On the Configuration Spaces of Grassmannian Manifolds

Let's define the i-th ordered configuration space as the space of all distinct points H1,...,Hh in the complex Grassmannian Gr(k,n) whose sum is a subspace of dimension i. We prove that this space is (when non empty) a complex submanifold of Gr(k,n)h of dimension i(n-i)+hk(i-k) and its fundamental group is trivial if i=min(n,hk), hk is different from n and n>2 and equal to the braid group of the sphere if n=2. Eventually we compute the fundamental group in the special case of hyperplane arrangements, i.e. k=n-1.