Logarithmic Cartan geometry on complex manifolds with trivial logarithmic tangent bundle

TL;DR

This paper studies complex manifolds with trivial logarithmic tangent bundles and shows that certain Cartan geometries are invariant under automorphisms if they admit compatible logarithmic connections.

Contribution

It establishes a criterion linking the invariance of Cartan geometries under automorphisms to the existence of compatible logarithmic connections.

Findings

Cartan geometries are invariant under automorphisms preserving the divisor

Existence of compatible logarithmic connections characterizes invariance

Provides a classification framework for such geometries on trivial tangent bundles

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 $(E_H,\,\Theta)$ 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 $\rho\, \in\, \mathbb A$ if and only if the principal $H$--bundle $E_H$ admits a logarithmic connection $\Delta$ singular on $D$ such that $\Theta$ is preserved by the connection $\Delta$.