Categorical Torelli theorem for smooth Fano varieties

Abstract

Let $X$ be a smooth Fano variety and $H$ an ample line bundle with $-K_X=rH$ and $r\le\dim X-2$. We prove that the appropriately defined Kuznetsov component of $X$ with respect to $H$, together with its rotation functor, determines $(X,H)$ when $H$ is very ample, and more generally when $H$ is globally generated and the image of the morphism defined by $|H|$ has a hypersurface as projective dual. As a consequence we prove Perry's conjecture, which predicts that a smooth Fano variety $X$ is determined up to isomorphism by the orthogonal complement $\mathcal{O}_X^{\perp}$ of its structure sheaf in $\mathrm{D}^b(X)$, for all smooth Fano varieties of dimension at least three with very ample anticanonical bundle, and more generally with globally generated anticanonical bundle whose image has a hypersurface as dual. We also prove it when a multiple $-qK_X$ is very ample and the corresponding embedding is cut out by equations of degree at most $(\mathrm{dim} X-1)/q$. The idea is to read off $X$ from its hyperplane sections. We show that the Kuznetsov component with its rotation functor contains the residual categories of all hyperplane sections of $X$, singular ones included. The sections whose categories are not smooth form the projective dual of $X$, and classical projective duality recovers $X$. When $H$ is not very ample, the nodes of a single tangent hyperplane section, defined over a function field and detected by its singularity category, recover $X$ as a finite cover of its image. Applications include Fano hypersurfaces, all prime Fano threefolds of index one and genus at least three, cyclic covers of quadrics, Fano complete intersections and Gushel--Mukai varieties.