This talk concerns Gabrielov’s rank Theorem, a fundamental result in local complex and real-analytic geometry, proved in the 1970’s. Contrasting with the algebraic case, it is not in general true that the analytic rank of an analytic map (that is, the dimension of the analytic-Zariski closure of its image) is equal to the generic rank of the map (that is, the generic dimension of its image). This phenomenon is involved in several pathological examples in local real-analytic geometry. Gabrielov’s rank Theorem provides a formal condition for the equality to hold. Despite its importance, the original proof is considered very difficult. There is no alternative proof in the literature, besides a work from Tougeron, which is itself considered very difficult. I will present a new work in collaboration with André Belotto da Silva and Guillaume Rond, where we provide a complete proof of Gabrielov’s rank Theorem, for which we develop formal-geometric techniques, inspired by ideas from Gabrielov and Tougeron, which clarify the proof. I will start with some fundamental examples of the phenomenon at hand, and expose the main ingredients of the strategy of this difficult proof.

Étudier les variétés à travers des dégénérescences est une technique standard en géométrie algébrique. Dans ce projet avec Böhning et von Bothmer, nous appliquons cette technique aux anneaux de Chow. Pour des dégénérescences semistables, nous construisons un anneaux de Chow prelog associé à la fibre spécial qui reçoit une flèche de l'anneaux de Chow de la fibre générique. Comme application, nous démontrons que une décomposition de la diagonale se spécialise en une décomposition prelog de la diagonale. En ce faisant, nous rendons applicables à des familles semistables le critère de Colliot-Thélène et Voisin.

https://www.lebesgue.fr/fr/content/seminars-jrna2021

A consequence of Thom Isotopy Lemma is that the set of solutions of a regular smooth equation is stable under C^1-small perturbations (it remains isotopic to the original one), but what happens if the perturbation is just C^0-small? In this case, the topology of the set of solution may change, but it turns out that the Homology groups cannot "decrease". In this talk I will present such result and some related examples and applications. This theorem is useful in those contexts where the price to pay to approximate something in C^1 is higher than in C^0. For instance in the search for quantitative bounds (here the price can be the degree of an algebraic approximation) or in combination with Eliashberg's and Mishachev's holonomic approximation Theorem (which is C^0 at most).