# Atlas of Complex Projective Space

In the context of Quantum Mechanics I'm trying to verify that the complex projective space $P(\mathbb{C}^n):=\mathbb{C}^n / \sim$ with $x \sim y :\iff x = \lambda \cdot y$ for some $\lambda \in \mathbb{C}$ where $x,y\in \mathbb{C}^n$ is a Kähler manifold.

It's an exercise that I've done in my differential geometry class but in the quantum mechanical context the construction of the atlas is a bit different. Basically I'm just experiencing one technial difficulty which I'm going to describe now:

In the following ( , ) always denotes the euclidean scalar product on $\mathbb{C}^n$. Take $\phi \in S(\mathbb{C}^n) := \{ \psi \in \mathbb{C}^n : (\psi,\psi)=1 \}$ . Consider $[\phi] \in P(\mathbb{C}^n)$ the corresponding equivalence class in the complex projective space. Now consider: $V_\phi := \{ x \in P(\mathbb{C}^n) : (\phi,x)\ \neq 0 \}$, $\{\phi\}^{\perp}:=\{ x \in \mathbb{C}^n : (\phi,x)=0 \}$ and the map $b_{\phi} : V_\phi \to \{\phi\}^{\perp}$ given by $[x] \mapsto \frac{x}{(\phi,x)} - \phi$ which is just the renormalized orthogonal projection onto the orthogonal complement of $\phi$. The claim is that $\{ (V_{\phi}, b_{\phi}) \}_{\phi \in S(\mathbb{C}^n)}$ is an atlas for $P(\mathbb{C}^n)$.

The technical difficulty that I'm having is the following. When computing the transition maps between different charts I want to find an explicit form of the set $b_{\phi}(V_{\phi} \cap V_{\psi})$ for some $\phi,\psi \in S(\mathbb{C}^n)$. My guess is that we have $b_{\phi}(V_{\phi} \cap V_{\psi})$ = $\{\phi\}^{\perp}$ $\cap \{\psi\}^{\perp}$. However I'm unable to show that $b_{\phi}(V_{\phi} \cap V_{\psi})$ $\supset$ $\{\phi\}^{\perp}$ $\cap \{\psi\}^{\perp}$ and here I'm particularly worried about the case when we have $(\phi,\psi) = 0$.

I'm not sure if there might be a mistake in my guess for the set $b_{\phi}(V_{\phi} \cap V_{\psi})$. In any case I'd be more than happy if someone could provide me with some help with this (rather technical) issue.

Best regards.

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$\mathbb{C}P^n$ is Kähler with respect to Fubini-Study metric en.wikipedia.org/wiki/K%C3%A4hler_manifold#Examples – john mangual Oct 9 '12 at 13:42