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Torelli theorems for singularities

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Summary

In this paper we study how μ=const deformations of an isolated hypersurface singularity affect the mixed Hodge structure (MHS) on the cohomology of the Milnor fiber. Using Scherk and Steenbrink'sD description of this MHS and the related approach of Varchenko, we give a necessary and sufficient condition for the infinitesimal Torelli theorem to hold and identify the μ=const stratum in the discriminant of the miniversal deformation. After examining the geometric reasons causing Torelli to fail for certain deformations, we define a local moduli space and prove a local Torelli theorem for singularities. In the process we complete the work of M. Saito and Malgrange on K. Saito's conjecture about primitive forms.

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Author notes

  1. Yakov Karpishpan

    Present address: Department of Mathematics, The Johns Hopkins University, 21218, Baltimore, MD, USA

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  1. Department of Mathematics, Massachusetts Institute of Technology, 02139, Cambridge, MA, USA

    Yakov Karpishpan

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  1. Yakov Karpishpan
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Research partially supported by the Sloan Foundation Doctoral Dissertation Fellowship and by NSF Grant DMS 8801743

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Karpishpan, Y. Torelli theorems for singularities. Invent Math 100, 97–141 (1990). https://doi.org/10.1007/BF01231182

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