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Equivariant Kuznetsov Components of Certain Cubic Fourfolds

Xianyu Hu

Abstract

Let $M$ denote a specific cubic fourfold that accommodates a group action by $\mathbb{Z}/3\mathbb{Z}$. Through utilization of derived Mckay correspondence, we present a new proof establishing the identification of the equivariant Kuznetsov component in the equivariant derived category of $M$ with the derived category of certain abelian surface. This surface naturally emerges from the defining equation of the cubic fourfold $M$.

Equivariant Kuznetsov Components of Certain Cubic Fourfolds

Abstract

Let denote a specific cubic fourfold that accommodates a group action by . Through utilization of derived Mckay correspondence, we present a new proof establishing the identification of the equivariant Kuznetsov component in the equivariant derived category of with the derived category of certain abelian surface. This surface naturally emerges from the defining equation of the cubic fourfold .
Paper Structure (21 sections, 30 theorems, 103 equations, 1 table)

This paper contains 21 sections, 30 theorems, 103 equations, 1 table.

Table of Contents

  1. Preliminaries
  2. Semiorthogonal decomposition
  3. Mutations
  4. Equivariant derived categories
  5. Categorical actions
  6. Triangulated equivariant categories
  7. Equivariant sheaves and derived categories
  8. Derived Mckay correspondence and G-Hilbert schemes
  9. Equivariant Kuznetsov component
  10. Step I: The geometry of the G-Hilbert scheme Y
  11. The main goal and central idea
  12. The geometry of $\text{Bl}_{M^{G}}(M)/G$
  13. The identification of $G\text{-Hilb}_{{\mathbb{C}}}(M)$ with $\text{Bl}_{M^{G}}(M)/G$
  14. Step II: The equivariant Kuznetsov component of M
  15. An overview of the strategy of Step II
  16. ...and 6 more sections

Key Result

Theorem 1

(Theorem main theorem) $\mathcal{K}u_{M}^{G}\cong D^{b}(E_{0}\times E_{1})$.

Theorems & Definitions (61)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Proposition 4
  • Definition 1.1
  • Definition 1.2
  • Proposition 1.3
  • Lemma 1.4
  • proof
  • Lemma 1.5
  • ...and 51 more