  We show that the cohomology of the perfect cone (also called first Voronoi)
toroidal compactification of the moduli space of complex principally polarized
abelian varieties stabilizes, in close to the top degree. Moreover, we show
that this stable cohomology is purely algebraic, and we compute it in degree up
to 13. Our explicit computations and stabilization results apply in greater
generality to various toroidal compactifications and partial compactifications,
and in particular we show that the cohomology of the matroidal partial
compactification stabilizes (in low degree). For degree up to 8, we describe
explicitly the generators of the cohomology. We also discuss various approaches
to computing all of the stable cohomology in arbitrary degree.
