arXiv subjects

Charles Vial

Publications and source records attributed to Charles Vial.

43 records · Page 3Linked to original sources

arXiv2013

Non-commutative resolutions and Grothendieck groups

Let $R$ be a noetherian normal domain. We investigate when $R$ admits a faithful module whose endomorphism ring has finite global dimension. This can be viewed as a non-commutative desingularization of $\Spec(R)$. We show that the existence of such modules forces stringent conditions on the Grothendieck group of finitely generated modules over $R$. In some cases those conditions are enough to imply that $\Spec(R)$ has only rational singularities.

math.AC↗

arXiv2012

Chow-Kuenneth decomposition for 3- and 4-folds fibred by varieties with small Chow group of zero-cycles

Let $k$ be a field and let $Ω$ be a universal domain over $k$. Let $f:X \r S$ be a dominant morphism defined over $k$ from a smooth projective variety $X$ to a smooth projective variety $S$ of dimension $\leq 2$ such that the general fibre of $f_Ω$ has trivial Chow group of zero-cycles. For example, $X$ could be the total space of a two-dimensional family of varieties whose general member is rationally connected. Suppose that $X$ has dimension $\leq 4$. Then we prove that $X$ has a self-dual Murre decomposition, i.e. that $X$ has a self-dual Chow--Kuenneth decomposition which satisfies Murre's conjectures (B) and (D). Moreover we prove that the motivic Lefschetz conjecture holds for $X$ and hence so does the Lefschetz standard conjecture. We also give new examples of threefolds of general type which are Kimura finite-dimensional, new examples of fourfolds of general type having a self-dual Murre decomposition, as well as new examples of varieties with finite degree three unramified cohomology.

math.AG↗

arXiv2012

Niveau and coniveau filtrations on cohomology groups and Chow groups

The Bloch-Beilinson-Murre conjectures predict the existence of a descending filtration on Chow groups of smooth projective varieties which is functorial with respect to the action of correspondences and whose graded parts depend solely on the topology -- i.e. the cohomology -- of $X$. In this paper, we wish to explore, at the cost of having to assume general conjectures about algebraic cycles, how the coniveau filtration on the cohomology of $X$ has an incidence on the Chow groups of $X$. However, by keeping such assumptions minimal, we are able to prove some of these conjectures either in low-dimensional cases or when a variety is known to have small Chow groups. For instance, we give a new example of a fourfold of general type with trivial Chow group of zero-cycles and we prove Murre's conjectures for threefolds dominated by a product of curves, for threefolds rationally dominated by the product of three curves, for rationally connected fourfolds and for complete intersections of low degree. The BBM conjectures are closely related to Kimura-O'Sullivan's notion of finite-dimensionality. Assuming the standard conjectures on algebraic cycles the former is known to imply the latter. We show that the missing ingredient for finite-dimensionality to imply the BBM conjectures is the coincidence of a certain niveau filtration with the coniveau filtration on Chow groups.

math.AG↗

arXiv2012

Chow groups of smooth varieties fibred by quadrics

Let $f : X \rightarrow B$ be a proper flat dominant morphism between two smooth quasi-projective complex varieties $X$ and $B$. Assume that there exists an integer $l$ such that all closed fibres $X_b$ of $f$ satisfy $CH_j(X_b) = \Q$ for all $j \leq l$. Then we prove an analogue of the projective bundle formula for $CH_i(X)$ for $i \leq l$. When $B$ is a surface, $X$ is projective and $l = \lfloor \frac{\dim X - 3}{2} \rfloor$, this makes it possible to construct a Chow-Künneth decomposition for $X$ that satisfies Murre's conjectures. For instance we prove Murre's conjectures for complex smooth projective varieties $X$ fibred over a surface (via a flat morphism) by quadrics, or by complete intersections of dimension 4 of bidegree $(2,2)$.

math.AG↗

arXiv2011

Projectors on the intermediate algebraic Jacobians

Let $X$ be a complex smooth projective variety of dimension $d$. Under some assumption on the cohomology of $X$, we construct mutually orthogonal idempotents in $CH_d(X \times X) \otimes \Q$ whose action on algebraically trivial cycles coincides with the Abel-Jacobi map. Such a construction generalizes Murre's construction of the Albanese and Picard idempotents and makes it possible to give new examples of varieties admitting a self-dual Chow-Künneth decomposition satisfying the motivic Lefschetz conjecture as well as new examples of varieties having a Kimura finite dimensional Chow motive. For instance, we prove that fourfolds with Chow group of zero-cycles supported on a curve (e.g. rationally connected fourfolds) have a self-dual Chow-Künneth decomposition which satisfies the motivic Lefschetz conjecture and consequently Grothendieck's standard conjectures. We also prove that hypersurfaces of very low degree are Kimura finite dimensional.

math.AG↗

arXiv2011

Pure motives with representable Chow groups

Let $k$ be an algebraically closed field. We show using Kahn's and Sujatha's theory of birational motives that a Chow motive over $k$ whose Chow groups are all representable belongs to the full and thick subcategory of motives generated by the twisted motives of curves. -- Motifs purs dont les groupes de Chow sont représentables. Soit $k$ un corps algébriquement clos. Nous prouvons, en nous servant de la théorie des motifs birationnels développée par Kahn et Sujatha, qu'un motif de Chow défini sur $k$ dont les groupes de Chow sont tous représentables appartient à la sous-catégorie pleine et épaisse des motifs engendrée par les motifs de courbes tordus.

math.AG↗

arXiv2008

Operations in Milnor K-theory

We show that operations in Milnor K-theory mod $p$ of a field are spanned by divided power operations. After giving an explicit formula for divided power operations and extending them to some new cases, we determine for all fields $k$ and all prime numbers $p$, all the operations $K^M_i/p \to K^M_j/p$ commuting with field extensions over the base field $k$. Moreover, the integral case is discussed and we determine the operations $K^M_i/p \to K^M_j/p$ for smooth schemes over a field.

math.KT↗