Homogeneous Principal Bundles over Manifolds with Trivial Logarithmic Tangent Bundle

Winkelmann considered compact complex manifolds X equipped with a reduced effective normal crossing divisor D subset of 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 C satisfying certain conditions (see J. Winkelmann, On manifolds with trivial logarithmic tangent bundle, Osaka J. Math. 41 (2004) 473-484; and On manifolds with trivial logarithmic tangent bundle: the non-Kdhler case, Transform. Groups 13 (2008) 195-209); this 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 is an element of G) is infinitesimally rigid at the identity element of the group G.