[Paper Review] Weighted projective embeddings, stability of orbifolds and constant scalar curvature Kähler metrics

Julius Ross, Richard Thomas|arXiv (Cornell University)|Jul 30, 2009

Geometry and complex manifolds32 references16 citations

TL;DR

This paper establishes an orbifold version of the Yau-Tian-Donaldson conjecture by introducing weighted projective embeddings for orbifolds with cyclic stabilizers, enabling a GIT-style stability notion via reductive quotients. It proves that the existence of a constant scalar curvature Kähler (cscK) metric on an orbifold implies K-semistability, and extends slope stability to orbifolds, providing a concrete obstruction to cscK 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.

Motivation & Objective

• To extend the Yau-Tian-Donaldson conjecture to orbifolds by formulating a stability condition compatible with constant scalar curvature Kähler (cscK) metrics.
• To resolve the non-reductive automorphism group of weighted projective space by reducing to a reductive quotient, enabling a Geometric Invariant Theory (GIT) framework.
• To define balanced metrics and Fubini-Study metrics on weighted projective spaces, crucial for approximating cscK metrics via asymptotic expansions.
• To generalize slope stability to orbifolds and use it as an obstruction to the existence of cscK metrics, extending results from smooth manifolds.
• To unify known obstructions (e.g., GMSY, Troyanov, Ghigi-Kollár, Rollin-Singer) within a single orbifold stability framework.

Proposed method

• Embed polarized orbifolds with cyclic stabilizers into weighted projective space using a weighted form of the Kodaira embedding, leveraging sections of multiple powers $L^k$ to preserve orbifold data.
• Replace the non-reductive automorphism group of weighted projective space with its reductive part (a product of general linear groups), enabling a GIT quotient and a well-defined stability notion.
• Define Fubini-Study metrics on weighted projective spaces via moment maps, accounting for curvature differences from the standard Fubini-Study metric.
• Use asymptotic Bergman kernel expansions with weighted sections to ensure convergence to cscK metrics, where weights are determined by the moment map's ambiguity.
• Introduce orbifold Riemann-Roch and orbifold Chern classes to compute degrees and slopes in the orbifold setting, essential for stability criteria.
• Establish a correspondence between orbifold vector bundles and parabolic bundles, translating orbifold stability into parabolic stability for known results.

Experimental results

Research questions

• RQ1 Can the Yau-Tian-Donaldson conjecture be extended to orbifolds using weighted projective embeddings and GIT stability?
• RQ2 How can one define a well-behaved moduli problem for orbifolds when the automorphism group of weighted projective space is non-reductive?
• RQ3 What is the correct notion of stability (K-stability, slope stability) for orbifolds, and how does it obstruct the existence of cscK metrics?
• RQ4 How do known obstructions to cscK metrics (e.g., GMSY, Troyanov) fit into this unified framework?
• RQ5 To what extent does the stability of an orbifold bundle $E$ on a curve determine the stability of $\mathbb{P}(E)$ with respect to a polarizing line bundle $L_m$?

Key findings

• The existence of a constant scalar curvature Kähler metric on a polarized orbifold implies K-semistability, extending Donaldson’s theorem to the orbifold setting.
• Orbifold K-stability is well-defined via reductive quotients of weighted projective space, resolving the non-reductive automorphism issue.
• Slope stability for orbifolds provides a concrete obstruction to the existence of cscK metrics, generalizing the manifold case.
• For orbifold ruled surfaces $\mathbb{P}(E)$, if the underlying vector bundle $E$ is unstable, then $\mathbb{P}(E)$ is slope unstable and thus does not admit a cscK metric.
• The correspondence between orbifold bundles and parabolic bundles preserves stability, allowing known results on parabolic stability to be reinterpreted in the orbifold context.
• Canonically polarized orbifolds (with $K_{\text{orb}}$ numerically trivial or $L = K_{\text{orb}}$) are slope stable, confirming their cscK metric existence via the orbifold Aubin-Yau theorems.