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.
