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Cycle Spaces

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Book cover Several Complex Variables VII

Part of the book series: Encyclopaedia of Mathematical Sciences ((EMS,volume 74))

Abstract

Given a complex space X, the Douady space D(X) parametrizes all puredimensional compact complex subspaces of X. D(X) carries a natural complex structure and moreover there is an universal family over it (see § 1). When X is projective, D(X) is just the Hilbert scheme of X, as constructed by A. Grothendieck. Clearly the construction of the Douady space for a general complex space is harder than in the algebraic situation. The cycle space C(X) or Barlet space of a reduced complex space X in contrast parametrizes linear combinations (with positive integer coefficients) of irreducible compact analytic sets, all of the same dimension (these are called cycles). The construction of C(X) is somehow easier than that of C(X) and is sketched in § 2. The algebraic counterpart (of C(X)) is the Chow scheme.

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Authors
  1. F. Campana
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  2. Th. Peternell
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Editors and Affiliations

  1. Mathematisches Institut, Universität Göttingen, Bunsenstr. 3-5, 37073, Göttingen, Germany

    H. Grauert

  2. Mathematisches Institut, Universität Bayreuth, Universitätsstr. 30, 95440, Bayreuth, Germany

    Th. Peternell

  3. Mathematisches Institut, Universität Münster, Einsteinstr. 62, 48149, Münster, Germany

    R. Remmert

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© 1994 Springer-Verlag Berlin Heidelberg

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Campana, F., Peternell, T. (1994). Cycle Spaces. In: Grauert, H., Peternell, T., Remmert, R. (eds) Several Complex Variables VII. Encyclopaedia of Mathematical Sciences, vol 74. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-09873-8_9

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  • DOI: https://doi.org/10.1007/978-3-662-09873-8_9

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-08150-7

  • Online ISBN: 978-3-662-09873-8

  • eBook Packages: Springer Book Archive

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