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 (Xs) s∈S of compact n−cycles in Z parametrized by a compact normal complex space S.Is the algebraic dimension of Z ≥ p ? *

Compact subvarieties with ample normal bundles, algebraicity, and cones of cycles

“…If d = dim X − 1, then it is classically known that X is projective, but in higher codimension there are only a few results, [BM04], [OP04]. These results will be explicity discussed in Section 3.…”