Quotient singularities, eta invariants, and self-dual metrics

Abstract

There are three main components to this article:

1. A formula for the $\eta$–invariant of the signature complex for any finite subgroup of $SO\left(4\right)$ acting freely on $S^3$ is given. An application of this is a nonexistence result for Ricci-flat ALE metrics on certain spaces.

2. A formula for the orbifold correction term that arises in the index of the self-dual deformation complex is proved for all finite subgroups of $SO\left(4\right)$ which act freely on $S^3$. Some applications of this formula to the realm of self-dual and scalar-flat Kähler metrics are also discussed.

3. Two infinite families of scalar-flat anti-self-dual ALE spaces with groups at infinity not contained in $U\left(2\right)$ are constructed. Using these spaces, examples of self-dual metrics on $n\#{ℂ\mathbb{P}}^2$ are obtained for $n\ge 3$. These examples admit an $S^1$–action, but are not of LeBrun type.