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

Cerveau, Dominique; Lins-Neto, Alcides; Loray, Frank; Pereira, Jorge Vitório; Touzet, Frédéric

Comment. Math. Helv. 81, No. 1, 157-169 (2006).

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.

Citations:

Zbl 0158.27604