  We define a tautological projection operator for algebraic cycle classes on
the moduli space of principally polarized abelian varieties $\mathcal{A}_g$:
every cycle class decomposes canonically as a sum of a tautological and a
non-tautological part. The main new result required for the definition of the
projection operator is the vanishing of the top Chern class of the Hodge bundle
over the boundary $\bar{\mathcal{A}}_g\smallsetminus \mathcal{A}_g$ of any
toroidal compactification $\bar{\mathcal{A}}_g$ of the moduli space
$\mathcal{A}_g$. We prove the vanishing by a careful study of residues in the
boundary geometry.
  The existence of the projection operator raises many natural questions about
cycles on $\mathcal{A}_g$. We calculate the projections of all product cycles
$\mathcal{A}_{g_1}\times \ldots \times \mathcal{A}_{g_\ell}$ in terms of Schur
determinants, discuss Faber's earlier calculations related to the Torelli
locus, and state several open questions. The Appendix contains a conjecture
about the projection of the locus of abelian varieties with real
multiplication.
