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Sandro Manfredini; Simona Settepanella
On the Configuration Spaces of Grassmannian Manifolds
Annales de la faculté des sciences de Toulouse Sér. 6, 23 no. 2: Numéro Spécial à l’occasion de la conférence Arrangements in Pyrénées, Pau 11-15 juin 2012 (2014), p. 353-359, doi: 10.5802/afst.1410
Article PDF | Reviews MR 3205597 | Zbl 06297896

Résumé - 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 $\scriptstyle {\mathbb{C}}^n$, whose sum is a subspace of dimension $i$. We prove that ${\mathcal{F}}_h^i(k,n)$ 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 $i=min(n,hk)$, $hk \ne n$ and $n>2$ and equal to the braid group of the sphere $\scriptstyle {\mathbb{C}}$$P^1$ if $n=2$. Eventually we compute the fundamental group in the special case of hyperplane arrangements, i.e. $k=n-1$.

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