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

By J. Ross, Richard P. Thomas. Journal of differential geometry, 2009

114 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/723be845503425d4ae036a834b393ffa7d0ba82b

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In Applied Mathematics

1. Chi Li
2. Robert J. Berman
3. Song Sun
4. G'abor Sz'ekelyhidi
5. Yanir A. Rubinstein
6. G. Tian
7. Simon Donaldson
8. Xiuxiong Chen

In Geometry and Topology

1. Chi Li
2. Robert J. Berman
3. G'abor Sz'ekelyhidi
4. Song Sun
5. Yanir A. Rubinstein
6. G. Tian
7. Xiuxiong Chen
8. Simon Donaldson

Citations in the corpus

• Ricci flow on surfaces with conic singularities (2013, background): “Following [Ross and Thomas 2011], these conditions can also be reinterpreted as saying that the twisted Futaki invariant of the pair ( M, ∑ (1−βi )pi ) is nonnegative, or, equivalently, that this pair is logarithmically K-stable.”
• Sasaki–Einstein metrics and K–stability (2015, methodology): “ continuity method. We will set up this continuity method in Section 2, where we also give the precise definition of K-stability based on our previous work [27], extending the definition of Ross-Thomas [67] from the quasi-regular case. The main technical results are contained in Sections 3 and 4. In S”
• A thermodynamical formalism for Monge–Ampère equations, Moser–Trudinger inequalities and Kähler–Einstein metrics (2010, background): “metric on Xwhich is smooth in the orbifold sense, which from a differential geometric point of view means that ωhas cone angles 2π1/min the directions transverse to D(see for example the discussion in [67]). In our general setting we define the alpha-invariant of the pair (X,∆) by α(X,∆) := α(−c1(KX+∆”
• Frankel conjecture and Sasaki geometry (2012, unknown): “llary 1.1. A compact polarized orbifold (M,J,g,L) with positive bisectional curvature is bi-holomorphic to a finite quotient of a weighted projective space. The notion polarized orbifold is taken from [50]. By this we mean there is an orbi-line bundle L, and in any orbifold chart (Up,Lp,Gp) the actio”
• Integral bounds on curvature and Gromov–Hausdorff limits (2012, result): “This phenomenon—that near to a singularity ρ gets larger or smaller depending on divisibility—is similar to the orbifold situation considered by Ross and Thomas in [17].”
• Asymptotically conical Calabi-Yau manifolds, III (2012, background): “(i) There exists a test configuration [7, 28] π : (Y,L) → C with general fibre (Y,L) such that • there exists a homeomorphism F from Y0 onto the central fibre Y0 such that if v denotes the vertex of the projective cone Y0, then F |Y0\{v} is biholomorphic onto its image, and • L is the Q-line bundle ”