On Yau's Theorem for Asymptotically Conical Orbifolds

TL;DR

This paper introduces a new class of asymptotically conical K"ahler orbifolds and proves the existence of solutions to a complex Monge-Ampère equation, enabling the construction of Ricci-flat and Calabi-Yau metrics on these orbifolds.

Contribution

It extends existence results for complex Monge-Ampère equations to asymptotically conical orbifolds, allowing for new Ricci-flat and Calabi-Yau metric constructions.

Findings

Existence of rapidly decaying solutions to the Monge-Ampère equation on orbifolds.

Construction of Ricci-flat metrics on orbifold crepant resolutions.

Presence of a family of Calabi-Yau metrics in each K"ahler class.

Abstract

A notion of asymptotically conical K\"ahler orbifold is introduced, and, following previous existence results in the setting of asymptotically conical manifolds, it is shown that a certain complex Monge-Amp\'ere equation admits a rapidly decaying solution (which is unique for certain intervals of decay rates), allowing one to construct K\"ahler 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 K\"ahler class that contains positive (1,1)-forms.