Espace des twisteurs d'une variété quaternionique Kähler généralisée

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To give an almost quaternionic structure on a 4n-manifold M is equivalent to give its bundle of twistors Z(Q) M. When Q is invariant under a torsion free connection, Z(Q) can be provided with an almost complex structure J. In the case n = 1 Atiyah, Hitchin and Singer have related the integrability of J to the geometry of (M, Q). For n> 1 Salamon showed that the almost complex structure 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, Q). We will see that we can associate a twistor space denoted by Z(Q) which is a S²-bundle over M. When Q is invariant under a generalized torsion free connection, then Z(Q) comes with an almost generalized complex structure J. Whatever the dimension of M is, we give a criterion for integrability of the almost generalized complex structure J on Z(Q). In the particular case where (M, g,Q) is a generalized quaternionic Kähler manifold, we show that J is always integrable as soon as n>1. We illustrate this work by giving several examples of generalized quaternionic Kähler manifolds for which the almost generalized complex structure J on the twistor space Z(Q) is integrable.