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Equivariant Kuznetsov components for cubic fourfolds with a symplectic involution

Laure Flapan, Sarah Frei, Lisa Marquand

Abstract

We study the equivariant Kuznetsov component $\mathrm{Ku}_G(X)$ of a general cubic fourfold $X$ with a symplectic involution. We show that $\mathrm{Ku}_G(X)$ is equivalent to the derived category $D^b(S)$ of a $K3$ surface $S$, where $S$ is given as a component of the fixed locus of the induced symplectic action on the Fano variety of lines on $X$.

Equivariant Kuznetsov components for cubic fourfolds with a symplectic involution

Abstract

We study the equivariant Kuznetsov component of a general cubic fourfold with a symplectic involution. We show that is equivalent to the derived category of a surface , where is given as a component of the fixed locus of the induced symplectic action on the Fano variety of lines on .

Paper Structure

This paper contains 11 sections, 20 theorems, 34 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Calabi--Yau categories and Hochschild homology
  4. $G$-equivariant categories
  5. Root stacks
  6. Hochschild homology of $\mathrm{Ku}_G(X)$
  7. The geometry of $X$
  8. The geometry of the quotient
  9. Two decompositions for $D^b_G(\widetilde{X})$
  10. Hochschild homology of $\mathrm{Ku}_G(X)$
  11. The equivalence

Key Result

Theorem 1.1

Let $X$ be a general cubic fourfold with a symplectic involution $\phi\in \mathrm{Aut}(X)$ and let $G:=\langle \phi\rangle \cong \mathbb{Z}/2\mathbb{Z}$. Then there is an equivalence of categories where $S\subset F(X)$ is the $K3$ component of the fixed locus of the induced action of $G$ on the Fano variety of lines of $X$.

Theorems & Definitions (35)

  • Theorem 1.1
  • Proposition 1.2: \ref{['prop: Z']}
  • Remark 1.3
  • Proposition 2.1
  • Lemma 2.2
  • Proposition 2.3
  • Lemma 2.4: Bridgeland's trick
  • Definition 2.5
  • Definition 2.6
  • Remark 2.7
  • ...and 25 more