Holomorphic Quotient Maps on Projective Space

[Kreizhn]

Homework Statement

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.

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?

 

[mighty2000]

I would first clarify the question with the person who posted it to make sure I understand what they are asking. It seems like they are asking for a proof that the map \pi : \mathbb{C}^{n+1}\setminus\{0\} \to \mathbb{CP}^n is a smooth map between smooth manifolds. In this context, "smooth" means that the map is infinitely differentiable. This is different from holomorphic/analytic functions, which are specifically functions on complex manifolds. In this case, we are working with a map between smooth manifolds, not just complex functions.

To prove that the map \pi is smooth, we need to show that it is infinitely differentiable. This means that for any point p \in \mathbb{C}^{n+1}\setminus\{0\}, the composition \phi_i \circ \pi is smooth at p , where \phi_i is the coordinate chart given in the question. This can be done by showing that all partial derivatives of \phi_i \circ \pi are continuous at p .

In summary, to prove that the map \pi is smooth, we need to show that for any point p \in \mathbb{C}^{n+1}\setminus\{0\}, the composition \phi_i \circ \pi is smooth at p , meaning that all partial derivatives of \phi_i \circ \pi are continuous at p .