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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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