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Lifting derived equivalences of abelian surfaces to generalized Kummer varieties

Yuxuan Yang

Abstract

In this article, we study the $G$-autoequivalences of the derived category $\mathbf{D}^b_G(A)$ of $G$-equivariant objects for an abelian variety $A$ with $G$ being a finite subgroup of $\mathrm{Pic}^0(A)$. We provide a result analogue to Orlov's short exact sequence for derived equivalences of abelian varieties. It can be generalized to the derived equivalences of abelian varieties for a same $G$ in general. Furthermore, we find derived equivalences of generalized Kummer varieties by lifting derived equivalences of abelian surfaces using the $G$-equivariant version of Orlov's short exact sequence and some ``splitting" propositions.

Lifting derived equivalences of abelian surfaces to generalized Kummer varieties

Abstract

In this article, we study the -autoequivalences of the derived category of -equivariant objects for an abelian variety with being a finite subgroup of . We provide a result analogue to Orlov's short exact sequence for derived equivalences of abelian varieties. It can be generalized to the derived equivalences of abelian varieties for a same in general. Furthermore, we find derived equivalences of generalized Kummer varieties by lifting derived equivalences of abelian surfaces using the -equivariant version of Orlov's short exact sequence and some ``splitting" propositions.

Paper Structure

This paper contains 16 sections, 29 theorems, 256 equations.

Table of Contents

  1. Relating derived equivalences of isogeneous abelian varieties
  2. Group actions of categories, $G$-functors and equivariant categories.
  3. $G$-functors between equivariant categories of abelian varieties
  4. Relating the images of $\lambda_q(f,\sigma)$ (and $\lambda_{q,q'}(f,\sigma)$) and $f$ via Orlov's representation.
  5. Relating the cohomological actions of $\lambda_q(f,\sigma)$ (and $\lambda_{q,q'}(f,\sigma)$) and $f$ on the spin representation.
  6. Surjectivity in Theorem \ref{['Th_GrhoA']} and Theorem \ref{["Th_GrhoAA'"]}
  7. Final part of the proof of Theorem \ref{['Th_GrhoA']} and Theorem \ref{["Th_GrhoAA'"]}
  8. Finite index properties for Theorem \ref{['Th_GrhoA']}
  9. Symplectic maps Versus Hodge isometries
  10. Generalized Kummer varieties
  11. Lifting $\mathcal{G}$-autoequivalences of $\mathbf{D}^b_{\mathcal{G}}(A)$ to autoequivalences of $\mathbf{D}^b(\mathrm{Kum}^{n-1}(A)\times A)$
  12. Lifting $\mathcal{G}$-autoequivalences of $\mathbf{D}^b_{\mathcal{G}}(A)$ to autoequivalences of $\mathbf{D}^b(\mathrm{Kum}^{n-1}(A))$
  13. Lifting some autoequivalences of $\mathbf{D}^b(A)$ to autoequivalences of $\mathbf{D}^b(\mathrm{Kum}^{n-1}(A))$
  14. Derived equivalences between generalized Kummer varieties from Theorem \ref{["split_NAANA'A'"]}
  15. Derived autoequivalences of Kummer K3 surfaces
  16. ...and 1 more sections

Key Result

Theorem 1

Let the notation be as above over an algebraically closed field of characteristic $0$. We have a short exact sequence It fits into the commutative diagram with exact rows below, where the first and the third exact rows come from Orlov's Theorem.

Theorems & Definitions (64)

  • Theorem 1: Theorem \ref{['Th_GrhoA']}
  • Lemma 2: Diagram (\ref{['delta_EqA']}) from Proposition \ref{["delta_AA' Eq"]}
  • Theorem 3: Corollary \ref{['split_NAA']}
  • Corollary 4: Corollary \ref{['split_NAA index n2']}
  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Remark 1.8
  • Remark 1.9
  • ...and 54 more