Infinitesimal categorical Torelli theorems for Fano threefolds

Let X be a smooth Fano variety and Ku(X) its Kuznetsov component. A Torelli theorem for Ku(X) states that Ku(X) is uniquely determined by a certain polarized abelian variety associated to it. An infinitesimal Torelli theorem for X states that the differential of the period map is injective. A catego...

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Bibliographic Details
Published in:Journal of pure and applied algebra 2023-12, Vol.227 (12), p.107418, Article 107418
Main Authors: Jacovskis, Augustinas, Lin, Xun, Liu, Zhiyu, Zhang, Shizhuo
Format: Article
Language:English
Subjects:
Bridgeland moduli spaces
Brill-Noether locus
Categorical Torelli theorems
Derived categories
Fano threefolds
Kuznetsov components
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Summary:Let X be a smooth Fano variety and Ku(X) its Kuznetsov component. A Torelli theorem for Ku(X) states that Ku(X) is uniquely determined by a certain polarized abelian variety associated to it. An infinitesimal Torelli theorem for X states that the differential of the period map is injective. A categorical variant of the infinitesimal Torelli theorem for X states that the morphism η:H1(X,TX)→HH2(Ku(X)) is injective. In the present article, we use the machinery of Hochschild (co)homology to relate the aforementioned three Torelli-type theorems for smooth Fano varieties via a commutative diagram. As an application, we prove infinitesimal categorical Torelli theorems for a class of prime Fano threefolds. We then prove, infinitesimally, a restatement of the Debarre–Iliev–Manivel conjecture regarding the general fibre of the period map for ordinary Gushel–Mukai threefolds.
ISSN:0022-4049
1873-1376
DOI:10.1016/j.jpaa.2023.107418