Holonomy classification of Lorentz-K\"ahler manifolds

TL;DR

This paper classifies the holonomy algebras of Lorentz-Kähler manifolds, introduces complex Walker coordinates, and characterizes complex pp-waves, advancing the understanding of their geometric structures.

Contribution

It provides a complete classification of holonomy algebras for Lorentz-Kähler manifolds and constructs explicit metrics for each case, including a review of symmetric spaces.

Findings

Holonomy algebras of Lorentz-Kähler manifolds are classified.

Explicit metrics are constructed for each holonomy algebra.

Complex pp-waves are characterized via curvature, holonomy, and potential.

Abstract

The classification problem for holonomy of pseudo-Riemannian manifolds is actual and open. In the present paper, holonomy algebras of Lorentz-K\"ahler manifolds are classified. A simple construction of a metric for each holonomy algebra is given. Complex Walker coordinates are introduced and described using the potential. Complex pp-waves are characterized in terms of the curvature, holonomy and the potential. Classification of Lorentz-K\"ahler symmetric spaces is reviewed.