  We bound the length of the periodic part of the orbit of a preperiodic
rational subvariety via good reduction information. This bound depends only on
the degree of the map, the degree of the subvariety, the dimension of the
projective space, the degree of the number field, and the prime of good
reduction. As part of the proof, we extend the corresponding good reduction
bound for points proven by the author for non-singular varieties to all
projective varieties. Toward proving an absolute bound on the period for a
given map, we study the canonical height of a subvariety via Chow forms and
compute the bound between the height and canonical height of a subvariety. This
gives the existence of a bound on the number of preperiodic rational
subvarieties of bounded degree for a given map. An explicit bound is given for
hypersurfaces.
