On the rationality of divisors and meromorphic functions

Author:
Chia Chi Tung

Journal:
Trans. Amer. Math. Soc. **239** (1978), 399-406

MSC:
Primary 32L05

DOI:
https://doi.org/10.1090/S0002-9947-1978-0463511-1

MathSciNet review:
0463511

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 *p*-dimensional 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 2*p*-measure in every branch of *X*.

**[1]**Aldo Andreotti and Wilhelm Stoll,*Analytic and algebraic dependence of meromorphic functions*, Lecture Notes in Mathematics, Vol. 234, Springer-Verlag, Berlin-New York, 1971. MR**0390298****[2]**Herbert Federer,*Geometric measure theory*, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York, 1969. MR**0257325****[3]**Robert C. Gunning and Hugo Rossi,*Analytic functions of several complex variables*, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1965. MR**0180696****[4]**Reinhold Remmert and Karl Stein,*Über dei wesentlichen Singularitäten analytischer Mengen*, Math. Ann.**126**(1953), 263–306 (German). MR**0060033**, https://doi.org/10.1007/BF01343164**[5]**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), 363-380.**[6]**Wilhelm Stoll,*Einige Bemerkungen zur Fortsetzbarkeit analytischer Mengen*, Math. Z.**60**(1954), 287–304 (German). MR**0065656**, https://doi.org/10.1007/BF01187378**[7]**Wilhelm Stoll,*The multiplicity of a holomorphic map*, Invent. Math.**2**(1966), 15–58. MR**0210947**, https://doi.org/10.1007/BF01403389**[8]**C. Tung,*The first main theorem of value distribution on complex spaces*, Thesis, Notre Dame, 1973.

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
32L05

Retrieve articles in all journals with MSC: 32L05

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1978-0463511-1

Keywords:
Complex space,
analytic subset,
meromorphic function,
divisor,
multiplicity,
Stein manifold,
Remmert-Stein theorem

Article copyright:
© Copyright 1978
American Mathematical Society