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)

 
 

 

Some degeneracy theorems for entire functions with values in an algebraic variety


Author: James A. Carlson
Journal: Trans. Amer. Math. Soc. 168 (1972), 273-301
MSC: Primary 32A05; Secondary 14F99
DOI: https://doi.org/10.1090/S0002-9947-1972-0296356-0
MathSciNet review: 0296356
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In the first part of this paper we prove the following extension theorem. Let $ P_q^ \ast $ be a $ q$-dimensional punctured polycylinder, i.e. a product of disks and punctured disks. Let $ {W_n}$ be a compact complex manifold such that the bundle of holomorphic $ q$-forms is positive in the sense of Grauert. Let $ f:P_q^ \ast \to {W_n}$ be a holomorphic map whose Jacobian determinant does not vanish identically. Then $ f$ extends as a rational map to the full polycylinder $ {P_q}$. In the second half of the paper we prove the following generalization of the little Picard theorem to several complex variables: Let $ V \subset {P_n}$ be a hypersurface of degree $ d \geqq n + 3$ whose singularities are locally normal crossings. Then any holomorphic map $ f:{C^n} \to {P_n} - V$ has identically vanishing Jacobian determinant.


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

  • [1] S. S. Chern, Differential geometry, its past and future, Internat. Congress Math., Nice, France, 1970.
  • [2] P. Griffiths, Hermitian differential geometry, Chern classes, and positive vector bundles, Global Analysis (Papers in Honor of K. Kodaira), Univ. Tokyo Press, Tokyo, 1969, pp. 185-251. MR 41 #2717. MR 0258070 (41:2717)
  • [3] -, Holomorphic mappings into canonical algebraic varieties, Ann. of Math. 93 (1971), 439-458. MR 0281954 (43:7668)
  • [4] -, On the periods of certain rational integrals. I, II, Ann. of Math. (2) 90 (1969), 460-495; ibid., 496-541. MR 41 #5357. MR 0260733 (41:5357)
  • [5] H. Grauert, Über Modifikationen und exzeptionelle analytische Mengen, Math. Ann. 146 (1962), 331-368. MR 25 #583. MR 0137127 (25:583)
  • [6] F. Hirzebruch, Topological methods in algebraic geometry, 2nd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete, Heft 9, Springer-Verlag, Berlin, 1962; English transl., Die Grundlehren der math. Wissenschaften, Band 131, Springer-Verlag, New York, 1966. MR 25 #1155; MR 34 #2573. MR 0202713 (34:2573)
  • [7] S. Kobayashi, On the automorphism group of a certain class of algebraic manifolds, Tôhoku Math. J. (2) 11 (1959), 184-190. MR 22 #3014. MR 0112159 (22:3014)
  • [8] K. Kodaira, On holomorphic mappings of polydiscs into compact complex manifolds (to appear). MR 0301228 (46:386)
  • [9] P. Lelong, Plurisubharmonic functions and differential forms, Gordon and Breach, New York, 1969.
  • [10] W. Rudin, Real and complex analysis, McGraw-Hill, New York, 1966. MR 35 #1420. MR 0210528 (35:1420)
  • [11] H. Wu, The equidistribution theory of holomorphic curves, Ann. of Math. Studies, no. 64, Princeton Univ. Press, Princeton, N. J., 1970. MR 0273070 (42:7951)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 32A05, 14F99

Retrieve articles in all journals with MSC: 32A05, 14F99


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1972-0296356-0
Keywords: Several complex variables, extension of holomorphic maps, generalization of Picard theorem, Schottky-Landau theorem, branched cover
Article copyright: © Copyright 1972 American Mathematical Society

American Mathematical Society