  We investigate complex surfaces that fiber over Teichm\"uller curves where
the generic fiber is a Veech surface. When the fiber has genus one, these
surfaces are elliptic fibrations; for higher genus fibers, they are typically
minimal surfaces of general type. We compute the topological and
complex-geometric invariants of these surfaces via the monodromy action on the
mod-$m$ homology of the fiber. We get exact values of the invariants for all
known algebraically primitive Teichm\"uller curves.
