Espace des twisteurs d'une vari\'et\'e quaternionique K\"ahler g\'en\'eralis\'ee

TL;DR

This paper extends classical twistor theory for quaternionic manifolds to the realm of generalized complex geometry, establishing integrability criteria for the associated twistor spaces and providing examples of generalized quaternionic Kähler manifolds.

Contribution

It introduces the concept of almost generalized quaternionic manifolds and proves the integrability of their twistor spaces in higher dimensions, generalizing previous results.

Findings

The twistor space $ ext{Z}( ext{Q})$ admits an almost complex structure $ extbf{J}$ under certain invariance conditions.

For generalized quaternionic Kähler manifolds with $n>1$, the almost generalized complex structure $ extbf{J}$ is always integrable.

Several examples of such manifolds are provided where $ extbf{J}$ is integrable.

Abstract

To give an almost quaternionic structure on a 4n-manifold $M$ is equivalent to give its bundle of twistors $Z(Q)\longrightarrow M$. When $Q$ is invariant under a torsion free connection, $Z(Q) $ can be provided with an almost complex structure $ \mathbb J $. In the case $ n = 1 $ Atiyah, Hitchin and Singer have related the integrability of $ \mathbb J $ to the geometry of $ (M, Q) $. For $ n> 1 $ Salamon showed that the almost complex structure $ \mathbb J $ on $ Z (Q) $ is always integrable. The purpose of this article is to extend these results to the generalized complex geometry. We begin by defining the concept of almost generalized quaternionic manifolds $ (M, g, \mathcal Q ) $. We will see that we can associate a twistor space denoted by $ \mathcal Z( \mathcal Q) $ which is a $ \mathbb S^2$-bundle over $ M $. When $\mathcal Q$ is invariant under a generalized torsion free connection, then $ \mathcal Z(\mathcal Q) $ comes with an almost generalized complex structure $\mathbb J$. Whatever the dimension of $ M $ is, we give a criterion for integrability of the almost generalized complex structure $ \mathbb J $ on $ \mathcal Z(\mathcal Q) $. In the particular case where $ (M, g,\mathcal Q) $ is a generalized quaternionic K\"ahler manifold, we show that $\mathbb J$ is always integrable as soon as $n>1$. We illustrate this work by giving several examples of generalized quaternionic K\"ahler manifolds for which the almost generalized complex structure $ \mathbb J $ on the twistor space $ \mathcal Z(\mathcal Q) $ is integrable.