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 hyperkähler limit X (in the $C^2$ pointed topology) then X cannot contain a holomorphic 2-sphere (for any of its hyperkähler complex structures). In particular, this rules out the formation of bubbles modelled on ALE gravitational instantons in such families of metrics.

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 LMS

References in corpus (2)

• Einstein metrics and complex singularities
• Toric selfdual Einstein metrics on compact orbifolds