An Index Theorem on Anti-Self-Dual Orbifolds

An index theorem for the anti-self-dual deformation complex on anti-self-dual orbifolds with singularities conjugate to ADE-type is proved. In 1988, Claude Lebrun gave examples of scalar-flat Kähler ALE metrics with negative mass, on the total space of the bundle
$\mathcal O(-n)$
over S
²
. A corollary of this index theorem is that the moduli space of anti-self-dual ALE metrics near each of these metrics has dimension at least 4n−12, and thus for n≥4 the LeBrun metrics admit a plethora of nontrivial anti-self-dual deformations.