Holomorphic Jet Modules and Holomorphic Connections for Noncommutative Complex Curves
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arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| author | Biswas, Indranil Guin, Satyajit Kumar, Pradip |
| author_facet | Biswas, Indranil Guin, Satyajit Kumar, Pradip |
| contents | We extend Atiyah's holomorphic jet bundle formalism to holomorphic vector bundles over noncommutative algebras endowed with a bigraded differential calculus truncated at bidegree $(1,1)$; we refer to such structures as noncommutative complex curves. For a holomorphic vector bundle $(E,\overline{\nabla}_E)$ over such an algebra $\mathcal{A}$, we construct a canonical holomorphic structure $\overline{\nabla}_J$ on the first jet module $J_E^1\,$, making the jet sequence \[ 0\longrightarrow Ω^{1,0}(\mathcal{A})\otimes_{\mathcal A}E\longrightarrow J_E^1\longrightarrow E\longrightarrow 0 \] exact in the holomorphic category. The association $(E,\overline\nabla_E)\rightsquigarrow(J_E^1\,,\overline\nabla_J)$ defines an endofunctor on the category of holomorphic vector bundles over $\mathcal{A}$. We define the notion of holomorphic connection in this setting and prove that a holomorphic vector bundle admits a holomorphic connection if and only if the jet sequence splits in the holomorphic category, or equivalently, if and only if its Atiyah class vanishes. This yields a noncommutative analogue of Atiyah's classical correspondence for Riemann surfaces. Finally, we specialize to the quantum projective line $\mathbb{CP}_q^1\,$ and determine when $\overline{\nabla}_J$ defines a bimodule connection, assuming that $\overline{\nabla}_E$ does. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_27481 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Holomorphic Jet Modules and Holomorphic Connections for Noncommutative Complex Curves Biswas, Indranil Guin, Satyajit Kumar, Pradip Quantum Algebra Algebraic Geometry 58B34, 46L87, 14A22 We extend Atiyah's holomorphic jet bundle formalism to holomorphic vector bundles over noncommutative algebras endowed with a bigraded differential calculus truncated at bidegree $(1,1)$; we refer to such structures as noncommutative complex curves. For a holomorphic vector bundle $(E,\overline{\nabla}_E)$ over such an algebra $\mathcal{A}$, we construct a canonical holomorphic structure $\overline{\nabla}_J$ on the first jet module $J_E^1\,$, making the jet sequence \[ 0\longrightarrow Ω^{1,0}(\mathcal{A})\otimes_{\mathcal A}E\longrightarrow J_E^1\longrightarrow E\longrightarrow 0 \] exact in the holomorphic category. The association $(E,\overline\nabla_E)\rightsquigarrow(J_E^1\,,\overline\nabla_J)$ defines an endofunctor on the category of holomorphic vector bundles over $\mathcal{A}$. We define the notion of holomorphic connection in this setting and prove that a holomorphic vector bundle admits a holomorphic connection if and only if the jet sequence splits in the holomorphic category, or equivalently, if and only if its Atiyah class vanishes. This yields a noncommutative analogue of Atiyah's classical correspondence for Riemann surfaces. Finally, we specialize to the quantum projective line $\mathbb{CP}_q^1\,$ and determine when $\overline{\nabla}_J$ defines a bimodule connection, assuming that $\overline{\nabla}_E$ does. |
| title | Holomorphic Jet Modules and Holomorphic Connections for Noncommutative Complex Curves |
| topic | Quantum Algebra Algebraic Geometry 58B34, 46L87, 14A22 |
| url | https://arxiv.org/abs/2604.27481 |