Classification results for conformally K\"ahler gravitational instantons

We investigate the asymptotic geometry of Hermitian non-K\"ahler Ricci-flat metrics with finite $\int|Rm|^2$ at infinity. Specifically, we prove: 1. Any such metric is asymptotic to an ALE, ALF-A, AF, skewed special Kasner, ALH* model at infinity. 2. Any Hermitian non-K\"ahler gravitational instanton with non-Euclidean volume growth is one of the following: the Kerr family, the Chen-Teo family, the Taub-bolt space, the reversed Taub-NUT space. This particularly confirms a conjecture by Aksteiner-Andersson. It includes the well-known Kerr family from general relativity. 3. All Hermitian non-K\"ahler gravitational instantons can be compactified to log del Pezzo surfaces. This explains a curious relation to compact Hermitian non-K\"ahler Einstein 4-manifolds. For a 4-dimensional Ricci-flat metric, being Hermitian non-K\"ahler is equivalent to being non-trivially conformally K\"ahler.