Integral decomposition of the diagonal (Chow motives)

Let $k$ be a field of characteristic zero and let $X$ be a smooth proper varity over $k$ of dimension $d$. The Künneth standard conjecture conjectures that there exist projectors $e_0, e_1, \ldots, e_n$ in the Chow group $\mathrm{CH}^d(X\times X)\otimes \mathbb Q$ which sum up to the diagonal $\Delta \subseteq X\times X$, and induce the projections $H^\ast(X,F) \to H^i(X,F)$ in any Weil cohomology theory with coefficients in a field of characteristic zero $F$ (or any $\mathbb Q$-algebra $F$ for that matter).

Although still a conjecture in general it is known to be true for curves (trivially) for surfaces (Murre) and for abelian varieties (Shermenev, and Deninger-Murre). The constructions of these decompositions make essential use of the fact that we are working with $\mathrm{CH}^d(X\times X)\otimes \mathbb Q$ and not just $\mathrm{CH}^d(X\times X)$. For instance, an essential ingredient for the construction of the decomposition of the diagonal of an abelian variety is Fourier-Mukai transform, so there is somewhere an exponential with $\frac{1}{m!}$ coefficients.

Question: Is there any reason why the Künneth standard conjecture should fail with integral coefficients? That is, why should there not exist a decomposition of the diagonal in a sum of idempotents $e_i \in \mathrm{CH}^d(X\times X)$ inducing the projections $H^\ast(X,F) \to H^i(X,F)$ in any reasonable cohomology, say singular cohomology with $F=\mathbb Z$ and $\ell$-adic cohomology with $F = \mathbb Z_\ell$?

• I will write something longer if I get a chance, but one thing to say is that if $k$ is not algebraically closed there are simple counterexamples - take $X$ to be a conic with no points. – Tony Scholl Oct 23 '10 at 10:45
• True, I should have asked this for $k$ algebraically closed. Conic with no rational points = Conic with no 0-cycle of degree 1, so $h^0$ doesn't split off. – Xandi Tuni Oct 25 '10 at 9:23
• even over an algebraically closed field, this may not hold in general. See for instance, the examples of Janos Kollar: if $X$ is a general hypersurface of degree $3m^2$ in projective space (and $(6,m) =1)$), then the class of every curve on $X$ is $m$-times the class of a line. Kollar's result is referred to here www.math.ist.utl.pt/geometry09/kollar.pdf – SGP Apr 15 '11 at 23:51