On Yau's theorem for effective orbifolds

TL;DR

This paper extends Yau's theorem to effective orbifolds, providing a detailed proof using a continuity method and demonstrating the existence of Kähler-Einstein and Ricci-flat metrics on orbifolds.

Contribution

It adapts Yau's original proof technique to the orbifold setting, establishing the existence of Kähler-Einstein and Ricci-flat metrics on effective orbifolds.

Findings

Proof of Yau's theorem for effective orbifolds

Existence of Kähler-Einstein metrics on orbifolds with negative first Chern class

Examples of Calabi-Yau orbifolds with Ricci-flat metrics

Abstract

In 1978, Yau confirmed a conjecture due to Calabi stating the existence of K\"ahler metrics with prescribed Ricci forms on compact K\"ahler 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-Amp\`ere equation. We then outline how to obtain K\"ahler-Einstein metrics on orbifolds with $c_1(\mathcal{X}) < 0$ by solving a slightly different Monge-Amp\`ere 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.