We study some global aspects of differential complex 2-forms and 3-forms on complex manifolds. We compute the cohomology classes represented by the sets of points on a manifold where such a form degenerates in various senses, together with other similar cohomological obstructions. Based on these results and a formula for projective representations, we calculate the degree of the projectivization of certain orbits of the representation $\Lambda^k\C^n$.