
Journal of Lie Theory 29 (2019), No. 4, 941956 Copyright Heldermann Verlag 2019 Homogeneous Principal Bundles over Manifolds with Trivial Logarithmic Tangent Bundle Hassan Azad Abdus Salam School of Mathematical Sciences, GC University, Lahore 54600, Pakistan hassan.azad@sms.edu.pk Indranil Biswas School of Mathematics, Tata Institute of Fundamental Research, Mumbai 400005, India indranil@math.tifr.res.in M. Azeem Khadam Abdus Salam School of Mathematical Sciences, GC University, Lahore 54600, Pakistan azeem.khadam@sms.edu.pk [Abstractpdf] 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 (see J.\,Winkelmann, {\it On manifolds with trivial logarithmic tangent bundle}, Osaka J. Math. 41 (2004) 473484; and {\it On manifolds with trivial logarithmic tangent bundle: the nonK\"ahler case}, Transform. Groups 13 (2008) 195209); 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 statements are equivalent: \par (1)\ \ $E_H$ is homogeneous. \par (2)\ \ $E_H$ admits a logarithmic connection singular over $D$. \par (3)\ \ The family of principal $H$bundles $\{g^*E_H\}_{g\in \mathbb G}$ is infinitesimally rigid at the identity element of the group $\mathbb G$. Keywords: Logarithmic connection, homogeneous bundle, semitorus, infinitesimal rigidity. MSC: 32M12, 32L05, 32G08. [ Fulltextpdf (142 KB)] for subscribers only. 