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Serre algebra, matrix factorization and categorical Torelli theorem for hypersurfaces

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Lin,  Xun
Max Planck Institute for Mathematics, Max Planck Society;

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Zhang,  Shizhuo
Max Planck Institute for Mathematics, Max Planck Society;

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2310.09927.pdf
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Citation

Lin, X., & Zhang, S. (2025). Serre algebra, matrix factorization and categorical Torelli theorem for hypersurfaces. Mathematische Annalen, 391(1), 163-177. doi:10.1007/s00208-024-02915-8.


Cite as: https://hdl.handle.net/21.11116/0000-000F-590E-F
Abstract
Let X be a smooth Fano variety. We attach a bi-graded associative algebra AS=⨁i,j∈ZHom(Id,SiKu(X)[j]) to the Kuznetsov component Ku(X) whenever it is defined. Then we construct a natural sub-algebra of AS 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⊂Pn(n≥2) of degree d if gcd(n+1,d)=1. In addition, we give a new proof of the [Pir22,Theorem1.2] using a similar idea.