Serre algebra, matrix factorization and categorical Torelli theorem for hypersurfaces

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