In compactifications over smooth geometrical spaces, closed differential forms can take on a prominent role. For instance, closed forms can represent the geometrical structure of special holonomy manifolds and also fluxes that are present in the compactifications. In this talk, we will describe novel geometrical invariants that arise on manifolds with a distinguished closed form. In particular, we will show that there are natural cohomologies of mapping cone type that in general are dependent on the distinguished closed form. These cohomologies provide another tool to help count the massless scalars that arise in compactifications.