INTEGRATION OF MEROMORPHIC COHOMOLOGY CLASSES AND APPLICATIONS

DANIEL BARLET, JON MAGNÚSSON
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