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 .
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As an application of our results we prove that the following problem has a positive answer in
many cases :
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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.
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Is the algebraic dimension of Z ≥ p ?