Infinitesimal categorical Torelli theorems for Fano threefolds

TL;DR

This paper establishes infinitesimal categorical Torelli theorems for certain Fano threefolds by relating Hochschild (co)homology to classical Torelli results, advancing understanding of derived categories and period maps.

Contribution

It introduces a categorical infinitesimal Torelli theorem for Fano threefolds and relates it to classical and Kuznetsov component Torelli theorems using Hochschild (co)homology.

Findings

Proves infinitesimal categorical Torelli theorem for prime Fano threefolds.

Relates classical and categorical Torelli theorems via Hochschild (co)homology.

Restates the Debarre-Iliev-Manivel conjecture infinitesimally.

Abstract

Let $X$ be a smooth Fano variety and $\mathcal{K}u(X)$ the Kuznetsov component. Torelli theorems for $\mathcal{K}u(X)$ says that it is uniquely determined by a polarized abelian variety attached to it. An infinitesimal Torelli theorem for $X$ says that the differential of the period map is injective. A categorical variant of infinitesimal Torelli theorem for $X$ says that the morphism $H^1(X,T_X)\xrightarrow{\eta} HH^2(\mathcal{K}u(X))$ is injective. In the present article, we use the machinery of Hochschild (co)homology to relate the three Torelli-type theorems for smooth Fano varieties via a commutative diagram. As an application, we first prove infinitesimal categorical Torelli theorem for a class of prime Fano threefolds. Then we prove a restatement of the Debarre-Iliev-Manivel conjecture infinitesimally.