IVHS via Kuznetsov components and categorical Torelli theorems for weighted hypersurfaces

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 $\mathbb{P}(a_0,\dots ,a_n)$ reconstructs the graded subalgebra of the Jacobian ring generated by the degree $t:=\mathrm{g}\mathrm{c}\mathrm{d}(d,{\mathrm{\Sigma }}_{i=0}^n{a}_i)$ 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 ${\mathbb{P}}^n$.