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

• Main Author: Deschamps, Guillaume
• Format: Journal Article
• Language: English
• Published: 22.01.2014
• Subjects: Mathematics - Differential Geometry
• DOI: 10.48550/arxiv.1401.5605

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.