A torsor under an abelian variety is twisted derived equivalent to another smooth variety exactly when that variety is a moduli space of simple semi-homogeneous sheaves on the torsor, with the universal obstruction as the twist.
Point Objects on Abelian Varieties
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abstract
We classify the point objects in the derived category $D(X)$ of a torsor under an abelian variety over a field of characteristic $0$.
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Semi-Homogeneous Sheaves and Twisted Derived Categories
A torsor under an abelian variety is twisted derived equivalent to another smooth variety exactly when that variety is a moduli space of simple semi-homogeneous sheaves on the torsor, with the universal obstruction as the twist.