Talk:Hodge theory
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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?
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)
[edit] 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
[edit] p-adic Hodge theory
What is p-adic Hodge theory?
