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

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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 $A(D)$ be the set of all $x \in X$ for which $({\text {supp}}\;D) \cap {E_x}$ is a proper algebraic set in ${E_x}$. The purpose of this paper is to prove that the following conditions are equivalent: (i) $A(D)$ 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 $S \cap {E_x}$ is a proper algebraic set for all *x* in a set of positive 2*p*-measure in every branch of *X*.

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

Article copyright:
© Copyright 1978
American Mathematical Society