We introduce a weak Lefschetz-type result on Chow groups of complete intersections. As an application, we can reproduce some of the results in [P]. The purpose of this paper is not to reproduce all of [P] but rather illustrate why the aforementioned weak Lefschetz result is an interesting idea worth exploiting in itself. We hope the reader agrees.

This note is about certain locally complete families of Calabi–Yau varieties constructed by Cynk and Hulek, and certain varieties constructed by Schreieder. We prove that the cycle class map on the Chow ring of powers of these varieties admits a section, and that these varieties admit a multiplicative self-dual Chow–Künneth decomposition. As a consequence of both results, we prove that the subring of the Chow ring generated by divisors, Chern classes, and intersections of two cycles of positive codimension injects into cohomology via the cycle class map. We also prove that the small diagonal of Schreieder surfaces admits a decomposition similar to that of K3 surfaces. As a by-product of our main result, we verify a conjecture of Voisin concerning zero-cycles on the self-product of Cynk–Hulek Calabi–Yau varieties, and in the odd-dimensional case we verify a conjecture of Voevodsky concerning smash-equivalence. Finally, in positive characteristic, we show that the supersingular Cynk–Hulek Calabi–Yau varieties provide examples of Calabi–Yau varieties with “degenerate” motive.

Let U/ℂ be a smooth quasi-projective variety of dimension d, CHr (U,m) Bloch's higher Chow group, and

clr,m: CHr (U,m) ⊗ ℚ → homMHS (ℚ(0), H2r−m (U, ℚ(r)))

the cycle class map. Beilinson once conjectured clr,m to be surjective [Be]; however, Jannsen was the first to find a counterexample in the case m = 1 [Ja1]. In this paper we study the image of clr,m in more detail (as well as at the “generic point” of U) in terms of kernels of Abel-Jacobi mappings. When r = m, we deduce from the Bloch-Kato conjecture (now a theorem) various results, in particular that the cokernel of clm,m at the generic point is the same for integral or rational coefficients.

Based on the algebraic cobordism theory of Levine and Morel, we develop a theory of algebraic cobordism modulo algebraic equivalence.

We prove that this theory can reproduce Chow groups modulo algebraic equivalence and the semi-topological K0-groups. We also show that with finite coefficients, this theory agrees with the algebraic cobordism theory.

We compute our cobordism theory for some low dimensional varieties. The results on infinite generation of some Griffiths groups by Clemens and on smash-nilpotence by Voevodsky and Voisin are also lifted and reinterpreted in terms of this cobordism theory.

For a perfect field k, we use the techniques of Bondal-Kapranov and Hanamura to construct a tensor triangulated category of mixed motives over the truncated polynomial ring k[t]/(tm+1). The extension groups in this category are given by Bloch's higher Chow groups and the additive higher Chow groups. The main new ingredient is the moving lemma for additive higher Chow groups by the authors and its refinements.

We show that the limit of a one-parameter admissible normal function with no singularities lies in a non-classical sub-object of the limiting intermediate Jacobian. Using this, we construct a Hausdorff slit analytic space, with complex Lie group fibres, which ‘graphs’ such normal functions. For singular normal functions, an extension of the sub-object by a finite group leads to the Néron models. When the normal function comes from geometry, that is, a family of algebraic cycles on a semistably degenerating family of varieties, its limit may be interpreted via the Abel–Jacobi map on motivic cohomology of the singular fibre, hence via regulators on K-groups of its substrata. Two examples are worked out in detail, for families of 1-cycles on CY and abelian 3-folds, where this produces interesting arithmetic constraints on such limits. We also show how to compute the finite ‘singularity group’ in the geometric setting.

By using the Grothendieck–Riemann–Roch theorem we derive cycle relations modulo algebraic equivalence in the Jacobian of a curve. The relations generalize the relations found by Colombo and van Geemen and are analogous to but simpler than the relations recently found by Herbaut. In an appendix by Zagier, it is shown that these sets of relations are equivalent.

