Talk:Complex manifold

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Mathematics rating:	Start Class	Mid Priority	Field: Geometry
Please update this rating as the article progresses, or if the rating is inaccurate. Click to show/hide comments. Please add to or update the comments to suggest improvements to the article. Geometry guy 18:43, 21 May 2007 (UTC)

[edit] Complex structure vs. homeomorphic to R²

(title added Nbarth 23:05, 10 November 2007 (UTC))

A complex topological manifold would simply be a topological manifold of even dimension, because C is homeomorphic to R^2. But what about a complex C^k-manifold? Since complex differentiabillity doesn't imply complex differentiabillity it would be different. Maybe, as in the real case, all these have a compatible smooth atlas. This stuff would be worth mentioning though. --MarSch 30 June 2005 15:47 (UTC)

You meant to say "real differentiabillity doesn't imply complex differentiabillity". But it goes the other way around. A complex C^k-manifold of dimension n is an analytic real manifold of dimension 2n, so I see no problem. But it is you who is the expert in this stuff, am I getting something wrong?

In the sentence

A complex topological manifold would simply be a topological manifold of even dimension

you should have specified that when you say topological manifold of even dimension you mean the complex manifold viewed as a real manifold has even dimension, otherwise that sentence does not make much sence. Oleg Alexandrov 30 June 2005 16:08 (UTC)

I'm sure we all know this, but just to be clear, there are even dimensional real manifolds which do not admit complex structures, so the sentence "a complex manifold would simply be a topological manifold of even [real] dimension" is not quite true. Also, since C^1 implies analytic in the complex case, there is no need to distinguish between different C^ks when it comes to complex manifolds. -Lethe | Talk June 30, 2005 16:19 (UTC)

Everything both of you are saying is correct, although I am not an expert on this topic and my complex analysis seems to have faded a little. What I was talking about is a complex topological manifold or a topological space which is locally homeomorphic to C^n. That would just be a topological manifold of even dimension, which explains why you don't bother defining it. I'll try and say a few words about this stuff. --MarSch 30 June 2005 17:41 (UTC)

In what sense would such a beast be considered "complex"? -Lethe | Talk June 30, 2005 19:07 (UTC)

It isn't. A complex manifold has, by definition, complex analytic (or holomorphic) transition functions. These are always real analytic (and so smooth). Nobody ever talks about C^k complex manifolds. Such discussion only applies to real differentiable manifolds. The article needs attention. -- Fropuff 30 June 2005 19:31 (UTC)

I was trying to explain why nobody ever talks about those things. Why does nobody talk about complex topological manifolds? Because they aren't really new things and arguably not complex. This is not abvious though, but a consequence of results in complex analysis. --MarSch 1 July 2005 10:58 (UTC)

I think this article is not the right place for such a discussion. Perhaps an article on complex differentiablity or complex analytic functions. -- Fropuff 1 July 2005 14:25 (UTC)

Talking about homeomorphisms to the complex plane is like talking about about group homomorphisms between vector spaces; it's just silly. Anyone who's studied a bit of category theory knows that what you really want is to consider the underlying topological space of the complexes. Or the underlying abelian group of a vector space. See forgetful functor for an explanation. So normal complex manifolds use C as their model because C has a complex structure. Homeomorphisms to C ignore this complex structure, and really, on some pedantic level, I might claim that homeomorphisms to C don't exist. You either have biholomorphisms to C, or you have homeomorphisms to R^2.

In any event, I'm not sure that the intro paragraph is the place to clear up a weird misconception that no one has, by introducing a terminology that is found in no textbook. -Lethe | Talk July 1, 2005 14:52 (UTC)

[edit] Kähler and Calabi-Yau manifolds

I pressed "Return" by mistake when editing that section in the article. My point is that we don't need even to say that Kähler manifold is the main article, that is obvious from the very first sentence in the section. Oleg Alexandrov 1 July 2005 02:13 (UTC)

[edit] patches are local domains in Cⁿ

A differential manifold can be defined as glued patches being each homeomorphic to Rⁿ or either a domain in Rⁿ. Both definitions are in fact equivalent. In complex analysis however, it is important to use domains in Cⁿ as patches and not the whole space Cⁿ. I think that this does not become clear in the definition, it is even wrong there, imho. Hottiger 14:36, 6 March 2006 (UTC)