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Fibrations, divisors and transcendental leaves. (English) Zbl 1089.32027
Let $$M$$ be a compact complex manifold and $$T(M)$$ the quotient of the group of rational divisors by the group of rational $$S^1$$-flat divisors. The author proves that $$\dim_{\mathbb{Q}} T(M) < \infty$$ and the following results involving $$T(M)$$.
1) If $$D_i$$, $$1 \leq i \leq r \geq 3$$, are connected effective divisors which are pairwise disjoint and whose $$T$$-classes span a linear subspace of $$T(M)$$ of dimension at most $$r-2$$. Then there exists a map $$\rho: M \rightarrow C$$ with connected fibers to a smooth curve which maps the $$D_i$$’s to points.
2) If $$D_1$$ and $$D_2$$ are smooth connected disjoint hyper-surfaces such that $$[D_1]$$ and $$[D_2]$$ lie in the same line of $$T(M)$$, then there exist étale $$\mathbb{Z}/n$$ and $$\mathbb{Z}/m$$-coverings of $$D_1$$ and $$D_2$$ which are diffeomorphic and such that $$m[D_1]=n[D_2]$$, $$m$$ and $$n$$ being positive integers.
3) Let $$F$$ be a codimension-one holomorphic foliation of $$M$$ and $$L$$ a transcendental leaf of $$F$$. Then the cardinality of the set of compact complex irreducible hyper-surfaces of $$M$$ which do not intersect the topological closure of $$L$$ is at most $$\dim_{\mathbb{Q}} T(M) +1$$ and when the equality holds, $$F$$ is given by a closed logarithmic 1-form.
Finally, in the paper one can find an appendix by Meersseman where it is proved that $$T$$ is the suitable object for those theorems, since at least 1) and 2) do not hold for the Néron-Severi group of $$M$$.

##### MSC:
 32S65 Singularities of holomorphic vector fields and foliations
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##### References:
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