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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.

##### MSC:
 32S65 Singularities of holomorphic vector fields and foliations
##### Keywords:
complex projective manifold; pluripolar set; entire curve
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