On the canonical bundle of complex solvmanifolds and applications to hypercomplex geometry
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866907973526224896 |
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| author | Andrada, Adrián Tolcachier, Alejandro |
| author_facet | Andrada, Adrián Tolcachier, Alejandro |
| contents | We study complex solvmanifolds $Γ\backslash G$ with holomorphically trivial canonical bundle. We show that the trivializing section of this bundle can be either invariant or non-invariant by the action of $G$. First we characterize the existence of invariant trivializing sections in terms of the Koszul 1-form $ψ$ canonically associated to $(\mathfrak{g},J)$, where $\mathfrak{g}$ is the Lie algebra of $G$, and we use this characterization to produce new examples of complex solvmanifolds with trivial canonical bundle. Moreover, we provide an algebraic obstruction, also in terms of $ψ$, for a complex solvmanifold to have trivial (or more generally holomorphically torsion) canonical bundle. Finally, we exhibit a compact hypercomplex solvmanifold $(M^{4n},\{J_1,J_2,J_3\})$ such that the canonical bundle of $(M,J_α)$ is trivial only for $α=1$, so that $M$ is not an $\operatorname{SL}(n,\mathbb{H})$-manifold. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_16673 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the canonical bundle of complex solvmanifolds and applications to hypercomplex geometry Andrada, Adrián Tolcachier, Alejandro Differential Geometry 53C15, 32M10, 22E25, 22E40 We study complex solvmanifolds $Γ\backslash G$ with holomorphically trivial canonical bundle. We show that the trivializing section of this bundle can be either invariant or non-invariant by the action of $G$. First we characterize the existence of invariant trivializing sections in terms of the Koszul 1-form $ψ$ canonically associated to $(\mathfrak{g},J)$, where $\mathfrak{g}$ is the Lie algebra of $G$, and we use this characterization to produce new examples of complex solvmanifolds with trivial canonical bundle. Moreover, we provide an algebraic obstruction, also in terms of $ψ$, for a complex solvmanifold to have trivial (or more generally holomorphically torsion) canonical bundle. Finally, we exhibit a compact hypercomplex solvmanifold $(M^{4n},\{J_1,J_2,J_3\})$ such that the canonical bundle of $(M,J_α)$ is trivial only for $α=1$, so that $M$ is not an $\operatorname{SL}(n,\mathbb{H})$-manifold. |
| title | On the canonical bundle of complex solvmanifolds and applications to hypercomplex geometry |
| topic | Differential Geometry 53C15, 32M10, 22E25, 22E40 |
| url | https://arxiv.org/abs/2307.16673 |