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.

中文翻译：

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非 Kähler 全纯辛流形的几何和自同构

我们考虑在 D. Guan 和第一作者的作品中描述的唯一一类已知的非 Kähler 不可约全纯辛流形。任何这样的维度为 $2n-2$ 的流形 $Q$ 都是作为某个非 Kähler 流形 $W_F$ 的有限度 $n^2$ 覆盖获得的，我们称之为 $Q$ 的底。我们证明了 $Q$ 及其基的代数约简是维度 $n-1$ 的射影空间。此外，给出了$Q$中子流形的部分分类，描述了其代数约简的简并轨迹，证明了$Q$的自同构群满足Jordan性质。