Related papers: Zero cycles on singular varieties and their desing…

Let X be a separated scheme of finite type over a field k and D a non-reduced effective Cartier divisor on it. We attach to the pair (X, D) a cycle complex with modulus, whose homotopy groups - called higher Chow groups with modulus -…

For a smooth projective variety X over an arbitrary field k, we discuss the surjectivity of the Albanese map from the Chow group of zero-cycles of degree zero on X to the group of rational points of the Albanese variety of X. Over…

Let $X$ be a complete intersection inside a variety $M$ with finite dimensional motive and for which the Lefschetz-type conjecture $B(M)$ holds. We show how conditions on the niveau filtration on the homology of $X$ influence directly the…

An old conjecture of Voisin describes how zero-cycles on a variety $X$ should behave when pulled-back to the self-product $X^m$ for $m$ larger than the geometric genus of $X$. Using complete intersections of quadrics, we give examples of…

Let $X$ be a double EPW sextic, and $\iota$ its anti-symplectic involution. We relate the $\iota$-anti-invariant part of the Chow group of zero-cycles of $X$ with Voisin's rational orbit filtration. For a general double EPW sextic $X$, we…

Let X be a K3 surface. We show that the Chow group CH_0(X) of 0-cycles contains a "fundamental class" c_X of degree 1 with remarkable properties: any product of divisors is proportional to this class, and so is the second Chern class…

Let K be a finite extension of Q_p and X a smooth projective variety over K. We define the notion of totally degenerate reduction of such an X and the associated Chow complexes of the special fibre of a suitable regular proper model of X…

We study a local to global principle for certain higher zero-cycles over global fields. We thereby verify a conjecture of Colliot-Th\'el\`ene for these cycles. Our main tool are the Kato conjectures proved by Jannsen, Kerz and Saito. Our…

Let $X$ be a hyperk\"ahler variety, and assume $X$ has a non-symplectic automorphism $\sigma$ of order $>{1\over 2}\dim X$. Bloch's conjecture predicts that the quotient $X/<\sigma>$ should have trivial Chow group of $0$-cycles. We verify…

Let $f: X \to \mathbb{A}^1$ be a regular function on a smooth complex algebraic variety $X$. We formulate and prove an equivalence between the algebraic formal twisted de Rham complex of $f$ and the vanishing cycles with respect to $f$ as…

We define a filtration on the Chow groups of a smooth projective variety X over a field k by using the cycle map into continuous l-adic etale cohomology. The main theorem says that if k is a function field in one variable over a finite…

For a generalized Kummer variety X of dimension 2n, we will construct for each 0 < i < n some co-isotropic subvarieties in X foliated by i-dimensional constant cycle subvarieties. These subvarieties serve to prove that the rational orbit…

Connections between the second Chow group of a smooth projective variety and its third unramified cohomology group, with coefficients the roots of unity twisted twice, feature in several recent works. In this note we revisit a 1996 paper by…

We prove Bloch's formula for 0-cycles on affine schemes over algebraically closed fields. We prove this formula also for projective schemes over algebraically closed fields which are regular in codimension one. Several applications,…

We apply the classical technique on cyclic objects of Alain Connes to various objects, in particular to the higher Chow complex of S. Bloch to prove a Connes periodicity long exact sequence involving motivic cohomology groups. The Cyclic…

Let $k$ be a field of characteristic zero, and let $X$ be a projective variety embedded into a projective space over $k$. For two natural numbers $r$ and $d$ let $C_{r,d}(X)$ be the Chow scheme parametrizing effective cycles of dimension…

Building on our previous work, we investigate an analogue of the differential symbol map used in the Bloch-Gabber-Kato theorem. Within this framework, for an appropriate variety over a field, the higher Chow group corresponds to the 0-th…

We give a description of the tensor product of SC-reciprocity presheaves with transfers in terms of $K$-group of geometric type, and we study a structure of the tensor product of $\mathbb{G}_a$ and $\mathbb{G}_a$. We apply our description…

Auel-Bigazzi-B\"ohning-Graf von Bothmer proved that if a proper smooth variety $X$ over a field $k$ of characteristic $p>0$ has universally trivial Chow group of $0$-cycles, the cohomological Brauer group of $X$ is universally trivial as…

We study the monodromy of vanishing cycles for map-germs $f:(C^{2n},0) \to (\CM^k,0)$ whose components are in involution. Although the singular fibres of such maps have non-isolated singularities, it is shown that the regular fibres are…