We embed polarised orbifolds with cyclic stabiliser groups into weighted projective space via a weighted form of Kodaira embedding. Dividing by the (non-reductive) automorphisms of weighted projective space then formally gives a moduli space of orbifolds.
We show how to express this as a reductive quotient and so a GIT problem, thus defining a notion of stability for orbifolds.
We then prove an orbifold version of Donaldson’s theorem: the existence of an orbifold K¨ahler metric of constant scalar curvature
implies K-semistability.
By extending the notion of slope stability to orbifolds, we therefore get an explicit obstruction to the existence of constant scalar
curvature orbifold K¨ahler metrics. We describe the manifold applications of this orbifold result, and show how many previously
known results (Troyanov, Ghigi-Koll´ar, Rollin-Singer, the AdSCFT Sasaki-Einstein obstructions of Gauntlett-Martelli-Sparks-Yau) fit into this framework.