Harmonic morphisms in Hermitian geometry

We prove several results on holomorphic harmonic morphisms between Hermitian manifolds. In the non-compact case, we find conditions on holomorphic maps from domains in C-2 to C to retain the harmonicity when the metric is conformally changed. We conclude that there are no non-constant harmonic morphisms from S-4 minus a point to a Riemann surface. As for the compact case, we show that holomorphic harmonic morphisms from compact Kahler manifolds of non-negative sectional curvature to Kahler manifolds which are not surfaces, are totally geodesic maps. We also provide restrictions on the Hodge numbers when such maps exist. Finally, we prove that flag manifolds carry cosymplectic structures which turn homogeneous projections into holomorphic harmonic morphisms.