Un phénomène de Hartogs dans les variétés projectives

Abstract

Let V be a projective manifold, dimV ≥ 2. Let Z be an open subset in V which is pseudoconcave (see [1]). LetM(Z) be the field of meromorphic functions on Z . According to Andreotti [1],M(Z) is a finite extension ofM(V ). Hence meromorphic functions on a pseudoconcave domain in a projective manifold are algebraic. In [3], Theorem 3.2.1, we claimed that they are rationals. This claim is false. Counterexamples can be found in [6], Chapter 5, page 199 or in [2], chapter 9, example 9.1. In order to state aHartogs’ theoremon a projectivemanifold,wemust take care of algebraic multivaluedness. We state corrected versions of main theorem 3.2.4 in [3] and of Corollary 6 of [4]. First, we work with a pseudoconcave domain Z such that M(Z) M(V ). Then, we use a primitive element to reduce to the first case. Theorem Let V be a projective manifold, dimV ≥ 2. Let U be an open subset in V such that V \Ū is pseudoconcave in the sense of Andreotti and the boundary of U is connected. Let H be the maximal compact reduced divisor in U (see [3]). Assume meromorphic functions on V \Ū are rationals. Let F → V be a holomorphic vector bundle. Then any meromorphic section of F defined on a connected neighborhood W of ∂U extends as a meromorphic section of F over U. Moreover if that section is holomorphic on W , then its extension is holomorphic on U\H.