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About transversely projective holomorphic foliations. (Sur les feuilletages holomorphes transversalement projectifs.) (French) Zbl 1032.32020

Let \(M\) be a complex manifold of dimension \(n\). A transversely projective holomorphic foliation on \(M\) is given by a covering of \(M\) by local submersions in the projective space \(\mathbb P^1\) with cocycles in the group of projective transformations \(\text{SL}(2,\mathbb C)\). Standard examples consist in foliations which are generically transverse to a fibration by compact connected holomorphic curves (which the author calls “Riccati foliations”) and foliations obtained from these by meromorphic pull-back.
In this paper, the author addresses the question of whether any transversely projective holomorphic foliation can be obtained in this manner. He shows that the answer is negative by giving examples of foliations on the projective space \(\mathbb P^2\) which are transversely holomorphic in the complement of an analytic set and which cannot be obtained by a rational pull-back from a Riccati foliation of an algebraic surface. The paper contains also local information on transversely projective holomorphic foliations. The author gives an example of a germ of a foliation near the origin of the complex 2-dimensional plane which is transversely projective in the complement of five smooth curves but which cannot obtained as a meromorphic pull-back of a Riccati foliation.

MSC:

32S65 Singularities of holomorphic vector fields and foliations
32S40 Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects)
37F75 Dynamical aspects of holomorphic foliations and vector fields
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