Variation Of Mixed Hodge Structures

TL;DR

This paper explores the properties and degenerations of variation of mixed Hodge structures (VMHS), a mathematical framework that captures geometric information of algebraic family fibers, extending classical Hodge theory.

Contribution

It provides a detailed analysis of degenerating behaviors, the existence of relative monodromy filtrations, and formalizes the concept of abstract admissible VMHS.

Findings

Degeneration properties of VMHS are characterized.

Existence of relative monodromy filtration is established.

Framework for abstract admissible VMHS is developed.

Abstract

Variation of mixed Hodge structures(VMHS), introduced by P. Deligne, is a linear structure reflecting the geometry on cohomology of the fibers of an algebraic family, generalizing variation of Hodge structures for smooth proper families, introduced by P. Griffiths. Hence, it is a strong tool to study the variation of the geometric structure of fibers of a morphism. We describe here the degenerating properties of a VMHS of geometric origin and the existence of a relative monodromy filtration, as well the definition and properties of abstract admissible VMHS.