Complete non-compact $\operatorname{Spin}(7)$-manifolds from $T^2$-bundles over AC Calabi Yau manifolds

TL;DR

This paper introduces a new method to construct complete non-compact 8-manifolds with Spin(7) holonomy using T^2-bundles over asymptotically conical Calabi-Yau manifolds, revealing novel geometric structures at infinity.

Contribution

The authors develop a novel construction of Spin(7) manifolds from T^2-bundles over AC Calabi-Yau spaces, including the first examples of complete toric Spin(7) manifolds.

Findings

Constructed infinitely many diffeomorphism types of AT^2C Spin(7)-manifolds.

Produced the first known complete toric Spin(7)-manifolds.

Introduced the asymptotically T^2-fibred conical (AT^2C) geometry.

Abstract

We develop a new construction of complete non-compact 8-manifolds with Riemannian holonomy equal to $\operatorname{Spin}(7)$. As a consequence of the holonomy reduction, these manifolds are Ricci-flat. These metrics are built on the total spaces of principal $T^2$-bundles over asymptotically conical Calabi Yau manifolds, and the result is generalized to orbifolds. The resulting metrics have a new geometry at infinity that we call asymptotically $T^2$-fibred conical ($AT^2C$) 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 $AT^2C$ $\operatorname{Spin}(7)$-manifolds and the first known examples of complete toric $\operatorname{Spin}(7)$-manifold.