The deformation theory of anti-self-dual conformal structures
Riemannian geometry in dimension 4 has some special features which are not present in any other dimension. In dimensions \(n\geq 5\) the group SO\((n)\) is simple, whereas SO\((4)\) is not simple. The bundle of 2-forms on an oriented 4-Riemannian manifold is decomposed into its self dual (SD) and anti-self-dual (ASD) parts under the action of the Hodge operator. For a real 4-dimensional Kähler manifold, the Bochner tensor is equal to the ASD part of the Weyl tensor. In the paper under review, the authors begin a systematic study of all solutions of the ASD equation on a given smooth closed 4-manifold. A theory of moduli spaces analogous to that of \textit{M. F. Atiyah}, \textit{N. J. Hitchin} and \textit{I. M. Singer} [Proc. R. Soc. Lond., Ser. A 362, 425- 461 (1978; Zbl 0389.53011)] in the case of connections is developed. The relationship with scalar curvature is investigated. Deformations of \((M,G)\)-structures are considered and interesting examples are given. The moduli spaces of conformal structures can be used to obtain information about the differential topology of 4-manifolds, in the spirit of the Donaldson invariants.
scientific article; zbMATH DE number 86221
scientific article