  We study the coherent cohomology of generalized flag supervarieties. Our main
observation is that these groups are closely related to the free resolutions of
(certain generalizations of) determinantal ideals. In the case of super
Grassmannians, we completely compute the cohomology of the structure sheaf: it
is composed of the singular cohomology of a Grassmannian and the syzygies of a
determinantal variety. The majority of the work involves studying the geometry
of an analog of the Grothendieck-Springer resolution associated to the super
Grassmannian; this takes place in the world of ordinary (non-super) algebraic
geometry. Our work gives a conceptual explanation of the result of
Pragacz-Weyman that the syzygies of determinantal ideals admit an action of the
general linear supergroup. In a subsequent paper, we will treat other flag
supervarieties in detail.
