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

TL;DR

The paper develops a twisted extension of Mukai’s theory for abelian varieties by studying $1$-twisted semi-homogeneous vector bundles on $G_m$-gerbes and their complexes. It introduces and analyzes representability for moduli of twisted complexes, constructs the functor $oldsymbol{ extPhi}_{X}(E)$ for simple and non-simple twists, and proves that point objects in $D(X)^{(1)}$ arise from torsors under sub-abelian varieties with semi-homogeneous data. The main contributions include a sharp classification of point objects, a Krull–Schmidt style decomposition in the twisted setting, and a complete description of twisted Fourier–Mukai partners, showing that any twisted FM partner of an abelian variety is again an abelian variety of the same dimension. These results extend the classical non-twisted theory to the twisted setting, providing a robust framework for understanding derived-equivalences in the presence of Brauer twists and gerbes, with implications for moduli, descent, and isogeny techniques in the study of abelian varieties.

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.

Theorems & Definitions (99)

• Definition 1.1
• Remark 1.2
• Remark 1.3
• Theorem 1.4: Theorem \ref{['linebundle']}
• Theorem 1.5: Theorem \ref{['summand']}
• Remark 1.6
• Theorem 1.7: Section 5
• Definition 1.8
• Theorem 1.9: Section 6
• Remark 1.10