The Fano of lines, the Kuznetsov component, and a flop

TL;DR

This paper proves that the Kuznetsov component of a cubic fourfold's derived category is equivalent to the derived category of the Fano variety of lines, confirming a conjecture and introducing a new derived equivalence for a specific flop.

Contribution

It establishes a derived equivalence between the Kuznetsov component and the Fano variety of lines, confirming Galkin's conjecture and introducing a novel flop equivalence.

Findings

Kuznetsov component is equivalent to the Fano of lines

Confirmed Galkin's conjecture about the non-commutative K3 surface

Introduced a new derived equivalence for a 12-dimensional flop

Abstract

The Kuznetsov component of the derived category of a cubic fourfold is a `non-commutative K3 surface'. Its symmetric square is hence a `non-commutative hyperkaehler fourfold'. We prove that this category is equivalent to the derived category of an actual hyperkaehler fourfold: the Fano of lines in the cubic. This verifies a conjecture of Galkin. One of the key steps in our proof is a new derived equivalence for a specific 12-dimensional flop.