  In this paper the authors study the behavior of the sheaf cohomology functors
$R^{\bullet}\text{ind}_{B}^{G}(-)$ where $G$ is an algebraic group scheme
corresponding to a simple classical Lie superalgebra and $B$ is a BBW parabolic
subgroup as defined by D. Grantcharov, N. Grantcharov, Nakano and Wu. We
provide a systematic treatment that allows us to study the behavior of these
cohomology groups $R^{\bullet}\text{ind}_{B}^{G}L_{\mathfrak f}(\lambda)$ where
$L_{\mathfrak f}(\lambda)$ is an irreducible representation for the detecting
subalgebra ${\mathfrak f}$. In particular, we prove an analog of Kempf's
vanishing theorem and the Bott-Borel-Weil theorem for large weights.
