Singularities of variations of mixed Hodge structure

TL;DR

This paper establishes a criterion for the existence of a limiting mixed Hodge structure in variations of mixed Hodge structures over punctured disks, extending Schmid's theorem to the mixed case and linking it to the extension of certain gradings.

Contribution

It provides a Hodge-theoretic condition for the extension of gradings that determine the limiting mixed Hodge structure in mixed Hodge variations.

Findings

Limiting mixed Hodge structures exist if certain gradings extend smoothly.

The result generalizes Schmid's theorem from pure to mixed Hodge structures.

Provides a criterion for the existence of the relative monodromy weight filtration.

Abstract

We prove that a variation of graded-polarizable mixed Hodge structure over a punctured disk with unipotent monodromy, has a limiting mixed Hodge structure at the puncture (i.e., it is admissible in the sense of [SZ]) which splits over $\R$, if and only if certain grading of the complexified weight filtration, depending smoothly on the Hodge filtration, extends across the puncture. In particular, the result exactly supplements Schmid's Theorem for pure structures, which holds for the graded variation, and gives a Hodge-theoretic condition for the relative monodromy weight filtration to exist.