Generalized twistor spaces for hyperkähler manifolds

Abstract

Let M be a hyperkähler manifold. The S 2 -family of complex structures compatible with the hyperkähler metric can be assembled into a single complex structure on Z = M × S 2; the resulting complex manifold is known as the twistor space of M. We describe the analogous construction for generalized complex structures in the sense of Hitchin. Specifically, we exhibit a natural S 2 × S 2 -family of generalized complex structures compatible with the hyperkähler metric, and assemble them into a single generalized complex structure on X = M × S 2 × S 2. We call the resulting generalized complex manifold the generalized twistor space of M.