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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Stein manifolds on which the strong Poincaré problem can be solved


Author: Robert Ephraim
Journal: Proc. Amer. Math. Soc. 70 (1978), 136-138
MSC: Primary 32E10
DOI: https://doi.org/10.1090/S0002-9939-1978-0481104-2
MathSciNet review: 0481104
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Abstract: Let M be a Stein manifold. Suppose every meromorphic function on M may be written as the quotient of two holomorphic functions which are pointwise relatively prime at every point of M. Then it will be shown that $ {H^2}(M,Z) = 0$. Thus, the solvability of the Strong Poincaré Problem is equivalent to the vanishing of the second integral cohomology, which in turn is equivalent to the solvability of Cousin II, all on a Stein manifold M. This closes a gap in the classically known theory.


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

  • [1] H. G. Dales, The ring of holomorphic functions on a Stein compact set as a unique factorization domain, Proc. Amer. Math. Soc. 44 (1974), 88-92. MR 0333245 (48:11570)
  • [2] L. Hormander, An introduction to complex analysis in several variables, Van Nostrand, Princeton, N. J., 1966. MR 0203075 (34:2933)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1978-0481104-2
Keywords: Stein manifold, Strong Poincaré Problem, meromorphic function, positive divisors, Cousin II, integral cohomology
Article copyright: © Copyright 1978 American Mathematical Society

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