Pieri-type intersection formulas and primary obstructions for decomposing 2-forms

We study the homological intersection behaviour for the Chern cells of the universal bundle of G(d,Qn), the space of [d]-planes in the smooth quadric Qn in ℙn+1 over the field of complex numbers. For this purpose we define some auxiliary cells in terms of which the intersection properties of the Chern cells can be described. This is then applied to obtain some new necessary conditions for the global decomposability of a 2-form of constant rank.