Is Smoothness the Key to Understanding Complex Projective Space?

[Kreizhn]

So I have been tasked with what is likely a very simple problem, but have forgotten so much complex analysis that I would like to very the problem.

Let $\mathbb{CP}^n$ denote the n-dimensional complex projective space. We want to show that the quotient map $\pi: \mathbb C^{n+1}\setminus\{0\} \to \mathbb{CP}^n$ is smooth.

Essentially, I just want to ensure that nothing tricky is going on when we talk about "smooth" complex functions.

1) Having identified $\mathbb{CP}^n$ as an n dimensional space as compared to a 2n dimensional would imply we are looking at it with a complex structure rather than a real one. Does this cause any problems? I'm thinking I cannot just work the solution for the real structure and directly apply it to the complex one, since something about Cauchy-Riemann equations is jumping out at me.

2) I assume smooth in this context is infinitely complex differentiable. For this would it be sufficient to show that the function is entire on $\mathbb C^{n+1}$?

3) How must I adjust working with a function of multiple complex variables?

 

[fresh_42]

Are you sure $\pi$ is well-defined, since you cannot say which coordinate is unequal zero?

The answer depends a bit on how you define the complex projective spaces, which determines how the analytical structure is written. E.g. $\mathbb{CP}^n = \mathbb{S}^{2n+1}/\mathbb{S}^1$
(https://ncatlab.org/nlab/show/complex+projective+space)

Smooth complex projective varieties can be found here (p.23 f):
http://page.mi.fu-berlin.de/groemich/complex.pdf

For a more general introduction see
http://homepages.math.uic.edu/~ddumas/work/survey/survey.pdf