Here is the spell-out of Will's comment, that there is no free linear action of $\mathbb{Z}/n$ on $\mathbb{CP}^N$, for $1\leq N\leq\infty$. View $\mathbb{CP}^N$ as the set of 1-dimensional linear subspaces in a complex Hilbert space $\cal{H}$ of dimension $N+1$. Any linear action $\pi$ of $\mathbb{Z}/n$ on $\cal{H}$ decomposes as a sum of isotypic components: fix a primitive $n$-th root of unit $\omega\in\mathbb{C}$; then $\cal{H}=\bigoplus_{i=0}^{n-1}\cal{H}_i$, where $\cal{H}_i={v\in\cal{H}:\pi(k)v=\omega^{ik}v, \cal{H}_i=\{v\in\cal{H}:\pi(k)v=\omega^{ik}v, \forall k\in\mathbb{Z}/n}$k\in\mathbb{Z}/n\}$. Any one-dimensional subspace contained in some non-zero$\cal{H}_i$, will then be a fixed point for the corresponding action on$\mathbb{CP}^N$. 1 Here is the spell-out of Will's comment, that there is no free linear action of$\mathbb{Z}/n$on$\mathbb{CP}^N$, for$1\leq N\leq\infty$. View$\mathbb{CP}^N$as the set of 1-dimensional linear subspaces in a complex Hilbert space$\cal{H}$of dimension$N+1$. Any linear action$\pi$of$\mathbb{Z}/n$on$\cal{H}$decomposes as a sum of isotypic components: fix a primitive$n$-th root of unit$\omega\in\mathbb{C}$; then$\cal{H}=\bigoplus_{i=0}^{n-1}\cal{H}_i$, where$\cal{H}_i={v\in\cal{H}:\pi(k)v=\omega^{ik}v, \forall k\in\mathbb{Z}/n}$. Any one-dimensional subspace contained in some non-zero$\cal{H}_i$, will then be a fixed point for the corresponding action on$\mathbb{CP}^N\$.