  Boe, Kujawa and Nakano recently investigated relative cohomology for
classical Lie superalgebras and developed a theory of support varieties. The
dimensions of these support varieties give a geometric interpretation of the
combinatorial notions of defect and atypicality due to Kac, Wakimoto, and
Serganova. In this paper we calculate the cohomology ring of the Cartan type
Lie superalgebra W(n) relative to the degree zero component W(n)_{0} and show
that this ring is a finitely generated polynomial ring. This allows one to
define support varieties for finite dimensional W(n)-supermodules which are
completely reducible over W(n)_{0}. We calculate the support varieties of all
simple supermodules in this category. Remarkably our computations coincide with
the prior notion of atypicality for Cartan type superalgebras due to Serganova.
We also present new results on the realizability of support varieties which
hold for both classical and Cartan type superalgebras.
