kodaira-type vanishing theorem for singular varieties originally due to Jaffe and the first author. Specifically, we show that if $L$ is a nef and big line bundle on a projective variety of characteristic zero, the $i^{\text{th}}$ cohomology of $L^{-1}$ vanishes for $i$ in a range determined by the depth and dimension of the singular locus.