We say smooth projective varities are derived equivalent if their bounded derived categories of coherent sheaves are equivalent. Thanks to Orlov's work, we know a lot of facts about derived equivalent abelian varieties. In paticular, we know that there are only finitely many abelian varities (up to isomorphism) that are derived equivalent to an abelain variety. On the other hand, Favero showed that there are only finitely many smooth projective varieties (up to isomorphism) that are derived equivalent to an abelian variety in this paper (https://arxiv.org/pdf/0712.0201.pdf).
This led me to a question whether there is a known counter-example of a smooth projective non-abelian variety that is derived equivalent to an abelian variety. (We know if dimension $\leq 2$, then there is no such counter-example.)
Thank you in advance.
Yes. This is proven (for instance) in
Theorem 0.4 of https://link.springer.com/article/10.1007/s00209-009-0655-z