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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
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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
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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
171:, 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
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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).
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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?
393:. 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). --
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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?
370:, 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?
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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.
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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.
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221:-- Helpful it would be if more specific constructive criticism you offered. Otherwise delete both our comments will the moderators. -Yoda
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I'm not an expert, but the split seems reasonable, with a summary retained in this article. What should be done with
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Well, it's both, isn't it? Hodge structures do more than point at the Hodge conjecture; they have moduli and provide
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objection, I will be doing the split-off in a couple of days and turns this page into a broad-concept article. --
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As an afterthought, maybe Hodge structure could be moved to a separate article with some sort of segue into it?
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I propose we split off the Hodge theory for complex projective varieties and other two related sections to
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as a generalization of the de Rham complex. The Hodge theorem extends to this setting, as follows. Let
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It does, sort of. See the section on de Rham cohomology where it defines the space of harmonic forms.
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on
Knowledge. If you would like to participate, please visit the project page, where you can join
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Maybe the author could be a great deal more generous to the reader by explaining — at least
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749:{\displaystyle 0\to \Gamma (E_{0})\to \Gamma (E_{1})\to \cdots \to \Gamma (E_{N})\to 0}
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Could someone try to simplify the first sentence? I find it really difficult to parse.
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774:. Thereby answering an obvious question that 97% of readers will probably have.
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And complex projective varieties are all examples of Kähler manifolds.
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sections of these vector bundles, and that the induced sequence
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Instead of merely providing a link to an article about it.
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The information in the sentence could probably be written
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and the omission glares at you. Obviously it should say
637:{\displaystyle L_{i}:\Gamma (E_{i})\to \Gamma (E_{i+1})}
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Split off Hodge theory for complex projective varieties
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Hodge theory of elliptic complexes" begins as follows:
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page, in that
Kodaira really tidied up the analysis.
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469:Please state how an "elliptic complex" is defined
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188:So, I still find this article imbalanced.
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544:{\displaystyle E_{0},E_{1},\ldots ,E_{N}}
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38:It is of interest to the following
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115:Knowledge:WikiProject Mathematics
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362:Link to Helmholtz decomposition
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233:Harmonic form redirects here
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414:Mark viking
308:mathematics
286:mathematics
112:Mathematics
103:mathematics
59:Mathematics
841:Categories
804:These are
651:acting on
810:Are they?
488:defined
239:Billlion
205:Jholland
197:Jholland
162:Jholland
768:briefly
372:Spiri82
340:Fixed.
139:on the
30:C-class
826:Right?
820:Right?
482:Atiyah
36:scale.
486:Bott
484:and
460:talk
437:talk
433:Taku
418:talk
399:talk
395:Taku
376:talk
350:talk
331:talk
262:talk
243:talk
806:not
551:be
306:In
284:In
131:Low
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770:—
741:→
722:Γ
719:→
716:⋯
713:→
694:Γ
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672:Γ
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610:Γ
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561:dV
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Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
