2019/07/31 by Hassan Azad, Indranil Biswas, Azad, Hassan +3
Mathematics · #32G08 #32L05 #32M12 #Algebraic Geometry (math.AG) #Complex Variables (math.CV) #FOS: Mathematics #math.AG #math.CV #msc:32G08 #msc:32L05 #msc:32M12
paper · pdf · doi:10.48550/arxiv.1908.00522
arXiv admin note: text overlap with arXiv:1907.13006
arxiv created 2019/07/31 · arxiv updated 2019/08/02
Winkelmann considered compact complex manifolds X equipped with a reduced effective normal crossing divisor D ⊂ 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 \citeWi1, \citeWi2; 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)~ EH is homogeneous. (2)~ EH admits a logarithmic connection singular over D. (3)~ The family of principal H--bundles \g^*EH\g∈ \mathbb G is infinitesimally rigid at the identity element of the group \mathbb G.