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On entire curves tangent to foliation. (English) Zbl 1147.32019
Let \(X\) be a complex projective manifold of dimension \(n\), equipped with a holomorphic foliation by curves of dimension one \(\mathcal{F}\), with a possibly nonempty singular set Sing(\(\mathcal{F})\).
The paper is devoted to prove that if \(T \subset X^0\) is an embedded \((n-1)\)-disc transverse to \(\mathcal{F}\), and \(S \subset T\) is an embedded \(k\)-disk, \(1 \leq k \leq n-1\), then there exists a (canonical) splitting of \(\varepsilon(S) := S \cap \{ p \in X^0 | \) there exists a non constant \(f: \mathbb{C} \rightarrow X, f(0)=p, \) tangent to \(\mathcal{F}\}\), \(\varepsilon(S) = \mathcal{P}(S) \cup \mathcal{Z}(S)\), where \(\mathcal{Z}(S)\) is thin in \(S\) and \(\mathcal{P}(S)\) is either complete pluripolar in \(S\) or full. Moreover, in the last case (\(\mathcal{P}(S) =S\)) there exists a meromorphic map \(F: S \times \mathbb{C} \rightarrow X\) such that for every \(s \in S\) the restriction \(F(s,\cdot): \mathbb{C} \rightarrow X\) is (after removing indeterminacies) an entire curve tangent to \(\mathcal{F}\), and it sends \(0\) to \(s \in S \subset X\).
The author also includes an interesting appendix where the interest of the theorem is explained to prove a weak version of a Lang conjecture (for surfaces) what asserts that if \(Y\) is a smooth complex projective surface of general type, then there exists a pluripolar subset of \(Y\) which contains all the entire curves.

32S65 Singularities of holomorphic vector fields and foliations
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