Remarks on approximate decompositions of the diagonal

In this paper, we investigate, for varieties over $\mathbb{C}$ with trivial group of $0$-cycles, the gap between essential ${\mathrm{C}\mathrm{H}}_0$-dimension $2$ and essential ${\mathrm{C}\mathrm{H}}_0$-dimension $0$. In particular, we present sufficient (and necessary) conditions for a variety with trivial group of $0$-cycles and essential ${\mathrm{C}\mathrm{H}}_0$-dimension $\le 2$ to have, in fact, essential ${\mathrm{C}\mathrm{H}}_0$-dimension $0$.