Talk:Kähler manifold

message: Kähler manifolds are usually thought of as finite dimensional manifolds. There is a generalized theory of infinite dimensional manifold, but it would be good to give at least one reference to literature dealing primarily with finite dimensional Kähler manifold. Hottiger 17:26, 6 March 2006 (UTC)
thread_title: Untitled

message: I think this entry would benefit if the importance of Kahler manifolds is explained.  I don't know it.
:I second this comment. Kähler manifolds need to be motivated. LTX2mny (talk) 10:55, 22 April 2008 (UTC)
thread_title:  Why are they important? 

message: Many sources define Kahler manifolds as symplectic manifolds with integrable compatible almost complex structures. The relation between this definition and the definition via Kahler metrics should be made explicit. I would be tempted to give the definition mentioned here first in the article, and then later mention the relation with Kahler metrics. Of course, being a symplectic geometer I may be biased. What do other thinks? 136.152.180.51 23:23, 19 April 2007 (UTC)
thread_title:  Definition 

message: I'm really not a specialist, but is there a reason why the article makes no mention of the fundamental Kahler identity? Does it only hold for a certain class of Kahler manifold? Just wondering -- Taku (talk) 22:53, 15 June 2008 (UTC)
thread_title:  Kahler identity 

message: I have rewritten a fair amount of the article and tried to take into consideration all of the points addressed on this talk page. It is thus far only a cursory rewrite. 
I have removed the original author's definition of a Kähler manifold, since it seemed incredibly wishy-washy. For example, the article often referred to "an additional integrability condition." What does it mean for a Hermitian metric to be integrable? This is not addressed in the linked article. Did the author mean that the  U(n)-subbundle of the tangent frame bundle was integrable? Is this equivalent to the (in my opinion) more common definitions which I have included? I could not find this definition in the references I was able to access.
I would love to re-incorporate the definition wherein we endow the tangent frame bundle with an integral  U(n) subbundle. If the article on G-structures made the idea more explicit that would be sufficient. If anyone could provide a reference for this, I would be more than happy to do the write-up. Kreizhn (talk) 21:52, 3 August 2012 (UTC)
thread_title:  Partial Rewrite 

message: I recognized some entanglements between 3 articles, Hermitian mfd., Kahler mfd. and Ricci tensor. Firstly, I found a unexpected translation, Kahler metric (redirect)-> Hermitian manifold. The Kahler condition is stronger than the Hermitian condition because a closed hermitian form is Kahler. I think that Kahler metric should translate to Kahler manifold, not to Hermitian manifold. But I don't know the way to release or delete the redirection Kahler metric to Hermitian manifold. Another entanglments might are between them.--Enyokoyama (talk) 17:32, 9 February 2013 (UTC)
thread_title:  entanglement between 3 articles 

message: 15 Feb 2013 someone, whose address is 24.154.111.112, has written on this article and the article "Hodge conjecture."
:This is incorrect. Kahler manifold is not absolute, as proven in the Hodge conjecture when one cannot assume X is a Kahler manifold due to decomposition not being constant. Hp, q(X) is not a  subgroup of cohomology classes being that (X is not a Kahler manifold) and cannot be represented by harmonic forms of (p, q).
On this article, "Kahler manifold," the definition from the symplectic viewpoint is not dependent on harmonic forms as someone wrote.
On another article, "Hodge conjecture," the original author as saying at the head of this paragraph assumes Kahler manifold in order to be clear for the origin of Hodge conjecure, which is the properties of Kahler manifolds and fortunately on compact Kahler manifolds there exist such harmonic forms.
I'll undo them in a few days.--Enyokoyama (talk) 06:53, 16 February 2013 (UTC)
thread_title:  paragraph written by someone