Frédéric Déglise

This work revisits a path opened in my thesis, aimed at understanding generic motives, pro-mixed motives associated with models of a function field over a fixed base field. The theory is first linked to Beilinson's dual approach to the same objects, highlighting its relevance for understanding the motivic t-structure. New calculations are obtained that better explain the morphisms of generic motives. They are also applied to obtain new results on the motivic cohomology of function fields: the latter increases with transcendental elements, but likely in a very controllable manner. One of the merits of this paper is also to provide a direct determination of the motivic cohomology of number fields based on Borel's results and a detailed study of the lambda-filtration in rational K-theory. This work is in its final version, intended for publication in the special edition in honor of Jacob Murre.

The theory of orientation in stable motivic homotopy has been extended by Ivan Panin and Charles Walter. We have studied this theory in collaboration with Jean Fasel by introducing the notion of ternary formal law, as well as the Borel character, which is a quadratic analog of the Chern character. In this preprint, we demonstrate a Grothendieck-Riemann-Roch formula for oriented virtual fundamental classes (as defined in this work), which in turn leads to several variants of the Riemann-Roch formula, including the usual version for the Gysin map in cohomology associated with a projective lci morphism. An explicit calculation is given for K3 surfaces.

This work originates from the desire to calculate the degree of quadratic cycles over arbitrary fields. Two problems can arise in the theory: inseparable extensions and characteristic 2. The main idea to solve them is to use the rug provided by Grothendieck duality. Ultimately, the notes aim to fully describe the theory of Milnor-Witt K-theory, following the axiomatics of Feld on Milnor-Witt modules. These notes serve as the basis for upcoming notes on quadratic cycles. They are based on a series of lectures given at the spring school "Invariants in Algebraic Geometry".

This preprint, currently unfinished, aims to complete the program of understanding the perverse homotopy t-structure defined by J. Ayoub, and generalized by M. Bondarko and the first author, along with the associated Leray spectral sequence. The key point is the calculation of the heart of this t-structure in terms of Cousin complexes, a concept inherited from Grothendieck—himself inspired by Cartan, Zariski, and Samuel. With stronger assumptions, we show that this heart is equivalent to a generalization of Rost cycle modules for the stable A1-derived category. [At present, it remains to be shown that one of the two adjunction morphisms is indeed an isomorphism.]

This work proposes a motivic version of several notions from differential geometry: link of a singularity, punctured tubular neighborhood, and end space. Using the six functors formalism, we provide a definition of the punctured tubular neighborhood and stable homotopy type at infinity that align with considerations by Marc Levine for the former, Hughes-Ranicki, and Jörg Wildeshaus for the latter. These two notions coincide when considering a compactified situation. We develop methods for calculating these two invariants, including a general formula in the case of a normal crossing singularity (or more generally "h-smooth") and an interpretation in terms of fundamental class. We apply these calculation methods to obtain a motivic homotopic version of Mumford’s plumbing calculation. This calculation is expressed using a quadratic intersection matrix. Along the way, we develop some new methods in the theory of the six functors, such as an alternating cdh resolution notion that allows us to provide a motivic homotopic version of the Rapoport-Zink complex, used to calculate vanishing cycles in a semi-stable situation. These methods are applicable and used in various contexts, such as Artin motives or Nori motives.

According to its very construction, Voevodsky's theory of motivic complexes over a perfect field comes with a t-structure he called the homotopy t-structure (since it does not coincide with the motivic t-structure). In my thesis, I showed how to define the stable version (i.e., non-effective) of this t-structure and identified its heart with the theory of Rost cycle modules (cf. here).
In the axiomatic framework of stable homotopy functors, Ayoub introduced a relative version, over a base scheme S, of this t-structure, later generalized by M. Bondarko and the author. He had conjectured a few years ago, during presentations, that the heart of this t-structure, applied to the correct version of DM(S).
This note aims to outline a strategy for proving this result. This strategy has largely been surpassed in the preprint of 10/2022, but I leave this version for reference.