Cohomology of the structure sheaf of real and complex supermanifolds

The cohomology of the structure sheaf of real and complex supermanifolds is studied. It is found to be nontrivial (also in the real case), unless the supermanifold is De Witt, i.e., it is a fiber bundle on an ordinary manifold with a vector fiber. As a consequence, the Dolbeault theorem can be extended only to complex De Witt supermanifolds. The relevance of the nontrivial cohomology of the structure sheaf to the classification of complex line superbundles is discussed. The relationship between the Picard group of a complex De Witt supermanifold and the Picard group of its body is shown.