Seminar

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Let $X$ be a smooth Fano threefold such that the Kuznetsov component $Ku(X)$ is defined as the right orthogonal complement of an appropriate exceptional collection. It is a categorical variant of the intermediate Jacobian of Fano threefolds. I will introduce the abstract intermediate Jacobian for Kuznetsov component (given by Alex Perry) and show that it recovers classical intermediate Jacobian for smooth Fano threefolds.I will talk about Torelli problems, categorical Torelli problems and their infinitesimal versions. Then I will use the machinery of Hochschild (co)homology to relate three Torelli-type theorems for smooth Fano varieties via a commutative diagram. As an application, I first prove the infinitesimal categorical Torelli theorem for a class of prime Fano threefolds. Then I will prove a restatement of the Debarre-Iliev-Manivel conjecture infinitesimally.

Infinitesimal categorical Torelli theorems for Fano threefolds