Reduction of the singularities of foliations and applications
Final forms for a three-dimensional vector field under blowing-up
We study the final situations which may be obtained for a singular vector field by permissible blowing-ups of the ambient space (in dimension three). These situations are preserved by permissible blowing-ups and its structure is simple from the view-point of the integral branches. Technically, we take a logarithmic approach, by marking in each step the exceptional divisor of the transformation.
Dicritical logarithmic foliations.
We show the existence of weak logarithmic models for any (dicritical or not) holomorphic foliation F of (C2,0) without saddle-nodes in its desingularization. The models are written in terms of a representative set of separatrices, whose equisingularity types are controlled by the Milnor number of the foliation.
Hypersurfaces intégrales des feuilletages holomorphes
Soit un germe en de 1-forme différentielle holomorphe, satisfaisant la condition d’intégrabilité et non dicritique, i.e. sur toute surface non intégrale de , on ne peut tracer, au voisinage de 0, qu’un nombre fini de germes de courbes analytiques , intégrales de , avec . Alors possède un germe d’hypersurface analytique intégrale.
Singular foliations of toric type
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