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Derived isogenies between abelian varieties

Zhiyuan Li, Ziwei Lu, Zhichao Tang

TL;DR

This work establishes a twisted derived Torelli theorem for abelian varieties by linking twisted derived categories $\mathop{\mathrm{D^b}}(X,\alpha)$ to symplectic abelian data $A_{(X,\alpha)}$ and an associated symplectic isomorphism. It defines and analyzes derived isogeny via twisted Fourier–Mukai equivalences, showing that in characteristic zero two abelian varieties of dimension $g>1$ are derived isogenous precisely when they are principally isogenous (with $g=1$ yielding isomorphism). The authors develop a twisted Orlov framework, using equivariant categories and enhanced symplectic structures to transport autoequivalences into symplectic geometry, and they decompose principal isogenies into spectrally paired steps to connect derived equivalences with classical isogenies. These results illuminate the relationship between derived categories and isogeny theory, with Hodge-theoretic realizations over $\mathbb{C}$ and applications to Kuga–Satake correspondences. Overall, the paper provides a comprehensive bridge between twisted derived categories, symplectic abelian geometry, and classical isogeny theory, yielding precise characterizations of derived isogeny classes and their invariants.

Abstract

In this paper, we establish a derived Torelli Theorem for twisted abelian varieties. Starting from this, we explore the relation between derived isogenies and classical isogenies. We show that two abelian varieties of dimension $\geq 2$ are derived isogenous if and only if they are principally isogenous over fields of characteristic zero. This generalized the result for abelian surfaces and completely solves the question raised in [arXiv:2108.08710].

Derived isogenies between abelian varieties

TL;DR

This work establishes a twisted derived Torelli theorem for abelian varieties by linking twisted derived categories to symplectic abelian data and an associated symplectic isomorphism. It defines and analyzes derived isogeny via twisted Fourier–Mukai equivalences, showing that in characteristic zero two abelian varieties of dimension are derived isogenous precisely when they are principally isogenous (with yielding isomorphism). The authors develop a twisted Orlov framework, using equivariant categories and enhanced symplectic structures to transport autoequivalences into symplectic geometry, and they decompose principal isogenies into spectrally paired steps to connect derived equivalences with classical isogenies. These results illuminate the relationship between derived categories and isogeny theory, with Hodge-theoretic realizations over and applications to Kuga–Satake correspondences. Overall, the paper provides a comprehensive bridge between twisted derived categories, symplectic abelian geometry, and classical isogeny theory, yielding precise characterizations of derived isogeny classes and their invariants.

Abstract

In this paper, we establish a derived Torelli Theorem for twisted abelian varieties. Starting from this, we explore the relation between derived isogenies and classical isogenies. We show that two abelian varieties of dimension are derived isogenous if and only if they are principally isogenous over fields of characteristic zero. This generalized the result for abelian surfaces and completely solves the question raised in [arXiv:2108.08710].
Paper Structure (33 sections, 24 theorems, 148 equations)

This paper contains 33 sections, 24 theorems, 148 equations.

Table of Contents

  1. Introduction
  2. Derived Torelli theorem
  3. Derived Isogenies
  4. Idea of the Proof
  5. Conventions
  6. Twisted derived categories and equivariant categories
  7. Twisted Sheaves
  8. Equivariant Categories
  9. Geometric Convention:
  10. Inflation and Forgetful Functors:
  11. Equivariant Categories for Galois Covers
  12. Twisted Sheaves on Abelian Varieties
  13. Symplectic abelian varieties
  14. Symplectic Biextensions and Lagrangian Subvarieties
  15. Symplectic Isomorphisms and Derived Equivalences
  16. ...and 18 more sections

Key Result

Theorem 1.1

Let $(X_1,\alpha_1)$ and $(X_2,\alpha_2)$ be twisted abelian varieties over an algebraically closed field $k$ with $\mathop{\mathrm{ord}}\nolimits(\alpha_i)$ coprime to $\mathop{\mathrm{char}}\nolimits(k)\neq2$. If there exists a derived equivalence then there is a symplectic isomorphism Moreover, when $\mathop{\mathrm{char}}\nolimits(k) = 0$, the converse holds and thus these two conditions are

Theorems & Definitions (59)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Definition 1.4
  • Definition 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Corollary 1.8
  • Corollary 1.9
  • Remark 1.10
  • ...and 49 more