IVHS via Kuznetsov components and categorical Torelli theorems for weighted hypersurfaces

Abstract

We study the categorical Torelli theorem for smooth (weighted) hypersurfaces in (weighted) projective spaces via the Hochschild--Serre algebra of its Kuznetsov component. In the first part of the paper, we show that a natural graded subalgebra of the Hochschild--Serre algebra of the Kuznetsov component of a degree $d$ weighted hypersurface in $\mathbb{P}(a_0,\ldots,a_n)$ reconstructs the graded subalgebra of the Jacobian ring generated by the degree $t:=\mathrm{gcd}(d,Σ_{i=0}^na_i)$ piece under mild assumptions. Using results of Donagi and Cox--Green, this gives a categorical Torelli theorem for most smooth hypersurfaces $Y$ of degree $d \le n$ in $\mathbb{P}^n$ such that $d$ does not divide $n+1$ (the exception being the cases of the form $(d,n) = (4, 4k + 2)$, for which a result of Voisin lets us deduce a generic categorical Torelli theorem when $k \ge 150$). Next, we show that the Jacobian ring of the Veronese double cone can be reconstructed from its graded subalgebra of even degree, thus proving a categorical Torelli theorem for the Veronese double cone. In the second part, we rebuild the infinitesimal Variation of Hodge structures of a series of (weighted) hypersurfaces from their Kuznetsov components via the Hochschild--Serre algebra. As a result, we prove categorical Torelli theorems for two classes of (weighted) hypersurfaces: $(1):$ Generalized Veronese double cone; $(2):$ Certain $k$-sheeted covering of $\mathbb{P}^n$, when they are generic. Then, we prove a refined categorical Torelli theorem for a Fano variety whose Kuznetsov component is a Calabi--Yau category of dimension $2m+1$. Finally, we prove the actual categorical Torelli theorem for generalized Veronese double cone and $k$-sheeted covering of $\mathbb{P}^n$.