Differential Topology Question on Complex Projective Space 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. 
