Rencontre de clôture de l'ANR HQDiag

Mardi 18 novembre

Lieu : Salle F116, Bâtiment F, UFR IM2AG

  • 12h00 – 13h45 : Lunch à l’Institut Fourier (cafétéria)
  • 14h00 – 14h45 : Raphaël Ruimy
    Title : La filtration par le poids sur la cohomologie réelle est motivique
    Abstract : Dans cet exposé, on présentera la filtration par le poids sur l'homologie des points réels d'une variété réelle à coefficients dans F_2. Celle-ci a été construite par Totaro-McCrory-Parusinski comme suit : on munit l'homologie d'une variété projective lisse de la filtration triviale, puis on étend formellement à toutes les variétés lisses en utilisant le triangle de localisation, puis à toutes les variétés en utilisant la résolution des singularités. Cette méthode rappelle fortement les propriétés du motif d'un k-schéma sur un corps construit par Voevodsky. J'expliquerai comment re-construire la filtration par le poids par des méthodes motiviques en utilisant la théorie développée par Bondarko.
  • 15h00 – 15h45 : Pietro Gigli
    Title : Some computations of the symplectic bordism ring
    Abstract : The symplectic bordism spectrum $MSp$, constructed by Panin and Walter, represents the universal symplectically oriented cohomology theory, but its coefficient ring has no clear presentation. I will give a sketchy overview of what is known about the symplectic bordism ring. In particular, I will include a computation of the coefficient ring of the $\eta$-completed part $MSp^\wedge_\eta$, obtained through a modified version of the Pontryagin-Thom construction, which was part of my PhD thesis. Time permitting, I will also discuss a conjectural comparison with the complex bordism ring in the context of real isotropic spectra, following some ideas of Vishik and Tanania.
  • 16h00 – 16h45 : William Hornslien
    Title : Homotopy classes of endomorphisms of the projective line
    Abstract : A fundamental problem in algebraic topology is the study of homotopy groups of spheres. The projective line is a sphere in motivic homotopy theory, and its A^1-homotopy classes of endomorphisms is analogue to the fundamental group of the circle in classical topology. Morel computed this group using abstract methods. In this talk we will describe the group of A^1-homotopy classes of endomorphisms of the projective line by using basic algebraic geometry. This is joint work with Viktor Balch Barth, Gereon Quick, and Glen Matthew Wilson.

Mercredi 19 novembre

Lieu : Salle B29, Institut Fourier

  • 9h15 – 10h00 : Pierre Martinez
    Title : Bigraded cohomology for real algebraic varieties and its arithmetic variant
    Abstract : I will first introduce the bigraded cohomology for real algebraic varieties developed by Johannes Huisman and Dewi Gleuher. This is a cohomology theory that refines the equivariant cohomology "à la Kahn-Krasnov" of the complex points of a real variety, the latter often being preferred in the cohomological study of real algebraic varieties. Since the construction of this bigraded cohomology and its associated characteristic classes relies on the sheaf exponential morphism, I will explain how to produce an arithmetic (or algebraic) variant of these cohomology groups, whose main advantage is to eliminate topological or transcendental conditions. I will conclude by comparing these two versions of bigraded cohomology.
  • 10h15 – 11h00 : Victor Chachay
    Title : An enriched count for lines in a degree 2 del Pezzo surface
    Abstract : The number of lines in a del Pezzo surface is a classical invariant if we work over an algebraically closed field. To recover an invariant in general, we look at the induced problem in Chow-Witt groups. We can define an Euler class even without orientation in a Chow-Witt group in the case of degree 2 del Pezzo surfaces. Computing and giving a meaning of this will be the goal of the talk.
  • 11h15 – 12h00 : Ivan Rosas-Soto
    Title : Algebraic cycles of some Fano varieties with Hodge structure of level one
    Abstract : In this talk I will review the Chow and étale motivic cohomology groups of smooth complete intersections with Hodge structures of level one, classified by Deligne and Rapoport, with particular attention to fivefolds. We will see how these results can be extended to algebraic cycles on other smooth Fano manifolds with Hodge structures of level one. As an application of this, we will prove the integral Hodge conjecture for smooth quartic double fivefolds using the étale motivic approach. Specifically, we will examine their unramified cohomology groups with torsion coefficients. This is based on joint work with Pedro Montero.
  • 12h00 – 13h30 : Lunch à l’Institut Fourier (cafétéria)
  • 13h30 – 17h00 : Discussions libres