Title: Isotropic Killing vector fields and structures on complex surfaces

Abstract: In a 4-dimensional vector space with scalar product of signature (2,2) and fixed orientation, two independent vectors spanning a maximal isotropic (null) plane determine a canonical action of the para-quaternioins. We noticed that on an oriented 4-manifold with such pseudo-Riemannian metric, existence of two isotropic (null) Killing vector fields leads to integrability of the induced structure - called para-hypercomplex, and the metric is anti-selfdual. Using the Kodaira classification one can describe the topology of the underlying 4-manifold in the compact case. In the talk examples of such structures on several of the 4-manifolds will be provided and some restrictions for a compact complex surface to admit split signature Hermitian metric with one non-vanishing null Killing vector field will be established. The talk is based on a joint project with J. Davidov and O. Mushkarov.