Let $X$ be a smooth (quasi-affine) complex variety; suppose that its cohomology (say, with integral coefficients) is trivial in degrees $0 < i\le s$ (for some $s>0$). What can one say about such varieties; are there any 'classification' results? In particular, could one say something about the lower motivic cohomology/Chow groups of $X$?
$\begingroup$
$\endgroup$
5
asked Feb 26, 2012 at 13:06
-
$\begingroup$ Do you assume $s < \mathrm{dim}(X)$ or something like that? $\endgroup$– Martin BrandenburgCommented Feb 26, 2012 at 14:07
-
$\begingroup$ Yes; unfortunately.:) Certainly, if the dimension is too small, then $X$ could be replaced by $X\times \mathbb{A}^1$ without changing $s$. $\endgroup$– Mikhail BondarkoCommented Feb 26, 2012 at 14:14
-
$\begingroup$ About "classification": even if one assumes that the motive of $X$ is the same as the motive of an affine space, I don't think there is a reasonable classification. Also, there exist varieties whose motive is that of an affine space, but its Picard group is non-trivial. $\endgroup$– Alexander BravermanCommented Feb 26, 2012 at 16:39
-
$\begingroup$ Are you sure? I have always believed that the Picard group of a motif (of a variety) $M$ is just $Hom_{DM}(M,Z(1)[2])$. $\endgroup$– Mikhail BondarkoCommented Feb 26, 2012 at 17:38
-
$\begingroup$ And yes, I am interested in 'motivic' information on $X$.:) $\endgroup$– Mikhail BondarkoCommented Feb 26, 2012 at 19:57
Add a comment
|