Vector bundles on smooth real affine varieties

Samuel LERBET (ENS Paris)

Let $X$ be an affine scheme. To classify vector bundles on $X$, one usually divides the problem in two subquestions: the stable computation, namely the calculation of the reduced $K$-theory of $X$, and the comparison between this abelianised situation and the unstable one, which is usually more difficult to understand. In the unstable situation, there are two relevant problems: the splitting problem—under which condition does a vector bundle split a trivial rank $1$ summand?—and the cancellation problem—are stably isomorphic vector bundles in fact isomorphic? Thanks to classic theorems of Bass and Serre, these two problems are very well-understood in negative corank, where the corank is the difference between the dimension and the rank. When $X$ is smooth over an algebraically closed base field of characteristic $0$, for vector bundles of corank at most $1$, splitting is controlled by the vanishing of the top Chern class by work of Murthy, Asok–Fasel and Asok–Bachmann–Hopkins, and cancellation holds thanks to work of Suslin and Fasel. Our aim in this talk will be to explain these results in more detail and what counterparts can be formulated when $k$ has more arithmetic complexity in $k$, namely when $k$ is the field of real numbers. This is joint work with A. Asok and J. Fasel, and with J. Fasel and S. Banerjee.

Towards a bigraded (co)homology theory for real algebraic varieties

Pierre MARTINEZ (Brest) and Santiago Toro Oquendo (Lens)

We will introduce the bigraded cohomology for real algebraic varieties developed by Johannes Huisman and Dewi Gleuher. This theory simultaneously encodes the topology of the complex points of a real algebraic variety, the orbit space under the natural Galois action, and its real locus. It provides a refinement of the equivariant cohomology "à la Kahn-Krasnov" of the complex points of a real variety, the latter often being preferred (by the algebraic geometers) in the cohomological study of real algebraic varieties. These bigraded cohomology groups enjoy several notable features. In particular, they admit characteristic classes of Real vector bundles, thereby providing a unified setting for both Chern and Stiefel–Whitney classes. Moreover, algebraic subvarieties of a smooth real algebraic variety carry Thom classes and fundamental classes in the bigraded cohomology of the ambient space. We will present explicit computations in simple examples and discuss a range of formal properties of the theory, including Thom and suspension isomorphisms, associated characteristic classes, Euler classes and fundamental classes. Finally, if time permits, we will also consider the homological counterpart of this theory, which refines, in particular, the Borel–Moore homology (with mod 2 coefficients) of the real locus of a real algebraic variety.

Raphaël RUIMY (Grenoble)

I will explain how to construct the motivic $t$-structure on $1$-motives with integral coefficients over a scheme of characteristic zero or a Dedekind scheme, building on previous works of Orgogozo, Barbieri-Viale and Kahn, Pepin Lehalleur and Haas among others. When we invert the residue characteristic exponents of the base, this t-structure induces a $t$-structure on the category of smooth $1$-motives whose heart is the category of Deligne $1$-motives with torsion which we prove to be abelian. This relies on the fact that over a normal scheme, Deligne $1$-motives with torsion are determined by their fiber on the generic point and on an explicit description of good reduction Deligne $1$-motives. If I have enough time, I will try to sketch how the proof could be adapted in characteristic $p$ as well as how to add a $p$-torsion part over a field.

Nilpotence in étale motivic homotopy theory

Swann TUBACH (IMJ-PRG)

The motivic homotopy category, as introduced by Morel and Voevodsky, is a version of the category of spectra in algebraic geometry. Many of the properties of spectra remain true in the motivic homotopy category, but some analogies require very heavy machinery to be made. Moreover there are some discrepancies between the two worlds: for example a corollary of a theorem of Morel implies that the algebraic Hopf map is never nilpotent whereas in topology we always have that the fourth power of the Hopf map vanishes. In this talk, I will explain a joint work with Klaus Mattis in which we observe that in the étale-local version of the motivic homotopy category, most of the need for heavy machinery and most of the discrepancies, vanish.

The motivic fundamental groupoid at tangential basepoints

Sofian TUR-DORVAULT (Montpellier)

If $U$ is a smooth scheme over a subfield of the field of complex numbers, it is known from the work of Pierre Deligne and Alexander Goncharov that the prounipotent completion of the fundamental group of $U^{\mathrm{an}}$ based at any point has a motivic incarnation. More precisely, its coordinate ring arises as the degree-zero homology of the Betti realization of a Hopf algebra object in Voevodsky's triangulated category of motives. More generally, he conjectured a similar property for the fundamental group based at a "point at infinity", i.e. the datum of a point $x$ on a smooth compactification of $U$ with normal crossings boundary, together with the datum of a tangent vector at $x$, normal to the boundary. While Deligne and Goncharov proved this conjecture in the case of the projective line minus three points, the general case remained still open. In this talk, I will explain how logarithmic geometry, together with the notion of virtual morphisms between log schemes, allows one to construct the motivic fundamental group in full generality and to compute its realizations.

Algebraic cycles of some Fano varieties with Hodge structure of level one

Ivan Rosas-Soto (UMPA Ens de Lyon)

In this talk I will review Chow groups 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 extend to algebraic cycles on other smooth Fano manifolds with Hodge structures of level one and, as an application, we prove the integral Hodge conjecture for smooth quartic double fivefolds by means of the étale motivic approach. This is based on a joint work with Pedro Montero.

Algebraic symplectic cobordism

Pietro Gigli (IMB)

The combinatorial (MW-)motive

Keyao Peng (IMB)

The Motivic Picard--Lefschetz Formula

Ran Azouri (IMJ-PRG)