Spin(7) metrics from Kähler Geometry
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2020
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| _version_ | 1866914995444383744 |
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| author | Fowdar, Udhav |
| author_facet | Fowdar, Udhav |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2002_03449 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Spin(7) metrics from Kähler Geometry Fowdar, Udhav Differential Geometry 53C10, 53C29, 53C55 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. |
| title | Spin(7) metrics from Kähler Geometry |
| topic | Differential Geometry 53C10, 53C29, 53C55 |
| url | https://arxiv.org/abs/2002.03449 |