Harmonic morphisms and moment maps on hyper-Kähler manifolds

Abstract

We characterise the actions by holomorphic isometries on a Kähler manifold, of an abelian Lie group, admitting a moment map which is horizontally weakly conformal (with respect to some Euclidean structure on the Lie algebra of the group). Furthermore, let $$\varphi $$ be a hyper-Kähler moment map of an abelian Lie group T acting by triholomorphic isometries on a hyper-Kähler manifold M. If $$\dim T=1$$ then $$\varphi $$ is a harmonic morphism. Moreover, we illustrate this on the tangent bundle of the complex projective space equipped with the Calabi hyper-Kähler structure, and we obtain an explicit global formula for the map. If $$\dim T\ge 2$$ and either $$\varphi $$ has critical points, or M is nonflat and $$\dim M=4\dim T$$ then $$\varphi $$ cannot be horizontally weakly conformal.