Ressource pédagogique : Franc Forstneri? - Non singular holomorphic foliations on Stein manifolds (Part 1)

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A nonsingular holomorphic foliation of codimension on a complex manifold is locally given by the level sets of a holomorphic submersion to the Euclidean space . If is a Stein manifold, there also exist plenty of global foliations of this form, so long as there are no topological obstructions. More precisely, if then any -tuple of pointwise linearly independent (1,0)-forms can be continuously deformed to a -tuple of differentials where is a holomorphic submersion of to . Such a submersion always exists if is no more than the integer part of . More generally, if is a complex vector subbundle of the tangent bundle such that is a flat bundle, then is homotopic (through complex vector subbundles of ) to an integrable subbundle, i.e., to the tangent bundle of a nonsingular holomorphic foliation on . I will prove these results and discuss open problems, the most interesting one of them being related to a conjecture of Bogomolov.

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