Complete non-compact $\operatorname{Spin}(7)$-manifolds from $T^2$-bundles over AC Calabi Yau manifolds
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866929677706199040 |
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| author | Cavalleri, Nicolò |
| author_facet | Cavalleri, Nicolò |
| contents | 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 hyperkähler 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_19486 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Complete non-compact $\operatorname{Spin}(7)$-manifolds from $T^2$-bundles over AC Calabi Yau manifolds Cavalleri, Nicolò Differential Geometry 53C10, 53C25, 53C29, 53C80 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 hyperkähler 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. |
| title | Complete non-compact $\operatorname{Spin}(7)$-manifolds from $T^2$-bundles over AC Calabi Yau manifolds |
| topic | Differential Geometry 53C10, 53C25, 53C29, 53C80 |
| url | https://arxiv.org/abs/2407.19486 |