Talk:Hodge conjecture

message: 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)
thread_title: Announcement of counterexample

message: 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)
thread_title:  References 

message: 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)
thread_title: integral Hodge conjecture section needs improvement

message: 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)
replies: text: 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)
thread_title:  Hodge loci 

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

message: 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)
thread_title:  paragraph written by someone 

message: 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)
replies: text: 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)

text: 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)
thread_title:  Adding a technical tag. 

message: 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)
thread_title:  More specific references to counterexamples 

message: 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)
thread_title:  No explanation in the introduction for the layman 

message: 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, \Complex) = \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)
thread_title:  Is this right?