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

Foscolo, Lorenzo

doi:10.2140/gt.2021.25.339

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

Lorenzo Foscolo

Geometry & Topology 25 (2021) 339–408

DOI: 10.2140/gt.2021.25.339

Abstract

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 $G_{2}$ –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 $G_{2}$ –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.

Keywords

exceptional holonomy, complete noncompact Ricci-flat manifolds, self-dual Einstein 4-orbifolds

Mathematical Subject Classification 2010

Primary: 53C25

Secondary: 53C10, 53C29, 53C80

References

Publication

Received: 5 April 2019

Revised: 20 December 2019

Accepted: 17 February 2020

Published: 2 March 2021

Proposed: Simon Donaldson

Seconded: Mark Gross, Yasha Eliashberg

Authors

Lorenzo Foscolo
Department of Mathematics University College London London United Kingdom

Volume 25, issue 1 (2021)