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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

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2 Answers 2

up vote 4 down vote accepted

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.

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Thank you for your answer. Is your $X$ projective? –  unknown Sep 11 '11 at 17:44
    
Yes, I assume $X$ is projective and by motives I mean Chow motives. I am not sure how to formulate a meaningful decomposition statement in Voevodsky's category for arbitrary varieties. –  ulrich Sep 12 '11 at 5:37
    
I think I have something in mind. The answer definitely depends on what one wants to do with such a decomposition statement. But let me thank you very much for your competent answer. –  unknown Sep 15 '11 at 18:55
    
This argument works in any additive category where we have a similar formula. One can also consider Voevodsky's motives with rational coefficients, or Voevodsky-Suslin finite correspondences with rational coefficients modulo homotopy equivalence here. –  Mikhail Bondarko Apr 18 '13 at 17:10

Sebastian del Ba˜no. On the Chow motive of some moduli spaces. J. Reine Angew. Math., 532:105–132, 2001 contains a simple proof of this fact

It constructs sections to some motive morphisms geometrically. I believe the contribution of Brosnan paper above is to make the construction work when the decomposition is not defined over the base field.

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