Real Seifert forms and polarizing forms of Steenbrink mixed Hodge structures

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.