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

TL;DR

This paper extends index theorems to anti-self-dual orbifold-cone metrics in four dimensions, providing new tools to analyze their moduli spaces and geometric properties.

Contribution

It computes the index of the anti-self-dual deformation complex for orbifold-cone metrics, enabling deeper understanding of their moduli space structure.

Findings

Index formula for anti-self-dual orbifold-cone metrics

Applications to moduli space analysis

Connections to edge-cone singularity geometry

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.