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On the density of algebraic foliations without algebraic invariant sets. (English) Zbl 1116.32023
Let $$X$$ be a smooth complex projective variety of dimension greater than or equal to 2, $${\mathcal L}$$ an ample line bundle and $$k\gg 0$$ an integer.
The authors prove that a very generic global section of the twisted tangent sheaf $$\Theta_X \otimes {\mathcal L} ^{\otimes k}$$ gives rise to a one dimensional, singular foliation of $$X$$ without any proper algebraic invariant subvarieties of nonzero dimension. As a corollary the authors obtain a dynamical characterization of ampleness for line bundles $${\mathcal L}$$ over smooth projective surfaces, as the line bundles such that the generic section of $$\Theta _X \otimes {\mathcal L}^{\otimes k}$$ induces a foliation without invariant algebraic curves.

##### MSC:
 32S65 Singularities of holomorphic vector fields and foliations 32L10 Sheaves and cohomology of sections of holomorphic vector bundles, general results
##### Keywords:
holomorphic foliations; algebraic invariant sets
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##### References:
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