On the rationality of divisors and meromorphic functions
Author:
Chia Chi Tung
Journal:
Trans. Amer. Math. Soc. 239 (1978), 399406
MSC:
Primary 32L05
MathSciNet review:
0463511
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Abstract: Let E be a holomorphic vector bundle over a connected complex manifold X and D a divisor on E. Let be the set of all for which is a proper algebraic set in . The purpose of this paper is to prove that the following conditions are equivalent: (i) has positive measure in X; (ii) D extends to a unique divisor on the projective completion Ē of E; (iii) D is locally given by the divisors of rational meromorphic functions defined over open sets in X. Similar results for meromorphic functions are derived. The proof requires an extension theorem for analytic set: Assume E is a holomorphic vector bundle over a pure pdimensional complex space X and S an analytic set in E of pure codimension 1. Then the closure S of S in E is analytic if and only if is a proper algebraic set for all x in a set of positive 2pmeasure in every branch of X.
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 H. Federer, Geometric measure theory, Springer, Berlin and New York, 1969. MR 0257325 (41:1976)
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 R. C. Gunning and H. Rossi, Analytic functions of several complex variables, PrenticeHall, Englewood Cliffs, N. J., 1965. MR 0180696 (31:4927)
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 R. Remmert and K. Stein, Über die wesentlichen Singularitäten analytischer Mengen, Math. Ann. 126 (1953), 263306. MR 0060033 (15:615e)
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 L. I. Ronkin, Some questions on the distribution of zeros of entire functions of several variables, Mat. Sb. 16 (1972), Math. USSR Sb. 16 (1972), 363380.
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 W. Stoll, Einige Bemerkungen zur Fortsetzbarkeit analytischer Mengen, Math. Z. 60 (1954), 287304. MR 0065656 (16:463b)
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 , The multiplicity of a holomorphic map, Invent. Math. 2 (1966), 1538. MR 0210947 (35:1832)
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 C. Tung, The first main theorem of value distribution on complex spaces, Thesis, Notre Dame, 1973.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947197804635111
PII:
S 00029947(1978)04635111
Keywords:
Complex space,
analytic subset,
meromorphic function,
divisor,
multiplicity,
Stein manifold,
RemmertStein theorem
Article copyright:
© Copyright 1978
American Mathematical Society
