Complete non-compact Spin (7) -manifolds from T²-bundles over AC Calabi Yau manifolds

We develop a new construction of complete non-compact 8-manifolds with Riemannian holonomy equal to Spin (7). As a consequence of the holonomy reduction, these manifolds are Ricci-flat. These metrics are built on the total spaces of principal T²-bundles over asymptotically conical Calabi Yau manifolds. The resulting metrics have a new geometry at infinity that we call asymptotically T²-fibred conical (AT²C) and which generalizes to higher dimensions the ALG metrics of 4-dimensional hyperk\"ahler geometry, analogously to how ALC metrics generalize ALF metrics. As an application of this construction, we produce infinitely many diffeomorphism types of AT²C Spin (7) -manifolds and the first known example of complete toric Spin (7) -manifold.