Limits of Riemannian 4-manifolds and the symplectic geometry of their twistor spaces

Abstract

The twistor space of a Riemannian 4-manifold carries two almost complex structures, $J_+$ and $J_-$, and a natural closed 2-form $\omega$. This article studies limits of manifolds for which $\omega$ tames either $J_+$ or $J_-$. This amounts to a curvature inequality involving self-dual Weyl curvature and Ricci curvature, and which is satisfied, for example, by all anti-self-dual Einstein manifolds with non-zero scalar curvature. We prove that if a sequence of manifolds satisfying the curvature inequality converges to a hyperk\"ahler limit X (in the $C^2$ pointed topology) then X cannot contain a holomorphic 2-sphere (for any of its hyperk\"ahler complex structures). In particular, this rules out the formation of bubbles modelled on ALE gravitational instantons in such families of metrics.Comment: 12 pages. v2 corollary 6 has been removed (a partial answer to a question of Biquard) since the "proof" was false. Some remarks have been added, some typos removed. This is the version accepted for publication by Transactions of the LM