Vanishing of Local Cohomology with Applications to Hodge Theory

TL;DR

This paper investigates the vanishing properties of local cohomology in the context of mixed Hodge modules and applies these results to understand the Hodge structure of cohomology groups on quasi-projective varieties.

Contribution

It introduces a novel approach using local cohomology and mixed Hodge module theory to analyze the Hodge structure of cohomology in algebraic geometry.

Findings

Analysis of graded pieces of the de Rham complex

Results on the Hodge structure of low-degree cohomology

Application of local cohomology to Hodge theory

Abstract

Let $\textbf{H} = ((H, F^{\bullet}), L)$ be a polarized variation of Hodge structure on a smooth quasi-projective variety $U.$ By M. Saito's theory of mixed Hodge modules, the variation of Hodge structure $\textbf{H}$ can be viewed as a polarized Hodge module $\mathcal{M} \in HM(U).$ Let $X$ be a compactification of $U,$ and $j:U \hookrightarrow X$ is the natural map. In this paper, we use local cohomology with mixed Hodge module theory to study $j_{+}\mathcal{M} \in D^{b}MHM(X).$ In particular, we study the graded pieces of the de Rham complex $Gr^{F}_{p}DR(j_{+}\mathcal{M}) \in D^{b}_{coh}(X),$ and the Hodge structure of $H^{i}(U,L)$ for $i$ in sufficiently low degrees.