Cohomology of supermanifolds

Abstract

The aim of this Chapter is to unfold a basic cohomological theory for supermanifolds, which will be exploited in the next Chapter to study the structure of superbundles; in particular to build a theory of characteristic classes. This cohomology theory does not embody only trivial extensions of results valid for differentiable manifolds. For instance, the natural analogue of the de Rham theorem does not hold in general and, similarly, in the case of complex supermanifolds there is, generally speaking, no analogue of the Dolbeault theorem. These features are consequences of the fact that the structure sheaf of a supermanifold does not need to be cohomologically trivial. Related to this is also the fact that the cohomology of the complex of global graded differential forms on a G-supermanifold (M, A) (i.e. the’ super de Rham cohomology’ of (M, A)) depends on the G-supermanifold structure of (M, A), so that homeomorphic and even smoothly diffeomorphic G-supermanifolds may have a different super de Rham cohomology; that is, super de Rham cohomology is a fine invariant of the supermanifold structure.

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