Quotient singularities, eta invariants, and self-dual metrics

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ISSN 1364-0380 (online)

ISSN 1465-3060 (print)

Geometry & Topology 20 (2016) 1773–1806

DOI: 10.2140/gt.2016.20.1773

There are three main components to this article:

A formula for the $η$ –invariant of the signature complex for any finite subgroup of $SO (4)$ 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 $SO (4)$ 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 $U (2)$ are constructed. Using these spaces, examples of self-dual metrics on $n # {ℂ ℙ}^{2}$ are obtained for $n \geq 3$ . These examples admit an $S^{1}$ –action, but are not of LeBrun type.

quotient singularities, eta invariants, self-dual, ALE, orbifold

Primary: 53C25, 58J20

Received: 11 May 2015

Accepted: 21 August 2015

Published: 4 July 2016

Proposed: Simon Donaldson

Seconded: Tobias H Colding, Ciprian Manolescu

Michael T Lock
Department of Mathematics University of Texas Austin, TX 78712 United States

Jeff A Viaclovsky
Department of Mathematics University of Wisconsin Madison, WI 53706 United States