Spin(7) metrics from K\"ahler Geometry

TL;DR

This paper explores the construction of new $Spin(7)$ holonomy metrics via K"ahler geometry and symmetry reductions, extending known ansatzes and providing explicit examples.

Contribution

It introduces a method to generate $Spin(7)$ metrics from K"ahler quotients with symmetry, producing infinitely many explicit examples.

Findings

Existence of Hamiltonian $S^1$ or $bT^2$ actions preserving complex structure.

Reduction of the problem to PDEs on lower-dimensional manifolds.

Construction of new explicit $Spin(7)$ holonomy metrics.

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 K\"ahler. We show that there exists either a Hamiltonian $S^1$ or $\mathbb{T}^2$ action on the quotient preserving the complex structure. Performing a K\"ahler 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.