Serre algebra, matrix factorization and categorical Torelli theorem for hypersurfaces

Xun Lin; Shizhuo Zhang

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arXiv · 2310.09927

Serre algebra, matrix factorization and categorical Torelli theorem for hypersurfaces

Xun Lin·Shizhuo Zhang

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Abstract

Let $X$ be a smooth Fano variety. We attach a bi-graded associative algebra $\mathrm{HS}(\mathcal{K}u(X))=\bigoplus_{i,j\in \mathbb{Z}} \mathrm{Hom}(\mathrm{Id},S_{\mathcal{K}u(X)}^{i}[j])$ to the Kuznetsov component $\mathcal{K}u(X)$ whenever it is defined. Then we construct a natural sub-algebra of $\mathrm{HS}(\mathcal{K}u(X))$ 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 $X\subset\mathbb{P}^n(n\geq 2)$ 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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BibTeX RIS

Xun Lin, Shizhuo Zhang. 2023-10-15. Serre algebra, matrix factorization and categorical Torelli theorem for hypersurfaces. https://arxiv.org/abs/2310.09927

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