Generalized twistor spaces for hyperk\"ahler manifolds

Abstract

Let M be a hyperk\"ahler manifold. The S^2-family of complex structures compatible with the hyperk\"ahler metric can be assembled into a single complex structure on Z=MxS^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^2xS^2-family of generalized complex structures compatible with the hyperk\"ahler metric, and assemble them into a single generalized complex structure on X=MxS^2xS^2. We call the resulting generalized complex manifold the generalized twistor space of M.