Positivity of the diagonal

We study how the geometry of a projective variety $X$ is reflected in the positivity properties of the diagonal ${\mathrm{\Delta }}_X$ considered as a cycle on $X \times X$. We analyze when the diagonal is big, when it is nef, and when it is rigid. In each case, we give several implications for the geometric properties of $X$. For example, when the diagonal is big, we prove that the Hodge groups $H^{k,0}(X)$ vanish for $k>0$. We also classify varieties of low dimension where the diagonal is nef and big.