On K\"ahler conformal compactifications of $U(n)$-invariant ALE spaces

TL;DR

This paper demonstrates that a class of ALE spaces admits explicit Kähler conformal compactifications, enabling new constructions of Bochner-Kähler metrics and Kähler edge-cone metrics, with implications for Einstein metrics.

Contribution

It provides explicit formulas for conformal factors and Kähler potentials, and introduces new constructions of Kähler metrics on weighted projective spaces and $ ext{CP}^2$ with cone singularities.

Findings

Explicit conformal compactifications for ALE spaces.

New construction of Bochner-Kähler metrics on weighted projective spaces.

Explicit Kähler edge-cone metrics on $ ext{CP}^2$ with cone angles.

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-K\"ahler 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.