On the configuration spaces of grassmannian manifolds

Let Fi h(k, n) be the ith ordered configuration space of all distinct points H1, . . . ,Hh in the Grassmannian Gr(k, n) of k-dimensional sub-spaces of Cn, whose sum is a subspace of dimension i. We prove that Fi h(k, n) 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 /= n and n > 2 and equal to the braid group of the sphere CP1 if n = 2. Eventually we compute the fundamental group in the special case of hyperplane arrangements, i.e. k = n - 1.