  We construct an affine stratification for $\mathcal A_4(\mathbb C)$ of length
6, in the sense of Roth and Vakil, which gives us an upper bound of 6 for the
cohomological dimension of $\mathcal A_4(\mathbb C)$. We conjecture that in
general for arbitrary $g$ the cohomological dimension of $\mathcal A_g$ is
equal to $g(g-1)/2$.
