Talk:Hodge theory

message: This seems to be a definition of Hodge structure, which is what the self-proclaiimed "Hodge theorists" talk about -- namely those working on the Hodge conjecture.  But from my point of view, Hodge _theory_ is really the study of finding "harmonic" representatives of cohomology classes on various sorts of structures (e.g., Riemannian manifolds, complex manifolds, algebraic varieties, homogeneous vector bundles on Lie groups, etc).   Maybe there should be some discussion about the origins and applications of Hodge theory (from this point of view).
Jholland 01:52, 15 Sep 2004 (UTC)
replies: text: As an afterthought, maybe Hodge structure could be moved to a separate article with some sort of segue into it?
Jholland
I have posted some more detailed discussion of Hodge theory in the de Rham case, and for elliptic complexes.  It still needs a lot of work, but I'm having some Wiki problems, so editing is slow.
Jholland 03:24, 7 May 2005 (UTC)

message: ..but it does not define a harmonic form!Billlion (talk) 20:17, 12 November 2008 (UTC)
replies: text: It does, sort of.  See the section on de Rham cohomology where it defines the space of harmonic forms. siℓℓy rabbit (talk) 18:46, 9 December 2008 (UTC)
thread_title: Harmonic form redirects here

message: The first line of this article says:
: 
:: In mathematics, Hodge theory is one aspect of
: 
and the omission glares at you.  Obviously it should say
: 
:: In mathematics, Hodge theory, named after ?????? Hodge, is one aspect of
: 
Michael Hardy (talk) 18:42, 9 December 2008 (UTC)
replies: text: Fixed. siℓℓy rabbit (talk) 18:44, 9 December 2008 (UTC)
thread_title:  Glaring omission 

message: I clicked through from the page on Helmholtz decomposition, hoping to find information about its multidimensional extension (as stated on that page).
On the present page, I do not see any mention of Helmholtz decomposition though. What is the link between both?
Spiri82 (talk) 06:57, 2 May 2017 (UTC)
thread_title:  Link to Helmholtz decomposition 

message: I propose we split off the Hodge theory for complex projective varieties and other two related sections to Hodge theory in complex algebraic geometry. The reason is because, as I understand, Hodge theory for differential manifolds and one in complex geometry are distinct theories, if spiritually similar, (and thus should be covered by separate articles). -- Taku (talk) 22:57, 22 March 2018 (UTC)
replies: text: I'm not an expert, but the split seems reasonable, with a summary retained in this article. What should be done with Hodge structure, which is claimed as the main article for the complex projective variety section? --Mark viking (talk) 00:52, 25 March 2018 (UTC)
replies: text: As I understand, "Hodge structure" is an axiomatization of Hodge theory (on Kähler manifolds); the usual Hodge theory simply says there exists Hodge structure. Since the Hodge structure article is well-developed and is lengthy, I think it's ok to have two separate articles. Since I'm not seeing an objection, I will be doing the split-off in a couple of days and turns this page into a broad-concept article. -- Taku (talk) 03:06, 29 March 2018 (UTC)
thread_title:  Split off Hodge theory for complex projective varieties 

message: The article contains a seeming non-sequitur: "On the other hand, the integral can be written as the cap product of the homology class of Z...". There is no previous mention of an integral or of Z. Can someone elaborate? Sloth sisyphos (talk) 16:30, 29 March 2023 (UTC)
thread_title:  cap product?