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Twisted Arinkin transforms and derived categories of moduli spaces on Kuznetsov components

Moritz Hartlieb, Saket Shah

Abstract

In this note, we generalize results of Donagi and Pantev on twisted derived equivalences between elliptically fibered surfaces to higher dimensions. First, we establish a twisted derived equivalence between torsors under abelian schemes satisfying a certain compatibility condition. Then, relying on the work of Arinkin on compactified Jacobians, we extend the equivalence to twisted compactified Jacobians associated to curves on K3 surfaces. This positively answers a question stated by Mattei and Meinsma. We then extend a result of Bottini and Huybrechts for Fano varieties of lines on cubic fourfolds to general moduli spaces of Bridgeland-stable objects on Kuznetsov components admitting rational Lagrangian fibrations.

Twisted Arinkin transforms and derived categories of moduli spaces on Kuznetsov components

Abstract

In this note, we generalize results of Donagi and Pantev on twisted derived equivalences between elliptically fibered surfaces to higher dimensions. First, we establish a twisted derived equivalence between torsors under abelian schemes satisfying a certain compatibility condition. Then, relying on the work of Arinkin on compactified Jacobians, we extend the equivalence to twisted compactified Jacobians associated to curves on K3 surfaces. This positively answers a question stated by Mattei and Meinsma. We then extend a result of Bottini and Huybrechts for Fano varieties of lines on cubic fourfolds to general moduli spaces of Bridgeland-stable objects on Kuznetsov components admitting rational Lagrangian fibrations.
Paper Structure (13 sections, 36 theorems, 88 equations)

This paper contains 13 sections, 36 theorems, 88 equations.

Table of Contents

  1. Introduction
  2. Acknowledgments
  3. Conventions
  4. Background
  5. Good abelian fibrations and Arinkin sheaves
  6. Twisted sheaves
  7. Twisted compactified Jacobians
  8. Twisting Arinkin sheaves
  9. Twisted Fourier--Mukai transforms
  10. Identifying certain Brauer classes
  11. Brauer classes on twisted compactified Jacobians
  12. Markman--Mukai lattices and derived equivalences of K3 categories
  13. Twisted derived categories of moduli spaces on Kuznetsov components

Key Result

Theorem 1.1

Let $A$ be an abelian scheme over a quasi-projective non-singular variety $B$. Let $X$ be a torsor under $A$ and $Y$ a torsor under the dual abelian scheme $\check{A}$. Assume that $[X] \in H^1(B, \mathcal{A})$ is $n$-torsion and $[Y] \in H^1(B, \check{\mathcal{A}})$ is $n$-divisible. Then, there ar for all $\gamma \in \operatorname{Br}(B).$

Theorems & Definitions (72)

  • Theorem 1.1: Thm. \ref{['thm:main_abelian_schemes']}
  • Remark 1.2
  • Theorem 1.3: donagipantev
  • Theorem 1.4: Thm. \ref{['thm:main_hk']}
  • Corollary 1.5: Cor. \ref{['cor:generallangrangianfibered']}
  • Conjecture 1.6: Cf. zhang
  • Theorem 1.7: kemboisegal
  • Theorem 1.8: Thm. \ref{['thm:main_kuznetsov_component']}
  • Definition 2.1
  • Theorem 2.2
  • ...and 62 more