## Quotient singularities, eta invariants, and self-dual metrics.(English)Zbl 1348.53054

The paper under review focuses on questions arising from the study of four-dimensional spaces that have isolated singularities or noncompact ends which are modeled, respectively, on neighbourhoods of the origin and of infinity of $$\mathbb{R}^4\setminus\Gamma,$$ where $$\Gamma\subset \mathrm{SO}(4)$$ is a finite subgroup which acts freely on $$S^3.$$ In particular:
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 nonexistence result for Ricci-flat ALE metrics on certain spaces.
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.
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, examples of self-dual metrics on $$n\#\mathbb{CP}^2$$ are obtained for $$n\geq 3$$. These examples admit an $$S^1$$-action, but are not of LeBrun type.

### MSC:

 53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.) 58J20 Index theory and related fixed-point theorems on manifolds 58J28 Eta-invariants, Chern-Simons invariants
Full Text: