  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.
