Circle-invariant fat bundles and symplectic Fano 6-manifolds

Abstract

We prove that a compact 4-manifold which supports a circle-invariant fat SO ( 3 ) -bundle is diffeomorphic to either S 4 or C P ¯ 2 . The proof involves studying the resulting Hamiltonian circle action on an associated symplectic 6-manifold. Applying our result to the twistor bundle of Riemannian 4-manifolds shows that S 4 and C P ¯ 2 are the only 4-manifolds admitting circle-invariant metrics solving a certain curvature inequality. This can be seen as an analogue of Hsiang–Kleiner's theorem that only S 4 and C P 2 admit circle-invariant metrics of positive sectional curvature.