Hochschild cohomology versus the Jacobian ring and the Torelli theorem for cubic fourfolds

The Jacobian ring J(X) of a smooth hypersurface X subset of Pn+1 determines the isomorphism type of X. This has been used by Donagi and others to prove the generic global Torelli theorem for hypersurfaces in many cases. However, in Voisin's original proof (and, in fact, in all other proofs) of the global Torelli theorem for smooth cubic four-folds X subset of P-5, the Jacobian ring does not intervene. In this paper, we present a proof of the global Torelli theorem for cubic fourfolds that relies on the Jacobian ring and the (derived) global Torelli theorem for K3 surfaces. It emphasizes, once again, the close and still mysterious relation between K3 surfaces and smooth cubic fourfolds. More generally, for a variant of Hochschild cohomology IIH*(A(X), (1)) of Kuznetsov's category A(X) (together with the degree-shift functor (1)) associated with an arbitrary smooth hypersurface X subset of Pn+1 of degree d <= n + 2, we construct a graded-ring homomorphism J(X) (sic) HH*(A(X), (1)), which is shown to be bijective whenever A(X) is a Calabi-Yau category.