2016/10/13 by Daniel Huybrechts, Huybrechts, Daniel, Jørgen Vold Rennemo +1 · 2 citations
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.1610.04128
openalex publication_date 2016/10/13 · openalex created_date 2022/10/03 · openalex updated_date 2026/07/28
The Jacobian ring J(X) of a smooth hypersurface determines its isomorphism\ntype. This has been used by Donagi and others to prove the generic global\nTorelli theorem for hypersurfaces in many cases. In Voisin's original proof of\nthe global Torelli theorem for smooth cubic fourfolds the Jacobian ring does\nnot intervene. In this paper we present a proof of the global Torelli theorem\nfor cubic fourfolds that relies on the Jacobian ring and the (derived) global\nTorelli theorem for K3 surfaces. It emphasizes, once again, the relation\nbetween K3 surfaces and smooth cubic fourfolds. More generally, for a variant\nof Hochschild cohomology of Kuznetsov's category (together with the degree\nshift functor) associated with an arbitrary smooth hypersurface we construct a\ngraded ring homomorphism from the Jacobian ring to it, which is shown to be\nbijective whenever Kuznetsov's category is a Calabi-Yau category.\n