Curvature at infinity of scalar-flat ALE four-manifolds

We study refined asymptotics of scalar-flat ALE four-manifolds in the Tian–Viaclovsky setting, namely for self-dual or anti-self-dual metrics and for metrics with harmonic curvature. Starting from the ALE coordinates obtained by Tian–Viaclovsky, we construct preferred coordinates at infinity and identify the homogeneous $\mathrm{\mid }x{\mathrm{\mid }}^{-2}$ term in the metric expansion. This term splits canonically into a scalar part determined by the ALE ADM mass and an algebraic Weyl tensor at infinity. As an application, we consider scalar-flat Kähler ALE metrics on minimal resolutions $\pi :X\to {\mathbb{C}}^2\mathrm{/}\mathrm{\Gamma }$ of quotient surface singularities. In this case, the leading Weyl tensor at infinity vanishes exactly when the minimal resolution is crepant.