On holomorphic principal bundles over a compact Riemann surface admitting a flat connection

Hassan Azad; Indranil Biswas

doi:10.1007/s002080100273

On holomorphic principal bundles over a compact Riemann surface admitting a flat connection

Hassan Azad^*, Indranil Biswas

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

27 Scopus citations

Abstract

Let G be a connected reductive linear algebraic group over ℂ, and X a compact connected Riemann surface. Let L ⊂ G be a Levi factor of some parabolic subgroup of G, with L₀ = L/[L, L] its maximal abelian quotient. We prove that a holomorphic G-bundle E_G over X admits a flat connection if and only if for every such L and every reduction E_L ⊂ E_G of the structure group of E_G to L, the L₀-bundle obtained by extending the structure group of E_L is topologically trivial. For G = GL(n, ℂ), this is a well-known result of A. Weil.

Original language	English
Pages (from-to)	333-346
Number of pages	14
Journal	Mathematische Annalen
Volume	322
Issue number	2
DOIs	https://doi.org/10.1007/s002080100273
State	Published - 2002
Externally published	Yes

ASJC Scopus subject areas

General Mathematics

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10.1007/s002080100273

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title = "On holomorphic principal bundles over a compact Riemann surface admitting a flat connection",

abstract = "Let G be a connected reductive linear algebraic group over ℂ, and X a compact connected Riemann surface. Let L ⊂ G be a Levi factor of some parabolic subgroup of G, with L0 = L/[L, L] its maximal abelian quotient. We prove that a holomorphic G-bundle EG over X admits a flat connection if and only if for every such L and every reduction EL ⊂ EG of the structure group of EG to L, the L0-bundle obtained by extending the structure group of EL is topologically trivial. For G = GL(n, ℂ), this is a well-known result of A. Weil.",

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T1 - On holomorphic principal bundles over a compact Riemann surface admitting a flat connection

AU - Azad, Hassan

AU - Biswas, Indranil

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N2 - Let G be a connected reductive linear algebraic group over ℂ, and X a compact connected Riemann surface. Let L ⊂ G be a Levi factor of some parabolic subgroup of G, with L0 = L/[L, L] its maximal abelian quotient. We prove that a holomorphic G-bundle EG over X admits a flat connection if and only if for every such L and every reduction EL ⊂ EG of the structure group of EG to L, the L0-bundle obtained by extending the structure group of EL is topologically trivial. For G = GL(n, ℂ), this is a well-known result of A. Weil.

AB - Let G be a connected reductive linear algebraic group over ℂ, and X a compact connected Riemann surface. Let L ⊂ G be a Levi factor of some parabolic subgroup of G, with L0 = L/[L, L] its maximal abelian quotient. We prove that a holomorphic G-bundle EG over X admits a flat connection if and only if for every such L and every reduction EL ⊂ EG of the structure group of EG to L, the L0-bundle obtained by extending the structure group of EL is topologically trivial. For G = GL(n, ℂ), this is a well-known result of A. Weil.

UR - https://www.scopus.com/pages/publications/0036002807

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DO - 10.1007/s002080100273

M3 - Article

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JO - Mathematische Annalen

JF - Mathematische Annalen

IS - 2

ER -

On holomorphic principal bundles over a compact Riemann surface admitting a flat connection

Abstract

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