Logarithmic Cartan geometry on complex manifolds with trivial logarithmic tangent bundle
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909388216729600 |
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| author | Biswas, Indranil Dumitrescu, Sorin Morye, Archana S. |
| author_facet | Biswas, Indranil Dumitrescu, Sorin Morye, Archana S. |
| contents | Let $M$ be a compact complex manifold, and $D\, \subset\, M$ a reduced normal crossing divisor on it, such that the logarithmic tangent bundle $TM(-\log D)$ is holomorphically trivial. Let ${\mathbb A}$ denote the maximal connected subgroup of the group of all holomorphic automorphisms of $M$ that preserve the divisor $D$. Take a holomorphic Cartan geometry $(E_H,\,Θ)$ of type $(G,\, H)$ on $M$, where $H\, \subset\, G$ are complex Lie groups. We prove that $(E_H,\,Θ)$ is isomorphic to $(ρ^* E_H,\,ρ^* Θ)$ for every $ρ\, \in\, \mathbb A$ if and only if the principal $H$--bundle $E_H$ admits a logarithmic connection $Δ$ singular on $D$ such that $Θ$ is preserved by the connection $Δ$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_08593 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Logarithmic Cartan geometry on complex manifolds with trivial logarithmic tangent bundle Biswas, Indranil Dumitrescu, Sorin Morye, Archana S. Complex Variables Algebraic Geometry Differential Geometry Let $M$ be a compact complex manifold, and $D\, \subset\, M$ a reduced normal crossing divisor on it, such that the logarithmic tangent bundle $TM(-\log D)$ is holomorphically trivial. Let ${\mathbb A}$ denote the maximal connected subgroup of the group of all holomorphic automorphisms of $M$ that preserve the divisor $D$. Take a holomorphic Cartan geometry $(E_H,\,Θ)$ of type $(G,\, H)$ on $M$, where $H\, \subset\, G$ are complex Lie groups. We prove that $(E_H,\,Θ)$ is isomorphic to $(ρ^* E_H,\,ρ^* Θ)$ for every $ρ\, \in\, \mathbb A$ if and only if the principal $H$--bundle $E_H$ admits a logarithmic connection $Δ$ singular on $D$ such that $Θ$ is preserved by the connection $Δ$. |
| title | Logarithmic Cartan geometry on complex manifolds with trivial logarithmic tangent bundle |
| topic | Complex Variables Algebraic Geometry Differential Geometry |
| url | https://arxiv.org/abs/2411.08593 |