Limiting mixed Hodge structures on the relative log de Rham cohomology groups of a projective semistable log smooth degeneration

TL;DR

This paper establishes that the relative log de Rham cohomology groups in a projective semistable log smooth degeneration naturally admit a limiting mixed Hodge structure, extending Hodge theory to a broader class of degenerations.

Contribution

It constructs a natural limiting mixed Hodge structure on the relative log de Rham cohomology groups for semistable log smooth degenerations, with explicit filtrations and nilpotent endomorphisms.

Findings

Existence of a natural limiting mixed Hodge structure

Construction of filtrations and nilpotent endomorphisms

Partial properties of a nilpotent orbit in multiple variables

Abstract

We prove that the relative log de Rham cohomology groups of a projective semistable log smooth degeneration admit a natural \textit{limiting} mixed Hodge structure. More precisely, we construct a family of increasing filtrations and a family of nilpotent endomorphisms on the relative log de Rham cohomology groups and show that they satisfy a part of good properties of a nilpotnet orbit in several variables.