Local Brunella's Alternative I. RICH Foliations
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Publication:5252102
DOI10.1093/IMRN/RNU011zbMATH Open1317.32064arXiv1309.6421OpenAlexW2962773818MaRDI QIDQ5252102
Author name not available (Why is that?)
Publication date: 29 May 2015
Published in: (Search for Journal in Brave)
Abstract: This paper is devoted to studying the structure of codimension one singular holomorphic foliations on without invariant germs of analytic surface. We focus on the so-called CH-foliations, that is, foliations without saddle nodes in two dimensional sections. Considering a reduction of singularities, we detect the possible existence of "nodal components", which are a higher dimensional version of the nodal separators in dimension two. If the foliation is without nodal components, we prove that all the leaves in a neighborhood of the origin contain at least one germ of analytic curve at the origin. We also study the structure of nodal components for the case of "Relatively Isolated CH-foliations" and we show that they cut the dicritical components or they exit the origin through a non compact invariant curve. This allows us to give a precise statement of a local version of Brunella's alternative: if we do not have an invariant surface, all the leaves contain a germ of analytic curve or it is possible to detect the nodal components in the generic points of the singular curves before doing the reduction of singularities.
Full work available at URL: https://arxiv.org/abs/1309.6421
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