Limit mixed Hodge structures of hyperk\"ahler manifolds

TL;DR

This paper investigates the behavior of limit mixed Hodge structures in degenerating families of hyperk"ahler manifolds, revealing that maximal unipotency in monodromy leads to Hodge-Tate type structures across all cohomology groups.

Contribution

It demonstrates that under maximal unipotency, the limit mixed Hodge structures on all cohomology groups are of Hodge-Tate type, extending understanding of degenerations in hyperk"ahler geometry.

Findings

Limit mixed Hodge structures are of Hodge-Tate type under maximal unipotency.

Monodromy action on H^2 determines the structure of limits.

Extends Deligne's work to hyperk"ahler degenerations.

Abstract

This note is inspired by the work of Deligne on the local behavior of Hodge structures at infinity. We study limit mixed Hodge structures of degenerating families of compact hyperk\"ahler manifolds. We show that when the monodromy action on $H^2$ has maximal index of unipotency, the limit mixed Hodge structures on all cohomology groups are of Hodge-Tate type.