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Complete noncompact Spin(7) manifolds from self-dual Einstein $4$–orbifolds

Lorenzo Foscolo

Geometry & Topology 25 (2021) 339–408

We present an analytic construction of complete noncompact 8–dimensional Ricci-flat manifolds with holonomy Spin(7). The construction relies on the study of the adiabatic limit of metrics with holonomy Spin(7) on principal Seifert circle bundles over asymptotically conical G2–orbifolds. The metrics we produce have an asymptotic geometry, so-called ALC geometry, that generalises to higher dimensions the geometry of 4–dimensional ALF hyperkähler metrics.

We apply our construction to asymptotically conical G2–metrics arising from self-dual Einstein 4–orbifolds with positive scalar curvature. As illustrative examples of the power of our construction, we produce complete noncompact Spin(7)–manifolds with arbitrarily large second Betti number and infinitely many distinct families of ALC Spin(7)–metrics on the same smooth 8–manifold.

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exceptional holonomy, complete noncompact Ricci-flat manifolds, self-dual Einstein 4-orbifolds
Mathematical Subject Classification 2010
Primary: 53C25
Secondary: 53C10, 53C29, 53C80
Received: 5 April 2019
Revised: 20 December 2019
Accepted: 17 February 2020
Published: 2 March 2021
Proposed: Simon Donaldson
Seconded: Mark Gross, Yasha Eliashberg
Lorenzo Foscolo
Department of Mathematics
University College London
United Kingdom