  We prove that all points of a toroidal compactification lying over
0-dimensional cusps are rationally equivalent in the integral Chow group for
most classical modular varieties (Siegel, Hilbert, orthogonal, Hermitian,
quaternionic). This gives a generalization, and even strengthening, of the
Manin-Drinfeld theorem in higher dimension from the viewpoint of algebraic
cycles. The same result no longer holds for Picard modular varieties, but for
them we prove that the difference of any two "special" boundary points, which
are dense in the boundary, is torsion.
