Strongly degenerate homogeneous pseudo-K\"ahler structures of linear type and complex plane waves

Abstract

We study the class $\mathcal{K}_{2}+\mathcal{K}_{4}$ of homogeneous pseudo-K\"ahler structures in the strongly degenerate case. The local form and the holonomy of a pseudo-K\"ahler manifold admitting such a structure is obtained, leading to a possible complex generalization of homogeneous plane waves. The same question is tackled in the case of pseudo-hyper-K\"ahler and pseudo-quaternion K\"ahler manifolds.