Real Seifert forms and polarizing forms of Steenbrink mixed Hodge structures

TL;DR

This paper systematically describes various pairings associated with isolated hypersurface singularities, including Seifert and polarizing forms, and introduces new corrections to Steenbrink's Hodge filtration using Fourier-Laplace transforms.

Contribution

It provides a systematic and abstract framework for understanding pairings in singularity theory, including new corrections to Steenbrink's Hodge filtration and classifications of Seifert form pairs.

Findings

Classification of irreducible Seifert form pairs

Correction of Thom-Sebastiani formula for Steenbrink's Hodge filtration

Relation between Seifert form and polarizing form via Fourier-Laplace transform

Abstract

An isolated hypersurface singularity comes equipped with many different pairings on different spaces, the intersection form and the Seifert form on the Milnor lattice, a polarizing form for a mixed Hodge structure on a dual space, and a flat pairing on the cohomology bundle. This paper describes them and their relations systematically in an abstract setting. We expect applications also in other areas than singularity theory. A good part of the paper is elementary, but not well known: the classification of irreducible Seifert form pairs, the polarizing form on the generalized eigenspace with eigenvalue 1, an automorphism from a Fourier-Laplace transformation which involves the Gamma function and which relates Seifert form and polarizing form and a flat pairing on the cohomology bundle. New is a correction of a Thom-Sebastiani formula for Steenbrink's Hodge filtration in the case of singularities. It uses the Fourier-Laplace transformation. A special case is a square root of a Tate twist for Steenbrink mixed Hodge structures.