- #1
Kreizhn
- 714
- 1
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?
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?