On the rationality of divisors and meromorphic functions

Abstract

Let E be a holomorphic vector bundle over a connected complex manifold X and D a divisor on E. Let A ( D ) A(D) be the set of all x ∈ X x \in X for which ( supp D ) ∩ E x ({\text {supp}}\;D) \cap {E_x} is a proper algebraic set in E x {E_x} . The purpose of this paper is to prove that the following conditions are equivalent: (i) A ( D ) 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 ∩ E x S \cap {E_x} is a proper algebraic set for all x in a set of positive 2p-measure in every branch of X.