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

TL;DR

This paper proves the global Torelli theorem for smooth cubic fourfolds using the Jacobian ring and Hochschild cohomology, highlighting the link between cubic fourfolds and K3 surfaces, and extends the approach to hypersurfaces via derived categories.

Contribution

It introduces a new proof of the Torelli theorem for cubic fourfolds leveraging the Jacobian ring and Hochschild cohomology, and constructs a graded ring homomorphism for hypersurfaces.

Findings

The Jacobian ring determines the isomorphism type of smooth hypersurfaces.

A bijective correspondence exists between the Jacobian ring and Hochschild cohomology for Calabi-Yau categories.

The proof emphasizes the relation between cubic fourfolds and K3 surfaces.

Abstract

The Jacobian ring J(X) of a smooth hypersurface determines its isomorphism type. This has been used by Donagi and others to prove the generic global Torelli theorem for hypersurfaces in many cases. In Voisin's original proof of the global Torelli theorem for smooth cubic fourfolds 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 relation between K3 surfaces and smooth cubic fourfolds. More generally, for a variant of Hochschild cohomology of Kuznetsov's category (together with the degree shift functor) associated with an arbitrary smooth hypersurface we construct a graded ring homomorphism from the Jacobian ring to it, which is shown to be bijective whenever Kuznetsov's category is a Calabi-Yau category.