Serre algebra, matrix factorization and categorical Torelli theorem for hypersurfaces

2023/10/15 by Xun Lin, Lin, Xun, Shizhuo Zhang +1
Physics and Astronomy · Mathematics · #Nonlinear Waves and Solitons #Algebraic structures and combinatorial models #Advanced Differential Equations and Dynamical Systems

paper · pdf · doi:10.48550/arxiv.2310.09927

Abstract

Let X be a smooth Fano variety. We attach a bi-graded associative algebra HS(Ku(X))=\bigoplus_{i,j∈ ℤ} Hom(Id,S_Ku(X)ⁱ[j]) to the Kuznetsov component Ku(X) whenever it is defined. Then we construct a natural sub-algebra of HS(Ku(X)) when X is a Fano hypersurface and establish its relation with Jacobian ring Jac(X). As an application, we prove a categorical Torelli theorem for Fano hypersurface X⊂ℙⁿ(n≥ 2) of degree d if gcd(n+1,d)=1. In addition, we give a new proof of the [Pir22, Theorem 1.2] using a similar idea.

Related

• IVHS via Kuznetsov components and categorical Torelli theorems for weighted hypersurfaces
• Categorical Torelli theorem for hypersurfaces
• Infinitesimal categorical Torelli theorems for Fano threefolds
• Kuznetsov's Fano threefold conjecture via Hochschild-Serre algebra
• Moduli spaces on the Kuznetsov component of Fano threefolds of index 2