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## 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 of D. Guan and the first 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.