Weighted projective embeddings, stability of orbifolds, and constant scalar curvature Kähler metrics
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Publication:661934
DOI10.4310/JDG/1317758871zbMATH Open1244.32013arXiv0907.5214OpenAlexW1827173265WikidataQ115170721 ScholiaQ115170721MaRDI QIDQ661934
Author name not available (Why is that?)
Publication date: 11 February 2012
Published in: (Search for Journal in Brave)
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.
Full work available at URL: https://arxiv.org/abs/0907.5214
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