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We initiate a study of punctured tubular neighborhoods and homotopy theory at infinity in motivic settings. We use the six functors formalism to give an intrinsic definition of the stable motivic homotopy type at infinity of an algebraic variety. Our main computational tools include cdh-descent for normal crossing divisors, Euler classes, Gysin maps, and homotopy purity. Under-adic realization, the motive at infinity recovers a formula for vanishing cycles due to Rapoport-Zink; similar results hold for Steenbrink's limiting Hodge structures and Wildeshaus' boundary motives. Under the topological Betti realization, the stable motivic homotopy type at infinity of an algebraic variety recovers the singular complex at infinity of the corresponding topological space. We coin the notion of homotopically smooth morphisms with respect to a motivic ∞-category and use it to show a generalization to virtual vector bundles of Morel-Voevodsky's purity theorem, which yields an escalated form of Atiyah duality with compact support. Further, we study a quadratic refinement of intersection degrees, taking values in motivic cohomotopy groups. For relative surfaces, we show the stable motivic homotopy type at infinity witnesses a quadratic version of Mumford's plumbing construction for smooth complex algebraic surfaces. Our construction and computation of stable motivic links of Du Val singularities on normal surfaces is expressed entirely in terms of Dynkin diagrams. In characteristic p > 0, this improves Artin's analysis on Du Val singularities through étale local fundamental groups. The main results in the paper are also valid for-adic sheaves, mixed Hodge modules, and more generally motivic ∞-categories., Nous commençons une étude des voisinages tubulaires épointés et de la théorie de l'homotopie à l'infini dans un cadre motivique. Nous utilisons le formalisme des six foncteurs pour donner une définition intrinsèque du type d'homotopie motivique stable à l'infini d'une variété algébrique. Nos principaux outils de calcul incluent la descente cdh pour les diviseurs à croisement normaux, les classes d'Euler, les morphismes de Gysin et la pureté homotopique. Dans le cadre de la réalisation adique, le motif à l'infini redonne une formule pour les cycles évanescents due à Rapoport-Zink ; des résultats similaires s'appliquent aux structures de Hodge limites de Steenbrink et aux motifs bord de Wildeshaus. La réalisation topologique de Betti du type d'homotopie motivique stable à l'infini d'une variété algébrique redonne le complexe singulier à l'infini de l'espace topologique correspondant. Nous introduisons la notion de morphismes homotopiquement lisses par rapport à une ∞-catégorie motivique et nous l'utilisons pour montrer une généralisation du théorème de pureté de Morel-Voevodsky, qui donne une forme généralisée de la dualité d'Atiyah à support compact. De plus, nous étudions un raffinement quadratique des degrés d'intersection, prenant des valeurs dans les groupes de cohomotopies motiviques. Pour les surfaces relatives, nous montrons que le type d'homotopie motivique stable à l'infini témoigne d'une version quadratique de la construction de plomberie de Mumford pour les surfaces algébriques complexes lisses. Notre construction et le calcul des liens motiviques stables des singularités de Du Val sur les surfaces normales sont exprimés entièrement en termes de diagrammes de Dynkin. En caractéristique p > 0, cela améliore l'analyse d'Artin sur les singularités Du Val à travers les groupes fondamentaux locaux étales. Les principaux résultats de l'article sont également valables pour les faisceaux l-adiques, les modules de Hodge mixtes, et plus généralement les ∞-catégories motiviques.

We apply Wildeshaus's theory of motivic intermediate extensions to the motivic decomposition conjecture, formulated by Deninger-Murre and Corti-Hanamura. We first obtain a general motivic decomposition for the Chow motive of an arbitrary smooth projective family $f:X \rightarrow S$ whose geometric fibers are Tate. Using Voevodsky's motives with rational coefficients, the formula is valid for an arbitrary regular base $S$, without assuming the existence of a base field or even of a prime integer $\ell$ invertible on $S$. This result, and some of Bondarko' ideas, lead us to a generalized formulation of Corti-Hanamura's conjecture. Secondly we establish the existence of the motivic decomposition when $f:X \rightarrow S$ is a projective quadric bundle over a characteristic $0$ base, which is either sufficiently general or whose discriminant locus is a normal crossing divisor. This provides a motivic lift of the Bernstein-Beilinson-Deligne decomposition in this setting., Comment: Final version, 37 pages (introduction rewritten, details added in some proofs). To appear in IMRN

