zbMATH — the first resource for mathematics

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.
53C30 Differential geometry of homogeneous manifolds
32C25 Analytic subsets and submanifolds
PDF BibTeX Cite
Full Text: DOI EuDML
[1] Abe, K.: A generalization of the Hopf fibration I and II. Tohoku Math J.29 and30, 335-374 and 177-210 (1977 and 1978). · Zbl 0393.53018
[2] Abe, K.: On a class of Hermitian manifolds. Inventiones Math.51, 103-121 (1979). · Zbl 0422.53034
[3] Calabi, E., Eckmann, B.: A class of complex manifolds which are not algebraic. Annals Math.53, 494-500 (1953). · Zbl 0051.40304
[4] Goto, M.: On algebraic homogeneous spaces. Amer. J. Math.76, 811-818 (1954). · Zbl 0056.39803
[5] Griffiths, P.: Some geometric and analytic properties of homogeneous complex manifold, Part 1. Acta Math.110, 116-155 (1963). · Zbl 0171.44601
[6] Griffiths, P., Harris, J.: Principles of Algebraic Geometry. New York: Wiley-Interscience. 1978. · Zbl 0408.14001
[7] Helgason, S.: Differential Geometry and Symmetric Spaces. New York: Academic Press. 1962. · Zbl 0111.18101
[8] Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry. Vols. I and II. New York: Wiley-Interscience. 1969. · Zbl 0175.48504
[9] Stolzenberg, G.: Volumes, Limits and Extensions of Analytic Varieties, Lecture Notes Math.19. Berlin-Heidelberg-New York: Springer. 1966. · Zbl 0142.33801
[10] Whitney, H.: Complex Analytic Varieties. Reading, Mass.: Addison-Wesley 1972. · Zbl 0265.32008
[11] Wang, H.-C.: Closed manifolds with homogeneous complex structure. Amer. J. Math.76, 1-32 (1954). · Zbl 0055.16603
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.