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

Normal crossing singularities and Hodge theory over Artin rings