This question seems like it would be very hard to do directly. I wouldn't know where to begin. I was wondering if anyone had a very slick proof of this. The only thing I think is easy is that its connected. The rest seems like the only way I know how to do it would be way to hard to construct easily. I would appreciate it if someone had an easy solution as I am studying for a qual. Thank you.
Recall that the complex projective space $\mathbb{C}P^d$ is the quotient space of $\mathbb{C}^{d+1} \backslash \{0\}$ under the equivalence relation $x \sim y$ if and only if there is a $\lambda \in \mathbb{C}$ with $v = \lambda $w. Prove that $\mathbb{C}^ d$ is a compact, connected, orientable manifold of dimension $2^d$.
Hints:
Compactness: You can alternatively realize $\mathbb{P}^n(\mathbb{C})$ as the coset space $S^{2n+1}/U(1)$.
Connectedness: Same hint.
Orientable: Complex projective space has the structure of a complex manifold--this about why this allows you to create a chart whose overlaps have positive Jacobians.
Dimension: same hint as for compactness and connectedness.