Spin(7) metrics from Kähler Geometry

Abstract

We investigate the ${\mathbb{T}}^2$-quotient of a torsion free $Spin(7)$-structure on an $8$-manifold under the assumption that the quotient $6$-manifold is Kähler. We show that there exists either a Hamiltonian $S^1$ or ${\mathbb{T}}^2$ action on the quotient preserving the complex structure. Performing a Kähler reduction in each case reduces the problem of finding $Spin(7)$ metrics to studying a system of PDEs on either a $4$- or $2$-manifold with trivial canonical bundle, which in the compact case corresponds to either ${\mathbb{T}}^4$, a K3 surface or an elliptic curve. By reversing this construction we give infinitely many new explicit examples of $Spin(7)$ holonomy metrics. In the simplest case, our result can be viewed as an extension of the Gibbons-Hawking ansatz.

27 pages, to appear in Communications in Analysis and Geometry

References in corpus (2)

• $S^1$-invariant Laplacian flow
• $S^1$-quotient of $Spin(7)$-structures

Cited by in corpus (1)

• Explicit abelian instantons on $S^1$-invariant Kähler Einstein $6$-manifolds