Talk:Complex manifold

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)

Certainly you can have homeomorphisms to the complex plane C. I understand that its importance is not its topological structure so much as its complex structure. But, at the same time, it is occasionally convenient notation to denote the plane by C rather than R^2, even if you are interested only in its topology and not in its complex structure. For example, if you wish to give a parametrisation of some particular curve in the plane, it may be practical to describe it by some combination of exponentials which reads easier than a horrible mash of trig functions. 99.231.65.91 (talk) 02:01, 24 January 2009 (UTC)

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)

patches are local domains in $C^{n}$

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

Canonical orientation???

The article states:

"Since the transition maps between charts are biholomorphic, complex manifolds are, in particular, smooth and canonically oriented (not just orientable: a biholomorphic map to (a subset of) $C^{n}$ gives an orientation, as biholomorphic maps are orientation-preserving)."

I am not understanding why a complex manifold has a "canonical orientation".

Isn't this orientation equivalent to choosing between i and -i, or choosing between clockwise and counterclockwise? If so, I am not sure why one would call it "canonical". Please explain. Thanks.Daqu (talk) 06:14, 2 December 2008 (UTC)

Codswallop!

Codswallop! The article says that "...the Whitney embedding theorem tells us that every smooth manifold can be embedded as a smooth submanifold of Rⁿ ". But this is clearly untrue! Is the person that wrote that part even a mathematician?! The Whitney embedding theorem was even linked, and with one click we can see that the strong version says "...any smooth m-dimensional manifold (required also to be Hausdorff and second-countable) can be smoothly embedded in Euclidean $2m$-space, if m>0. So, given some technical details, we can embed a two-manifold in four-space, a three-manifold in six-space, etc. BUT "...every smooth manifold can be embedded as a smooth submanifold of Rⁿ"????  Δεκλαν Δαφισ   (talk)  19:26, 6 July 2009 (UTC)

Take it easy, there! First of all, manifolds are often presumed to be Hausdorff and second countable. And the statement you quote isn't necessarily nonsense, just incompletely quantified. For example, if the word "some" were added after the word "of", it would read:

"...the Whitney embedding theorem tells us that every smooth manifold can be embedded as a smooth submanifold of some Rⁿ ."

which is clearly correct.Daqu (talk) 01:10, 29 March 2010 (UTC)

Complex structure on a 6-sphere

Isn't the information that the existence of a complex structure on a 6-sphere is open, outdated? It seems to me that this was recently published and is more or less accepted to be correct by the mathematical community, although I'm not able to check details... Franp9am (talk) 20:29, 23 December 2015 (UTC)

I added a very conservative, minimalist note. 67.198.37.16 (talk) 21:00, 7 May 2019 (UTC)

liouville's theorem or max modulus principle?

I am refferring to the sentence: "Consider for example any compact connected complex manifold M: any holomorphic function on it [i.e.: $M\to \mathbb {C} ^{n}$, I suppose] is locally constant by Liouville's theorem"

I don't understand how to use liovulle theorem in that case! Instead, if you use the maximum modulus principle (http://www.encyclopediaofmath.org/index.php/Maximum-modulus_principle) it is all clear: suppose that the modulus of a holomorphic function $f:M\to \mathbb {C} ^{n}$ has a local maximum in $y_{0}\in M$. Use a local chart $\varphi :(U,0)\to (V,y_{0})$ (where $U$ is the disc in $\mathbb {C} ^{n}$). Then the modulus of $f\circ \varphi :U\to R$ has a local maximum in $0$. Hence $f\circ \varphi$ is constant on U. Hence $f$ is locally constant!

Moreover, the entire paragraph is rather confusing: who is $n$? It seems that it is proved that an $n$ dimensional complex manifolds cannot be holomorphically embedded in $\mathbb {C} ^{2n}$ (but maybe it could be embedded in some $\mathbb {C} ^{N}$).. --151.29.230.177 (talk) 21:04, 8 April 2015 (UTC)

Liouville's theorem states that every holomorphic function with bounded image is constant. Since any map from a compact space must have compact, and thus bounded image, it follows that every holomorphic map from a compact complex manifold must be constant. Why they say "locally constant" is a mystery to me, in fact I'm going to fix that right now. — Preceding unsigned comment added by 108.13.11.67 (talk) 18:59, 20 September 2016 (UTC)

Higgs and spontaneous symmetry breaking

The previously-mentioned link in the 6-sphere has this (to me) astounding paragraph:

A complex structure on a complex manifold can always be re-interpreted as a spontaneously broken classical vacuum solution in a non-linear version of a Yang-Mills-Higgs theory formulated on the underlying manifold. This is because an almost complex tensor field J is mathematically the same as a Higgs field Φ. Although the terminology comes from theoretical physics, the corresponding mathematical structures are well-defined.

I find this to be quite remarkable, and worth a detailed exposition here, or somewhere. I have a good-enough imagination that I can imagine how this might be true (plus I know what Higgs fields are, I'm unclear why J is the Higgs field, instead of being just the classical solution to one, or some patch near the minimum...) But I find that imagination is unsuitable, when something more explicit can be stated. 67.198.37.16 (talk) 21:39, 7 May 2019 (UTC)

Overly technical

The article is currently too technical for non-experts to understand; I am adding a tag to suggest the article be improved to be understandable to non-experts. Betanote4 (talk) 18:16, 5 August 2020 (UTC)

Quaternionic generalisation

There was a claim about quaternions projective planes having a complex strucutere, which is wrong, see "NON-EXISTENCE OF ALMOST-COMPLEX STRUCTURES ON QUATERNIONIC PROJECTIVE SPACES" By Messey. Simiminis (talk) 13:53, 27 November 2023 (UTC)