On the mixed Hodge structure associated to hypersurface singularities

TL;DR

This paper explores the mixed Hodge structure linked to hypersurface singularities, extending it over degenerations and analyzing the associated pairings and signatures.

Contribution

It establishes a new mixed Hodge structure on the module of relative differentials and analyzes the deformation of polarization and bilinear relations.

Findings

Extension of mixed Hodge structure over degenerate points

Deformation of polarization to Grothendieck residue pairing

Calculation of Hodge signature of Grothendieck pairing

Abstract

Let $f:\mathbb{C}^{n+1} \to \mathbb{C}$ be a germ of hypersurface with isolated singularity. One can associate to $f$ a polarized variation of mixed Hodge structure $\mathcal{H}$ over the punctured disc, where the Hodge filtration is the limit Hodge filtration of W. Schmid and J. Steenbrink. By the work of M. Saito and P. Deligne the VMHS associated to cohomologies of the fibers of $f$ can be extended over the degenerate point $0$ of disc. The new fiber obtained in this way is isomorphic to the module of relative differentials of $f$ denoted $\Omega_f$. A mixed Hodge structure can be defined on $\Omega_f$ in this way. The polarization on $\mathcal{H}$ deforms to Grothendieck residue pairing modified by a varying sign on the Hodge graded pieces in this process. This also proves the existence of a Riemann-Hodge bilinear relation for Grothendieck pairing and allow to calculate the Hodge signature of Grothendieck pairing.