Symmetric and skew-symmetric complex structures

TL;DR

This paper explores complex symplectic and pseudo-Kähler structures on complex manifolds, classifies such structures on 4-dimensional Lie algebras, and introduces a method to construct hypersymplectic structures, including an example on a 4-step nilmanifold.

Contribution

It provides a classification of complex symplectic structures on 4-dimensional Lie algebras and develops a new method for constructing hypersymplectic structures from these data.

Findings

Classified complex symplectic structures on 4-dimensional Lie algebras.

Developed a method to construct hypersymplectic structures.

Provided an example of a hypersymplectic structure on a 4-step nilmanifold.

Abstract

On a complex manifold $(M,J)$, we interpret complex symplectic and pseudo-K\"ahler 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 an example of a hypersymplectic structure on a 4-step nilmanifold.