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.


32S65 Singularities of holomorphic vector fields and foliations
32L10 Sheaves and cohomology of sections of holomorphic vector bundles, general results
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