Uwe Jannsen

Mixed motives and algebraic K-theory

Mixed motives for absolute hodge cycles.- Algebraic cycles, K-theory, and extension classes.- K-theory and ?-adic cohomology.

On the ℓ-adic cohomology of varieties over number fields and its Galois cohomology

If X is a smooth, projective variety over a number field k, then the absolute Galois group Gk = Gal(k/k) acts on the etale cohomology groups Hi(X, ℚl/ℤl(n)), where X = X Xk k for an algebraic closure k of k. In this paper I study some properties of these Gk-modules; in particular, I am interested in the corank of the Galois cohomology groups

On finite-dimensional motives and Murre's conjecture

{H^v}\,({G_k},{H^i}(\bar X,\,{Q_\ell }/{Z_\ell }(n))).

Principe de Hasse cohomologique

Algebraic cycles give rise to important invariants of algebraic varieties, and it is common to study the groups of algebraic cycles via so-called adequate equivalence relations. For example, the basic Chow groups are defined by considering cycles modulo rational equivalence. Rational, algebraic, homological and numerical equivalence have been considered since long time, and it is still a most interesting task to understand the precise relationship between them. But there are other adequate equivalence relations, like the l-cubical equivalence which coincides with algebraic equivalence for l = 1.

Algebraic K-Theory and Motivic Cohomology

The conjectures of Bloch, Beilinson, and Murre predict the existence of a certain functorial filtration on the Chow groups (with Q-coefficients) of all smooth projective varieties, whose graded quotients only depends on cycles modulo homological equivalence. This filtration would offer a rather good understanding of these Chow groups, and would allow to prove several other conjectures, like Bloch’s conjecture on surfaces of geometric genus 0. In Murre’s formulation (cf. 4.1 below) one can check the validity of the conjecture for particular smooth projective varieties, and in fact, a slightly weaker form of the conjecture has been proved for several cases, e.g., for surfaces [Mu1] and several threefolds [GM] (proving parts (A), (B) and (D) of the conjecture, and giving evidence for (C)). But to my knowledge, there are few results for higher-dimensional varieties, and the strongest form of Murre’s conjecture (including part (C)) is only known for curves, rational surfaces, and, trivially, for Brauer-Severi varieties. The first aim of this paper is to exhibit cases, where the full Murre conjecture can be shown. The positive aspect is that we get this for some non-trivial cases of varieties of higher (in fact arbitrarily high) dimension, the negative aspect is that we get this just for some special varieties and special ground fields. In particular, not over some universal domain. As a sample, we get the following:

Continuous étale cohomology

Good hyperplane sections, whose existence is assured by Bertini’s theorem, and good families of hyperplane sections, so-called Lefschetz pencils, are well-known constructions and powerwful tools in classical geometry, i.e., for varieties over a field. But for arithmetic questions one is naturally led to the consideration of models over Dedekind rings and, for local questions, to schemes over discrete valuation rings. It is the aim of this note to provide extensions of the mentioned constructions to this situation. We point out some new phenomena, and give an application to the class field theory of varieties over local fields with good reduction. See also [JS] for more arithmetic applications. Let A be a discrete valuation ring with fraction field K, maximal ideal m and residue field F = A/m. Let η = Spec(K) and s = Spec(F ) be the generic and closed point of Spec(A), respectively. For any scheme X over A we let Xη = X ×A K and Xs = X ×A F be its generic and special fibre, respectively.

Motives, numerical equivalence, and semi-simplicity

Abstract.The concept of weights on the cohomology of algebraic varieties was initiated by fundamental ideas and work of A. Grothendieck and P. Deligne. It is deeply connected with the concept of motives and appeared first on the singular cohomology as the weights of (possibly mixed) Hodge structures and on the etale cohomology as the weights of eigenvalues of Frobenius. But weights also appear on algebraic fundamental groups and in p-adic Hodge theory, where they become only visible after applying the comparison functors of Fontaine. After rehearsing various versions of weights, we explain some more recent applications of weights, e.g. to Hasse principles and the computation of motivic cohomology, and discuss some open questions.