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.

In this article, we apply the methods of our work on Fontaine’s theory in equal characteristics to the φ/𝔖-modules of Breuil and Kisin. Thanks to a previous article of Kisin, this yields a new and rather elementary proof of the theorem ‘weakly admissible implies admissible’ of Colmez and Fontaine.

In this paper, we investigate the action of the ${\Bbb Q}$-cohomology of the compact dual $\widehat {X}$ of a compact Shimura Variety $S(Γ)$ on the ${\Bbb Q}$-cohomology of $S(Γ)$ under a cup product. We use this to split the cohomology of $S(Γ)$ into a direct sum of (not necessarily irreducible) ${{\Bbb Q}}$-Hodge structures. As an application, we prove that for the class of arithmetic subgroups of the unitary groups ${\rm U}(p,q)$ arising from Hermitian forms over CM fields, the Mumford–Tate groups associated to certain holomorphic cohomology classes on $S(Γ)$ are Abelian. As another application, we show that all classes of Hodge type (1,1) in H2 of unitary four-folds associated to the group ${\rm U}(2,2)$ are algebraic.