arXiv:1204.6661 Normal crossing singularities and Hodge theory over Artin rings from arXiv Front: math.AG by Christian Lehn 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.
nLab page on Hodge theory generalized