Derived isogenies between abelian varieties

TL;DR

This paper proves a derived Torelli Theorem for twisted abelian varieties, linking derived isogenies to classical isogenies, and fully resolving a previously open question for abelian varieties of dimension at least two.

Contribution

It establishes a comprehensive connection between derived and classical isogenies for higher-dimensional abelian varieties, extending previous results for surfaces.

Findings

Derived isogenies coincide with classical isogenies for dimension ≥ 2.

The result generalizes known theorems for abelian surfaces.

The paper fully answers an open question from prior research.

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].