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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.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.


Compact manifolds in hyperbolicity
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by Robert Brody PDF
Trans. Amer. Math. Soc. 235 (1978), 213-219 Request permission


In this paper we establish the strongest possible criterion for the hyperbolicity of a compact complex manifold: such a manifold is hyperbolic if and only if it contains no (nontrivial) complex lines. In addition, we study the behavior of such manifolds under deformation and, in particular, answer the two most natural questions about such deformations: Is the space of hyperbolic complex structures on a given ${C^\infty }$ manifold open in the space of all its complex structures? (Yes.) Is it closed? (Not in general.) These results answer questions first posed by Kobayashi in [4] and [5].
  • Salomon Bochner and Deane Montgomery, Groups on analytic manifolds, Ann. of Math. (2) 48 (1947), 659–669. MR 22223, DOI 10.2307/1969133
  • A. Douady, Le problème des modules pour les variétés analytiques complexes (d’après M. Kuranishi), Séminaire Bourbaki: 1964/65, Exposé 277, Benjamin, New York and Amsterdam, 1966. MR 33 #54201.
  • Lars Hörmander, An introduction to complex analysis in several variables, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1966. MR 0203075
  • Shoshichi Kobayashi, Hyperbolic manifolds and holomorphic mappings, Pure and Applied Mathematics, vol. 2, Marcel Dekker, Inc., New York, 1970. MR 0277770
  • —, Some problems in intrinsic distances and measures, The Greek Math. Soc., C. Carathéodory Symposium, (September, 1973), pp. 306-317.
  • H. L. Royden, Remarks on the Kobayashi metric, Several complex variables, II (Proc. Internat. Conf., Univ. Maryland, College Park, Md., 1970) Lecture Notes in Math., Vol. 185, Springer, Berlin, 1971, pp. 125–137. MR 0304694
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Additional Information
  • © Copyright 1978 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 235 (1978), 213-219
  • MSC: Primary 32H20
  • DOI:
  • MathSciNet review: 0470252