Generalized pseudo Kaehler structures

TL;DR

This paper explores pseudo bihermitian structures and their relation to generalized pseudo Kaehler geometry, providing classifications, examples, and geometric structures on 4-manifolds.

Contribution

It extends the theory of bihermitian and generalized Kaehler structures to the pseudo-Riemannian setting, including classifications and new geometric insights.

Findings

Classification of compact complex surfaces admitting such structures

Examples of bihermitian structures on these surfaces

Null plane distribution induces an Engel structure under certain conditions

Abstract

In the paper we consider pseudo bihermitian structures - a pair of complex structures compatible with a pseudo Riemannian metric. As in the positive definite case we establish its relations with generalized (pseudo) Kaehler geometry and holomorphic Poisson structures. We provide a list of compact complex surfaces which could admit such structure and also examples of bihermitian structures on some of them. We also consider a naturally defined null plane distribution on generalized pseudo Kaehler 4-manifold and show that under a mild restriction it determines an Engel structure.