Advertisement

Local Properties of Analytic Varieties

  • Hassler Whitney
Part of the Contemporary Mathematicians book series (CM)

Abstract

Algebraic and analytic varieties have become increasingly important in recent years, both in the complex and the real case. Their local structure has been intensively investigated, by algebraic and by analytic means. Local geometric properties are less well understood. Our principal purpose here is to study properties of tangent vectors and tangent planes in the neighborhood of singular points. We study stratifications of an analytic variety into analytic manifolds; in particular, we may require that a transversal to a stratum is also tranversal to the higher dimensional strata near a given point. A conjecture on possible fiberings of the variety (and of surrounding space) is stated; it is proved at points of strata of codimension 2 in the surrounding space. In the last sections, we see that a variety may have numbers or functions intrinsically attached at points or along strata; also an analytic variety may be locally unlike an algebraic variety.

Keywords

Singular Point Analytic Variety Open Subset Local Property Algebraic Variety 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    S. Abhyankar Concepts of order and rank on a complex space, and a condition for normality, Math. Annalen, 141 (1960), P. 171–192.CrossRefGoogle Scholar
  2. [2]
    E. Bishop, Partially analytic spaces, Am. Journal of Math., 83 (1961). p. 669–692.CrossRefGoogle Scholar
  3. [3]
    H. Cartan, Séminaire 1951–52, École Normal Sup., Paris.Google Scholar
  4. [4]
    H. Cartain, Variétés analytiques-réelles et variétés analytiques-complexes, Bull. Soc. Math. France, 85 (1957), p. 77–100.Google Scholar
  5. [5]
    H. Grauert and R. Remmert, Zur Theorie der Modifikationen. I. Stetige und eigentliche Modiiikationen komplexer Räume., Math. Annalen, 129 (1955), p. 274–296.CrossRefGoogle Scholar
  6. [6]
    H. Grauert, Komplexe Räume, Math. Annalen, 136 (1958), p. 245–318.CrossRefGoogle Scholar
  7. [7]
    N. Levinson, A polynomial canonical form for certain analytic functions of two variables at a critical point of two variables at a critical point, Bull. Am. Math. Soc., 66 (1960), p. 366–368.CrossRefGoogle Scholar
  8. [8]
    N. Levinson, Transformation of an analytic function of several complex variables to a canonical form, Duke Math. Journal, 28 (1961), p. 345–354.CrossRefGoogle Scholar
  9. [9]
    R. Remmert and K. Stein, Über die wesentlichen Singularitäten analytischer Mengen, Math. Annalen, 126 (1953), p. 263–306.CrossRefGoogle Scholar
  10. [10]
    H. Rossi, Vector fields on analytic spaces, Annals of Math., 78 (1963), 455–467.CrossRefGoogle Scholar
  11. [11]
    P. Samuel, Méthodes d’algèbre abstraite en géométrie algébrique, Ergebnisse der Math., (N. F.) Heft 4, Springer, Berlin, 1955.Google Scholar
  12. [12]
    J.-P. Serre, Géométrie algébrique et géométrie analytique, Annales de L’Institut Fourier, VI (1955), p. 1–42.Google Scholar
  13. [13]
    R. Thom, Sur l’homologie des variétés algébriques réelles, this volume.Google Scholar
  14. [14]
    H. Whitney, Elementary structure of real algebraic varieties, Annals of Math., 66 (1957), p. 545–556.CrossRefGoogle Scholar
  15. [15]
    H. Whitney, Tangents to an analytic variety (to appear in Annals of Math.).Google Scholar
  16. [16]
    H. Whitney and F. Bruhat, Quelques propriétés fondamentales des ensembles analytiques-réels, Comm. Math. Helvetici, 33 (1959), p. 132–160.CrossRefGoogle Scholar
  17. [17]
    O. Zariski and P. Samuel, Commutative Algebra, vol. II. Princeton, N.J.: D. van Nostrand Co., 1960.Google Scholar

Copyright information

© Birkhäuser Boston 1992

Authors and Affiliations

  • Hassler Whitney

There are no affiliations available

Personalised recommendations