Normal crossing singularities and Hodge theory over Artin rings

TL;DR

This paper develops a Hodge theory for simple normal crossing varieties over Artinian rings, introducing mixed Hodge structures over Artin rings and proving properties of Hodge bundle maps.

Contribution

It introduces the concept of mixed Hodge structures over Artin rings and applies it to analyze the cohomology of normal crossing varieties over Artinian bases.

Findings

Maps between graded Hodge bundles have constant rank

Established a Hodge theory framework over Artinian rings

Defined mixed Hodge structures over Artin rings

Abstract

We develop a Hodge theory for relative simple normal crossing varieties over an Artinian base scheme. We introduce the notion of a mixed Hodge structure over an Artin ring, which axiomatizes the structure that is found on the cohomology of such a variety. As an application we prove that the maps between the graded pieces of the Hodge bundles have constant rank.