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Twisted Derived Equivalences Between Abelian Varieties

Tyler Lane

Abstract

We generalize a result of Popa-Schnell and show that the isogeny class of the Picard variety is twisted derived invariant. Using this, we prove that any twisted Fourier-Mukai partner of an abelian variety is an abelian variety. We then provide a necessary and sufficient isogeny-based condition for two abelian varieties to be twisted derived equivalent.

Twisted Derived Equivalences Between Abelian Varieties

Abstract

We generalize a result of Popa-Schnell and show that the isogeny class of the Picard variety is twisted derived invariant. Using this, we prove that any twisted Fourier-Mukai partner of an abelian variety is an abelian variety. We then provide a necessary and sufficient isogeny-based condition for two abelian varieties to be twisted derived equivalent.
Paper Structure (10 sections, 27 theorems, 36 equations)

This paper contains 10 sections, 27 theorems, 36 equations.

Table of Contents

  1. Preliminaries
  2. Invariance of the Picard Variety
  3. A Twisted Derived Equivalence Criterion
  4. Perspectives on the Brauer Group
  5. Semi-Homogeneous Bundles
  6. Symplectic Isomorphisms
  7. Symplectic Abelian Varieties
  8. Symplectic Isogenies
  9. Symplectic AVs associated to Brauer classes
  10. Twisted Derived Equivalence

Key Result

Theorem A

Let $X$ and $Y$ be smooth projective varieties over $\mathbb{C}$. If $X$ and $Y$ are twisted derived equivalent, then $\text{Pic}^0_X$ and $\text{Pic}^0_Y$ are isogenous.

Theorems & Definitions (49)

  • Theorem A: \ref{['t']}
  • Theorem B: \ref{['t2']}
  • Theorem C: \ref{['t criterion']}, \ref{['4: t main']}
  • Theorem 1.1: Ols25
  • Lemma 1.2
  • Proposition 1.3: Ols25
  • Corollary 1.4
  • proof
  • Theorem 2.1
  • proof
  • ...and 39 more