= Chow Künneth failure for an elliptic self-product
{c}
On $E\times E$ for a complex <elliptic curve>, the two factor curves and the diagonal have <intersection matrix> $\begin{pmatrix}0&1&1\\1&0&1\\1&1&0\end{pmatrix}$, of determinant two. Their cohomology classes are therefore independent. The degree-one image of the <external product of Chow classes> has cohomology contained in the span of the two factor classes. The <cycle class map> thus shows that the diagonal is outside that image, proving failure of surjectivity.
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