Quasi-ALE metrics with holonomy \(\text{SU}(m)\) and \(\text{Sp}(m)\) (Q5936997)

Assume \(G\) is a finite subgroup of \(U(m)\), and \(X\) a resolution of \(\mathbb{C}^m/G\). The author defines a special class of Kähler metrics \(g\) on \(X\) called Quasi Asymptotically Locally Euclidean (QALE) metrics. In the paper it is proved an existence theorem for Ricci-flat QALE Kähler metrics: if \(G\) is a finite subgroup of \(SU(m)\) and \(X\) a crepant resolution of \(C^m/G\), then there is a unique Ricci-flat QALE Kähler metric on \(X\) in each Kähler class. The author proves the result by applying a version of the Calabi conjecture for QALE manifolds. It is also determined the holonomy group of the metrics in terms of \(G\). The author states in the Introduction of the paper that: ``The author does not know of any previous papers on QALE manifolds at all, at the time of writing; this may be the first paper on the subject''.