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Strongly degenerate homogeneous pseudo-Kähler structures of linear type and complex plane waves. (English) Zbl 1280.53047

Summary: We study the class \(\mathcal K_2 + \mathcal K_4\) of homogeneous pseudo-Kähler structures in the strongly degenerate case. The local form and the holonomy of a pseudo-Kähler manifold admitting such a structure are obtained, leading to a possible complex generalization of homogeneous plane waves. The same question is tackled in the case of pseudo-hyper-Kähler and pseudo-quaternion Kähler manifolds.

MSC:

53C30 Differential geometry of homogeneous manifolds
53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
53C55 Global differential geometry of Hermitian and Kählerian manifolds
53C80 Applications of global differential geometry to the sciences
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