# 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:

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