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Search: MSC category 14C25 ( Algebraic cycles )

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1. CMB Online first

Laterveer, Robert
A brief note concerning hard Lefschetz for Chow groups
We formulate a conjectural hard Lefschetz property for Chow groups, and prove this in some special cases: roughly speaking, for varieties with finite-dimensional motive, and for varieties whose self-product has vanishing middle-dimensional Griffiths group. An appendix includes related statements that follow from results of Vial.

Keywords:algebraic cycles, Chow groups, finite-dimensional motives
Categories:14C15, 14C25, 14C30

2. CMB Online first

Kang, Su-Jeong
Refined Motivic Dimension
We define a refined motivic dimension for an algebraic variety by modifying the definition of motivic dimension by Arapura. We apply this to check and recheck the generalized Hodge conjecture for certain varieties, such as uniruled, rationally connected varieties and a rational surface fibration.

Keywords:motivic dimension, generalized Hodge conjecture
Categories:14C30, 14C25

3. CMB 2012 (vol 56 pp. 640)

Türkmen, İnan Utku
Regulator Indecomposable Cycles on a Product of Elliptic Curves
We provide a novel proof of the existence of regulator indecomposables in the cycle group $CH^2(X,1)$, where $X$ is a sufficiently general product of two elliptic curves. In particular, the nature of our proof provides an illustration of Beilinson rigidity.

Keywords:real regulator, regulator indecomposable, higher Chow group, indecomposable cycle

4. CMB 2008 (vol 51 pp. 283)

Ravindra, G. V.
The Noether--Lefschetz Theorem Via Vanishing of Coherent Cohomology
We prove that for a generic hypersurface in $\mathbb P^{2n+1}$ of degree at least $2+2/n$, the $n$-th Picard number is one. The proof is algebraic in nature and follows from certain coherent cohomology vanishing.

Keywords:Noether--Lefschetz, algebraic cycles, Picard number
Categories:14C15, 14C25

5. CMB 2006 (vol 49 pp. 464)

Ravindra, G. V.
A Note on Detecting Algebraic Cycles
The purpose of this note is to show that the homologically trivial cycles contructed by Clemens and their generalisations due to Paranjape can be detected by the technique of spreading out. More precisely, we associate to these cycles (and the ambient varieties in which they live) certain families which arise naturally and which are defined over $\bbC$ and show that these cycles, along with their relations, can be detected in the singular cohomology of the total space of these families.


6. CMB 2005 (vol 48 pp. 237)

Kimura, Kenichiro
Indecomposable Higher Chow Cycles
Let $X$ be a projective smooth variety over a field $k$. In the first part we show that an indecomposable element in $CH^2(X,1)$ can be lifted to an indecomposable element in $CH^3(X_K,2)$ where $K$ is the function field of 1 variable over $k$. We also show that if $X$ is the self-product of an elliptic curve over $\Q$ then the $\Q$-vector space of indecomposable cycles $CH^3_{ind}(X_\C,2)_\Q$ is infinite dimensional. In the second part we give a new definition of the group of indecomposable cycles of $CH^3(X,2)$ and give an example of non-torsion cycle in this group.

Categories:14C25, 19D45

7. CMB 2002 (vol 45 pp. 204)

Fakhruddin, Najmuddin
On the Chow Groups of Supersingular Varieties
We compute the rational Chow groups of supersingular abelian varieties and some other related varieties, such as supersingular Fermat varieties and supersingular $K3$ surfaces. These computations are concordant with the conjectural relationship, for a smooth projective variety, between the structure of Chow groups and the coniveau filtration on the cohomology.

Categories:14C25, 14K99

8. CMB 2002 (vol 45 pp. 213)

Gordon, B. Brent; Joshi, Kirti
Griffiths Groups of Supersingular Abelian Varieties
The Griffiths group $\Gr^r(X)$ of a smooth projective variety $X$ over an algebraically closed field is defined to be the group of homologically trivial algebraic cycles of codimension $r$ on $X$ modulo the subgroup of algebraically trivial algebraic cycles. The main result of this paper is that the Griffiths group $\Gr^2 (A_{\bar{k}})$ of a supersingular abelian variety $A_{\bar{k}}$ over the algebraic closure of a finite field of characteristic $p$ is at most a $p$-primary torsion group. As a corollary the same conclusion holds for supersingular Fermat threefolds. In contrast, using methods of C.~Schoen it is also shown that if the Tate conjecture is valid for all smooth projective surfaces and all finite extensions of the finite ground field $k$ of characteristic $p>2$, then the Griffiths group of any ordinary abelian threefold $A_{\bar{k}}$ over the algebraic closure of $k$ is non-trivial; in fact, for all but a finite number of primes $\ell\ne p$ it is the case that $\Gr^2 (A_{\bar{k}}) \otimes \Z_\ell \neq 0$.

Keywords:Griffiths group, Beauville conjecture, supersingular Abelian variety, Chow group
Categories:14J20, 14C25

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