On holomorphic harmonic morphisms (Q1598168)

Holomorphic harmonic morphisms from Kähler manifolds to almost Hermitian manifolds are investigated. When the codomain is also a Kähler manifold, restrictions are obtained of such maps in the case of constant holomorphic curvature. Further is proved a Bochner-type formula for holomorphic harmonic morphisms. This gives insight -- under certain curvature restrictions on the domain, to the structure of the vertical distribution; in this respect, the author proves that when the domain is compact and non-negatively curved, the vertical distribution is totally geodesic.