Espace des twisteurs d’une variété quaternionique Kähler généralisée

Abstract

Specifying an almost quaternionic structure Q on a 4n-manifold M is equivalent to specifying a twistor bundle Z(Q)⟶M. When Q is invariant under a torsion free connection, Z(Q) can be endowed with an almost complex structure 𝒥. For n=1 Atiyah, Hitchin and Singer [2] have related the integrability of 𝒥 to the geometry of (M,Q). For n>1 Salamon [23, 24] showed that the almost complex structure 𝒥 on Z(Q) is always integrable. Recently, Pantilie [21] introduced the concept of a generalized quaternionic Kähler structure 𝒬 on M; defined a generalized twistor space 𝒵(𝒬); and showed that 𝒵(𝒬) comes naturally equiped with a tautological almost generalized complex structure, but leaves open the problem of the integrability. The purpose of this paper is precisely to fill this gap by showing that the almost generalized complex structure on 𝒵(𝒬) is always integrable for n>1. We will conclude by giving several examples.