Categorical representability and intermediate Jacobians of Fano threefolds

TL;DR

This paper explores the concept of categorical representability of projective varieties, especially complex threefolds, and demonstrates how it relates to classical properties and the reconstruction of intermediate Jacobians, with implications for birational geometry.

Contribution

It introduces a new notion of categorical representability based on semiorthogonal decompositions and connects it to classical geometric invariants of threefolds.

Findings

Reconstruction of intermediate Jacobians from semiorthogonal decompositions.

Establishment of relations between categorical and classical representability.

Examples illustrating how categorical representability informs birational properties.

Abstract

We define, basing upon semiorthogonal decompositions of $\Db(X)$, categorical representability of a projective variety $X$ and describe its relation with classical representabilities of the Chow ring. For complex threefolds satisfying both classical and categorical representability assumptions, we reconstruct the intermediate Jacobian from the semiorthogonal decomposition. We discuss finally how categorical representability can give useful information on the birational properties of $X$ by providing examples and stating open questions.