Serre algebra, matrix factorization and categorical Torelli theorem for hypersurfaces

Abstract

Let $X$ be a smooth Fano variety. We attach a bi-graded associative algebra $\mathrm{HS}(\mathcal{K}u(X))={⨁}_{i,j\in \mathbb{Z}}\mathrm{Hom}(\mathrm{Id},S_{\mathcal{K}u(X)}^i[j])$ to the Kuznetsov component $\mathcal{K}u(X)$ whenever it is defined. Then we construct a natural sub-algebra of $\mathrm{HS}(\mathcal{K}u(X))$ when $X$ is a Fano hypersurface and establish its relation with Jacobian ring $\mathrm{Jac}(X)$. As an application, we prove a categorical Torelli theorem for Fano hypersurface $X\subset {\mathbb{P}}^n(n\ge 2)$ of degree $d$ if $\gcd (n+1,d)=1.$ In addition, we give a new proof of the $[Pir22,Theorem1.2]$ using a similar idea.

13 pages, final version. To appear in Math Ann