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Spin(7) metrics from Kähler Geometry

Udhav Fowdar

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.

Spin(7) metrics from Kähler Geometry

Abstract

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

Paper Structure

This paper contains 24 sections, 14 theorems, 152 equations.

Table of Contents

  1. Introduction
  2. $\mathbb{T}^2$-reduction of torsion free $Spin(7)$-structures
  3. The basics
  4. The general setup
  5. The Kähler reduction
  6. The first reduction
  7. A second reduction
  8. Further reduction I
  9. $S^1$ Kähler reduction
  10. The $Spin(7)$ condition
  11. A general local existence result
  12. Constant solutions $\mathrm{I}$
  13. Examples with holonomy $Spin(7)$, $G_2$, $SU(3)$ and $SU(4)$.
  14. The GLPS examples
  15. $Spin(7)$ metrics from Gibbons-Hawking ansatz
  16. ...and 9 more sections

Key Result

Lemma 2.2

The condition $d\Phi=0$ is equivalent to $d\omega=0$ and the system with

Theorems & Definitions (27)

  • Remark 2.1
  • Lemma 2.2
  • Proposition 2.3
  • proof
  • Remark 2.4
  • Remark 3.1
  • Proposition 3.2
  • Proposition 3.3
  • Proposition 3.4
  • proof
  • ...and 17 more