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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Some degeneracy theorems for entire functions with values in an algebraic variety
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by James A. Carlson PDF
Trans. Amer. Math. Soc. 168 (1972), 273-301 Request permission


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.
    S. S. Chern, Differential geometry, its past and future, Internat. Congress Math., Nice, France, 1970.
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Additional Information
  • © Copyright 1972 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 168 (1972), 273-301
  • MSC: Primary 32A05; Secondary 14F99
  • DOI:
  • MathSciNet review: 0296356