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

Julius Ross University of Cambridge Richard Thomas Imperial College

Differential Geometry mathscidoc:1609.10221

Journal of Differential Geometry, 88, (1), 109-159, 2011
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.
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@inproceedings{julius2011weighted,
  title={WEIGHTED PROJECTIVE EMBEDDINGS, STABILITY OF ORBIFOLDS, AND CONSTANT SCALAR CURVATURE KAHLER METRICS},
  author={Julius Ross, and Richard Thomas},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160912074647188181880},
  booktitle={Journal of Differential Geometry},
  volume={88},
  number={1},
  pages={109-159},
  year={2011},
}
Julius Ross, and Richard Thomas. WEIGHTED PROJECTIVE EMBEDDINGS, STABILITY OF ORBIFOLDS, AND CONSTANT SCALAR CURVATURE KAHLER METRICS. 2011. Vol. 88. In Journal of Differential Geometry. pp.109-159. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160912074647188181880.
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