External product of Chow classes
= External product of Chow classes
{title2=$\alpha\boxtimes\beta=\pi_X^*\alpha\cdot\pi_Y^*\beta$}
For smooth projective varieties $X,Y$, pull back <Chow classes> from the two factors and take their <intersection product> on $X\times Y$. This gives the external product and a graded map $\operatorname{CH}^*(X)\otimes\operatorname{CH}^*(Y)\to\operatorname{CH}^*(X\times Y)$. Unlike the cohomological <Künneth theorem>, it need not generate every class of the product.