Holomorphic Lie algebroid connections on holomorphic principal bundles on compact Riemann surfaces
Fuente:
arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866917080065900544 |
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| author | Biswas, Indranil |
| author_facet | Biswas, Indranil |
| contents | For a $Γ$--equivariant holomorphic Lie algebroid $(V,\, ϕ)$, on a compact Riemann surface $X$ equipped with an action of a finite group $Γ$, we investigate the equivariant holomorphic Lie algebroid connections on holomorphic principal $G$--bundles over $X$, where $G$ is a connected affine complex reductive group. If $(V,\,ϕ)$ is nonsplit, then it is proved that every holomorphic principal $G$--bundle admits an equivariant holomorphic Lie algebroid connection. If $(V,\,ϕ)$ is split, then it is proved that the following four statements are equivalent:
An equivariant principal $G$--bundle $E_G$ admits an equivariant holomorphic Lie algebroid connection.
The equivariant principal $G$--bundle $E_G$ admits an equivariant holomorphic connection.
The principal $G$--bundle $E_G$ admits a holomorphic connection.
For every triple $(P,\, L(P),\, χ)$, where $L(P)$ is a Levi subgroup of a parabolic subgroup $P\, \subset\, G$ and $χ$ is a holomorphic character of $L(P)$, and every $Γ$--equivariant holomorphic reduction of structure group $E_{L(P)}$ of $E_G$ to $L(P)$, the degree of the line bundle over $X$ associated to $E_{L(P)}$ for $χ$ is zero.
The correspondence between $Γ$--equivariant principal $G$--bundles over $X$ and parabolic $G$--bundles on $X/Γ$ translates the above result to the context of parabolic $G$--bundles. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_10994 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Holomorphic Lie algebroid connections on holomorphic principal bundles on compact Riemann surfaces Biswas, Indranil Algebraic Geometry Differential Geometry For a $Γ$--equivariant holomorphic Lie algebroid $(V,\, ϕ)$, on a compact Riemann surface $X$ equipped with an action of a finite group $Γ$, we investigate the equivariant holomorphic Lie algebroid connections on holomorphic principal $G$--bundles over $X$, where $G$ is a connected affine complex reductive group. If $(V,\,ϕ)$ is nonsplit, then it is proved that every holomorphic principal $G$--bundle admits an equivariant holomorphic Lie algebroid connection. If $(V,\,ϕ)$ is split, then it is proved that the following four statements are equivalent: An equivariant principal $G$--bundle $E_G$ admits an equivariant holomorphic Lie algebroid connection. The equivariant principal $G$--bundle $E_G$ admits an equivariant holomorphic connection. The principal $G$--bundle $E_G$ admits a holomorphic connection. For every triple $(P,\, L(P),\, χ)$, where $L(P)$ is a Levi subgroup of a parabolic subgroup $P\, \subset\, G$ and $χ$ is a holomorphic character of $L(P)$, and every $Γ$--equivariant holomorphic reduction of structure group $E_{L(P)}$ of $E_G$ to $L(P)$, the degree of the line bundle over $X$ associated to $E_{L(P)}$ for $χ$ is zero. The correspondence between $Γ$--equivariant principal $G$--bundles over $X$ and parabolic $G$--bundles on $X/Γ$ translates the above result to the context of parabolic $G$--bundles. |
| title | Holomorphic Lie algebroid connections on holomorphic principal bundles on compact Riemann surfaces |
| topic | Algebraic Geometry Differential Geometry |
| url | https://arxiv.org/abs/2511.10994 |