  The moduli space of principally polarized abelian varieties $A_g$ of genus g
is defined over the integers and admits a minimal compactification $A_g^*$,
also defined over the integers. The Hodge bundle over $A_g$ has its Chern
classes in the Chow ring of $A_g$ with rational coefficients. We show that over
the prime field $F_p$, these Chern classes naturally lift to $A_g^*$ and do so
in the best possible way: despite the highly singular nature of $A_g^*$ they
are represented by algebraic cycles on $A_g^*\otimes F_p$ which define elements
in its bivariant Chow ring. This is in contrast to the situation in the
analytic topology, where these Chern classes have canonical lifts to the
complex cohomology of the minimal compactification as Goresky-Pardon classes,
which are known to define nontrivial Tate extensions inside the mixed Hodge
structure on this cohomology.
