## Algebraic reduction theorem for complex codimension one singular foliations.(English)Zbl 1095.37019

Summary: Let $$M$$ be a compact complex manifold equipped with $$n=\dim (M)$$ meromorphic vector fields that are linearly independent at a generic point. The main theorem is the following.
If $$M$$ is not bimeromorphic to an algebraic manifold, then any codimension-one complex foliation $${\mathcal F}$$ with a codimension $$\geq 2$$ singular set is the meromorphic pull-back of an algebraic foliation on a lower-dimensional algebraic manifold, or $${\mathcal F}$$ is transversely projective outside a proper analytic subset.
The two ingredients of the proof are the algebraic reduction theorem for the complex manifold $$M$$ and an algebraic version of Lie’s first theorem which is due to J. Tits [Bull. Soc. Math. Belg. 11, 100–115 (1959; Zbl 0158.27604)].

### MSC:

 37F75 Dynamical aspects of holomorphic foliations and vector fields 32S65 Singularities of holomorphic vector fields and foliations

### Keywords:

holomorphic foliations; transversal structure.

Zbl 0158.27604
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