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