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

par Fine, Joel
Référence Transactions of the London Mathematical Society, 4, 1, page (100-109)
Publication Publié, 2017-12

Article révisé par les pairs

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Résumé :

The twistor space of a Riemannian 4-manifold carries two almost complex structures, J+ and J-, and a natural closed 2-form ω. This article studies limits of manifolds for which ω 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 C2 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 asymptotically locally Euclidean gravitational instantons in such families of metrics.