On Kähler conformal compactifications of $U(n)$-invariant ALE spaces

Abstract

We prove that a certain class of ALE spaces always has a Kahler conformal compactification, and moreover provide explicit formulas for the conformal factor and the Kahler potential of said compactification. We then apply this to give a new and simple construction of the canonical Bochner-Kähler metric on certain weighted projective spaces, and also to explicitly construct a family Kahler edge-cone metrics on ${\mathbb{CP}}^2$, with singular set ${\mathbb{CP}}^1$, having cone angles $2\pi \beta$ for all $\beta >0$. We conclude by discussing how these results can be used to obtain certain well-known Einstein metrics.

14 pages

References in corpus (2)

• Curvature, Cones, and Characteristic Numbers
• Optimal Curvature Decays on Asymptotically Locally Euclidean Manifolds