Quotient singularities, eta invariants, and self-dual metrics

There are three main components to this article: (i) A formula for the eta-invariant of the signature complex for any finite subgroup of SO(4) acting freely on S-3 is given. An application of this is a nonexistence result for Ricci-flat ALE metrics on certain spaces. (ii) 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(4) which act freely on S-3. Some applications of this formula to the realm of self-dual and scalar-flat Kahler metrics are also discussed. (iii) Two infinite families of scalar-flat anti-self-dual ALE spaces with groups at infinity not contained in U(2) are constructed. Using these spaces, examples of self-dual metrics on n # CP2 are obtained for n >= 3. These examples admit an S-1-action, but are not of LeBrun type.