Schiffer variations and the generic Torelli theorem for hypersurfaces

TL;DR

This paper demonstrates how to recover certain hypersurfaces from their Hodge structure variations and extends the generic Torelli theorem to more cases with finitely many exceptions.

Contribution

It introduces a method to recover hypersurfaces from finite order Hodge variations and broadens the scope of the generic Torelli theorem for hypersurfaces.

Findings

Recovery of hypersurfaces from finite order Hodge variations.

Extension of the generic Torelli theorem to additional cases.

Identification of finitely many exceptions to the theorem.

Abstract

We show how to recover a general hypersurface in $\mathbb{P}^n$ of sufficiently large degree $d$ dividing $n+1$, from its finite order variation of Hodge structure. We also analyze the two other series of cases not covered by Donagi's generic Torelli theorem. Combined with Donagi's theorem, this shows that the generic Torelli theorem for hypersurfaces holds with finitely many exceptions.