Logarithmic Cartan geometry on complex manifolds with trivial logarithmic tangent bundle

2024/11/13 by Biswas, Indranil, Dumitrescu, Sorin, Morye, Archana S.
#Algebraic Geometry (math.AG) #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2411.08593

Abstract

Let M be a compact complex manifold, and D ⊂ M a reduced normal crossing divisor on it, such that the logarithmic tangent bundle TM(-log D) is holomorphically trivial. Let \mathbb A denote the maximal connected subgroup of the group of all holomorphic automorphisms of M that preserve the divisor D. Take a holomorphic Cartan geometry (E_H, Θ) of type (G, H) on M, where H ⊂ G are complex Lie groups. We prove that (E_H, Θ) is isomorphic to (ρ^* E_H, ρ^* Θ) for every ρ ∈ \mathbb A if and only if the principal H--bundle E_H admits a logarithmic connection Δ singular on D such that Θ is preserved by the connection Δ.

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