An index theorem on anti-self-dual orbifolds

TL;DR

This paper proves an index theorem for anti-self-dual deformation complexes on orbifolds with ADE-type singularities, revealing the existence of many non-trivial deformations of certain scalar-flat Kähler ALE metrics.

Contribution

It establishes an index theorem for anti-self-dual orbifolds with ADE singularities, enabling analysis of the moduli space of deformations of specific ALE metrics.

Findings

The moduli space dimension is at least 4n-12 near LeBrun metrics for n≥4.

LeBrun metrics admit numerous anti-self-dual deformations.

The index theorem applies to orbifolds with ADE-type singularities.

Abstract

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 K\"ahler 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 $n \geq 4$ the LeBrun metrics admit a plethora of non-trivial anti-self-dual deformations.