Volume 20, issue 3 (2016)

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Quotient singularities, eta invariants, and self-dual metrics

Michael T Lock and Jeff A Viaclovsky

Geometry & Topology 20 (2016) 1773–1806

There are three main components to this article:

  1. A formula for the η–invariant of the signature complex for any finite subgroup of SO(4) acting freely on S3 is given. An application of this is a nonexistence result for Ricci-flat ALE metrics on certain spaces.
  2. 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 S3. Some applications of this formula to the realm of self-dual and scalar-flat Kähler metrics are also discussed.
  3. 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 # 2 are obtained for n 3. These examples admit an S1–action, but are not of LeBrun type.

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quotient singularities, eta invariants, self-dual, ALE, orbifold
Mathematical Subject Classification 2010
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