Schiffer variations and the generic Torelli theorem for hypersurfaces

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.