On Yau's theorem for effective orbifolds

Abstract

In 1978, Yau confirmed a conjecture due to Calabi stating the existence of Kähler metrics with prescribed Ricci forms on compact Kähler manifolds. A version of this statement for effective orbifolds can be found in the literature. In this expository article, we provide details for a proof of this orbifold version of the statement by adapting Yau's original continuity method to the setting of effective orbifolds in order to solve a Monge-Ampère equation. We then outline how to obtain Kähler-Einstein metrics on orbifolds with $c_1(\mathcal{X})<0$ by solving a slightly different Monge-Ampère equation. We conclude by listing some explicit examples of Calabi-Yau orbifolds, which consequently admit Ricci flat metrics by Yau's theorem for effective orbifolds.

31 pages