Serre algebra, matrix factorization and categorical Torelli theorem for hypersurfaces

Author:

Xun Lin, Shizhuo Zhang

Keyword:

Mathematics, Algebraic Geometry, Algebraic Geometry (math.AG), Mathematical Physics (math-ph)

journal:

date:

2023-10-14 16:00:00

Abstract

Let $X$ be a smooth Fano variety. We attach a bi-graded associative algebra $\mathcal{A}_{S}=\bigoplus_{i,j\in \mathbb{Z}} \mathrm{Hom}(\mathrm{Id},S_{\mathcal{K}u(X)}^{i}[j])$ to the Kuznetsov component $\mathcal{K}u(X)$ whenever it is defined. Then we construct a natural sub-algebra of $\mathcal{A}_{S}$ when $X$ is a Fano hypersurface and establish its relation with Jacobian ring $J(X)$. As an application, we prove a categorical Torelli theorem for Fano hypersurface $X\subset\mathbb{P}^n(n\geq 2)$ of degree $d$ if $\mathrm{gcd}(n+1,d)=1.$ In addition, we give a new proof of the $[Pir22, Theorem 1.2]$ using a similar idea.

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