On Yau's Theorem for Asymptotically Conical Orbifolds

Abstract

A notion of asymptotically conical Kähler orbifold is introduced, and, following previous existence results in the setting of asymptotically conical manifolds, it is shown that a certain complex Monge-Ampére equation admits a rapidly decaying solution (which is unique for certain intervals of decay rates), allowing one to construct Kähler metrics with prescribed Ricci forms. In particular, if the orbifold has trivial canonical bundle, then Ricci-flat metrics can be constructed, provided certain additional hypotheses are met. This implies for example that orbifold crepant partial resolutions of varieties associated to Calabi-Yau cones admit a one-parameter family of Calabi-Yau metrics in each Kähler class that contains positive (1,1)-forms.