  Asymptotic syzygies of a normal crossing variety follow the same vanishing
behavior as one of its smooth components, unless there is a cohomological
obstruction arising from how the smooth components intersect each other. In
that case, we compute the asymptotic syzygies in terms of the cohomology of the
simplicial complex associated to the normal crossing variety.
  We combine our results on normal crossing varieties with knowledge of
degenerations of certain smooth projective varieties to obtain some results on
asymptotic syzygies of those smooth projective varieties.
