INTEGRATION OF MEROMORPHIC COHOMOLOGY CLASSES AND APPLICATIONS

Abstract

The main purpose of this article is to increase the efficiency of the tools introduced in [B.Mg. 98] and [B.Mg. 99], namely integration of meromorphic cohomology classes, and to generalize the results of [B.Mg. 99]. They describe how positivity conditions on the normal bundle of a compact complex submanifold Y of codimension n + 1 in a complex manifold Z can be transformed into positivity conditions for a Cartier divisor in a space parametrizing n-cycles in Z .

As an application of our results we prove that the following problem has a positive answer in many cases :

Let Z be a compact connected complex manifold of dimension n+p. Let Y ⊂ Z a submanifold of Z of dimension p-1 whose normal bundle N _Y|Z is (Griffiths) positive. We assume that there exists a covering analytic family (X _s ) _s∈S of compact n-cycles in Z parametrized by a compact normal complex space S.

Is the algebraic dimension of Z ≥ p ?