# What can one say about a smooth variety whose lower cohomology is trivial?

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 $X$?

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Do you assume $s < \mathrm{dim}(X)$ or something like that? –  Martin Brandenburg Feb 26 '12 at 14:07
Yes; unfortunately.:) Certainly, if the dimension is too small, then $X$ could be replaced by $X\times \mathbb{A}^1$ without changing $s$. –  Mikhail Bondarko Feb 26 '12 at 14:14
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. –  Alexander Braverman Feb 26 '12 at 16:39
Are you sure? I have always believed that the Picard group of a motif (of a variety) $M$ is just $Hom_{DM}(M,Z(1)[2])$. –  Mikhail Bondarko Feb 26 '12 at 17:38
And yes, I am interested in 'motivic' information on $X$.:) –  Mikhail Bondarko Feb 26 '12 at 19:57