Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

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

 
 

 

Parametrizations of analytic varieties


Author: Joseph Becker
Journal: Trans. Amer. Math. Soc. 183 (1973), 265-292
MSC: Primary 32B10
DOI: https://doi.org/10.1090/S0002-9947-1973-0344513-8
MathSciNet review: 0344513
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let V be an analytic subvariety of an open subset $ \Omega $ of $ {{\text{C}}^n}$ of pure dimension r; for any $ p \in V$, there exists an $ n - r$ dim plane T such that $ {\pi _T}:V \to {{\text{C}}^r}$, the projection along T to $ {{\text{C}}^r}$, is a branched covering of finite sheeting order $ \mu (V,p,T)$ in some neighborhood of V about p. $ {\pi _T}$ is called a global parametrization of V if $ {\pi _T}$ has all discrete fibers, e.g. $ {\dim _p}V \cap (T + p) = 0$ for all $ p \in V$.

Theorem. $ B = \{ (p,T) \in V \times G(n - r,n)\vert{\dim _p}V \cap (T + p) > 0\} $ is an analytic set. If $ {\pi _2}:V \times G \to G$ is the natural projection, then $ {\pi _2}(B)$ is a negligible set in G.

Theorem. $ \{ (p,T) \in V \times G\vert\mu (V,p,T) \geq k\} $ is an analytic set. For each $ p \in V$, there is a least $ \mu (V,p)$ and greatest $ m(V,p)$ sheeting multiplicity over all $ T \in G$.

If $ \Omega $ is Stein, V is the locus of finitely many holomorphic functions but its ideal in $ \mathcal{O}(\Omega )$ is not necessarily finitely generated.

Theorem. If $ \mu (V,p)$ is bounded on V, then its ideal is finitely generated.


References [Enhancements On Off] (What's this?)

  • [1] J. Becker, Continuing analytic sets across $ {{\mathbf{R}}^n}$, Math. Ann. 195 (1972), 103-106. MR 0298053 (45:7105)
  • [2] E. Bishop, Mappings of partially analytic spaces, Amer. J. Math. 83 (1961),209-242. MR 23 #A1054. MR 0123732 (23:A1054)
  • [3] -, Some global problems in the theory of functions of several complex variables, Amer. J. Math. 83 (1961), 479-498. MR 25 #4131. MR 0140717 (25:4131)
  • [4] -, Partially analytic spaces, Amer. J. Math. 83 (1961), 669-692. MR 25 #5191. MR 0141794 (25:5191)
  • [5] O. Forster and K. J. Ramspott, Über die Darstellung analytischer Mengen, Bayer. Akad. Wissenschaften Math.-Natur. Kl. S.-B. 1963, 89-99 (1964). MR 29 #4912. MR 0167640 (29:4912)
  • [6] R. C. Gunning, Lectures on complex analytic varieties: The local parametrization theorem, Princeton Univ. Press, Princeton, N. J.; Univ. of Tokyo Press, Tokyo, 1970. MR 42 #7941. MR 0273060 (42:7941)
  • [7] R. C. Gunning and H. Rossi, Analytic functions of several complex variables, Prentice-Hall Series in Modern Analysis, Prentice-Hall, Englewood Cliffs, N. J., 1965. MR 31 #4927. MR 0180696 (31:4927)
  • [8] H. Grauert, Charakterisierung der holomorphvolls tändiger komplexen Räume, Math. Ann. 129 (1955), 233-259. MR 17, 80. MR 0071084 (17:80d)
  • [9] B. Kripke, Finitely generated coherent analytic sheaves, Proc. Amer. Math. Soc. 21 (1969), 530-534. MR 39 #1681. MR 0240332 (39:1681)
  • [10] D. Mumford, Introduction to algebraic geometry, Harvard Univ. Press, Cambridge, Mass.
  • [11] R. Narasimhan, Introduction to the theory of analytic spaces, Lecture Notes in Math., no. 25, Springer-Verlag, Berlin and New York, 1966. MR 36 #428. MR 0217337 (36:428)
  • [12] R. Remmert, Holomorphe and meromorphe Abbildungen komplexer Räume, Math. Ann. 133 (1957), 328-370. MR 19, 1193. MR 0092996 (19:1193d)
  • [13] W. Rudin, A geometric criterion for algebraic varieties, J. Math. Mech. 17 (1967/68), 671-683. MR 36 #2829. MR 0219750 (36:2829)
  • [14] H. Whitney, Local properties of analytic varieties, Differential and Combinatorial Topology, Princeton Univ. Press, Princeton, N. J., 1965. MR 32 #5924. MR 0188486 (32:5924)
  • [15] -, Tangents to an analytic variety, Ann. of Math. (2) 81 (1965), 496-549. MR 33 #745. MR 0192520 (33:745)
  • [16] -, Complex analytic varieties, Addison-Wesley, Reading, Mass., 1972.
  • [17] F. S. Maculy, Algebraic theory of modular systems, Cambridge, Mass., 1916.
  • [18] R. Narasimhan, Imbeddings of holomorphically complete complex spaces, Amer. J. Math. 82 (1960), 917-934. MR 26 #6438. MR 0148942 (26:6438)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 32B10

Retrieve articles in all journals with MSC: 32B10


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1973-0344513-8
Keywords: Bounded multiplicity, finitely generated ideal
Article copyright: © Copyright 1973 American Mathematical Society

American Mathematical Society