Topological Restrictions For Circle Actions And Harmonic Morphisms

Let M m be a compact oriented smooth manifold which admits a smooth circle action with isolated fixed points which are isolated as singularities as well. Then all the Pontryagin numbers of M m are zero and its Euler number is nonnegative and even. In particular, M m has signature zero. Since a non-constant harmonic morphism with one-dimensional fibres gives rise to a circle action we have the following applications: (i) many compact manifolds, for example CP n , K3 surfaces, S 2n \Theta P g (n 2) where P g is the closed surface of genus g 2 can never be the total space of a non-constant harmonic morphism with one-dimensional fibres whatever metrics we put on them; (ii) let (M 4 ; g) be a compact orientable four-manifold and ' : (M 4 ; g) ! (N 3 ; h) a non-constant harmonic morphism. Suppose that one of the following assertions holds: ffl (M 4 ; g) is half-conformally flat and its scalar curvature is zero, ffl (M 4 ; g) is Einstein and half-conformally flat, ff..