On holomorphic harmonic morphisms

Abstract

We study holomorphic harmonic morphisms from Kahler manifolds to almost Hermitian manifolds. When the codomain is also Kahler we get restrictions on such maps in the case of constant holomorphic curvature. We also prove a Bochner-type formula for holomorphic harmonic morphisms which, under certain curvature conditions of the domain, gives insight to the structure of the vertical distribution. We thus prove that when the domain is compact and non-negatively curved, the vertical distribution is totally geodesic