Weighted projective embeddings, stability of orbifolds and constant scalar curvature Kähler metrics

Abstract

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 Kahler 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 Kahler metrics. We describe the manifold applications of this orbifold result, and show how many previously known results (Troyanov, Ghigi-Kollar, Rollin-Singer, the AdS/CFT Sasaki-Einstein obstructions of Gauntlett-Martelli-Sparks-Yau) fit into this framework.

49 pages. References updated

References in corpus (7)

• Extremal metrics and K-stability (PhD thesis)
• K-stability of constant scalar curvature Kähler manifolds
• Non-minimal scalar-flat Kaehler surfaces and parabolic stability
• Stable varieties with a twist
• An obstruction to the existence of constant scalar curvature Kähler metrics
• GIT stability of weighted pointed curves
• Kaehler-Einstein metrics on orbifolds and Einstein metrics on spheres

Cited by in corpus (6)

• Stable varieties with a twist
• Remarks on logarithmic K-stability
• A thermodynamical formalism for Monge-Ampere equations, Moser-Trudinger inequalities and Kahler-Einstein metrics
• The Sasaki-Ricci flow and compact Sasakian manifolds of positive transverse holomorphic bisectional curvature
• On the transverse scalar curvature of a compact Sasaki manifold
• Weighted Bergman kernels on orbifolds