Derived categories of Fano varieties of lines

TL;DR

This paper investigates the derived categories of Fano varieties of lines in smooth cubic fourfolds, providing evidence for a conjecture relating them to the Hilbert square of the Kuznetsov component, with proofs in specific cases.

Contribution

It offers new evidence supporting Galkin's conjecture and proves it for generic Fano varieties with a rational Lagrangian fibration, linking Hodge structures.

Findings

Confirmed the conjecture for certain Fano varieties

Established isometry of associated Hodge structures

Connected derived categories to Hilbert squares of Kuznetsov components

Abstract

We gather evidence for a conjecture of Galkin predicting the derived category of the Fano variety of lines contained in a smooth cubic fourfold to be equivalent to the Hilbert square of the Kuznetsov component of the derived category of the cubic. We prove the conjecture for generic Fano varieties admitting a rational Lagrangian fibration and show that the natural Hodge structures of weight two associated with the Fano variety and the Hilbert square are isometric.