# Holomorphic Quotient Maps on Projective Space

1. ### Kreizhn

743
1. The problem statement, all variables and given/known data
Let $\mathbb{CP}^n$ be n-dimensional complex projective space, and let $\pi: \mathbb C^{n+1}\setminus\{0\} \to \mathbb{CP}^n$ be the quotient map taking $\pi(z_1,\ldots,z_{n+1}) = [z_1,\ldots,z_{n+1}]$ where the square brackets represent the equivalence class of lines through this point.

Show that $\pi$ is smooth.

3. The attempt at a solution

There's no doubt in my mind I can do this, once I verify what the question is asking me to do. Here's my rationale, but I'm not sure if it's correct.

Since we've claimed that $\mathbb{CP}^n$ is n-dimensional, we must be considering it with a complex structure rather than a real structure. This means that, as a smooth manifold, it is sufficient to show that its composition with any coordinate chart is smooth. That is, $\phi_i \circ \pi : \mathbb{C}^{n+1}\setminus\{0\} \to \mathbb C^n$ is smooth, where
$$\phi_i([z_1,\ldots, z_{n+1}]) = \left( \frac{z_1}{z_i}, \ldots, \frac{z_{i-1}}{z_i}, \frac{z_{i+1}}{z_i}, \ldots, \frac{z_{n+1}}{z_i} \right)$$
with domain $z_i > 0$.

Okay, so assuming that's correct, we want to show that this function is smooth. But what is "smooth" in this context? I can't ever recall talking about "smooth" complex functions, only holomorphic/analytic complex function. Is it sufficient to show this is holomorphic, or do I need more?