Limits of Hodge structures via holonomic D-modules

TL;DR

This paper develops a new approach to constructing limiting mixed Hodge structures for degenerations of Kähler manifolds using holonomic D-modules, extending previous results and introducing sesquilinear pairings as a key tool.

Contribution

It generalizes Steenbrink's results by constructing limiting mixed Hodge structures without a Q-structure, employing sesquilinear pairings on D-modules, and proving a local invariant cycle theorem in this context.

Findings

Constructed limiting mixed Hodge structures via holonomic D-modules.

Replaced Q-structure with sesquilinear pairings on D-modules.

Proved the local invariant cycle theorem for this setting.

Abstract

We construct the limiting mixed Hodge structure of a degeneration of compact K\"ahler manifolds over the unit disk with a possibly non-reduced simple normal crossing singular central fiber via holonomic $\mathscr D$-modules, generalizing some results of Steenbrink. Our limiting mixed Hodge structure does not carry a $\mathbb Q$-structure; instead, we use sesquilinear pairings on $\mathscr D$-modules as a replacement. The limiting mixed Hodge structure can be computed by the cohomology of the cyclic coverings of certain intersections of components of the central fiber. Additionally, we prove the local invariant cycle theorem in this setting.