Line bundles on the moduli space of Lie algebroid connections over a curve
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866909170871042048 |
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| author | Biswas, Indranil Singh, Anoop |
| author_facet | Biswas, Indranil Singh, Anoop |
| contents | We explore algebro-geometric properties of the moduli space of holomorphic Lie algebroid ($ \mathcal{L} $) connections on a compact Riemann surface $X$ of genus $g \,\geq\, 3$. A smooth compactification of the moduli space of $\mathcal{L}$-connections, such that underlying vector bundle is stable, is constructed; the complement of the moduli space inside the compactification is a divisor. A criterion for the numerical effectiveness of the boundary divisor is given. We compute the Picard group of the moduli space, and analyze Lie algebroid Atiyah bundles associated with an ample line bundle. This enables us to conclude that regular functions on the space of certain Lie algebroid connections are constants. Moreover, under some condition, it is shown that the moduli space of $\mathcal{L}$-connections does not admit non-constant algebraic functions. Rationally connectedness of the moduli spaces is explored. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2208_00140 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Line bundles on the moduli space of Lie algebroid connections over a curve Biswas, Indranil Singh, Anoop Algebraic Geometry 14D20, 14C22, 14H05, 14M20 We explore algebro-geometric properties of the moduli space of holomorphic Lie algebroid ($ \mathcal{L} $) connections on a compact Riemann surface $X$ of genus $g \,\geq\, 3$. A smooth compactification of the moduli space of $\mathcal{L}$-connections, such that underlying vector bundle is stable, is constructed; the complement of the moduli space inside the compactification is a divisor. A criterion for the numerical effectiveness of the boundary divisor is given. We compute the Picard group of the moduli space, and analyze Lie algebroid Atiyah bundles associated with an ample line bundle. This enables us to conclude that regular functions on the space of certain Lie algebroid connections are constants. Moreover, under some condition, it is shown that the moduli space of $\mathcal{L}$-connections does not admit non-constant algebraic functions. Rationally connectedness of the moduli spaces is explored. |
| title | Line bundles on the moduli space of Lie algebroid connections over a curve |
| topic | Algebraic Geometry 14D20, 14C22, 14H05, 14M20 |
| url | https://arxiv.org/abs/2208.00140 |