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Bruno Kahn

Publications and source records attributed to Bruno Kahn.

At least 37 records · Page 2Linked to original sources
arXiv2021

On universal modular symbols

We clarify the relationship between works of Lee-Szczarba and Ash-Rudolph on the homology of the Steinberg module of a linear Tits building. This yields a simple proof of the Solomon-Tits theorem in this special case. We also give a (weak) relationship between this combinatorics and the one studied by van der Kallen, Suslin and Nesterenko to compute the homology of the general linear group with constant coefficients.

math.KT↗
arXiv2021

Motives with modulus, III: The categories of motives

We construct and study a triangulated category of motives with modulus $\mathbf{MDM}_{\mathrm{gm}}^{\mathrm{eff}}$ over a field $k$ that extends Voevodsky's category $\mathbf{DM}_{\mathrm{gm}}^{\mathrm{eff}}$ in such a way as to encompass non-homotopy invariant phenomena. In a similar way as $\mathbf{DM}_{\mathrm{gm}}^{\mathrm{eff}}$ is constructed out of smooth $k$-varieties, $\mathbf{MDM}_{\mathrm{gm}}^{\mathrm{eff}}$ is constructed out of proper modulus pairs, introduced in Part I of this work. To such a modulus pair we associate its motive in $\mathbf{MDM}_{\mathrm{gm}}^{\mathrm{eff}}$. In some cases the $\mathrm{Hom}$ group in $\mathbf{MDM}_{\mathrm{gm}}^{\mathrm{eff}}$ between the motives of two modulus pairs can be described in terms of Bloch's higher Chow groups.

math.AG↗
arXiv2021

Reciprocity sheaves, II

We exhibit an intimate relationship between "reciprocity sheaves" from arXiv:1402.4201 [math.AG] and "modulus sheaves with transfers" from arXiv:1908.02975 [math.AG] and arXiv:1910.14534 [math.AG].

math.AG↗
arXiv2020

Albanese kernels and Griffiths groups

We describe the Griffiths group of the product of a curve $C$ and a surface $S$ as a quotient of the Albanese kernel of $S$ over the function field of $C$. When $C$ is a hyperplane section of $S$ varying in a Lefschetz pencil, we prove the nonvanishing in $\text{Griff}(C\times S)$ of a modification of the graph of the embedding $C\hookrightarrow S$ for infinitely many members of the pencil, provided the ground field $k$ is of characteristic $0$, the geometric genus of $S$ is $>0$, and $k$ is large or $S$ is "of motivated abelian type".

math.AG↗
arXiv2020

Topologies on schemes and modulus pairs

We study relationships between the Nisnevich topology on smooth schemes and certain Grothendieck topologies on proper and not necessarily proper modulus pairs which were introduced respectively in [9] and [3]. Our results play an important role in the theory of sheaves with transfers on proper modulus pairs. This is a revised version of arXiv:1809.05851 [math.AG].

math.AG↗
arXiv2019

Motives with modulus

We construct and study a triangulated category of motives with modulus $\mathbf{MDM}_{\mathrm{gm}}^{\mathrm{eff}}$ over a field $k$ that extends Voevodsky's category $\mathbf{DM}_{\mathrm{gm}}^{\mathrm{eff}}$ in such a way as to encompass non-homotopy invariant phenomena. In a similar way as $\mathbf{DM}_{\mathrm{gm}}^{\mathrm{eff}}$ is constructed out of smooth $k$-varieties, $\mathbf{MDM}_{\mathrm{gm}}^{\mathrm{eff}}$ is constructed out of \emph{proper modulus pairs}, that is, pairs of a proper $k$-variety $X$ and an effective divisor $D$ on $X$ such that $X \setminus |D|$ is smooth. To a modulus pair $(X, D)$ we associate its motive $M(X, D) \in \mathbf{MDM}_{\mathrm{gm}}^{\mathrm{eff}}$. In some cases the Hom group in $\mathbf{MDM}_{\mathrm{gm}}^{\mathrm{eff}}$ between the motives of two modulus pairs can be described in terms of Bloch's higher Chow groups.

math.AG↗
arXiv2018

Mayer-Vietoris triangles for motives with modulus

We construct "MV squares" in the category $\mathbf{MCor}$ of modulus pairs which was introduced in arXiv:1511.07124 [math:AG]. They allow us to describe the category $\mathbf{MDM}_{\mathrm{gm}}^{\mathrm{eff}}$ of loc. cit. in a similar way as Voevodskys category $\mathbf{DM}_{\mathrm{gm}}^{\mathrm{eff}}$, thus sharpening the results of the quoted paper.

math.AG↗
arXiv2018

Divisibility properties of motivic cohomology

We extend results of Colliot-Thélène and Raskind on the $\mathcal{K}_2$-cohomology of smooth projective varieties over a separably closed field $k$ to the étale motivic cohomology of smooth, not necessarily projective, varieties over $k$. Some consequences are drawn, such as the degeneration of the Bloch-Lichtenbaum spectral sequence for any field containing $k$.

math.AG↗
arXiv2017

The derived functors of unramified cohomology

We study the first "derived functors of unramified cohomology" in the sense of arXiv:1506.08385 [math.AG], applied to the sheaves $\mathbf{G}_m$ and $\mathcal{K}_2$. We find interesting connections with classical cycle-theoretic invariants of smooth projective varieties, involving notably a version of the Griffiths group, and the indecomposable $(2,1)$-cycles.

math.AG↗
arXiv2017

Motifs et adjoints

We show in many cases the existence of adjoints to extension of scalars on categories of motivic nature, in the framework of field extensions. This is to be contrasted with the more classical situation where one deals with a finite type morphism of schemes. Among various applications, one is a functorial construction of the "Tate-Safarevic motive" introduced in arXiv:1401.6847 [math.NT]. We also deduce a possible approach to Bloch's conjecture on surfaces, by reduction to curves.

math.AG↗
arXiv2017

Torsion order of smooth projective surfaces

To a smooth projective variety $X$ whose Chow group of $0$-cycles is $\mathbf Q$-universally trivial one can associate its torsion index $\mathrm{Tor}(X)$, the smallest multiple of the diagonal appearing in a cycle-theoretic decomposition à la Bloch-Srinivas. We show that $\mathrm{Tor}(X)$ is the exponent of the torsion in the Néron-Severi-group of $X$ when $X$ is a surface over an algebraically closed field $k$, up to a power of the exponential characteristic of $k$.

math.AG↗
arXiv2016

On the derived category of 1-motives

This is the final version of the 2007 preprint titled "On the derived category of 1-motives, I". It has been substantially expanded to contain a motivic proof of (two thirds of) Deligne's conjecture on 1-motives with rational coefficients, hence the new title. Compared to the 2007 preprint, the additions mainly concern an abstract theory of realisations with weight filtrations; Deligne's conjecture is tackled though them by an adjunction game.

math.AG↗