# Cohomologie non ramifiée et conjecture de Hodge entière

@article{ColliotThelene2010CohomologieNR,
title={Cohomologie non ramifi{\'e}e et conjecture de Hodge enti{\e}re},
author={Jean-Louis Colliot-Th'elene and Claire Voisin},
journal={Duke Mathematical Journal},
year={2010},
volume={161},
pages={735-801}
}`
• Published 16 May 2010
• Mathematics
• Duke Mathematical Journal
Building upon the Bloch–Kato conjecture in Milnor K-theory, we relate the third unramified cohomology group with Q/Z coefficients with a group which measures the failure of the integral Hodge conjecture in degree 4. As a first consequence, a geometric theorem of the second-named author implies that the third unramified cohomology group with Q/Z coefficients vanishes on all uniruled threefolds. As a second consequence, a 1989 example by Ojanguren and the first named author implies that the…
Refined unramified homology of schemes
We introduce the notion of refined unramified cohomology of algebraic schemes and prove comparison theorems that identify some of these groups with cycle groups. This recovers for cycles of low
Algebraic cycles and refined unramified cohomology
We introduce refined unramified cohomology groups. This notion allows us to give in arbitrary degree a cohomological interpretation of the failure of integral Hodge- or Tate-type conjectures, of
Abel-Jacobi map , integral Hodge classes and decomposition of the diagonal
Given a smooth projective n-fold Y , with H(Y ) = 0, the Abel-Jacobi map induces a morphism from each smooth variety parameterizing codimension 2-cycles in Y to the intermediate Jacobian J(Y ), which
Birational invariants : cohomology, algebraic cycles and Hodge theory cohomologie
In this thesis, we study some birational invariants of smooth projective varieties, in view of rationality questions for these varieties. It consists of three parts, that can be read independently.In
La conjecture de Tate entière pour les cubiques de dimension quatre
• Mathematics
Compositio Mathematica
• 2014
Abstract We prove the integral Tate conjecture for cycles of codimension $2$ on smooth cubic fourfolds over an algebraic closure of a field finitely generated over its prime subfield and of
Troisième groupe de cohomologie non ramifiée des hypersurfaces de Fano
We establish the vanishing of the third unramified cohomology group for many types of Fano hypersurfaces in projective space over an algebraically closed field of arbitrary characteristic, and over a
Cohomologie non ramifiée dans le produit avec une courbe elliptique
RésuméUn théorème de Gabber (Enseign Math (2) 48(1–2):127–146, 2002) permet de construire des classes de cohomologie non ramifiée dans le produit de certaines variétés et d’une courbe elliptique. Le
Rationality problems and conjectures of Milnor and Bloch–Kato
• A. Asok
• Mathematics
Compositio Mathematica
• 2013
Abstract We show how the techniques of Voevodsky’s proof of the Milnor conjecture and the Voevodsky–Rost proof of its generalization the Bloch–Kato conjecture can be used to study counterexamples to
Unramified cohomology, integral coniveau filtration and Griffiths group
We prove that the degree k unramified cohomology with torsion coefficients of a smooth complex projective variety X with small CH_0(X) has a filtration of length [k/2], whose first filter is the
Hodge loci
The goal of this expository article is first of all to show that Hodge theory provides naturally defined subvarieties of any moduli space parameterizing smooth varieties, the “Hodge loci”, although

## References

SHOWING 1-10 OF 77 REFERENCES
Autour de la conjecture de Tatecoefficients Zpour les varietes sur les corps finis
• Mathematics
• 2009
This partly expository paper investigates versions of the Tate conjecture on the cycle map for varieties defined over finite fields with values in 'etale cohomology with Z_\ell-coefficients. The bulk
Torsion algebraic cycles and complex cobordism
Atiyah and Hirzebruch gave the first counterexamples to the Hodge conjecture with integral coefficients [3]. That conjecture predicted that every integral cohomology class of Hodge type (p, p) on a
Unramified cohomology and rationality problems
— The aim of this paper is to construct unirational function fields K over an algebraically closed field of characteristic 0 such that the unramified cohomology group H i nr(K,μ ⊗i p ) is not trivial
Unramified cohomology of degree 3 and Noether’s problem
Let G be a finite group and W be a faithful representation of G over C. The group G acts on the field of rational functions C(W). The question whether the field of invariant functions C(W)G is purely
Anticanonical divisors and curve classes on Fano manifolds
• Mathematics
• 2010
It is well known that the Hodge conjecture with rational coefficients holds for degree 2n-2 classes on complex projective n-folds. In this paper we study the more precise question if on a rationally
On the Periods of Certain Rational Integrals: II
In this section we want to re-prove the results of ?? 4 and 8 using sheaf cohomology. One reason for doing this is to clarify the discussion in those paragraphs and, in particular, to show how
The coniveau filtration and non-divisibility for algebraic cycles
• Mathematics
• 1996
(0.1) Let X be a smooth projective algebraic variety of dimension d defined over a number field K. Let BP(Xs C CHP(Xs denote the subgroup of the Chow group of Xs consisting of codimension p algebraic
On the Deligne–Beilinson cohomology sheaves
We are showing that the Deligne--Beilinson cohomology sheaves ${\cal H}^{q+1}({\bf Z}(q)_{\cal D})$ are torsion free by assuming Kato's conjectures hold true for function fields. This result is
On integral Hodge classes on uniruled or Calabi-Yau threefolds
Let X be a smooth complex projective variety of dimension n. The Hodge conjecture is then true for rational Hodge classes of degree 2n−2, that is, degree 2n−2 rational cohomology classes of Hodge