Categorical Torelli and Fourier-Mukai partners of hypersurfaces

Abstract

Let $X\subset\PP^{M-1}$ be a smooth hypersurface of degree $d<M$, and let $\Ku(X)$ be its Kuznetsov component, the orthogonal complement of the line bundles $\OO_X,\dots,\OO_X(M-d-1)$ in the derived category. We prove that $\Ku(X)$ determines $X$ for every smooth quartic in $\PP^{4k+1}$, $k\geq1$; together with earlier work, this gives categorical Torelli for every smooth hypersurface of degree $d<M$ with $d\nmid M$. The same algebraic reconstruction proves the classical generic Torelli theorem for these quartics, which completes generic Torelli for smooth hypersurfaces whose degree does not divide $M$, cubic surfaces excepted. When $d$ divides $M$, the Kuznetsov component is a Calabi--Yau category, as for cubic fourfolds, where categorical Torelli is known to fail. We show that, apart from cubic fourfolds, such a hypersurface has a number of Fourier--Mukai partners bounded by a constant depending only on $d$ and $M$, and that categorical Torelli holds in sufficiently large degree, in every dimension. The main tool is a deformation theorem: when geometric deformations surject onto the second Hochschild cohomology of the Kuznetsov components, an equivalence deforms together with the hypersurfaces, so statements proved for general hypersurfaces extend to all smooth ones.