Remarks on Kähler Orbifolds of non-negative Ricci curvature

TL;DR

This work extends Kobayashi's theorem from manifolds with positive Ricci curvature to compact Kähler orbifolds with non-negative Ricci curvature, under a condition on the fixed-point dimensions of local orbifold groups. It introduces orbifold charts, regular and singular strata, and the fixed-point spaces $F_x^{p,0}$ to control holomorphic forms, and proves that if $\sum_{p=1}^{n}\inf_{x}\dim F_x^{p,0}=0$, the orbifold is simply connected; in even dimensions the fundamental group is either trivial or $\mathbb{Z}_2$, with the latter implying $Ric\equiv 0$ (Calabi-Yau). The proofs adapt the Bochner technique and Kawasaki's orbifold Riemann-Roch, showing vanishing of $H^{p,0}$ for $p\ge1$, finiteness of $\pi_1(X)$, and, in the refined case, a holonomy reduction to $SU(2m)$ in the $\mathbb{Z}_2$ scenario. An explicit example $X=T^2_{\mathbb{C}}/\mathbb{Z}_2$ illustrates the distinction between topological and orbifold fundamental groups and highlights the role of singularities (ADE) in the broader context.

Abstract

This note proves orbifold versions of Kobayashi's theorem. The main result asserts that a compact Kähler orbifold with non-negative Ricci curvature, along with certain conditions regarding singularities, is simply connected.

Theorems & Definitions (8)

• Theorem 1
• Theorem 2
• Proposition 3
• proof
• Lemma 4
• proof
• proof : Proof of Theorem \ref{['main-th']}
• proof : Proof of Theorem \ref{['th-new']}