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We study the curvature of a manifold on which there can be defined acomplex-valued submersive harmonic morphism with either, totally geodesicfibers or that is holomorphic with respect to a complex structure which iscompatible with the second fundamental form.  We also give a necessary curvature condition for the existence ofcomplex-valued harmonic morphisms with totally geodesic fibers on Einsteinmanifolds.

Curvature conditions for complex-valued harmonic morphisms