Logarithmic Cartan geometry on complex manifolds with trivial logarithmic tangent bundle

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 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 (EH,Θ) of type (G,H) on M, where H⊂G are complex Lie groups. We prove that (EH,Θ) is isomorphic to (ρ⁎EH,ρ⁎Θ) for every ρ∈A if and only if the principal H–bundle EH admits a logarithmic connection Δ singular on D such that Θ is preserved by the connection Δ.