Une conjecture de Grothendieck prédit que le transport parallèle de toute classe de cycle algébrique invariante par monodromie est encore une classe de cycle algébrique. Nous prouvons qu'il s'agit d'une conséquence des conjectures standard.

Dans un travail antérieur, nous avions démontré, en caractéristique nulle, une variante de cette conjecture de déformation, en remplaçant la notion de cycle algébrique par celle de cycle motivé (adjonction formelle des inverses des isomorphismes de Lefschetz attachés à des polarisations). Nous étendons ici ce résultat en caractéristique arbitraire, sous diverses hypothèses (par exemple si la base est ‘presque complète’).

Nous définissons inconditionnellement des groupes de Galois motiviques dans ce contexte, et étudions leur variation en famille.

A conjecture due to Grothendieck predicts that the parallel transport of any monodromy-invariant algebraic cycle class remains an algebraic cycle class. We prove that this follows from the standard conjectures.

We have shown in a previous work a variant of this deformation conjecture in characteristic zero, in which algebraic cycles are replaced by motivated cycles (formal adjunction of inverses of Lefschetz isomorphisms attached to polarizations). We extend this result in any characteristic under various assumptions, for instance if the base is ‘almost complete’.

We give an unconditional construction of motivic Galois groups in this context, and study their variation in a family of projective smooth varieties.

La conjecture de Tate pour les variétés abéliennes sur les corps finis a été prouvée par Tate lui-même dans le cas d'un $H^2$, mais reste ouverte en degré supérieur. Nous démontrons ici une variante affaiblie de cette conjecture (en tout degré), qui décrit néanmoins tout cycle de Tate ‘en termes de’ cycles algébriques. Cette variante s'applique aussi aux variétés abéliennes $A$ sur un corps de type fini sur un corps fini, au moins dans le cas ‘de bonne réduction’, i.e. dans le cas où $A$ est fibre générique d'un schéma abélien sur une base projective normale sur un corps fini.

Tate's conjecture for abelian varieties over finite fields has been proven by Tate himself for $H^2$ but remains open in higher degree. We prove a weaker version of the conjecture in any degree, which describes every Tate cycle `in terms of' algebraic cycles. This variant also applies to abelian varieties $A$ over finitely generated fields of characteristic $p$, at least in the `good reduction' case, i.e. when $A$ is the generic fibre of an abelian scheme on a normal projective variety defined over a finite field.

Suppose that $K$ is a subfield of $\mathbb{C}$ for which the $\ell$-adic cyclotomic character has infinite image. Suppose that $C$ is a curve of genus $g\geq3$ defined over $K$, and that $\xi$ is a $K$-rational point of $C$. This paper considers the relation between the actions of the mapping class group of the pointed topological curve $(C^{\mathrm{an}},\xi)$ and the absolute Galois group $G_K$ of $K$ on the $\ell$-adic prounipotent fundamental group of $(C^{\mathrm{an}},\xi)$. A close relationship is established between

1. the image of the absolute Galois group of $K$ in the automorphism group of the $\ell$-adic unipotent fundamental group of $C$; and

2. the $\ell$-adic Galois cohomology classes associated to the algebraic $1$-cycle $C- C^{-}$ in the Jacobian of $C$, and to the algebraic $0$-cycle $(2g-2)\xi-K_C$ in $C$.

AMS 2000 Mathematics subject classification: Primary 11G30. Secondary 14H30; 12G05; 14C25; 14G32

On a real algebraic variety there may exist an algebraic cycle that is algebraically equivalent to zero and whose cohomology class is non-zero. The group of such cohomology classes can be highly non-trivial. It is interesting since it allows one to detect cohomology classes, in complementary dimension, which cannot be represented by algebraic cycles.

We investigate the relationship between the usual and general Hodge conjectures for abelian varieties. For certain abelian varieties $A$, we show that the usual Hodge conjecture for all powers of $A$ implies the general Hodge conjecture for $A$.