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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Complex structures on real product bundles with applications to differential geometry

Author: Richard S. Millman
Journal: Trans. Amer. Math. Soc. 166 (1972), 71-99
MSC: Primary 32L05; Secondary 53C10, 58A10
MathSciNet review: 0302943
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Abstract: The purpose of this paper is to classify holomorphic principal fibre bundles which admit a smooth section (i.e. are real product bundles). This is accomplished if the structure group is solvable of type (E). In the general case, a sufficient condition is obtained for a real product bundle to be equivalent to the complex product bundle. A necessary and sufficient condition for the existence of a holomorphic connection on a real product bundle is also obtained. Using this criterion in the case where the structure group is abelian, a generalization of a theorem due to Atiyah (in the case the structure group is $ {C^ \ast }$) is obtained.

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Keywords: Holomorphic principal fibre bundle, complex structure, connection, Kähler manifold, complex manifold
Article copyright: © Copyright 1972 American Mathematical Society