It's not to hard to see that the quotient map $\pi\colon \mathbb{C}^{n+1}\backslash \{0\} \to \mathbb{CP}^n$ is smooth and surjective. Does that imply that it is a submersion as well?
Quotient map $\pi\colon \mathbb{C}^{n+1}\backslash \{0\} \to \mathbb{CP}^n$ is a submersion
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differential-geometry
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3Generally, a surjective smooth map need not be a submersion. For instance $x\mapsto x^3$ is smooth surjective on the real line but isn't submersive over $0$. However, $\pi$ is submersive, and you can check this in local coordinates, i.e. using charts for the projective space. – 2012-10-07
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0Thanks for your comment! Could you please elaborate a little bit in terms of the charts? I understand that I have to prove that the differential is surjective, so the relation between these concepts is not clear to me. – 2012-10-07