The holonomy of the Obata connection on Joyce hypercomplex manifolds
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908527788818432 |
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| author | Brienza, Beatrice Fowdar, Udhav Gentili, Giovanni Vezzoni, Luigi |
| author_facet | Brienza, Beatrice Fowdar, Udhav Gentili, Giovanni Vezzoni, Luigi |
| contents | We study the holonomy of the Obata connection on Joyce hypercomplex manifolds. For all such group manifolds except $\mathrm{SU}(2n+1)$, we show that the holonomy group is strictly contained in the quaternionic general linear group. The case of $\mathrm{SU}(2n+1)$ is more subtle: for every $n>1$, we show that there exist infinitely many Joyce hypercomplex structures with Obata holonomy strictly contained in $\mathrm{GL}(n(n+1),\mathbb{H})$. On the other hand, Soldatenkov showed that $\mathrm{SU}(3)$ has Obata holonomy equal to $\mathrm{GL}(2,\mathbb{H})$ \cite{Sol}, and we present here a new example on $\mathrm{SU}(5)$ with holonomy equal to $\mathrm{GL}(6,\mathbb{H})$. Finally, we investigate Joyce hypercomplex manifolds whose restricted holonomy lie in $\mathrm{SL}(n, \mathbb{H})$, yielding new compact examples of twisted Calabi-Yau manifolds. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_07722 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The holonomy of the Obata connection on Joyce hypercomplex manifolds Brienza, Beatrice Fowdar, Udhav Gentili, Giovanni Vezzoni, Luigi Differential Geometry 22C05, 53C05, 53C26, 53C29 We study the holonomy of the Obata connection on Joyce hypercomplex manifolds. For all such group manifolds except $\mathrm{SU}(2n+1)$, we show that the holonomy group is strictly contained in the quaternionic general linear group. The case of $\mathrm{SU}(2n+1)$ is more subtle: for every $n>1$, we show that there exist infinitely many Joyce hypercomplex structures with Obata holonomy strictly contained in $\mathrm{GL}(n(n+1),\mathbb{H})$. On the other hand, Soldatenkov showed that $\mathrm{SU}(3)$ has Obata holonomy equal to $\mathrm{GL}(2,\mathbb{H})$ \cite{Sol}, and we present here a new example on $\mathrm{SU}(5)$ with holonomy equal to $\mathrm{GL}(6,\mathbb{H})$. Finally, we investigate Joyce hypercomplex manifolds whose restricted holonomy lie in $\mathrm{SL}(n, \mathbb{H})$, yielding new compact examples of twisted Calabi-Yau manifolds. |
| title | The holonomy of the Obata connection on Joyce hypercomplex manifolds |
| topic | Differential Geometry 22C05, 53C05, 53C26, 53C29 |
| url | https://arxiv.org/abs/2509.07722 |