On the Configuration Spaces of Grassmannian Manifolds

Abstract

Let $\mathcal{F}_h^i(k,n)$ be the $i$th ordered configuration space of all distinct points $H_1,\ldots,H_h$ in the Grassmannian $Gr(k,n)$ of $k$-dimensional subspaces of $\mc^n$, whose sum is a subspace of dimension $i$. We prove that $\mathcal{F}_h^i(k,n)$ is (when non empty) a complex sub\-ma\-ni\-fold of $Gr(k,n)^h$ of dimension $i(n-i)+hk(i-k)$ and its fundamental group is trivial if $i=min(n,hk)$, $hk \neq n$ and $n>2$ and equal to the braid group of the sphere $\mc P^1$ if $n=2$. Eventually we compute the fundamental group in the special case of hyperplane arrangements, i.e. $k=n-1$.