arXiv subjects
Publications and source records attributed to Bruno Kahn.
We prove the existence of divided powers in \'etale Chow groups of abelian varieties over a separably closed field, and hence of an integral lift of the Fourier transform, away from the characteristic and up to $2$-torsion. The method is to lift the Deninger-Murre Chow-K\"unneth projectors to integral ones, and draw consequences. Several techniques used here are new.
We give a reformuation of the Tate conjecture for a surface over a finite field in terms of suitable affine open subsets. We then present three attempts to prove this reformulation, each of them falling short. Interestingly, the last two are related to techniques used in proofs of Gersten's conjecture.
We point out a relationship between the norm residue isomorphism theorem of Suslin-Voevodsky-Rost and the theory of birational motives, as well as its generalisation to "higher jets".
We associate an $L$-function $L^{\mathrm{near}}(M,s)$ to any geometric motive over a global field $K$ in the sense of Voevodsky. This is a Dirichlet series which converges in some half-plane and has an Euler product factorisation. When $M$ is the dual of $M(X)$ for $X$ a smooth projective variety, $L^{\mathrm{near}}(M,s)$ differs from the alternating product of the zeta functions defined by Serre in 1969 only at places of bad reduction; in exchange, it is multiplicative with respect to exact triangles. If $K$ is a function field over $\mathbf{F}_q$, $L{\mathrm{near}}(M,s)$ is a rational function in $q^{-s}$ and enjoys a functional equation. The techniques use the full force of Ayoub's six (and even seven) operations.
We give a sheaf-theoretic version of the universal coefficient theorem.
We show that the continuous \'etale cohomology groups $H^n_{\mathrm{cont}}(X,\mathbf{Z}_l(n))$ of smooth varieties $X$ over a finite field $k$ are spanned as $\mathbf{Z}_l$-modules by the $n$-th Milnor $K$-sheaf locally for the Zariski topology, for all $n\ge 0$. Here $l$ is a prime invertible in $k$. This is the first general unconditional result towards the conjectures of arXiv:math/9801017 (math.AG) which put together the Tate and the Beilinson conjectures relative to algebraic cycles on smooth projective $k$-varieties.
We show that, for a polarised smooth projective variety $B \hookrightarrow \mathbb{P}^n_k$ of dimension $\geq 2$ over an infinite field $k$ and an abelian variety $A$ over the function field of $B$, there exists a dense Zariski open set of smooth geometrically connected hyperplane sections $h$ of $B$ such that $A$ has good reduction at $h$ and the specialisation homomorphism of Lang-N\'eron groups at $h$ is injective (up to a finite $p$-group in positive characteristic $p$). This gives a positive answer to a conjecture of the first author, which is used to deduce a negative definiteness result on his refined height pairing. This also sheds a new light on N\'eron's specialisation theorem.
We give a detailed proof of the B\'enabou-Roubaud theorem. As a byproduct it yields a weakening of its hypotheses: the base category does not need fibre products and the Beck-Chevalley condition, in the form of a natural transformation, can be weakened by only requiring the latter to be epi.
We give necessary conditions for a category fibred in pseudo-abelian additive categories over the classifying topos of a profinite group to be a stack; these conditions are sufficient when the coefficients are $\mathbf{Q}$-linear. This applies to pure motives over a field in the sense of Grothendieck, Deligne-Milne and Andr\'e, to mixed motives in the sense of Nori and to several motivic categories considered in arXiv:1506.08386 [math.AG]. We also give a simple proof of the exactness of a sequence of motivic Galois groups under a Galois extension of the base field, which applies to all the above (Tannakian) situations. Finally, we clarify the construction of the categories of Chow-Lefschetz motives given in arXiv:2302.08327 [math.AG] and simplify the computation of their motivic Galois group in the numerical case.
We discuss cases where Voevodsky's smash nilpotence conjecture is known, and give a few new ones. In particular we explain a theorem of the cube for $1$-cycles, which is due to Oussama Ouriachi.
We prove that, for any smooth and projective scheme $X$ over a field $k$ of char. $0$, the set of maps from Spec $k$ to $X$ in the $\mathbf{A}^1$-homotopy category of schemes $\mathcal{H}_{\mathbf{A}^1}(k)$ is in bijection with the quotient of $X(k)$ by $R$-equivalence, and is a birational invariant of $X$. This is achieved by establishing a precise relation between the localization of the category of smooth $k$-schemes by birational maps and the category $\mathcal{H}_{\mathbf{A}^1}(k)$, and by applying results of the second named author and R. Sujatha on birational invariants. This gives a new proof of results obtained by A. Asok and F. Morel.
We study the injectivity of the cycle class map with values in Jannsen's continuous \'etale cohomology, by using refinements that go through \'etale motivic cohomology and the ``tame'' version of Jannsen's cohomology. In particular, we use this to show that the Tate and the Beilinson conjectures imply that its kernel is torsion in positive characteristic, and to revisit recent counterexamples to injectivity.
We prove all conjectures from chapter 7 of Yves Andr\'e's book on motives in the case of products of elliptic curves. The proofs given here are simpler and more uniform than the previous proofs in known cases.
We develop Milne's theory of Lefschetz motives for general adequate equivalence relations and over a not necessarily algebraically closed base field. The corresponding categories turn out to enjoy all properties predicted by standard and less standard conjectures, in a stronger way: algebraic and numerical equivalences agree in this context. We also compute the Tannakian group associated to a Weil cohomology in a different and more conceptual way than Milne's case-by-case approach.
We prove that the Tate conjecture in codimension $1$ over a finitely generated field follows from the same conjecture for surfaces over its prime subfield. In positive characteristic, this is due to de Jong--Morrow over $\mathbf{F}_p$ and to Ambrosi for the reduction to $\mathbf{F}_p$. We give a different proof than Ambrosi's, which also works in characteristic $0$; over $\mathbf{Q}$, the reduction to surfaces follows from a simple argument using Lefschetz's $(1,1)$ theorem.
We study the construction of arXiv:2111.11217 [math:AG] in more detail, especially in the case of Schur-finite rigid $\otimes$-categories. This leads to some groundwork on the ideal structure of rigid additive and abelian $\otimes$-categories.
Inspired by recent work of Peter O'Sullivan (arXiv:2012.15703), we give a condition under which a faithful monoidal functor between abelian $\otimes$-categories is exact.