The complex projective line

Definition 1.1   The $n$-dimensional complex projective space ${\Bbb C}P^n$ is the set of all complex lines on ${\Bbb C}^{n+1}$ passing through the origin.

Notice that we may also define ${\Bbb C}P^n$ as the quotient of ${\Bbb C}^{n+1} \setminus \{0\}$ by the equivalence relation $v \sim w$ (or $[v] = [w]$) if $v$ and $w$ are complex multiples of each other.

If $(z_{1},\dots, z_{n+1})$ is a nonzero vector in ${\Bbb C}^{n+1}$ expressed in the canonical basis, we denote its equivalence class by $[z_{1}: \dots :z_{n+1}]$.

Since ${\Bbb C}^2$ is a four-dimensional vector space over the reals, it seems difficult at first sight to have an intuitive grasp of the complex projective line. The following exercises show that the complex projective line can be though of as the complex line ${\Bbb C}$ plus a point at infinity or as the unit sphere in ${\Bbb R}^3$.

Exercise 1.1 (00)   Show that the map from ${\Bbb C}P^{1} \setminus \{[1:0]\}$ to ${\Bbb C}$ defined by

$$[z_{1}:z_{2}] \longmapsto z_{1}/z_{2}$$

is a bijection.

Exercise 1.2 (10)   Let $F$ be the map that takes nonzero vectors in ${\Bbb C}^2$ to vectors in ${\Bbb R}^3$ by the following rule:

$$F(z_{1},z_{2}) := \left( \frac{z_{1}\bar{z}_{2} + \bar{z}_{1} z_... ...{z}_{1} - \bar{z}_{2} z_{2}}{z_{1}\bar{z}_{1} + \bar{z}_{2} z_{2}} \right) .$$

Show that $F$ defines a bijection between ${\Bbb C}P^1$ and the unit sphere in ${\Bbb R}^3$.

The relation between both representations of the complex projective line is given by the stereographic projection.

Exercise 1.3 (10)   Let $(x,y,z)$ be a point on $S^2$ different from the north pole $N := (0,0,1)$. Show that the line joining $(x,y,z)$ to the north pole intersects the $xy$-plane at the point $(1-z)^{-1}(x,y,0)$.

Definition 1.2   The stereographic projection is the map ${\cal S} : S^2 \setminus {N} \rightarrow {\Bbb C}$ defined by

$${\cal S}(x,y,z) := \frac{x}{1-z} + i \frac{y}{1-z}.$$

Exercise 1.4 (05)   Show that the inverse of the stereographic projection takes a complex number $u + iv$ to the point

$$\left(\frac{2u}{1 + u^2 + v^2}, \frac{2v}{1 + u^2 + v^2}, \frac{1 - u^2 - v^2}{1 + u^2 + v^2} \right)$$

on the unit sphere.

The sharp reader must have noted the similarity between the preceding formula and that of exercise 1.2. The precise relation is given by the following exercise:

Exercise 1.5 (05)   Let $[z_{1}: z_{2}]$ be a point on ${\Bbb C}P^1$ different from $[1:0]$. Show that ${\cal S}(F(z_{1},z_{2})) = z_{1}/z_{2}$.

The existence of these two geometric models for ${\Bbb C}P^1$ and their ties to complex numbers and spatial geometry is central to understanding the geometry of the complex projective line. In order to pass from one model to the other, we need to understand the properties of the stereographic projection.

Exercise 1.6 (20)   Show that the image of a circle on $S^2$ under the stereographic projection is either a circle or a straight line. Moreover, the image of a circle is a straight line if and only if the circle passes through the north pole.

Another important property of the stereographic projection is that it preserves angles. We will give a simple proof of this fact later on, but the reader is invited to verify this property in the simplest possible case:

Exercise 1.7 (05)   Show that the stereographic image of two curves on $S^2$ that intersect each other at the south pole with angle $\theta$ consists of two curves on the plane intersecting each other at the origin with angle $\theta$.