Dates
-
Location
Angers

La prochaine édition des journées de géométrie algébrique réelle se tiendra au LAREMA à Angers les 8 et 9 décembre 2022.

Orateurs confirmés :

  • Goulwen Fichou (Université Rennes 1)
  • Olivier de Gaay Fortman (Leibniz University Hannover)
  • Matilde Manzaroli (Universität Tübingen)
  • M'Hammed Oudrane (Université Côte d'Azur)
  • Adam Parusiński (Université Côte d'Azur)
  • Susanna Zimmermann (Université Paris-Saclay)

 

Programme du jeudi 8 décembre

  • 10h : Accueil
  • 10h30 : Adam Parusinski : Real motivic and topological Milnor fibres.
  • 12h : Déjeuner
  • 14h : M'hammed Oudrane : Sobolev sheaves on the definable topology.
  • 15h30 : Matilde Manzaroli : Betti numbers of real semi-stable degenerations via real logarithmic geometry
  • 17h : Olivier de Gaay Fortman : Espaces de modules en géométrie algébrique réelle

Le soir, un repas en commun est prévu au restaurant La Ferme.

 

Programme du vendredi 9 décembre

  • 9h15 : Susanna Zimmermann : Which algebraic groups act birationally on the real plane ?
  • 10h15 : Goulwen Fichou : Algebraic characterizations of homeomorphisms between algebraic varieties
  • 12h : Déjeuner

 

Titre et résumés des exposés :

  • Adam Parusinski : Real motivic and topological Milnor fibres

In a recent joint paper with J.-B. Campesato and G. Fichou we give two constructions that allow to compare directly the topological Milnor fibration and the motivic Milnor fibre of a regular complex function with normal crossing singularities. In this talk I discuss the real algebraic case.

 

 

  •  M'hammed Oudrane : Sobolev sheaves on the definable topology.

Sheaves on manifolds are good objects to deal with local problems, but from the point of view of tame geometry, the usual topology contains many open sets of pathological nature, which makes the family of open subanalytic sets (or definable sets in some fixed o-minimal structure) a good candidate for replacing the usual topology. On the subanalytic topology, sheaves that are defined by functional spaces are very important in the study of irregular holonomic D-modules, but unfortunately many functional spaces are not of local nature. In this talk we focus on Sobolev spaces. For negetive fractional Sobolev spaces, a sheafification (in the derived sense) was given by G. Lebeau. We present a method to construct this sheaves (in the usual sense) for s>= 0 in dimension 2, based on the geometric nature of open subanalytic sets in R^2. We give also some ideas about the higher dimensional case.

 

 

  • Matilde Manzaroli : Betti numbers of real semi-stable degenerations via real logarithmic geometry

In a work in progress with Emiliano Ambrosi, we study the real topology of totally real semistable degenerations, with certain technical conditions on the special fiber X0, and we give a bound for the individual real Betti numbers of a smooth fiber near 0 in terms of the complex geometry of X0. The main ingredient is the use of real logarithmic geometry, which allows to work with degenerations which are not necessarily toric and hence to go beyond the case of tropically smooth degenerations. This, in particular, generalises previous work of Renaudineau-Shaw, obtained via tropical techniques, to a more general setting.

 

  • Olivier de Gaay Fortman : Espaces de modules en géométrie algébrique réelle

Un espace de modules complexe (resp. réel) est un ensemble de classes d'isomorphisme d'objets algébriques définis sur des nombres complexes (resp. réels), enrichi d'une topologie ou d'une structure algébrique qui reflète la façon dont les objets paramétrés par l'espace se comportent en famille. Penser à l'espace de courbes complexes de genre g. Grâce aux travaux de Mumford, Deligne, Keel et Mori, d'excellentes techniques algébriques sont disponibles pour construire des espaces de modules complexes, et la théorie de Hodge permet de décrire leur géométrie. Ce qui se passe sur les réels est moins clair. Comment construire un espace de modules réel ? Comment utiliser la théorie de Hodge pour décrire sa géométrie ? Dans cet exposé, j'essaierai de répondre à ces questions.

 

  •  Susanna Zimmermann : Which algebraic groups act birationally on the real plane ?

A birational map of the plane is an isomorphism of the complement of a finite number of algebraic curves. Some algebraic groups act as birational maps on the plane and in this talk I will present the infinite ones. It turns out that there are not so many, up to inclusion, and they all act by automorphisms. on conic fibrations or on del Pezzo surfaces.A birational map of the plane is an isomorphism of the complement of a finite number of algebraic curves. Some algebraic groups act as birational maps on the plane and in this talk I will present the infinite ones. It turns out that there are not so many, up to inclusion, and they all act by automorphisms. on conic fibrations or on del Pezzo surfaces.

 

 

 

  • Goulwen Fichou : Algebraic characterizations of homeomorphisms between algebraic varieties

We address the question under which conditions a bijective morphism between complex algebraic varieties is an isomorphism. Our two answers involve the seminormalization and saturation for morphisms between varieties, together with an interpretation in terms of continuous rational functions on the complex points of the variety. We propose also a version for algebraic varieties defined on an algebraically closed field of characteristics zero. (Joint work with François Bernard, Jean-Philippe Monnier and Ronan Quarez).

 

 

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