Talk:Complex projective plane

The article is OK as far as it goes, but

The article is extremely skimpy on details. There is so much more to say about CP². No need to include highly technical details, but some things that might be included are various ways to visualize it, the fact that the 2-sphere S² that generates the second homology has a self-intersection number equal to 1, that it can be used to define a complex projective curve as the zero-set of a homogeneous polynomial in the 3 homogeneous coordinates of CP², that its cut locus is a 2-sphere, etc. — Preceding unsigned comment added by 173.8.212.241 (talk) 02:00, 6 April 2014 (UTC)

I forget, its also three tori glued together in a funny way, if I recall. 67.198.37.16 (talk) 04:40, 11 January 2016 (UTC)

And also, the homogeneous polynomial of degree 3 is a torus, as a (nearly trivial) example of a Calabi-Yau manifold, which is interesting because its analogous to the degree 4 polynomial in CP³ defines the K3 surface. This torus is thus sometimes called "the K2 surface". 67.198.37.17 (talk) 02:27, 7 May 2019 (UTC)

Atlas

Some notes-to-self about extending this article. This is remedial but useful as a quick-ref for other things :-) It would read something like this:

A covering of $\mathbb {CP} ^{2}$ is provided by the patches $U_{i}=\{Z\mid Z_{i}\neq 0\}$. An accompanying local trivialization is provided by the coordinate charts

$\varphi _{i}:\mathbb {CP} ^{2}\to \mathbb {C} ^{2}$

given by

$\varphi _{1}(Z_{1},Z_{2},Z_{3})\mapsto (Z_{2}/Z_{1},Z_{3}/Z_{1})$

and likewise for $\varphi _{2}$ and $\varphi _{3}$. The inverse is given by

$\varphi _{1}^{-1}(z_{2},z_{3})=(1,z_{2},z_{3})$

These can be used to define an atlas by setting the coordinate transition functions

$\tau _{ij}:\varphi _{i}(U_{i}\cap U_{j})\to \varphi _{j}(U_{i}\cap U_{j})$

to

$\tau _{ij}=\varphi _{j}\circ \varphi _{i}^{-1}$

Thus for example,

$\tau _{21}(z_{2},z_{3})=(\varphi _{2}\circ \varphi _{1}^{-1})(z_{2},z_{3})=\varphi _{2}(1,z_{2},z_{3})=(1/z_{2},z_{3}/z_{2})$

Note that these are holomorphic functions, and specifically fractional linear transformations ...

Note that these are tori, in that .... blah blah ...

This needs to be finished and probably belongs in the general complex projective space article. 67.198.37.16 (talk) 18:56, 3 November 2020 (UTC)