Talk:Complex projective space

{{multiple issues}}

It seems that there has been no attempt to explain or motivate the content of this article. It is exclusively pitched at the mathematician. This article needs a lot more explanation. The policy on what Wikipedia is not says:

"A Wikipedia article should not be presented on the assumption that the reader is well versed in the topic's field… While wikilinks should be provided for advanced terms and concepts in that field, articles should be written on the assumption that the reader will not or cannot follow these links, instead attempting to infer their meaning from the text." − WP:NOT PAPERS

I'm a mathematician myself, and I started to edit the article to make minor changes per the mathematical manual of style. But the more I edited, the more I realised how impenetrable the article would be to a non-graduate mathematician. As the general manual of style says:

"While some topics are intrinsically technical, editors should take every opportunity to make them accessible to an audience wider than the specialists in the field, and to a general audience where possible." − WP:JARGON

Wikipedia is not a text book, or an academic journal. It is our chance, as mathematicians, to explain out work to the world. These mathematics articles are our opportunity to entice people towards the beauty of mathematics, and not to scare them away with its symbolism. I'm not an expert on complex algebraic geometry, so I wouldn't be the best person to contribute towards this article; but I will offer my technical help in an editorial fashion. Just drop me a note on my talk page. I look forward to some input and some improvement. — Fly by Night (talk) 00:55, 14 December 2010 (UTC)

Is the article skewed too much towards complex geometry?

As far as I know Complex projective space can be viewed both as a manifold and as a projective variety. Which additional structures are natural depends on your POV. For example, from an algebraic pov the natural topology is the Zariski topology, rather than the cell-topology currently discussed in the article. The article seems to be written very much from the specific point of view of complex geometry, with algebraic geometry coming as an after thought. This seems very unbalanced to me.TimothyRias (talk) 16:35, 15 December 2010 (UTC)

The only thing really missing is a discussion of the coordinate ring and sheaf cohomology. The cell decomposition is identical in the algebraic category. As of yesterday, I've been moving the article in this direction. Please be patient. Sławomir Biały (talk) 17:01, 15 December 2010 (UTC)

railroad

The picture of the train tracks is confusing. If one wishes to illustrate the construction of the points at infinity, the figure should contain a clearly identifiable horizon (line at infinity). Meanwhile, the figure in the introduction section seems to have two parallel horizons. Tkuvho (talk) 15:42, 2 January 2011 (UTC)

I have no objection to substituting a better image. Best, Sławomir Biały (talk) 15:59, 2 January 2011 (UTC)

this is the best I could find:

Perhaps we can get rid of the comment somehow? Tkuvho (talk) 16:28, 2 January 2011 (UTC)

The image could be cropped, but that would get rid of the train as well which I think gives a helpful real-world context to the image. I personally think that an actual photograph would be ideal. But unfortunately it may be very difficult to find such a picture without at "extra" horizon. Sławomir Biały (talk) 13:50, 4 January 2011 (UTC)

The real confusing thing is that you use an image of a Nazi concentration camp (http://en.wikipedia.org/wiki/File:Birkenau_gate.JPG) as an illustration in a mathematical article. I find this disgusting and recommend to replace it quickly. 130.180.82.245 (talk) 20:24, 7 February 2013 (UTC)

I have reverted the recent edit that put that image in the article. Sławomir Biały (talk) 23:20, 7 February 2013 (UTC)

Construction

This article about a (complex) linear manifold says in the Construction section:

The coordinate transitions between two different such charts U_i and U_j are holomorphic functions (in fact they are fractional linear transformations). Thus CPⁿ carries the structure of a complex manifold of complex dimension n, and a fortiori the structure of a real differentiable manifold of real dimension 2n.

The mention of holomorphic functions introduces an analytic concept. Are not linear functions sufficient? The reference to Fractional linear transformations appears suspect.Rgdboer (talk) 23:08, 4 July 2014 (UTC)

The concepts of continuity, differentiability and analyticity are incidental in the context of projective spaces generally, and IMO should be confined to the Differential geometry section. The term fractional linear transformation should be replaced by homography. The reference to differentiability should also be removed from this section, IMO. —Quondum 02:26, 5 July 2014 (UTC)

No, complex projective space cannot be covered by linear charts. There is a topological obstruction (the Chern character) to the existence of a global flat affine connection. I do not think it makes sense to split out the discussion of regularity of the charts from the discussion of the charts themselves. That would seem to result in needless fragmentation of the article with no clear gain. (For one thing, whether the charts are regarded as (algebraically) regular, analytic, smooth or continuous depends on whether on is working in algebraic geometry, differential geometry, or topology. (This distinction is admittedly rather naive, which is also a reason not to discuss these cases separately.) Sławomir Biały (talk) 22:06, 5 July 2014 (UTC)

Yes, to support the link to Complex manifold the necessary patches (=charts) are mentioned. But this section of the article is to show construction of the complex projective space. The text runs ahead of a general reader. It is an example of a complex manifold but should not require this concept from differential geometry for its construction. Rather some motivation for complex projective space is necessary to establish it. Since C is an algebraically closed field it may be preferable to real projective space. The aim of this article should be to establish its concept firmly so that, when students turn to manifolds, this example is familiar. Moving from the general to the specific, as shown here, is a problem with mathematics educators that do not recall their learning. Only by developing a reservoir of instances are students prepared for generalities, such as manifolds in this case.Rgdboer (talk) 21:09, 8 July 2014 (UTC)

I think it makes sense to split out a separate section on the affine charts. Sławomir Biały (talk) 23:16, 8 July 2014 (UTC)

Seriously?

It is tremendously unhelpful to read in the section CW-decomposition the following:

"One useful way to construct the complex projective spaces $\mathbf {CP} ^{n}$ is through a recursive construction using CW-complexes. Recall that there is a homeomorphism $\mathbf {CP} ^{1}\cong S^{2}$ to the 2-sphere, giving the first space. We can then induct on the cells to get a pushout map"

which is followed by a diagram (that may or may not be commutative; the text does not say) containing four mappings, and which one is the "pushout map" the text does not say.

Then clicking on the link to the Pushout article, one reads the following:

"In category theory, a branch of mathematics, a pushout (also called a fibered coproduct or fibered sum or cocartesian square or amalgamated sum) is the colimit of a diagram consisting of two morphisms f : Z → X and g : Z → Y with a common domain."

Although all of this may be mathematically correct, it is exactly the opposite of the kind of writing that is appropriate for an encyclopedia article. 2601:200:C000:1A0:B85D:3EA8:7C2B:2D7C (talk) 17:54, 19 March 2021 (UTC)

What is a "nice" CW-complex?

I know what a CW-complex is, or at least I thought I did.

But the section Classifying space contains the phrase "nice CW-complex".

I have not heard of such a thing, nor was I able to find any definition by googling just now.

I hope that someone knowledgeable about the subject will clarify this point in the article (or else delete the word "nice" if it is not appropriate). 2601:200:C000:1A0:D4AD:44A7:8D0E:496F (talk) 23:34, 7 January 2023 (UTC)

Utter nonsense

The caption "Parallel lines in the plane intersect at the vanishing point in the line at infinity" (below a photo of railroad tracks converging as they go into the distance) is complete and total nonsense.

The photo shows no "parallel lines in the plane", as anyone who knows the definitions of the words "parallel", "lines", and "plane" ought to know.

This is even more misleading than a number of additional serious infelicities in this article.

If it's too much trouble to write clearly, then please do not inflict your writing on Wikipedia readers.