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29 March 2024
 
  » arxiv » 1908.0522

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Homogeneous principal bundles over manifolds with trivial logarithmic tangent bundle
Hassan Azad ; Indranil Biswas ; M. Azeem Khadam ;
Date 31 Jul 2019
AbstractWinkelmann 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 ${g^*E_H}_{gin mathbb G}$ is infinitesimally rigid at the identity element of the group $mathbb G$.
Source arXiv, 1908.0522
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