We initiate a study of punctured tubular neighborhoods and homotopy theory at infinity in motivic settings. We use the six functors formalism to give an intrinsic definition of the stable motivic homotopy type at infinity of an algebraic variety. Our main computational tools include cdh-descent for normal crossing divisors, Euler classes, Gysin maps, and homotopy purity. Under-adic realization, the motive at infinity recovers a formula for vanishing cycles due to Rapoport-Zink; similar results hold for Steenbrink's limiting Hodge structures and Wildeshaus' boundary motives. Under the topological Betti realization, the stable motivic homotopy type at infinity of an algebraic variety recovers the singular complex at infinity of the corresponding topological space. We coin the notion of homotopically smooth morphisms with respect to a motivic ∞-category and use it to show a generalization to virtual vector bundles of Morel-Voevodsky's purity theorem, which yields an escalated form of Atiyah duality with compact support. Further, we study a quadratic refinement of intersection degrees, taking values in motivic cohomotopy groups. For relative surfaces, we show the stable motivic homotopy type at infinity witnesses a quadratic version of Mumford's plumbing construction for smooth complex algebraic surfaces. Our construction and computation of stable motivic links of Du Val singularities on normal surfaces is expressed entirely in terms of Dynkin diagrams. In characteristic p > 0, this improves Artin's analysis on Du Val singularities through étale local fundamental groups. The main results in the paper are also valid for-adic sheaves, mixed Hodge modules, and more generally motivic ∞-categories.; Nous commençons une étude des voisinages tubulaires épointés et de la théorie de l'homotopie à l'infini dans un cadre motivique. Nous utilisons le formalisme des six foncteurs pour donner une définition intrinsèque du type d'homotopie motivique stable à l'infini d'une variété algébrique. Nos principaux outils de calcul incluent la descente cdh pour les diviseurs à croisement normaux, les classes d'Euler, les morphismes de Gysin et la pureté homotopique. Dans le cadre de la réalisation adique, le motif à l'infini redonne une formule pour les cycles évanescents due à Rapoport-Zink ; des résultats similaires s'appliquent aux structures de Hodge limites de Steenbrink et aux motifs bord de Wildeshaus. La réalisation topologique de Betti du type d'homotopie motivique stable à l'infini d'une variété algébrique redonne le complexe singulier à l'infini de l'espace topologique correspondant. Nous introduisons la notion de morphismes homotopiquement lisses par rapport à une ∞-catégorie motivique et nous l'utilisons pour montrer une généralisation du théorème de pureté de Morel-Voevodsky, qui donne une forme généralisée de la dualité d'Atiyah à support compact. De plus, nous étudions un raffinement quadratique des degrés d'intersection, prenant des valeurs dans les groupes de cohomotopies motiviques. Pour les surfaces relatives, nous montrons que le type d'homotopie motivique stable à l'infini témoigne d'une version quadratique de la construction de plomberie de Mumford pour les surfaces algébriques complexes lisses. Notre construction et le calcul des liens motiviques stables des singularités de Du Val sur les surfaces normales sont exprimés entièrement en termes de diagrammes de Dynkin. En caractéristique p > 0, cela améliore l'analyse d'Artin sur les singularités Du Val à travers les groupes fondamentaux locaux étales. Les principaux résultats de l'article sont également valables pour les faisceaux l-adiques, les modules de Hodge mixtes, et plus généralement les ∞-catégories motiviques.

We develop the theory of Milnor-Witt motives and motivic cohomology. Compared to Voevodsky's theory of motives and his motivic cohomology, the first difference appears in our definition of Milnor-Witt finite correspondences, where our cycles come equipped with quadratic forms. This yields a weaker notion of transfers and a derived category of motives that is closer to the stable homotopy theory of schemes. We prove a cancellation theorem when tensoring with the Tate object, we compare the diagonal part of our Milnor-Witt motivic cohomology to Minor-Witt K-theory and we provide spectra representing various versions of motivic cohomology in the $\mathbb{A}^1$-derived category or the stable homotopy category of schemes., Comment: 198 pages. This book is composed of updated versions of arXiv:1412.2989, arXiv:1708.06100, arXiv:1710.00594, arXiv:1708.06102, arXiv:1708.06098 and arXiv:1708.06095, together with an introduction, and an index

International audience; We develop the theory of fundamental classes in the setting of motivic homotopy theory. Using this we construct, for any motivic spectrum, an associated bivariant theory in the sense of Fulton-MacPherson. We import the tools of Fulton's intersection theory into this setting: (refined) Gysin maps, specialization maps, and formulas for excess intersections, self-intersections, and blow-ups. We also develop a theory of Euler classes of vector bundles in this setting. For the Milnor-Witt spectrum recently constructed by D\'eglise-Fasel, we get a bivariant theory extending the Chow-Witt groups of Barge-Morel, in the same way the higher Chow groups extend the classical Chow groups. As another application we prove a motivic Gauss-Bonnet formula, computing Euler characteristics in the motivic homotopy category.

