2

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 $2^d$.

Susan
  • 1,205
  • You might profit from a general perspective: $v\mapsto \lambda w$ is a free proper $\mathbb{C}^*$ action, hence its quotient is a manifold and in fact the map is a principal fiber bundle. Check out section 1.4 of http://www.math.toronto.edu/mein/teaching/action.pdf – Neal Apr 05 '13 at 03:46

1 Answers1

3

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.

Alex Youcis
  • 54,059
  • So Complex Manifolds are always orientable? – Susan Apr 05 '13 at 03:12
  • Also, what is $U(1)$? – Susan Apr 05 '13 at 03:13
  • @Susan Yes, and $U(1)$ is the 1-dimensional unitary group. – Alex Youcis Apr 05 '13 at 03:13
  • And the qoutient of a compact set by a closed set is compact? – Susan Apr 05 '13 at 03:18
  • @Susan The quotient of a compact set by ANYTHING is closed. Compactness is preserved under continuous maps. – Alex Youcis Apr 05 '13 at 03:21
  • I suppose you should also add hints for 2nd countability and Hausdorffness. Since we're taking quotients, it's not totally obvious. Asker: If this is for a qual, skipping those is going to get you docked points... – Potato Apr 05 '13 at 03:37
  • @Potato I assumed that the manifold part was assumed. That's just annoying to verify. The action of $U(1)$ on $S^{2n+1}$ is properly discontinuous, which is why the quotient is Hausdorff, and it's second countable is pretty easy. – Alex Youcis Apr 05 '13 at 03:45
  • Yeah, an open quotient of a second countable space is second countable. It's totally something I would be concerned above proving a qualifying exam, however. – Potato Apr 05 '13 at 03:49