Journal of Differential Geometry

Weighted Projective Embeddings, Stability of Orbifolds, and Constant Scarla Curvature Kär Metrics

Julius Ross and Richard Thomas

Full-text: Open access

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

Article information

Source
J. Differential Geom., Volume 88, Number 1 (2011), 109-159.

Dates
First available in Project Euclid: 4 October 2011

Permanent link to this document
https://projecteuclid.org/euclid.jdg/1317758871

Digital Object Identifier
doi:10.4310/jdg/1317758871

Mathematical Reviews number (MathSciNet)
MR2819757

Citation

Ross, Julius; Thomas, Richard. Weighted Projective Embeddings, Stability of Orbifolds, and Constant Scarla Curvature Kär Metrics. J. Differential Geom. 88 (2011), no. 1, 109--159. doi:10.4310/jdg/1317758871. https://projecteuclid.org/euclid.jdg/1317758871


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