On a complex manifold $(M,J)$, we interpret complex symplectic and pseudo-Kähler structures as symplectic forms with respect to which J is, respectively, symmetric and skew-symmetric. We classify complex symplectic structures on 4-dimensional Lie algebras. We develop a method for constructing hypersymplectic structures from the above data. This allows us to obtain two families of hypersymplectic structures on a 4-step nilmanifold.

中文翻译：

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对称和斜对称复杂结构

在复杂流形上 $(米,J)$，我们将复杂的辛和伪 Kähler 结构解释为关于J分别是对称和偏对称的辛形式。我们在 4 维李代数上对复杂的辛结构进行分类。我们开发了一种从上述数据构建超辛结构的方法。这使我们能够在 4 步 nilmanifold 上获得两个超辛结构家族。