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1. CMB Online first
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 |
2. CMB 2012 (vol 56 pp. 640)
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 Category:14C25 |
3. CMB 2008 (vol 51 pp. 283)
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 |
4. CMB 2006 (vol 49 pp. 464)
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.
Category:14C25 |
5. CMB 2005 (vol 48 pp. 237)
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 |
6. CMB 2002 (vol 45 pp. 204)
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 |
7. CMB 2002 (vol 45 pp. 213)
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 |