Derived categories of Fano varieties of lines

TL;DR

The paper investigates the relationship between the derived category of the Fano variety of lines $F_X$ of a smooth cubic fourfold and the Hilbert square of the Kuznetsov component $\mathcal{A}_X$ of $D^{\rm b}(X)$. It proves the conjectured equivalence $D^{\rm b}(F_X) \simeq \mathcal{A}_X^{[2]}$ for a dense class of cubics, notably when $F_X$ is birational to $S^{[2]}$ or when $F_X$ carries a rational Lagrangian fibration with genericity, by combining Birational- and Brauer-theoretic techniques with twisted moduli spaces on K3 surfaces. It provides Hodge-theoretic evidence via a weight-two isometry between the naive Mukai lattice of $F_X$ and the Mukai lattice of $\mathcal{A}_X^{[2]}$, and explains how Beckmann–Taelman twists reconcile naive and derived-invariant lattices. The approach relies on twisted BKR equivalences, twisted Jacobian moduli, and a twisted Kawamata–Namikawa framework to connect $F_X$ to moduli spaces of twisted sheaves on K3 curves, yielding a robust path to verify the conjecture on broader families and to describe the induced cohomological involution.

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.

Theorems & Definitions (31)

• Conjecture 1
• Theorem 2
• Theorem 3
• Theorem 4
• Remark 1.1
• Proposition 1.2
• proof
• Lemma 1.3
• Theorem 2.1
• Lemma 2.2