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A note on projective Levi flats and minimal sets of algebraic foliations. (English) Zbl 0963.32022

Let \(M\) be a complex compact manifold with a singular holomorphic foliation \(F\). A nontrivial minimal set of \(F\) is a nonempty, closed \(F\)-invariant subset of \(M\) that contains no singularities of \(F\).
The following is known for codimension one foliations of degree \(k\) on \(\mathbb{C} P^2\) [C. Camacho, A. Lins Neto and P. Sad, Inst. Hautes Étud. Sci., Publ. Math. 68, 187-203 (1988; Zbl 0682.57012)]: If \(k\geq 2\), there is an open non-empty subset of the space of such foliations that have nontrivial minimal sets; and for \(k=0,1\), there is no nontrivial minimal set.
Further progress on the subject is made in this paper: It is shown that codimension one holomorphic foliations on \(\mathbb{C}\mathbb{P}^n\) do not have nontrivial minimal sets for \(n \geq 3\); indeed, it is shown that any nontrivial closed invariant subset contains a singularity. It is also proved that there are no real analytic Levi flats in \(\mathbb{C}\mathbb{P}^n\) for \(n\geq 3\).

MSC:

32S65 Singularities of holomorphic vector fields and foliations
57R30 Foliations in differential topology; geometric theory

Citations:

Zbl 0682.57012

References:

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