A Semi-Smooth Kodaira Vanishing Theorem and Demi-Normal Cyclic Covering Lemma

TL;DR

This paper extends the Kodaira Vanishing Theorem to semi-smooth projective varieties over C, utilizing cyclic covers and demonstrating that cyclic covers of demi-normal varieties remain demi-normal.

Contribution

It introduces a semi-smooth version of the Kodaira Vanishing Theorem and proves that cyclic covers of demi-normal varieties are also demi-normal, expanding the theorem's applicability.

Findings

Proves a vanishing theorem for semi-smooth varieties.

Shows cyclic covers of demi-normal varieties are demi-normal.

Extends classical results to broader class of varieties.

Abstract

The classical Kodaira Vanishing Theorem states that Hi(X, {\omega}X \otimes L) = 0 for i > 0, where X is a smooth projective variety over C and L is an ample line bundle on X. We prove an analogous vanishing result under the assumption that X is a semi-smooth projective variety over C. This is done by passing to the normalization of X, which is a smooth projective variety when X is semi-smooth. The classical vanishing result is proved using a cyclic cover. Therefore we include a proof that the cyclic cover of a demi-normal variety is again demi-normal.