  On an orthogonal Shimura variety, one has a collection of special cycles in
the Gillet-Soule arithmetic Chow group. We describe how these cycles behave
under pullback to an embedded orthogonal Shimura variety of lower dimension.
The bulk of the paper is devoted to cases in which the special cycles intersect
the embedded Shimura variety improperly, for which we construct logarithmic
expansions of certain Green currents on the deformation to the normal bundle of
the embedding.
