Derived categories view on rationality problems

TL;DR

This paper explores how the structure of derived categories of smooth projective varieties relates to their birational properties, proposing a new birational invariant and discussing applications to specific classes of varieties.

Contribution

It introduces a new birational invariant based on derived categories and explores its potential applications to the geometry of Fano threefolds and cubic fourfolds.

Findings

Proposes a derived category analogue of the intermediate Jacobian.

Suggests a new birational invariant for smooth projective varieties.

Discusses potential applications to Fano threefolds and cubic fourfolds.

Abstract

We discuss a relation between the structure of derived categories of smooth projective varieties and their birational properties. We suggest a possible definition of a birational invariant, the derived category analogue of the intermediate Jacobian, and discuss its possible applications to the geometry of prime Fano threefolds and cubic fourfolds.