Point Objects and Derived Equivalences of Twisted Derived Categories of Abelian Varieties

TL;DR

This paper investigates twisted derived categories of abelian varieties, classifies point objects, and identifies their Fourier-Mukai partners, advancing understanding of derived equivalences in algebraic geometry.

Contribution

It introduces a classification of point objects and twisted Fourier-Mukai partners for abelian varieties, extending the theory of derived equivalences with new twisted structures.

Findings

Classification of point objects in twisted derived categories

Identification of twisted Fourier-Mukai partners of abelian varieties

Development of theory for 1-twisted semi-homogeneous vector bundles

Abstract

We study the notion of $1$-twisted semi-homogeneous vector bundles on $\mathbb{G}_m$-gerbes over abelian varieties, and classify point objects in the twisted derived categories of abelian varieties. As an application, we classify the twisted Fourier-Mukai partners of abelian varieties.