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?(adsbygoogle = window.adsbygoogle || []).push({});

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.

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

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