I would like that someone helps me to find an article on the net treating the following decomposition : $ \mathrm{CH}^k (X)\_{ \mathbb{Q} } = \displaystyle\bigoplus\_{ i + j = k } \mathrm{CH}^{i,j} (X) $. I don't remember exactly where i found that article on the net. Can anyone help me please ?


Thanks a lot to all of you.



About the decomposition of a Chow group of a variety
abx has basically answered your question in a comment, but let me add some detail. There is no natural such direct sum decomposition; you are probably thinking of the Bloch-Beilinson *filtration* on Chow groups, whose existence in general is only conjectural. A very nice reference is a paper by Jannsen in one of the "Motives" volumes. 


There are some examples where the conjectural filtration is expected to split, so that one really gets a direct sum decomposition. If $X$ is an abelian variety, then there is the Beauville decomposition of the Chow ring, which is defined unconditionally using the Fourier transform. It gives rise to a graduation (as opposed to a filtration) on Chow groups, satisfying all expected properties of the BB filtration except a conjectural vanishing property. 


A conjecture of Beauville predicts that for any hyperkähler variety $X$, the Bloch-Beilinson filtration on the Chow groups of $X$ splits. Here a direct sum decomposition satisfying all expected properties was constructed by Beauville and Voisin in the case of a K3 surface.


