  The complexity of a module is the rate of growth of the minimal projective
resolution of the module while the $z$-complexity is the rate of growth of the
number of indecomposable summands at each step in the resolution. Let
$\mathfrak{g}=\mathfrak{osp}(k|2)$ ($k>2$) be the type II orthosymplectic Lie
superalgebra of types $B$ or $D$. In this paper, we compute the complexity and
the $z$-complexity of the simple finite-dimensional
$\mathfrak{g}$-supermodules. We then give geometric interpretations using
support and associated varieties for these complexities.
