I am taking this class that follows John Lee's introduction to manifolds, but I have not yet taken a complex analysis class (I have just been pushing it to the last semester of my undergraduate). I am hence a little confused in how to deal with complex spaces and draw parallels to the real space. Here's the question:
A complex projective space, denoted by $\mathbb{CP}^n$, is the set of all 1-dimensional complex-linear subspaces of $\mathbb{C}^{n+1}$, with the quotient topology inherited from the natural projection $\pi:\mathbb{C}^{n+1}\backslash \{0\} \to \mathbb{CP}^n$. Show that $\mathbb{CP}^n$ is a compact 2n-dimensional topological manifold,and show how to give it a smooth structure analogous to one we constructed for $\mathbb{RP}^n$. (We use the correspondence: $(x^1+iy^1,....,x^{n+1}+iy^{n+1}) \leftrightarrow (x^1, y^1,....,x^{n+1}, y^{n+1})$ to identify $\mathbb{C}^{n+1}$ with $\mathbb{R}^{n+1}$.)
I got that $\pi(z) = \{\alpha*z | \alpha \in \mathbb{R}\}$ and the function restricted to a complex manifold $\pi:\{|z| = 1$& $y^{n+1} > 0 \} \to \mathbb{CP}^n$ is a bijection. But when writted as its correspondence to the (2n+2)-dim real space, this forms a semi-spherical shell in (2n+2)-dim space, which is a (2n+1)-dim manifold. Am I going wrong in some way treating the complex space as simply a 2x larger real space and how else can one approach it? Thanks!