Marín Pérez, David

Problemas de módulos para una clase de foliaciones holomorfas

Tesis Doctorals en Xarxa

Problemas de mo´dulos para una clase de foliacionesholomorfas.David Mar´ınGiven an holomorphic foliation F0 on a complex surface S, we want to find out howmany different analytic classes there are in the class of all the holomorphic foliationstopologically conjugated to F0. In other words, we want to identify the moduli space ofF0.We have studied two different contexts. The first one is the local context, where F0 is anholomorphic germ of foliation in C2 with a isolated singularity at the origin. The secondone is the global context, where F0 is an holomorphic singular foliation on a compactcomplex surface which is the total space of a fibre bundle over a non-singular complexcurve.The main results belong to the local context. There are several precedents to thiswork. Between the most important ones we can cite to D. Cerveau, X. Go´mez-Mont, A.Lins Neto, F. Loray, J-F. Mattei, R. Moussu, M. Nicolau and P. Sad. Although most ofthem treat on a weak moduli space because they use unfoldings or topologically trivialdeformations of F0.In this memoir we have obtained analogous counterparts and generalizations to the re-sults formulated by the authors cited above, in relation to the weak problem. For instante,we can stand out the generic rigidity for foliations defined by 1-forms of order two. There isanother remarkable result about the moduli space for foliations whose annulation order inthe singularity is three. This moduli space is a covering over C \ {0, 1} whose fundamentalgroup can be interpreted as symmetries of their projective holonomy representation. Themethod for the proof has been adapted to a more general situation: the quasi-homogeneousfoliations, for which we have obtained similar results.Concerning the global context we have obtained a rigidity theorem on fibred surfaces,similar to Ilyashenko’s Theorem on CP2. We particularize this result to the case of ruledsurfaces (in which the fibre is the projective line CP1), where we have studied too themoduli space for some non-rigid foliations (Riccati foliations).We emphasize that the techniques used in both contexts have a common point: theexistence of an holonomy representation which represents quite well the dynamics of thefoliation. According to this viewpoint, in the last part of the memoir we propose a newnotion of generalized holonomy which could play a similar role for foliations in CP2 orCP1 × CP1 other than Riccati foliations. This work finishes with a detailed study of thisnew notion in a explicit example.

Problemas de módulos para una clase de foliaciones holomorfas

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