#### Volume 20, issue 3 (2016)

 Recent Issues
 The Journal About the Journal Subscriptions Editorial Board Editorial Interests Editorial Procedure Submission Guidelines Submission Page Author Index To Appear ISSN (electronic): 1364-0380 ISSN (print): 1465-3060
Quotient singularities, eta invariants, and self-dual metrics

### Michael T Lock and Jeff A Viaclovsky

Geometry & Topology 20 (2016) 1773–1806
##### Abstract

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 ${S}^{3}\phantom{\rule{0.3em}{0ex}}$. 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 $n#{ℂℙ}^{2}$ are obtained for $n\ge 3$. These examples admit an ${S}^{1}\phantom{\rule{0.3em}{0ex}}$–action, but are not of LeBrun type.
##### Keywords
quotient singularities, eta invariants, self-dual, ALE, orbifold
##### Mathematical Subject Classification 2010
Primary: 53C25, 58J20