Spaces of geometrically generic configurations

Abstract

Let $X$ denote either ${\mathbb{CP}}^m$ or ${\mathbb{C}}^m$. We study certain analytic properties of the space ${\mathcal{E}}^n(X,gp)$ of ordered geometrically generic $n$-point configurations in $X$. This space consists of all $q=(q_1,...,q_n)\in X^n$ such that no $m+1$ of the points $q_1,...,q_n$ belong to a hyperplane in $X$. In particular, we show that for a big enough $n$ any holomorphic map $f:{\mathcal{E}}^n({\mathbb{CP}}^m,gp)\to {\mathcal{E}}^n({\mathbb{CP}}^m,gp)$ commuting with the natural action of the symmetric group $\mathbf{S}(n)$ in ${\mathcal{E}}^n({\mathbb{CP}}^m,gp)$ is of the form $f(q)=\tau (q)q=(\tau (q)q_1,...,\tau (q)q_n)$, $q\in {\mathcal{E}}^n({\mathbb{CP}}^m,gp)$, where $\tau :{\mathcal{E}}^n({\mathbb{CP}}^m,gp)\to \mathbf{PSL}(m+1,\mathbb{C})$ is an $\mathbf{S}(n)$-invariant holomorphic map. A similar result holds true for mappings of the configuration space ${\mathcal{E}}^n({\mathbb{C}}^m,gp)$.