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 \(\textrm{HS}(\mathcal {K}u(X))=\bigoplus _{i,j\in \mathbb {Z}} \textrm{Hom}(\textrm{Id},S_{\mathcal {K}u(X)}^{i}[j])\) to the Kuznetsov component \(\mathcal {K}u(X)\) whenever it is defined. Then we construct a natural sub-algebra of \(\textrm{HS}(\mathcal {K}u(X))\) when X is a Fano hypersurface and establish its relation with Jacobian ring \(\textrm{Jac}(X)\). As an application, we prove a categorical Torelli theorem for Fano hypersurface \(X\subset \mathbb {P}^n(n\ge 2)\) of degree d if \(\textrm{gcd}((n+1),d)=1.\) In addition, we give a new proof of the main theorem [15, Theorem 1.2] using a similar idea.