Holomorphic vector eld with one zero on the Grassmannian and cohomology

We consider a holomorphic vector eld on the complex Grassman- nian constructed from a nilpotent matrix. We show that this vector eld van- ishes only at a single point. Using the Baum-Bott localization theorem we give a Grothendieck residue formula for the intersection numbers of the Grassmannian. Knowing that Chern classes of the tautological bundle generate the cohomology ring of the Grassmannian we can compute the ideal of relations explicitly from the residue formula. This shows that the cohomology ring of the Grassmannian is determined by holomorphic vector eld around its only zero.