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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
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