Talk:Complex manifold

message: 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)
thread_title:  patches are local domains in <math>C^n</math> 

message: 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)
thread_title: Canonical orientation???

message: Isn't the information that the existence of a complex structure on a 6-sphere is open, outdated?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)
replies: text: I added a very conservative, minimalist note. 67.198.37.16 (talk) 21:00, 7 May 2019 (UTC)
thread_title:  Complex structure on a 6-sphere 

message: I am refferring to the sentence:
"Consider for example any compact connected complex manifold M: any holomorphic function on it [i.e.: M \to \Complex^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 \Complex^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 \Complex^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 \Complex^{2n} (but maybe it could be embedded in some \Complex^N).. --151.29.230.177 (talk) 21:04, 8 April 2015 (UTC)
replies: text: 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)
thread_title:  liouville's theorem or max modulus principle? 

message: 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)
thread_title:  Higgs and spontaneous symmetry breaking 

message: 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)
thread_title:  Overly technical 

message: 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)
thread_title:  Quaternionic generalisation