Čech and de Rham cohomology of integral forms

We present a study on the integral forms and their Čech and de Rham cohomology. We analyze the problem from a general perspective of sheaf theory and we explore examples in superprojective manifolds. Integral forms are fundamental in the theory of integration in a supermanifold. One can define the integral forms introducing a new sheaf containing, among other objects, the new basic forms δ(dθ)δ(dθ) where the symbol δδ has the usual formal properties of Dirac’s delta distribution and acts on functions and forms as a Dirac measure. They satisfy in addition some new relations on the sheaf. It turns out that the enlarged sheaf of integral and “ordinary” superforms contains also forms of “negative degree” and, moreover, due to the additional relations introduced it is, in a non trivial way, different from the usual superform cohomology.