Geometry and Automorphisms of Non-Kähler Holomorphic Symplectic Manifolds

Abstract 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 $n^2$ cover of some non-Kähler manifold $W_F$, 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.