# Math 401, Fall 2025: Thesis notes, S1, Complex projective space.

> [!CAUTION]
>
> In this section, without explicitly stated, all dimensions are in the complex field.

A complex projective space is a space that is the set of all lines through the origin in a complex vector space.

Described by that nature, there exists a natural definition of the complex projective space given as follows:

$$
\mathbb{C}P^n=\frac{\mathbb{C}^{n+1}\setminus\{0\}}{\sim}
$$

By this nature of ray-like properties, we can also describe the complex projective space as follows (in the math of QT, lecture 5)

$$
\mathbb{C}P^n=\left\{z=(z_0,z_1,\cdots,z_n)\in\mathbb{C}^{n+1}:|z_1|^2+\cdots+|z_n|^2=1\right\}/\sim
$$

