Compact complex surfaces with geometric structures related to split quaternions

TL;DR

This paper investigates the existence and properties of geometric structures related to split quaternions on compact complex surfaces, revealing new examples and conditions for para-hypercomplex, para-hyperhermitian, and para-hyperk"ahler structures.

Contribution

It characterizes when compact 4-manifolds admit para-hyperk"ahler structures and shows many complex surfaces support infinite families of para-hyperhermitian structures, unlike the definite case.

Findings

Para-hyperk"ahler structures exist iff the manifold has a split signature metric with specific vector fields.

Many compact complex surfaces with vanishing first Chern class admit infinite para-hyperhermitian structures.

Examples of complex surfaces with para-hyperhermitian structures not conformally para-hyperk"ahler are provided.

Abstract

We study the problem of existence of geometric structures on compact complex surfaces that are related to split quaternions. These structures, called para-hypercomplex, para-hyperhermitian and para-hyperk\"ahler are analogs of the hypercomplex, hyperhermitian and hyperk\"ahler structures in the definite case. We show that a compact oriented 4-manifold carries a para-hyperk\"ahler structure iff it has a metric of split signature together with two parallel, orthogonal and null vector fields. Every compact complex surface admiting a para-hyperhermitian structure has vanishing first Chern class and we show that, unlike the definite case, many of these surfaces carry infinite dimensional families of such structures. We provide also compact examples of complex surfaces with para-hyperhermitian structures which are not locally conformally para-hyperk\"ahler. Finally, we discuss the problem of non-existence of para-hyperhermitian structures on Inoue surfaces of type $S^0$ and provide a list of compact complex surfaces which could carry para-hypercomplex structures.