An index theorem for anti-self-dual orbifold-cone metrics

2012/09/14 by Michael T. Lock, Jeff A. Viaclovsky, Jeff Viaclovsky +2
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Geometry and complex manifolds #math.DG

paper · pdf · doi:10.48550/arxiv.1209.3243

18 pages

arxiv created 2012/09/14 · openalex publication_date 2012/09/14 · arxiv updated 2012/09/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Recently, Atiyah and LeBrun proved versions of the Gauss-Bonnet and Hirzebruch signature Theorems for metrics with edge-cone singularities in dimension four, which they applied to obtain an inequality of Hitchin-Thorpe type for Einstein edge-cone metrics. Interestingly, many natural examples of edge-cone metrics in dimension four are anti-self-dual (or self-dual depending upon choice of orientation). On such a space there is an important elliptic complex called the anti-self-dual deformation complex, whose index gives crucial information about the local structure of the moduli space of anti-self-dual metrics. In this paper, we compute the index of this complex in the orbifold case, and give several applications.

Citations

• Anti-self-dual orbifolds with cyclic quotient singularities
• Ricci flat Kahler metrics with edge singularities

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