Dear community,


in his 2005 Inventiones Paper "On motivic decompositions arising from the method
of Białynicki-Birula" P. Brosnan deduced from the classical (?) theorem of Bialynicki-Birula on decomposition of smooth projective varieties with $\mathbb{C}^\*$-action as a cellular variety by invoking a theorem of Karpenko (Theorem 3.1 in Brosnans paper) that the decomposition arising from Bialynicki-Birula also yields a decomposition of the corresponding motive (in the category of Chow-Motives) as a direct sum.


An expert in the field told me that the difficulty in Karpenkos theorem is mainly due to the fact that the motives are considered as motives with integral coefficients and a similar decomposition in the corresponding category with rational coefficients would be much easier (and in fact wouldn't even need the assumption that the variety is projective, perhaps if the motives are viewed in Voevodskys category).


My questions are: Can someone provide a reference (if the answer is not easy and short) or a sketch of a proof for that assertion? If there is an obvious reason why this statement isn't true I would very much welcome a comprehensive explanation.


Thank you all in advance



Decomposition of Motives of cellular varieties
The varieties considered by Brosnan and Karpenko are not cellular over the base field but become cellular over the algebraic closure. For a cellular variety over any field the Chow groups are freely generated by the closures of the cells and are "equal" to the cohomology. A Kunneth type formula also holds so the motive of a cellular variety is a direct sum of Tate motives.


Now if $X$ is a variety over a field $k$ such that $X\_{\bar k}$ is cellular, then the
$Gal({\bar k}/k)$ orbits of the projectors giving the decomposition of the motive of $X\_{\bar k}$ give rise to projectors over $k$ (i.e. take the sum over the orbit) if we use $\mathbb{Q}$ coefficients; this is because 
the map
$$CH^\*(X \times X) \otimes \mathbb{Q} \to (CH^\*((X\times X)\_{\bar k}) \otimes \mathbb{Q})^{Gal({\bar k}/k)}$$


is an isomorphism. 


These projectors give the decomposition of the motive of $X$. The summands in this decomposition are Artin-Tate motives i.e. over $\bar k$ they become sums of Tate motives.


