Complex surfaces and null conformal Killing vector fields

Abstract

We study the relation between the existence of null conformal Killing vector fields and existence of compatible complex and para-hypercomplex structures on a pseudo-Riemannian manifold with metric of signature (2, 2). We establish first the topological types of pseudo-Hermitian surfaces admitting a nowhere vanishing null vector field. Then we show that a pair of orthogonal, pointwise linearly independent, null, conformal Killing vector fields defines a para-hyperhermitian structure and use this fact for a classification of the smooth compact four-manifolds admitting such a pair of vector fields. We also provide examples of neutral metrics with two orthogonal, pointwise linearly independent, null Killing vector fields on most of these manifolds.There has also been some interest in compact 4-manifolds admitting neutral metrics and compatible complex or para-hypercomplex structures since topological information like the Kodaira classification of compact complex surfaces allows one to study global properties. Important results in this direction are the classifications of compact pseudo-Kähler Einstein and para-hyperkähler surfaces obtained by Petean [21] and Kamada [14], respectively. These structures appear in [18] as models for superstring theory with N=2 supersymmetry and [11] in relation to deformation spaces of harmonic maps from Riemann surfaces into Lie groups. Note also that such structures have been used recently by B. Klingler [15] in his proof of the Chern conjecture for affine manifolds.In our previous papers [6,7] we initiated the study of compact para-hyperhermitian surfaces, which are the neutral analog of the hyperhermitian 4-manifolds. These surfaces are anti-self-dual as in the positive definite case, but in contrast to the well-known classification of compact hyperhermitian surfaces [4], there are many more compact examples of para-hyperhermitian surfaces. In our study, we noticed also that any two pointwise independent, null, orthogonal and parallel vector fields on a 4-manifold with a neutral metric define a para-hyperkäher structure and conversely, any compact para-hyperkäher surface admits two vector fields with the above properties [14]. In this note, we weaken these conditions and study the neutral Hermitian surfaces admitting non-vanishing null conformal Killing vector fields. We observe first that the existence of a non-vanishing null vector field on a compact neutral Hermitian surface leads to a topological restriction which implies a rough classification of these surfaces. Next, we show that the existence of two point-wise independent, null and orthogonal conformal Killing vector fields leads to the existence of a para-hyperhermitian structure. These surfaces were studied in [6, 7] and here we consider the problem for existence of two Killing vector fields having the above properties with respect to the corresponding metrics. Now we describe shortly the content of the paper. In Section 2, some known facts about almost para-hyperhermitian structures are collected and in Lemma 2 they are c...