Journal of Lie Theory 29 (2019), No. 4, 941--956
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
[Abstract-pdf]
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) 473--484; and {\it On manifolds with trivial logarithmic tangent bundle: the non-K\"ahler case}, Transform. Groups 13 (2008) 195--209); 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, semi-torus, infinitesimal rigidity.
MSC: 32M12, 32L05, 32G08.
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