Workshop

Unstable real-étale motivic homotopy theory

Aravind ASOK (University of Southern California)

I will describe a model for unstable real-étale motivic homotopy theory over an arbitrary base scheme. Then, I will discuss the link between (unstable) real-étale homotopy theory and $\rho$-localization for connected motivic spaces. This is based on joint work with T. Bachmann, E. Elmanto and M. Hopkins.

Yonatan HARPAZ (Université Université Sorbonne Paris Nord)

Motivic knot theory

Clémentine LEMARIÉ-RIEUSSET (IMB Dijon)

In this thesis, we introduce a counterpart in algebraic geometry to knot theory. Since this new theory uses motivic homotopy theory (specifically, quadratic intersection theory), we name it motivic knot theory. We focus on motivic linking, which means that we study how two disjoint closed $F$-subschemes of an ambient $F$-scheme can be intertwined, i.e. linked together (where $F$ is a perfect field). In knot theory, the linking number of an oriented link with two components (i.e. of two disjoint oriented knots) is an integer which counts how many times one of the components turns around the other component. We define counterparts in algebraic geometry to oriented links with two components and to the linking number; we call these latter counterparts quadratic linking degrees. Our quadratic linking degrees are not necessarily integers; the ones we study the most take values in the Witt group of the ground field $F$, which is a group of equivalence classes of symmetric bilinear forms over $F$ (or equivalently, of quadratic forms over $F$, when the characteristic of $F$ is different from $2$). After answering questions which naturally arise from these quadratic linking degrees, we devise methods to compute them. These methods rely on explicit formulas for the residue morphisms of Milnor-Witt $K$-theory (from which boundary maps for the Rost-Schmid complexes are constructed) and for the intersection product of the Rost-Schmid ring (and in particular of the Chow-Witt ring). Using these methods, we explicitly compute our quadratic linking degrees on examples. Some of these examples are inspired by knot theory, specifically by torus links (including the Hopf and Solomon links).

The real cycle class map

Samuel LERBET (Université Grenoble-Alpes)

Let $X$ be a smooth complex algebraic variety. The complex cycle class map associates with any subvariety of $X$ a class in the singular cohomology of the complex locus of $X$ and this association descends to the Chow groups of $X$. However, the construction of this map relies on the orientation of complex analytic manifolds as differentiable manifolds through the use of fundamental classes and Poincaré duality. Thus following this construction for a real variety $X$ only allows one to define a cycle class map from the Chow groups of $X$ to the cohomology mod $2$ of its real locus. We shall explain how one may construct a reasonable analogue of the complex cycle class map by replacing Chow groups by Chow–Witt groups and describe some properties of this real cycle class map.

An interpolation between special and general linear algebraic bordism

Oliver RÖNDIGS (Universität Osnabrück)

This is a report on work in progress of my PhD student Ahina Nandy. She proved that a certain natural map $\mathrm{MSL} \wedge \mathbb{P}^\infty \to \mathrm{MGL} \wedge \mathbb{P}^1$ relating Thom spectra for the special and the general linear groups, respectively, is an equivalence over any base scheme. Besides illustrating the argument, the talk will indicate applications, such as the computation of the first homotopy module of $\mathrm{MSL}$ over fields.

Étale degree map

Ivan ROSAS-SOTO (IMB Dijon)

Using the triangulated category of étale motives over a field $k$, we define the group $\mathrm{CH}^\textrm{ét}_0(X)$ as an étale analogue of $0$-cycles of a smooth projective variety $X$ over $k$. In the talk, we will define the étale version of the degree map and give some examples of smooth projective varieties over a field $k$ without zero cycles of degree one but with étale zero cycles of degree one. Also, we will present some non-trivial examples where there is no étale zero cycle of degree 1.

Perverse $t$-structure and Lefschetz property on Artin motives

Raphaël RUIMY (ENS de Lyon)

The Affine Lefschetz Theorem (also known as Artin’s Vanishing Theorem) can be formulated as a t-exactness property with respect to the perverse $t$-structure. The stable $\infty$-category $DM_{\acute{e}t}(S)$ of étale motives is expected to be endowed with a perverse $t$-structure which should satisfy this property. In the talk, we will endow the subcategory $DM^A_{\acute{e}t}(S)$ of $DM_{\acute{e}t}(S)$ of Artin motives with a perverse $t$-structure and present the analog of the Affine Lefschetz Theorem when the dimension of the base scheme is $2$ or less. We will explain how this fails in dimension $4$.