#### Volume 25, issue 1 (2021)

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

### Lorenzo Foscolo

Geometry & Topology 25 (2021) 339–408
##### Abstract

We present an analytic construction of complete noncompact $8$–dimensional Ricci-flat manifolds with holonomy $Spin\left(7\right)$. The construction relies on the study of the adiabatic limit of metrics with holonomy $Spin\left(7\right)$ 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 $Spin\left(7\right)$–manifolds with arbitrarily large second Betti number and infinitely many distinct families of ALC $Spin\left(7\right)$–metrics on the same smooth $8$–manifold.

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##### Keywords
exceptional holonomy, complete noncompact Ricci-flat manifolds, self-dual Einstein 4-orbifolds
##### Mathematical Subject Classification 2010
Primary: 53C25
Secondary: 53C10, 53C29, 53C80