Invariant pseudo Kaehler metrics in dimension four

TL;DR

This paper classifies four-dimensional Lie groups with pseudo Kähler metrics, explores their geometric structures, and constructs Ricci flat examples, revealing that Ricci flat unimodular Kähler Lie algebras are flat in this dimension.

Contribution

It provides a complete classification of four-dimensional pseudo Kähler Lie groups and introduces a method to construct Ricci flat pseudo Kähler structures in higher dimensions.

Findings

Most complex homogeneous spaces admit pseudo Kähler Einstein metrics

Ricci flat and flat metrics are explicitly characterized

Ricci flat unimodular Kähler Lie algebras are flat in dimension four

Abstract

Four dimensional simply connected Lie groups admitting a pseudo K\"ahler metric are determined. The corresponding Lie algebras are modelized and the compatible pairs $(J,\omega)$ are parametrized up to complex isomorphism (where $J$ is a complex structure and $\omega$ is a symplectic structure). Such structure gives rise to a pseudo Riemannian metric $g$ for which $J$ is parallel. It is proved that most of these complex homogeneous spaces admit a pseudo K\"ahler Einstein metric. Ricci flat and flat metrics are determined. In particular Ricci flat unimodular K\"ahler Lie algebras are flat in dimension four. Other algebraic and geometric features are treated. A general construction of Ricci flat pseudo K\"ahler structures in higher dimensions on some affine Lie algebras is given. Walker and hypersymplectic metrics on Lie algebras are compared.