On 4-dimensional Ricci-flat ALE manifolds

In this paper, we prove: 1. There is a one-to-one correspondence between: Hermitian non-K\"ahler ALE gravitational instantons $(M,h)$, and Bach-flat K\"ahler orbifolds $(\widehat{M},\widehat{g})$ of complex dimension 2 with exactly one orbifold point $q$, such that the scalar curvature $s_{\widehat{g}}$ satisfies $s_{\widehat{g}}(q)=0$ while being positive elsewhere. 2. There is no Hermitian non-K\"ahler ALE gravitational instanton $(M,h)$ with structure group contained in $SU(2)$, except for the Eguchi-Hanson metric with reversed orientation. A 4-dimensional Ricci-flat metric being Hermitian non-K\"ahler is equivalent to being non-trivially conformally K\"ahler.