Construction of higher-dimensional ALF Calabi-Yau metrics

Abstract

Roughly speaking, an ALF metric of real dimension 4n should be a metric such that it has a (4n-1)-dimensional asymptotic cone, the volume growth of this metric is of order 4n-1 and its sectional curvature tends to 0 at infinity. In this paper, we first show that the Taub-NUT deformation of a hyperk\"ahler cone with respect to a locally free S1-symmetry is ALF hyperk\"ahler. Using this metric at infinity, we establish the existence of ALF Calabi-Yau metric on certain crepant resolutions. In particular, we prove that there exist ALF Calabi-Yau metrics on canonical bundles of classical homogeneous Fano contact manifolds.