Given a smooth variety X and an effective Cartier divisor D ⊂ X, we show that the cohomological Chow group of 0-cycles on the double of X along D has a canonical decomposition in terms of the Chow group of 0-cycles CH0(X) and the Chow group of 0cycles with modulus CH0(X|D) on X. When X is projective, we construct an Albanese variety with modulus and show that this is the universal regular quotient of CH0(X|D).As a consequence of the above decomposition, we prove the Roitman torsion theorem for the 0-cycles with modulus. We show that CH0(X|D) is torsion-free and there is an injective cycle class map CH0(X|D) ֒→ K0(X, D) if X is affine. For a smooth affine surface X, this is strengthened to show that K0(X, D) is an extension of CH1(X|D) by CH0(X|D). ± 17 4.1. The map ∆ * 17 4.2. The maps ι * ± 18 5. 0-Cycles on S X and 0-cycles with modulus on X 20 5.1. Overture 20

Let X be a separated scheme of finite type over a field k and D a non-reduced effective Cartier divisor on it. We attach to the pair (X, D) a cycle complex with modulus, those homotopy groups -called higher Chow groups with modulus -generalize additive higher Chow groups of Bloch-Esnault, Rülling, Park and Krishna-Levine, and that sheafified on X Zar gives a candidate definition for a relative motivic complex of the pair, that we compute in weight 1.When X is smooth over k and D is such that D red is a normal crossing divisor, we construct a fundamental class in the cohomology of relative differentials for a cycle satisfying the modulus condition, refining El-Zein's explicit construction of the fundamental class of a cycle. This is used to define a natural regulator map from the relative motivic complex of (X, D) to the relative de Rham complex. When X is defined over C, the same method leads to the construction of a regulator map to a relative version of Deligne cohomology, generalizing Bloch's regulator from higher Chow groups.Finally, when X is moreover connected and proper over C, we use relative Deligne cohomology to define relative intermediate Jacobians with modulus J r X|D of the pair (X, D). For r = dim X, we show that J r X|D is the universal regular quotient of the Chow group of 0-cycles with modulus.generalizing Bloch's computation in weight 1, Z X (1) ∼ = O × X [−1], and proving the first of the expected properties of the relative motivic cohomology groups (see Theorem 4.1). 1.7. Let (X, D) be as in 1.6. The second main technical result of this paper, presented in Section 7, is the construction, using the fundamental class in relative differentials, of regulator maps from the relative motivic complex Z X|D (r) to a the relative de Rham complex of XThe map φ dR is compatible with flat pullbacks and proper push forwards of pairs.When X is a smooth algebraic variety over the field of complex numbers, we can use the same technique to define regulator maps to a relative version of Deligne cohomology (see (8.10)) and to Betti cohomology with compact support, where ǫ is the morphism of sites and j : X → X is the open embedding of the complement of D in X, generalizing Bloch's regulator from higher Chow groups to Deligne cohomology, constructed in [7]. This regulator map is further studied in Section 9 in the case of additive Chow groups. Evaluated on suitable cycles, our regulator recovers Bloch-Esnault additive dilogarithm (introduced in [10]) and gives a refinement of [10, Proposition 5.1] (see (9.5)).1.8. Suppose that X is moreover connected and proper over C, and consider the induced maps in cohomology in degree 2r. We have a commutative diagram (see 10.1.1)and in analogy with the classical situation, we call the kernel J r X|D the r-th relative intermediate Jacobian of the pair (X, D). We note that they admit a description in terms of extensions groups Ext 1 in the abelian category of enriched Hodge structures defined by S.Bloch and V.Srinivas in [12].One can show that J r X|D fits into an exact sequen...

This document is a short user's guide to the theory of motives and homotopy theory in the setting of logarithmic geometry. We review some of the basic ideas and results in relation to other works on motives with modulus, motivic homotopy theory, and reciprocity sheaves.Résumé. Ce document est un petit guide d'utilisation de la théorie des motifs et de la théorie de l'homotopie dans le cadre de la géométrie logarithmique. Nous passons en revue certaines des idées de base et des résultats en relation avec d'autres travaux sur les motifs avec module, théorie de l'homotopie motivique, et les faisceaux de réciprocité.

Zero-cycle groups on algebraic varietiesTome 9 (2022), p. 281-325. © Les auteurs, 2022. Certains droits réservés. Cet article est mis à disposition selon les termes de la licence LICENCE INTERNATIONALE D'ATTRIBUTION CREATIVE COMMONS BY 4.0. https://creativecommons.org/licenses/by/4.0/ L'accès aux articles de la revue « Journal de l'École polytechnique -Mathématiques » (http://jep.centre-mersenne.org/), implique l'accord avec les conditions générales d'utilisation (http://jep.centre-mersenne.org/legal/). Publié avec le soutien du Centre National de la Recherche Scientifique Publication membre du Centre Mersenne pour l'édition scientifique ouverte www.centre-mersenne.org

We prove Bloch’s formula for the Chow group of 0-cycles with modulus on a smooth quasi-projective surface over a field. We use this formula to give a simple proof of the rank one case of a conjecture of Deligne and Drinfeld on lisse Q ¯ ℓ \overline {\mathbb {Q}}_{\ell } -sheaves. This was originally solved by Kerz and Saito in characteristic ≠ 2 \neq 2 .