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Connected sums of self-dual manifolds and deformations of singular spaces. (English) Zbl 0671.53029
The decomposition of the 2-forms on an oriented Riemannian 4-manifold into self-dual and anti-self-dual components (with respect to the *- operator) implies the existence of a special class of Riemannian manifolds, the “self-dual” or “half conformally flat” manifolds admitting a Riemannian metric with vanishing anti-self-dual part of the Weyl curvature. Only few examples of such manifolds have been known \((S^ 4\), \({\mathbb{C}}{\mathbb{P}}^ 2\), the Yau metrics on a K3 surface \(\bar K\) with reversed orientation). Recently Poon (for \(n=2,3)\) and Floer showed that the connected sum \(n{\mathbb{C}}{\mathbb{P}}^ 2\) of any number of copies of \({\mathbb{C}}{\mathbb{P}}^ 2\) admits self-dual metrics. The purpose of the present paper is to give a reasonably general theory for constructing self-dual structures on connected sums using twistor spaces.
Self-dual manifolds can be studied using techniques of complex geometry via the Penrose twistor construction. In the present work, the general theory of deformations of singular complex spaces is applied to the twistor situation to produce criteria under which a connected sum admits a self-dual structure: “If \(X_ 1\) and \(X_ 2\) are self-dual manifolds with twistor spaces \(Z_ 1\) and \(Z_ 2\) we look for metrics on the connected sum \(X_ 1\#X_ 2\) which are close to the given structures outside a small ‘neck’, where the connected sum is made. More precisely we find a twistor translation of this idea constructing a certain singular complex space Z using \(Z_ 1\) and \(Z_ 2\) and looking for twistor spaces made by small smoothings of Z. One of our main results is that if such smoothings exist then they always represent the twistor spaces of self-dual structures on the connected sum.”
Particular applications of the theory developed are Poon’s and Floer’s result mentioned above and the existence of self-dual metrics on connected sums NK # n\({\mathbb{C}}{\mathbb{P}}^ 2\), for \(n\geq 2N+1\). Also, as explained in the paper, the discussion can be carried over to the case of self-dual connections giving a twistor approach to some of that theory.
Reviewer: B.Zimmermann

53C20 Global Riemannian geometry, including pinching
32G99 Deformations of analytic structures
32L25 Twistor theory, double fibrations (complex-analytic aspects)
57R99 Differential topology
53C99 Global differential geometry
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