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Smooth Multivariate Complex Functions

  1. Sep 23, 2011 #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 [itex] \mathbb{CP}^n [/itex] denote the n-dimensional complex projective space. We want to show that the quotient map [itex] \pi: \mathbb C^{n+1}\setminus\{0\} \to \mathbb{CP}^n [/itex] is smooth.

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

    1) Having identified [itex] \mathbb{CP}^n [/itex] 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 [itex] \mathbb C^{n+1} [/itex]?

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