Geometry and Automorphisms of Non-Kähler Holomorphic Symplectic Manifolds

We consider the only one known class of non-Kähler irreducible holomorphic symplectic manifolds, described in the works by D. Guan and the 1st author. Any such manifold Q of dimension 2n-2 is obtained as a finite degree n2 cover of some non-Kähler manifold WF, which we call the base of Q. We show that the algebraic reduction of Q and its base is the projective space of dimension n-1. Besides, we give a partial classification of submanifolds in Q, describe the degeneracy locus of its algebraic reduction and prove that the automorphism group of Q satisfies the Jordan property