  Let $\mathfrak{g}=\mathfrak{g}_{\bar{0}}\oplus \mathfrak{g}_{\bar{1}}$ be a
classical Lie superalgebra and $\mathcal{F}$ be the category of finite
dimensional $\mathfrak{g}$-supermodules which are completely reducible over the
reductive Lie algebra $\mathfrak{g}_{\bar{0}}$. In an earlier paper the authors
demonstrated that for any module $M$ in $\mathcal{F}$ the rate of growth of the
minimal projective resolution (i.e., the complexity of $M$) is bounded by the
dimension of $\mathfrak{g}_{\bar{1}}$. In this paper we compute the complexity
of the simple modules and the Kac modules for the Lie superalgebra
$\mathfrak{gl}(m|n)$. In both cases we show that the complexity is related to
the atypicality of the block containing the module.
