Summary: In this paper we compute the cohomology groups of the second Voronoi and the perfect cone compactification $\AVOR 4$ and $\APERF 4$ respectively, of the moduli space of abelian fourfolds in degree $<= 9$. The main tool is the investigation of the strata of $\AVOR 4$ corresponding to semi-abelic varieties with constant torus rank.

14K10, 14F25, 14C15, 14D22

abelian varieties, Voronoi compactification, perfect cone compactification, Chow ring, cohomology ring