Fundamental groups of spaces of holomorphic maps and group actions

Abstract

For integers $d \geq 0$ and $1 \neq k \neq n$, let $\mathrm{Hol}_{d}(\mathbb{C}P^{k},\mathbb{C}P^{n})$ denote the space consisting of all holomorphic maps $f : \mathbb{C}P^{k}\to \mathbb{C}P^{n}$ of degree $d$. We shall compute the fundamental group of $\mathrm{Hol}_{d}(\mathbb{C}P^{k},\mathbb{C}P^{n})$ and study $\mathrm{PGL}_{n+1}(\mathbb{C})$-action on it.