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arXiv · 1501.03234

Quotient singularities, eta invariants, and self-dual metrics

Michael T. Lock·Jeff A. Viaclovsky
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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 Kähler 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 $n \geq 3$.

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Michael T. Lock, Jeff A. Viaclovsky. 2015-01-14. Quotient singularities, eta invariants, and self-dual metrics. https://doi.org/10.2140/gt.2016.20.1773

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