Geometry of BG-manifolds

In the late 90s Bogomolov and Guan were able to construct an example of holomorphic symplectic non-Kahler manifolds. I'm going to give a review of the construction of such manifolds and their deformation theory and geometry. In particular algebraic reduction of BG-manifolds is $\mathbb{P}^n$ with typical fiber being an abelian variety. Based on joint work with F. Bogomolov, A. Kuznetsova, M. Verbitsky and E. Yasinsky.