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Transactions of the American Mathematical Society

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Families of holomorphic maps into Riemann surfaces


Author: Theodore J. Barth
Journal: Trans. Amer. Math. Soc. 207 (1975), 175-187
MSC: Primary 32A17; Secondary 32H20
DOI: https://doi.org/10.1090/S0002-9947-1975-0374462-2
MathSciNet review: 0374462
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Abstract: In analogy with the Hartogs theorem that separate analyticity of a function implies analyticity, it is shown that a separately normal family of holomorphic maps from a polydisk into a Riemann surface is a normal family. This contrasts with examples of discontinuous separately analytic maps from a bidisk into the Riemann sphere. The proof uses a theorem on pseudoconvexity of normality domains, which is proved via the following convergence criterion: a sequence $ \{ {f_j}\} $ of holomorphic maps from a complex manifold into a Riemann surface converges to a nonconstant holomorphic map if and only if the sequence $ \{ f_j^{ - 1}\} $ of set-valued maps, defined on the Riemann surface, converges to a suitable set-valued map. Extending Osgood's theorem, it is also shown that a separately analytic map (resp. a separately normal family of holomorphic maps) from a polydisk into a hyperbolic complex space is analytic (resp. normal).


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1975-0374462-2
Keywords: Nonnegative divisor, pseudoconvex domain, normal family of maps, normal family of divisors, normality domain, Hartogs theorem, Osgood theorem, hyperbolic complex space, Kobayashi distance
Article copyright: © Copyright 1975 American Mathematical Society

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