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

We present an analytic construction of complete noncompact 8–dimensional Ricciflat 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 selfdual 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.