Hyperkahler 4n-Manifolds with n Commuting Quaternionic Killing Fields

Abstract

We consider a hyperk? hler 4n-manifold M. Using local holomorphic Darboux coordinates with respect to a compatible complex structure I on M, we find local necessary and sufficient conditions for a real smooth vector field X on M to be quaternionic Killing. We then apply this result to the case of a hyperk? hler manifold M admitting n commuting quaternionic Killing fields, X^1,..., X^n, the first n-1 of which are further assumed to be triholomorphic and quaternionically linearly independent pointwise. We then have two cases: if the self-dual part of DX^n vanishes, we get back the Hitchin-Karlhede-Lindstr??m-Roček result, and if the self-dual part of DX^n is non-zero, we obtain a partial generalization of the Boyer and Finley equation.