Study on Chohomology of supermanifolds

Abstract

In this present paper de Rham cohomology of graded manifolds, cohomology of graded differential forms and cohomology of DeWitt supermanifolds are studied. The structure of a G-supermanifold in general is not acyclic. So, the cohomology is defined via the complex of graded differential forms. This situation is investigated together with the graded Dolbeault cohomology of complex G-supermanifolds. A theorem is established that the structure sheaf of any DeWitt G-supermanifold is acyclic.