# Homogeneous principal bundles over manifolds with trivial logarithmic tangent bundle

## TL;DR

This paper characterizes homogeneous principal bundles over certain complex manifolds with trivial logarithmic tangent bundles, linking their homogeneity to the existence of logarithmic connections and infinitesimal rigidity.

## Contribution

It provides a new equivalence characterization of homogeneous principal bundles over manifolds with trivial logarithmic tangent bundles.

## Key findings

- Homogeneous principal bundles admit logarithmic connections over divisors.
- Homogeneity is equivalent to infinitesimal rigidity of the bundle family.
- Characterization applies to manifolds with trivial logarithmic tangent bundles.

## Abstract

Winkelmann considered compact complex manifolds $X$ equipped with a reduced effective normal crossing divisor $D\, \subset\, X$ such that the logarithmic tangent bundle $TX(-\log D)$ is holomorphically trivial. He characterized them as pairs $(X,\, D)$ admitting a holomorphic action of a complex Lie group $\mathbb G$ satisfying certain conditions \cite{Wi1}, \cite{Wi2}; this $\mathbb G$ is the connected component, containing the identity element, of the group of holomorphic automorphisms of $X$ that preserve $D$. We characterize the homogeneous holomorphic principal $H$--bundles over $X$, where $H$ is a connected complex Lie group. Our characterization says that the following three are equivalent: (1)~ $E_H$ is homogeneous. (2)~ $E_H$ admits a logarithmic connection singular over $D$. (3)~ The family of principal $H$--bundles $\{g^*E_H\}_{g\in \mathbb G}$ is infinitesimally rigid at the identity element of the group $\mathbb G$.