2022/11/06 by Jyoti Dasgupta, Dasgupta, Jyoti, Bivas Khan +3
Mathematics · Physics and Astronomy · #14J60 #14M25 #32L05 #53C05 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics #Nonlinear Waves and Solitons
paper · pdf · doi:10.48550/arxiv.2211.03047
openalex publication_date 2022/11/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let X be a normal projective variety over an algebraically closed field of characteristic zero. Let D be a reduced Weil divisor on X. Let G be a reductive linear algebraic group. We introduce the notion of a logarithmic connection on a principal G-bundle over X, which is singular along D. The existence of a logarithmic connection on the frame bundle associated with a vector bundle over X is shown to be equivalent to the existence of a logarithmic covariant derivative on the vector bundle if the logarithmic tangent sheaf of X is locally free. Additionally, when the algebraic group G is semisimple, we show that a principal G-bundle admits a logarithmic connection if and only if the associated adjoint bundle admits one. We also prove that the existence of a logarithmic connection on a principal bundle over a toric variety, singular along the boundary divisor, is equivalent to the existence of a torus equivariant structure on the bundle.