Categorical Torelli theorem for smooth Fano varieties

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 $\mathrm{\mid }H\mathrm{\mid }$ 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 $-q{K}_X$ is very ample and the corresponding embedding is cut out by equations of degree at most $(\mathrm{d}\mathrm{i}\mathrm{m}X-1)\mathrm{/}q$. 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.