Topological restrictions for circle actions and harmonic morphisms

Let $M^m$ be a compact oriented smooth manifold which admits a smooth circleaction 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 isnonnegative and even. In particular, $M^m$ has signature zero. Since anon-constant harmonic morphism with one-dimensional fibres gives rise to acircle action we have the following applications: (i) many compact manifolds,for example $CP^{n}$, $K3$ surfaces, $S^{2n}\times P_g$ ($n\geq2$) where $P_g$is the closed surface of genus $g\geq2$ can never be the domain of anon-constant harmonic morphism with one-dimensional fibres whatever metrics weput on them; (ii) let $(M^4,g)$ be a compact orientable four-manifold and$\phi:(M^4,g)\to(N^3,h)$ a non-constant harmonic morphism. Suppose that one ofthe following assertions holds: (1) $(M^4,g)$ is half-conformally flat and itsscalar curvature is zero, (2) $(M^4,g)$ is Einstein and half-conformally flat,(3) $(M^4,g,J)$ is Hermitian-Einstein. Then, up to homotheties and Riemanniancoverings, $\phi$ is the canonical projection $T^4\to T^3$ between flat tori.