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

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 $\mathbb{J}$. 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.