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 $\mathbb{A}$ denote the maximal connected subgroup of the group of all holomorphic automorphisms of $M$ that preserve the divisor $D$. Take a holomorphic Cartan geometry $(E_H,\hat{I}˜)$ of type $(G,H)$ on $M$, where $H\subset G$ are complex Lie groups. We prove that $(E_H,\hat{I}˜)$ is isomorphic to $(\ddot{I}{}^{\ast }{E}_H,\ddot{I}{}^{\ast }\hat{I}˜)$ for every $\ddot{I}\in \mathbb{A}$ if and only if the principal $H$--bundle $E_H$ admits a logarithmic connection $\hat{I}”$ singular on $D$ such that $\hat{I}˜$ is preserved by the connection $\hat{I}”$.

Final version to appear in Journal of Differential Geometry and its Applications