Talk:Hodge conjecture

Unsolved problems in mathematics

Please, can someone write a brief description of the conjecture? These two or three phrases (written as one or more questions) are needed for adding to the article the following template:

{{unsolved|mathematics|Brief description}}

That will put a tag with the brief description and a link to the list of Unsolved problems in mathematics, as well as putting this article in the Category:Unsolved problems in mathematics. There are more articles that must be tagged. Thanks! --surueña

The conjecture states that for a smooth projective complex algebraic variety, every Hodge cycle in a space H^p,p is a rational linear combination of classes of algebraic cycles of codimension p. Charles Matthews 15:06, 29 July 2005 (UTC)

Thank you, but I don't understand buttcheecks ... :-) Sorry for not get me explained correctly, but in my opinion the series of lists of Unsolved problems is intended for popular science, so the brief description of the unsolved problem should be understood more or less by everyone. Sometimes the text of the tag is not needed to explain the whole problem but only say why it is important and its consecuences. For example, see the tags in the following articles:

• Theory of everything
• Turbulence

Maybe the first step is to write the list of "popular science" questions regarding every unsolved problem in the list of Unsolved problems in mathematics, as done in Unsolved problems in physics. Anyway, maybe some of these mathematical problems are very, very difficult to explain to everyone, so thanks again for your brief description, and sorry for the inconvenience. --surueña 16:33:35, 2005-07-29 (UTC)

Well in mathematics some problems can be stated simply, and some can't. That hasn't much to do with their importance. The Hodge conjecture relates topology with algebraic geometry, i.e. polynomial equations in n variables. It won't mean much to anyone not conversant with both those fields. But your idea about 'popular science' may not be everybody's. Surely a list of unsolved problems is basically a list of problems that are important to solve, popular or not? Charles Matthews 22:00, 29 July 2005 (UTC)

Yes, I'm agree with you completely. But in my opinion the lists about unsolved problems can be useful for everyone: for experts because they will discover the list of other unsolved problems --experts that probably will contribute with high quality contents to those articles--, and to the unskilled, because they will understand a bit of the problem and why it is important, attracting more readers --and maybe they will help with maintenance works in that articles (fixing typos, adding wikis and fixing wrong links, improving the images, ...) like me--. In summary, they are good for Wikipedia, and, if done right, the brief explanations will not make the articles "less serious". For example, in the article about the theory of everything the tag doesn't explain anything about that complex theory, but gives useful info. --surueña 09:01:08, 2005-07-30 (UTC)

Announcement of counterexample

As far as I can see, the announcement by Fred W. Roush (with Kim) is from people who provided a counterexample to another conjecture, the Williams conjecture, nearly a decade ago (in symbolic dynamics). Roush is or was at Alabama State University. So, not an algebraic geometer, on the face of it. It would certainly be interesting to know what the technique applied is. Charles Matthews 15:08, 12 August 2006 (UTC)

200.199.213.124 18:11, 3 September 2006 (UTC) I have removed the countexample claim now they have recanted it; some time has passed since and it is not news anymore. I follow the precedent of the Riemann Hypothesis. In that case, only the current claim that goes unverified is mentioned; all others are reduced to a cursory note as they do not add anything to the current knowledge on the subject.

References

The only source for this article is to a non-online reference. Can anyone provide something clickable and checkable? wikipediatrix 19:46, 22 August 2006 (UTC)

phrase

Should the phrase:

" the conjecture says that certain de Rham cohomology classes are algebraic, that is, they are sums of Poincaré duals of the COHOMOLOGY classes of subvarieties "

be actualy written as:

" the conjecture says that certain de Rham cohomology classes are algebraic, that is, they are sums of Poincaré duals of the HOMOLOGY classes of subvarieties"

integral Hodge conjecture section needs improvement

The section on the integral Hodge conjecture has been oversimplified to the point of incorrectness - we need to distinguish between classes in integral cohomology (Hodge's original formulation) and classes in rational cohomology which are the images of classes in integral cohomology. Otherwise eg Kollar's example doesn't make sense. 131.111.24.26 (talk) 16:55, 14 December 2008 (UTC)

~~

Hodge loci

In the section on Hodge loci, should "the locus of all points on the base where the cohomology of a fiber is a Hodge class" actually read something like "the locus of all points on the base where a particular cohomology class in the fiber is a Hodge class", or something like that? That would seem to make more sense. Otherwise, what cohomology class is being talked about? I'm not familiar with the exact result, though. -- Spireguy (talk) 02:52, 10 June 2009 (UTC)

The variable here is a section of a given local system of free Z-modules (Hodge Theory and Complex Algebraic Geometry: Volume 2 by Claire Voisin, Leila Schneps p. 143, Google books. So what you say is pretty much what it is: trivialise locally topologically to a Cartesian product, and use that to mark a given integral cohomology class across the fibres. Charles Matthews (talk) 13:08, 10 June 2009 (UTC)

add an expository article by Claire Voisin as an external link.--Enyokoyama (talk) 01:37, 16 March 2013 (UTC)

abelian varieties paragraph needs some improvements

I think the paragraph abelian varieties needs some improvements. --Enyokoyama (talk) 04:24, 11 February 2013 (UTC)

paragraph written by someone

15 Feb 2013 someone, whose address is 24.154.111.112, has written on this article and the article "Kahler manifold."

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 another article, "Kahler manifold," the definition from the symplectic viewpoint is not dependent on harmonic forms as someone wrote.

