Talk:Hodge theory

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)

Well, it's both, isn't it? Hodge structures do more than point at the Hodge conjecture; they have moduli and provide Torelli theorems, and so on. In those applications one doesn't usually want to look closely at the harmonic representatives. Hodge was an algebraic geometer, no question. So far WP doesn't have the analytic theory described. I suppose there is some mention of the history on the Kunihiko Kodaira page, in that Kodaira really tidied up the analysis.

Charles Matthews 06:48, 15 Sep 2004 (UTC)

I guess what I was getting at is that the article moves immediately from a sort of vague undergrad level discussion to the level only of interest to an algebraic geometer. What is Hodge theory to an analyst, a common differential geometer, or a Lie theorist, for instance? Probably 99% of the papers I have seen actually using Hodge theory do not use, and are not primarily interested in, the abstract definition of a Hodge structure. What do harmonic representatives have to do with this abstract definition?

So, I still find this article imbalanced.

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)

first sentence

Could someone try to simplify the first sentence? I find it really difficult to parse.

-- Helpful it would be if more specific constructive criticism you offered. Otherwise delete both our comments will the moderators. -Yoda

p-adic Hodge theory

What is p-adic Hodge theory?

Harmonic form redirects here

..but it does not define a harmonic form!Billlion (talk) 20:17, 12 November 2008 (UTC)

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)

Glaring omission

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)

Fixed. siℓℓy rabbit (talk) 18:44, 9 December 2008 (UTC)

Link to Helmholtz decomposition

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)

Split off Hodge theory for complex projective varieties

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)

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)

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)

cap product?

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)

Please state how an "elliptic complex" is defined

The section Hodge theory of elliptic complexes" begins as follows:

"Atiyah and Bott defined elliptic complexes as a generalization of the de Rham complex. The Hodge theorem extends to this setting, as follows. Let $E_{0},E_{1},\ldots ,E_{N}$ be vector bundles, equipped with metrics, on a closed smooth manifold M with a volume form dV. Suppose that

$L_{i}:\Gamma (E_{i})\to \Gamma (E_{i+1})$

are linear differential operators acting on C^∞ sections of these vector bundles, and that the induced sequence

$0\to \Gamma (E_{0})\to \Gamma (E_{1})\to \cdots \to \Gamma (E_{N})\to 0$

is an elliptic complex."

Maybe the author could be a great deal more generous to the reader by explaining — at least briefly — what an elliptic complex is. Thereby answering an obvious question that 97% of readers will probably have.

Instead of merely providing a link to an article about it.

Confusing sentence

The section Hodge theory of complex projective varieties contains this sentence:

"The "Kähler package" is a powerful set of restrictions on the cohomology of smooth complex projective varieties (or compact Kähler manifolds), building on Hodge theory."

This is extremely unclear!!!

These are not "restrictions" placed on the cohomology in question, Are they?

Rather they are conditions satisfied by the cohomology of Kähler manifolds. Right?

And complex projective varieties are all examples of Kähler manifolds. Right?

The information in the sentence could probably be written even more confusingly than it is ... but I can't think of how.