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Derived categories of generalized Kummer varieties: extended Mukai vector

Yuxuan Yang

TL;DR

The paper extends the derived-equivalence framework from K3^{[n]}-type to hyper-Kähler manifolds deformation equivalent to generalized Kummer varieties by constructing the Kum^{n-1} lattice abla_X inside the extended Mukai lattice and proving its invariance under derived autoequivalences via the derived monodromy group. It develops a detailed lattice-theoretic and Hodge-structural toolkit, including Eichler transvections, to realize derived-automorphism actions as orthogonal transformations and to lift derived equivalences from abelian surfaces to Kum^{n-1}(A) via generalized BKR-type correspondences. The main results establish the invariance of abla_X under derived equivalences, constrain the image of the autoequivalence representation to a finite, well-described subgroup of the Kum-type orthogonal group, and, as a corollary, yield finiteness statements for Fourier–Mukai partners and a clear factorization through the Hodge-isometry automorphism group. Together, these findings deepen the understanding of how derived categories constrain the geometry of Kum-type hyper-Kähler manifolds and provide tools for classifying derived-equivalent Kum varieties via lattice-theoretic data.

Abstract

We use the extended Mukai vectors for hyper-Kähler manifolds to investigate the derived equivalences of the hyper-Kähler manifolds which are deformation equivalent to generalized Kummer varieties. Inspired by the idea for hyper-Kähler manifolds of $\mathrm{K3}^{[n]}$-type, we obtain an integral lattice which is proved to be invariant under the derived equivalences of generalized Kummer type varieties. Such results and applications are described using derived monodromy groups.

Derived categories of generalized Kummer varieties: extended Mukai vector

TL;DR

The paper extends the derived-equivalence framework from K3^{[n]}-type to hyper-Kähler manifolds deformation equivalent to generalized Kummer varieties by constructing the Kum^{n-1} lattice abla_X inside the extended Mukai lattice and proving its invariance under derived autoequivalences via the derived monodromy group. It develops a detailed lattice-theoretic and Hodge-structural toolkit, including Eichler transvections, to realize derived-automorphism actions as orthogonal transformations and to lift derived equivalences from abelian surfaces to Kum^{n-1}(A) via generalized BKR-type correspondences. The main results establish the invariance of abla_X under derived equivalences, constrain the image of the autoequivalence representation to a finite, well-described subgroup of the Kum-type orthogonal group, and, as a corollary, yield finiteness statements for Fourier–Mukai partners and a clear factorization through the Hodge-isometry automorphism group. Together, these findings deepen the understanding of how derived categories constrain the geometry of Kum-type hyper-Kähler manifolds and provide tools for classifying derived-equivalent Kum varieties via lattice-theoretic data.

Abstract

We use the extended Mukai vectors for hyper-Kähler manifolds to investigate the derived equivalences of the hyper-Kähler manifolds which are deformation equivalent to generalized Kummer varieties. Inspired by the idea for hyper-Kähler manifolds of -type, we obtain an integral lattice which is proved to be invariant under the derived equivalences of generalized Kummer type varieties. Such results and applications are described using derived monodromy groups.
Paper Structure (31 sections, 34 theorems, 187 equations)

This paper contains 31 sections, 34 theorems, 187 equations.

Table of Contents

  1. Introduction
  2. Background: Derived categories of abelian surfaces
  3. Integral structure
  4. Structure of the paper
  5. Acknowledgements
  6. Notation
  7. Review of hyper-Kähler manifolds and extended Mukai vectors Beckmann:22-2
  8. Notations and results
  9. Further examples of extended Mukai vectors
  10. Integral lattices for generalized Kummer type hyper-Kähler manifolds
  11. Lattices
  12. Hodge structures
  13. Derived monodromy group
  14. Derived monodromy group of projective irreducible holomorphic symplectic manifolds and abelian varieties
  15. Lifting derived equivalences of abelian surfaces to derived equivalences of corresponding generalized Kummer varieties
  16. ...and 16 more sections

Key Result

Theorem 1.2.1

Let $X$ and $Y$ be (projective) hyper-Kähler manifolds of generalized Kummer type of dimension $2(n-1),n\geq 3$, over an algebraically closed field of characteristic $0$ and $\Phi:\mathbf{D}^b(X)\xrightarrow[]{\simeq}\mathbf{D}^b(Y)$ a derived equivalence. Then $\Phi^{\widetilde{\mathrm{H}}}:\wideti

Theorems & Definitions (73)

  • Theorem 1.2.1: Theorem \ref{['th 9.2+']}
  • Theorem 1.2.2: Theorem \ref{['th 9.4+']}
  • Theorem 1.2.3: Theorem \ref{['th 9.8+']}
  • Proposition 2.1.2: Beckmann:22-2
  • Proposition 2.1.6: Beckmann:22-2
  • Definition 2.1.8: Beckmann:22-2
  • Lemma 2.1.9: Beckmann:22-2
  • Definition 2.1.10: Beckmann:22-2
  • Definition 2.1.11: Beckmann:22-2
  • Lemma 2.1.12: Beckmann:22-2
  • ...and 63 more