Cohomology of complex supertori

We consider supertori which are quotients of affine superspace by translations by algebraically independent odd parameters. Specifically, we describe the ring structure of its space of global sections by generators and relations and completely determine the coherent cohomology groups of its structure sheaf by reducing it to a problem of Lie algebra cohomology. We also show that Poincaré duality on sheaf cohomology is compatible with that of the group cohomology of the translation group and give explicit tables of examples. Finally, we compute the Picard groups of these supertori.