  Quantum homogeneous supervector bundles arising from the quantum general
linear supergoup are studied. The space of holomorphic sections is promoted to
a left exact covariant functor from a category of modules over a quantum
parabolic sub-supergroup to the category of locally finite modules of the
quantum general linear supergroup. The right derived functors of this functor
provides a form of Dolbeault cohomology for quantum homogeneous supervector
bundles. We explicitly compute the cohomology groups, which are given in terms
of well understood modules over the quantized universal enveloping algebra of
the general linear superalgebra.
