Workshop on Tame Geometry: Interactions Between O-minimal, Complex Analytic and Nonarchimedean Methods

Jun. 6-10 2022

Participación y/o asistencia de miembros del equipo: Fernando Sanz, Felipe Cano, Beatriz Molina y María Martín

Información del evento:

Conferencias impartidas por miembros del equipo:

Conferencia: Towards Reduction of Singularities of Generalized Analytic Functions.

Autora: Beatriz Molina

Resumen: Real generalized analytic functions are locally defined as sums of convergent generalized power series with coefficients in the real numbers; that is, power series whose exponents belong to a product of well-ordered sets of positive real numbers. In this talk we introduce generalized analytic manifolds and the blowing-up morphisms between them. We see that blowing-ups may exist or not depending on the existence of an ``standard analytic substructure''. We state several results of reduction of singularities for generalized analytic functions and we expose with detail the ``stratified version''. This is a work in collaboration with Jesús Palma-Márquez and Fernando Sanz-Sánchez.


Conferencia: Local Invariant hypersurfaces for codimension one foliations. The dicritical case.

Autor: Felipe Cano

Resumen: It is known that each germ of codimension one holomorphic foliation has at least one invariant hypersurface in the non dicritical case. The proof starts with the existence of invariant curve in the two dimensional situation, it continues in dimension three based on the reduction of singularities and a construction of the invariant hypersurface as a germ, after reduction of singularities, over the so called ``partial separatrices’’. In higher dimension the result is mainly cohomological , once we know the three-dimensional case. When the foliation is dicritical, the result is not true in general as it is shown by Jouanolou’s examples. Anyway, the three dimensional arguments may be extended for the dicritical case with a notion of ‘’extended partial separatrix’’ and, in this way, we obtain an equivalent notion of the existence of invariant hypersurface. The main positive result we know is for the class of toric type foliations, very close to the Newton nondegenerate foliations. We give details of the study of this dicritical situation. Finally, when there are no invariant hypersurface, we propose that a local version of Brunella’s alternative holds and we will show that this is so for the class of foliations with relatively isolated reduction of singularities. (Work in collaboration with Beatriz Molina-Samper)