Quotient singularities, eta invariants, and self-dual metrics

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 $\mathrm{SO}(4)$ 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 $\mathrm{SO}(4)$ 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. (iii) Two infinite families of scalar-flat anti-self-dual ALE spaces with groups at infinity not contained in $\mathrm{U}(2)$ are constructed. Using these spaces, new examples of self-dual metrics on $n\#{\mathbb{CP}}^2$ are obtained for $n\ge 3$.

29 pages

References in corpus (2)

• Anti-self-dual orbifolds with cyclic quotient singularities
• Deformation of LeBrun's ALE metrics with negative mass

Cited by in corpus (3)

• Curvature growth of some 4-dimensional gradient Ricci soliton singularity models
• Noncollapsed degeneration of Einstein 4-manifolds II
• On the geometry of asymptotically flat manifolds