Circle-invariant fat bundles and symplectic Fano 6-manifolds

TL;DR

This paper classifies 4-manifolds supporting circle-invariant fat SO(3)-bundles, showing they are diffeomorphic to S^4 or CP^2-bar, and explores implications for symplectic Fano 6-manifolds and circle-invariant metrics.

Contribution

It establishes a classification of 4-manifolds with circle-invariant fat SO(3)-bundles and connects this to symplectic geometry and curvature inequalities.

Findings

4-manifolds supporting such bundles are diffeomorphic to S^4 or CP^2-bar

Only these manifolds admit circle-invariant metrics satisfying the curvature inequality

Results extend the understanding of circle-invariant metrics in symplectic and Riemannian geometry

Abstract

We prove that a compact 4-manifold which supports a circle-invariant fat SO(3)-bundle is diffeomorphic to either S^4 or CP^2-bar. 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 CP^2-bar are the only 4-manifolds admitting circle-invariant metrics solving a certain curvature inequality. This can be seen as an analogue of Hsiang-Kliener's theorem that only S^4 and CP^2 admit circle-invariant metrics of positive sectional curvature.