# Atlas of complex projective 3-space $\mathbb{C}P^3$

1. Jan 11, 2013

### kloptok

I'm constructing an atlas for complex projective 3-space $\mathbb{C}P^3$. I use the construction with inhomogeneous coordinates $z_{i}/z_{j}$ and a chart is given by the points $\mathcal{U}_j=\{z_j\ne 0\}$. At the intersections $\mathcal{U}_i\cap \mathcal{U}_j$ I should specify (holomorphic) transition functions between coordinates. So far I'm fine. Now my problem is, what do I do with the intersections between three charts, $\mathcal{U}_i\cap \mathcal{U}_j\cap \mathcal{U}_k$ ? I have a feeling that I can skip them since $\mathcal{U}_i\cap\mathcal{U}_j \cap \mathcal{U}_k=\mathcal{U}_i \cap(\mathcal{U}_j\cap \mathcal{U}_k)$ and I know what to do at each of the intersections $\mathcal{U}_i\cap\mathcal{U}_j$. After all, isn't the point that at this intersection I can use either of the three coordinate systems and I only have to require holomorphic transitions between them?

Is this a correct way of thinking? It's probably a silly question but I can't get my head around this. All I can find when I search on the web is what to do at intersections between two charts, and nothing about multiple intersections.

2. Jan 11, 2013

### quasar987

You only need to check holomorphicity of the transition functions between each pair of charts since if you do that, then on the intersection of triplets, you're just restricting the domain of a map you already know is holomorphic.