Quotient singularities, eta invariants, and self-dual metrics

TL;DR

This paper derives formulas for eta invariants and orbifold correction terms related to quotient singularities, and constructs new scalar-flat anti-self-dual ALE spaces, advancing understanding of self-dual metrics and their geometric properties.

Contribution

It provides explicit formulas for eta invariants and correction terms for finite subgroups of SO(4), and constructs new scalar-flat anti-self-dual ALE spaces with novel group actions.

Findings

Formulas for eta invariants of signature complexes for finite subgroups of SO(4).

Formulas for orbifold correction terms in the index of self-dual deformation complexes.

Construction of new scalar-flat anti-self-dual ALE spaces with non-U(2) groups at infinity.

Abstract

There are three main components to this article: (i) A formula for the eta invariant of the signature complex for any finite subgroup of ${\rm{SO}}(4)$ acting freely on $S^3$ is given. An application of this is a non-existence 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 ${\rm{SO}}(4)$ which act freely on $S^3$. Some applications of this formula to the realm of self-dual and scalar-flat K\"ahler metrics are also discussed. (iii) Two infinite families of scalar-flat anti-self-dual ALE spaces with groups at infinity not contained in ${\rm{U}}(2)$ are constructed. Using these spaces, new examples of self-dual metrics on $n \# \mathbb{CP}^2$ are obtained for $n \geq 3$.