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Schiffer variations and the generic Torelli theorem for hypersurfaces

Part of: Cycles and subschemes Families, fibrations Surfaces and higher-dimensional varieties

Published online by Cambridge University Press:  08 February 2022

Claire Voisin
Claire Voisin*
Affiliation:
CNRS, IMJ-PRG, 4 Place Jussieu, 75005 Paris, France [email protected]
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Abstract

We prove the generic Torelli theorem for hypersurfaces in $\mathbb {P}^{n}$ of degree $d$ dividing $n+1$, for $d$ sufficiently large. Our proof involves the higher-order study of the variation of Hodge structure along particular one-parameter families of hypersurfaces that we call ‘Schiffer variations.’ We also analyze the case of degree $4$. Combined with Donagi's generic Torelli theorem and results of Cox and Green, this shows that the generic Torelli theorem for hypersurfaces holds with finitely many exceptions.

Keywords

Hodge structuresvariation of Hodge structurehypersurfacesTorelli theorem

MSC classification

Primary: 14C30: Transcendental methods, Hodge theory , Hodge conjecture 14C34: Torelli problem 14D07: Variation of Hodge structures 14J70: Hypersurfaces
Type
Research Article
Information
Compositio Mathematica , Volume 158 , Issue 1 , January 2022 , pp. 89 - 122
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Copyright
© 2022 The Author(s). The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence

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Footnotes

The author is supported by the ERC Synergy Grant HyperK (Grant agreement No. 854361).

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