Harmonic morphisms and moment maps on hyper-Kahler manifolds

We characterise the actions, by holomorphic isometries on a Kahler manifold with zero first Betti number, of an abelian Lie group of dimgeq 2, for which the moment map is horizontally weakly conformal (with respect to some Euclidean structure on the Lie algebra of the group). Furthermore, we study the hyper-Kahler moment map $phi$ induced by an abelian Lie group T acting by triholomorphic isometries on a hyper-Kahler manifold M, with zero first Betti number, thus obtaining the following: If dim T=1 then $phi$ is a harmonic morphism. Moreover, we illustrate this on the tangent bundle of the complex projective space equipped with the Calabi hyper-Kahler structure, and we obtain an explicit global formula for the map. If dim Tgeq 2 and either $phi$ has critical points, or M is nonflat and dim M=4 dim T then $phi$ cannot be horizontally weakly conformal.