Hodge adjacency conditions for singularities

TL;DR

This paper extends the understanding of Hodge theory by establishing compatibility relations for mixed Hodge numbers in families of singularities, broadening the scope beyond normal-crossing cases.

Contribution

It introduces generalized polarized relations for $k$-Du Bois singularities and explores their combinatorial properties, advancing the theoretical framework of asymptotic Hodge theory.

Findings

Compatibility relations between mixed Hodge numbers are established.

The notion of polarized relations is extended beyond normal-crossing boundaries.

Combinatorial properties of weak polarized relations graphs are studied.

Abstract

We prove compatibility relations between mixed Hodge numbers of $k$-Du Bois fibers in flat projective families and versal deformations of isolated $k$-Du Bois singularities. These extend the notion of polarized relations in asymptotic Hodge theory beyond the normal-crossing boundary case, and we study combinatorial properties of the resulting weak polarized relations graphs.