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On a generalization of the Hopf fibration. III. (Subvarieties in the C- spaces). (English) Zbl 0558.53034
[For parts I, II see TĂ´hoku Math. J., II. Ser. 29, 335-374 (1977; Zbl 0393.53018) and ibid. 30, 177-210 (1978; Zbl 0399.53009).]
Analytic subvarieties in C-spaces are discussed. First, a certain kind of closed 2-form \(d\omega\) is constructed. Then, the subvarieties are studied by means of this 2-form. \(d\omega\) may be considered as the curvature form of a connection when C-spaces are considered as the toral bundle space over an algebraic variety. This 2-form is horizontal in its nature with respect to the bundle structure and indicates, in general, how different the bundle is from the trivial bundle. Because of this twist in the bundle space, subvarieties in C-spaces tend to inherit the same structure. In this paper, the inherited fibration structure is studied. The most concrete results are obtained when the fiber torus has complex dimension 2.
MSC:
53C30 Differential geometry of homogeneous manifolds
32C25 Analytic subsets and submanifolds
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