Harmonic morphisms and moment maps on hyper-K\"ahler manifolds

We characterise the actions, by holomorphic isometries on a K\"ahler manifold with zero first Betti number, of an abelian Lie group of dim\geq 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-K\"ahler moment map $\phi$ induced by an abelian Lie group T acting by triholomorphic isometries on a hyper-K\"ahler 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-K\"ahler structure, and we obtain an explicit global formula for the map. If dim T\geq 2 and either $\phi$ has critical points, or M is nonflat and dim M=4 dim T then $\phi$ cannot be horizontally weakly conformal.