2004 Asian Journal of Mathematics  
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
more » ... e 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 (Xs) s∈S of compact n−cycles in Z parametrized by a compact normal complex space S. Is the algebraic dimension of Z ≥ p ? * .is). 1 In [V.85], J. Varouchas proved that this is equivalent to Z being bimeromorphic to a Kähler manifold. 2 The algebraic dimension of Z is the transcendence degree of the field of global meromorphic functions on Z. 3 We learnt from this problem from a talk given by Th. Peternell in Nancy; see [O.P.01].
doi:10.4310/ajm.2004.v8.n1.a13 fatcat:kte6h7giefb6fnqzjwrlad6f3a