Logarithmic Cartan geometry on complex manifolds with trivial logarithmic tangent bundle

Abstract

Let M be a compact complex manifold, and $D\subset M$ 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 $(E_H,\Theta )$ of type $(G,H)$ on M, where $H\subset G$ are complex Lie groups. We prove that $(E_H,\Theta )$ is isomorphic to $({\rho }^{⁎}{E}_H,{\rho }^{⁎}\Theta )$ for every $\rho \in A$ 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.