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**Definition 1.1**
The

-dimensional complex projective space

is the set of all
complex lines on

passing through the origin.

Notice that we may also define
as the quotient
of
by the equivalence relation
(or ) if and are *complex* multiples of
each other.

If
is a nonzero vector in
expressed in
the canonical basis, we denote its equivalence class by
.

Since
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 plus a *point at infinity*
or as the unit sphere in
.

**Exercise 1.1** (00)
Show that the map from

to

defined by

is a bijection.

**Exercise 1.2** (10)
Let

be the map that takes nonzero vectors in

to vectors
in

by the following rule:

Show that

defines a bijection between

and the unit sphere
in

.

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

**Exercise 1.3** (10)
Let

be a point on

different from the

*north pole*
. Show that the line joining

to the north pole
intersects the

-plane at the point

.

**Definition 1.2**
The

*stereographic projection* is the
map

defined by

**Exercise 1.4** (05)
Show that the inverse of the stereographic projection takes a complex
number

to the point

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

be a point on

different from

.
Show that

.

The existence of these two geometric models for
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

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

that intersect
each other at the south pole with angle

consists of two curves
on the plane intersecting each other at the origin with angle

.

** Next:** The action of on
** Up:** Geometry of the Complex
** Previous:** Geometry of the Complex
Juan Carlos Alvarez
2000-10-27