Complex Projective Space - Introduction

Introduction

The notion of a projective plane arises out of the idea of perspective in geometry and art: that it is sometimes useful to include in the Euclidean plane an additional "imaginary" line that represents the horizon that an artist painting the plane might see. Following each direction from the origin, there is a different point on the horizon, so the horizon can be thought of as the set of all directions from the origin. The Euclidean plane, together with its horizon, is called the real projective plane, and the horizon is sometimes called a line at infinity. By the same construction, projective spaces can be considered in higher dimensions. For instance, the real projective 3-space is a Euclidean space together with a plane at infinity that represents the horizon that an artist (who must, necessarily, live in four dimensions) would see.

These real projective spaces can be constructed in a slightly more rigorous way as follows. Here, let Rn+1 denote the real coordinate space of n+1 dimensions, and regard the landscape to be painted as a hyperplane in this space. Suppose that the eye of the artist is the origin in Rn+1. Then along each line through his eye, there is a point of the landscape or a point on its horizon. Thus the real projective space is the space of lines through the origin in Rn+1. Without reference to coordinates, this is the space of lines through the origin in an (n+1)-dimensional real vector space.

To describe the complex projective space in an analogous manner requires a generalization of the idea of vector, line, and direction. Imagine that instead of standing in a real Euclidean space, the artist is standing in a complex Euclidean space Cn+1 (which has real dimension 2n+2) and the landscape is a complex hyperplane (of real dimension 2n). Unlike the case of real Euclidean space, in the complex case there are directions in which the artist can look which do not see the landscape (because it does not have high enough dimension). However, in a complex space, there is an additional "phase" associated with the directions through a point, and by adjusting this phase the artist can guarantee that he typically sees the landscape. The "horizon" is then the space of directions, but such that two directions are regarded as "the same" if they differ only by a phase. The complex projective space is then the landscape (Cn) with the horizon attached "at infinity". Just like the real case, the complex projective space is the space of directions through the origin of Cn+1, where two directions are regarded as the same if they differ by a phase.