Generalized Kähler Taub-NUT metrics and two exceptional instantons (Q6101609)

In studying the local behaviour of metrics near singularities, or for studying how metrics behave as the manifold itself degenerates or is varied, it is often useful to have local models. In Kähler geometry, the metrics of primary interest are those with constant scalar curvature. The sorts of local models should then be scalar flat metrics in \(\mathbb C^n\). \textit{S. K. Donaldson} [Geom. Funct. Anal. 19, No. 1, 83--136 (2009; Zbl 1177.53067)] and the author [in preparation] have produced interesting local models, called generalised Kähler Taub-NUT metrics. This paper studies these metrics in detail. In order to use these sorts of local models for gluing arguments, essentially what one needs to know is their asymptotic behaviour. The main results of this paper explain the asymptotic geometry of these interesting generalised Kähler Taub-NUT metrics in a great deal of detail and in a very explicit manner. The results in the paper seem therefore very likely to be extremely useful for gluing problems. We know very little about the behaviour of constant scalar curvature metrics in (complex) dimension at least two, and so the results in the work under review seem likely to be one of the first steps in tackling this problem.