There is a conjectural relationship due to Yau-Tian-Donaldson between stability of projective manifolds and the existence of canonical Kahler metrics (e.g. Kahler-Einstein metrics). Embedding the projective manifold in a large projective space gives, on one hand, a Geometric Invariant Theory stability problem (by changing coordinates on the projective space) and, on the other, a notion of balanced metric which can be used to approximate the canonical Kahler metric in question. I shall discuss joint work with Richard Thomas that extends this framework to orbifolds with cyclic quotient singularities using embeddings in weighted projective space, and examples that show how several obstructions to constant scalar curvature orbifold metrics can be interpreted in terms of stability.
