Geometry & Topology

Quotient singularities, eta invariants, and self-dual metrics

Michael Lock and Jeff Viaclovsky

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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.

Article information

Geom. Topol., Volume 20, Number 3 (2016), 1773-1806.

Received: 11 May 2015
Accepted: 21 August 2015
First available in Project Euclid: 16 November 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53C25: Special Riemannian manifolds (Einstein, Sasakian, etc.) 58J20: Index theory and related fixed point theorems [See also 19K56, 46L80]

quotient singularities eta invariants self-dual ALE orbifold


Lock, Michael; Viaclovsky, Jeff. Quotient singularities, eta invariants, and self-dual metrics. Geom. Topol. 20 (2016), no. 3, 1773--1806. doi:10.2140/gt.2016.20.1773.

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