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.
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This work was supported by the DFG Grant He2287/4-1 (SISYPH).
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Balnojan, S., Hertling, C. Real Seifert Forms and Polarizing Forms of Steenbrink Mixed Hodge Structures. Bull Braz Math Soc, New Series 50, 233–274 (2019). https://doi.org/10.1007/s00574-018-0107-7
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DOI: https://doi.org/10.1007/s00574-018-0107-7
Keywords
- Real Seifert form
- Isometric structure
- Polarized mixed Hodge structure
- Isolated hypersurface singularity
- Brieskorn lattice
- Thom–Sebastiani formula