In this paper, we initiate a study of motivic homotopy theory at infinity. We use the six functor formalism to give an intrinsic definition of the stable motivic homotopy type at infinity of an algebraic variety. Our main computational tools include cdh-descent for normal crossing divisors, Euler classes, Gysin maps, and homotopy purity. Under $\ell$-adic realization, the motive at infinity recovers a formula for vanishing cycles due to Rapoport-Zink; similar results hold for Steenbrink's limiting Hodge structures and Wildeshaus' boundary motives. Under the topological Betti realization, the stable motivic homotopy type at infinity of an algebraic variety recovers the singular complex at infinity of the corresponding analytic space. We coin the notion of homotopically smooth morphisms with respect to a motivic $\infty$-category and use it to show a generalization to virtual vector bundles of Morel-Voevodsky's purity theorem, which yields an escalated form of Atiyah duality with compact support. Further we study a quadratic refinement of intersection degrees taking values in motivic cohomotopy groups. For relative surfaces, we show the stable motivic homotopy type at infinity witnesses a quadratic version of Mumford's plumbing construction for smooth complex algebraic surfaces. Our main results are also valid for $\ell$-adic sheaves, mixed Hodge modules, and more generally motivic $\infty$-categories., Comment: Draft, comments are welcome

This work is devoted to study orientation theory in arithmetic geometric within the motivic homotopy theory of Morel and Voevodsky. The main tool is a formulation of the absolute purity property for an \emph{arithmetic cohomology theory}, either represented by a cartesian section of the stable homotopy category or satisfying suitable axioms. We give many examples, formulate conjectures and prove a useful property of analytical invariance. Within this axiomatic, we thoroughly develop the theory of characteristic and fundamental classes, Gysin and residue morphisms. This is used to prove Riemann-Roch formulas, in Grothendieck style for arbitrary natural transformations of cohomologies, and a new one for residue morphisms. They are applied to rational motivic cohomology and \'etale rational $\ell$-adic cohomology, as expected by Grothendieck in \cite[XIV, 6.1]{SGA6}., Comment: 81 pages. Final version, to appear in the Actes of a 2016 conference in the Tata Institute. Thanks a lot goes to the referee for his enormous work (more than 100 comments) which was of great help. Among these corrections, he indicated to me a sign mistake in formula (3.2.14.a) which was very hard to detect

The aim of this work is to construct certain homotopy t-structures on various categories of motivic homotopy theory, extending works of Voevodsky, Morel, D\'eglise and Ayoub. We prove these $t$-structures possess many good properties, some analogous to that of the perverse $t$-structure of Beilinson, Bernstein and Deligne. We compute the homology of certain motives, notably in the case of relative curves. We also show that their hearts provide convenient extensions of the theory of homotopy invariant sheaves with transfers, extending some of the main results of Voevodsky. These t-structures are closely related to Gersten weight structures as defined by Bondarko., Comment: 66 pages. Third version. Many corrections; in particular a proof that the heart of the delta-homotopy t-structure is a Grothendieck abelian category. Comments are welcome

The proof of the coincidence of the Gysin morphism in motivic cohomology and the usual pushout on Chow groups has been improved (see Lemma 3.3 and Proposition 3.11)

In this work, we build a spectral sequence in motivic homotopy that is analogous to both the Serre spectral sequence in algebraic topology and the Leray spectral sequence in algebraic geometry. Here, we focus on laying the foundations necessary to build the spectral sequence and give a convenient description of its $E_2$-page. Our description of the $E_2$-page is in terms of homology of the local system of fibers, which is given using a theory similar to Rost's cycle modules. We close by providing some sample applications of the spectral sequence and some hints at future work., 35 pages; comments welcome!

We show that the logarithmic version of the syntomic cohomology of Fontaine and Messing for semistable varieties over $p$-adic rings extends uniquely to a cohomology theory for varieties over $p$-adic fields that satisfies $h$-descent. This new cohomology - syntomic cohomology - is a Bloch-Ogus cohomology theory, admits period map to \'etale cohomology, and has a syntomic descent spectral sequence (from an algebraic closure of the given field to the field itself) that is compatible with the Hochschild-Serre spectral sequence on the \'etale side. In relative dimension zero we recover the potentially semistable Selmer groups and, as an application, we prove that Soul\'e's \'etale regulators land in the potentially semistable Selmer groups. Our construction of syntomic cohomology is based on new ideas and techniques developed by Beilinson and Bhatt in their recent work on $p$-adic comparison, Comment: Final version; to appear in Algebra and Number Theory

We prove a conjecture of Morel identifying Voevodsky's homotopy invariant sheaves with transfers with spectra in the stable homotopy category which are concentrated in degree zero for the homotopy t-structure and have a trivial action of the Hopf map. This is done by relating these two kind of objects to Rost's cycle modules. Applications to algebraic cobordism and construction of cycle classes are given., Comment: 27 pp