On semi-finite vector bundles with connection over Kahler manifolds
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909750836330496 |
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| author | Amrutiya, Sanjay Biswas, Indranil |
| author_facet | Amrutiya, Sanjay Biswas, Indranil |
| contents | Let $X$ be a compact connected Kähler manifold. We consider the category $\mathcal{C}^\mathrm{EC}(X)$ of flat holomorphic connections $(E,\, \nabla^E)$ over $X$ satisfying the condition that the underlying holomorphic vector bundle $E$ admits a filtration of holomorphic subbundles preserved by the connection $\nabla^E$ such that the monodromy of the induced connection on each successive quotient has finite image. The category $\mathcal{C}^\mathrm{EC}(X)$, equipped with the neutral fiber functor that sends any object $(E,\, \nabla^E)$ to the fiber $E_{x_0}$, where $x_0\, \in\, X$ is a fixed point, defines a neutral Tannakian category over $\mathbb{C}$. Let $\varpi^{\mathrm{EC}}(X,\, x_0)$ denote the affine group scheme corresponding to this neutral Tannakian category $\mathcal{C}^\mathrm{EC}(X)$. Let $π^{\mathrm{EN}}(X,\, x_0)$ be an extension of the Nori fundamental group scheme over $\mathbb{C}$.
We show that $π^{\mathrm{EN}}(X,\, x_0)$ is a closed subgroup scheme of $\varpi^{\mathrm{EC}}(X,\, x_0)$. Finally, we discuss an example illustrating that if $X$ is not Kähler, then the natural homomorphism $π^{\mathrm{EN}}(X,\, x_0)\, \longrightarrow\, \varpi^{\mathrm{EC}}(X,\, x_0)$ might fail to be an embedding. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_17048 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On semi-finite vector bundles with connection over Kahler manifolds Amrutiya, Sanjay Biswas, Indranil Algebraic Geometry 53C07, 14C34, 16D90, 14K20 Let $X$ be a compact connected Kähler manifold. We consider the category $\mathcal{C}^\mathrm{EC}(X)$ of flat holomorphic connections $(E,\, \nabla^E)$ over $X$ satisfying the condition that the underlying holomorphic vector bundle $E$ admits a filtration of holomorphic subbundles preserved by the connection $\nabla^E$ such that the monodromy of the induced connection on each successive quotient has finite image. The category $\mathcal{C}^\mathrm{EC}(X)$, equipped with the neutral fiber functor that sends any object $(E,\, \nabla^E)$ to the fiber $E_{x_0}$, where $x_0\, \in\, X$ is a fixed point, defines a neutral Tannakian category over $\mathbb{C}$. Let $\varpi^{\mathrm{EC}}(X,\, x_0)$ denote the affine group scheme corresponding to this neutral Tannakian category $\mathcal{C}^\mathrm{EC}(X)$. Let $π^{\mathrm{EN}}(X,\, x_0)$ be an extension of the Nori fundamental group scheme over $\mathbb{C}$. We show that $π^{\mathrm{EN}}(X,\, x_0)$ is a closed subgroup scheme of $\varpi^{\mathrm{EC}}(X,\, x_0)$. Finally, we discuss an example illustrating that if $X$ is not Kähler, then the natural homomorphism $π^{\mathrm{EN}}(X,\, x_0)\, \longrightarrow\, \varpi^{\mathrm{EC}}(X,\, x_0)$ might fail to be an embedding. |
| title | On semi-finite vector bundles with connection over Kahler manifolds |
| topic | Algebraic Geometry 53C07, 14C34, 16D90, 14K20 |
| url | https://arxiv.org/abs/2508.17048 |