On this 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) 06:56, 16 February 2013 (UTC)

I think that the written paragraphs were irrelevant, but the assumption in the Hodge conjecture that X is algebraic cannot be weakened. In fact, though not introduced in this article, Zucker showed in 1977 that Complex Tori with non-analytic rational cohomology of type (p,p) turn to be an counterexample to Hodge conjecture. I'll add this fact into the relevant position in this article.--Enyokoyama (talk) 16:56, 16 February 2013 (UTC)

I've already add the above mention into the paragraph "Reformulation in terms of algebraic cycles." and add a reference with link for pdf.--Enyokoyama (talk) 18:28, 16 February 2013 (UTC)

Adding a technical tag.

I think that in order for this to be meaningful to someone who doesn't already understand the subject, there must be some explanation of Hodge classes, and of the formulas in general. Even though I have substantial interest in this area of mathematics, the article was virtually meaningless on the first time I read it. Gluons12 (talk) 21:07, 26 May 2016 (UTC).

I'm not sure what you're looking for. In a sense, nobody understands Hodge classes, or else they'd have proved the Hodge conjecture already. But I don't think that's what you mean to ask. Precisely what part of the article did you find confusing? Ozob (talk) 23:54, 26 May 2016 (UTC)

The issue is, the Hodge conjecture simply is not a subject that can be explained to those who do not have a deep understanding of the subject. I personally don't see how less technical this article can be without oversimplifying it. — Preceding unsigned comment added by 58.151.85.12 (talk) 07:51, 10 June 2016 (UTC)

More specific references to counterexamples

I'm thinking particularly of the reference to Atiyah and Hirzebruch's 'Vector bundles and homogeneous spaces'. I just read through that whole article and there was no mention of Hodge anything, let alone a specific example which could be said to be obviously relevant. This makes for a bad stand-alone reference. At the very least a section or example number should be given, or explanation that the counterexample can be extracted from their work, or best of all, a reference to a paper or book where it is spelled out. I haven't checked the other references, but this is particularly poor 218.215.42.121 (talk) 23:56, 26 December 2016 (UTC)

Further digging around suggests that it is in fact a different 1961 paper of Atiyah and Hirzbruch that provides the counterexample. I'm changing it 218.215.42.121 (talk) 23:59, 26 December 2016 (UTC)

No explanation in the introduction for the layman

Would it be okay to make the hodge conjecture understandable to the layman who knows little maths by saying something like "The hodge conjecture postulates that any surface of any imaginable shape (most shapes existing in more dimensions than the usual three spatial dimensions) can be described by sets of curved lines and straight lines joined together continuously." There must be a layman's definition! Carovingian (talk) 12:28, 15 July 2019 (UTC)

Consider removing new paragraph added to introduction

The paragraph in the introduction beginning "In simple terms, the Hodge conjecture asserts that..." was added recently. To me, this paragraph reads like pseudo-mathematical nonsense which doesn't actually contribute real understanding to anyone. Can we consider removing this paragraph? Benediamond (talk) 03:08, 29 December 2021 (UTC)

The Hodge conjecture is one of the most important but least generally understood problems in modern mathematics. It seems absurd that the lead to the article should convey exactly zero information to the non-specialist and be completely useless to someone without a graduate training in complex algebraic geometry. The paragraph added is mostly just a layman-aimed simplification of the first paragraph to Claire Voisin's recent discussion of the progress on the conjecture <https://webusers.imj-prg.fr/~claire.voisin/Articlesweb/voisinhodge.pdf. If your suggestion is to remove the paragraph, I would strongly suggest you come up with a comparably simplified explanation of at least the idea of the conjecture. It reflects quite poorly on the WikiProject the way people seem content with even the most important or central problems in our field to be completely and utterly impenetrable.

I am fully aware of the reality that to explain true mathematical detail of the conjecture is more or less impossible to the layperson, but a paragraph explaining in general terms the idea of a millenium prize problem is not "pseudo-mathematical nonsense" just because it does not precisely specify the details of the conjecture. Tazerenix (talk) 03:34, 29 December 2021 (UTC)

Is this right?

The section Motivation begins as follows:

"Let X be a compact complex manifold of complex dimension n. Then X is an orientable smooth manifold of real dimension $2n$, so its cohomology groups lie in degrees zero through $2n$. Assume X is a Kähler manifold, so that there is a decomposition on its cohomology with complex coefficients

$H^{n}(X,\mathbb {C} )=\bigoplus _{p+q=n}H^{p,q}(X),$

where $H^{p,q}(X)$ is the subgroup of cohomology classes which are represented by harmonic forms of type $(p,q)$."

Question: Why is this is stated only for the cohomology group of dimension n ???

Later in the article, this is (apparently) assumed for cohomology groups of other dimensions as well, and it certainly is true for them.

Why does this "motivation" discuss only the nth cohomology group?

Yes it should be written for all $k$, not just $k=n$. It's just a typo.Tazerenix (talk) 10:08, 30 October 2023 (UTC)