A Counterexample to the Hodge Conjecture?

A paper appeared last night on the arXiv by K.H. Kim and F.W. Roush entitled Counterexample to the Hodge Conjecture. The authors claim to construct an example using K3 surfaces for which the Hodge conjecture is false. If they’re right about this, this would be very shocking, and I would guess that most experts will be very skeptical about the result. Most likely someone soon will find a problem with the argument, but if not there will be a lot of excitement.

The Hodge conjecture is one of the Clay Millenium prize problems, so if this paper is right, the authors may very well be entitled to $1 million. For more about what the Hodge conjecture says, see the slides or video of a popular lecture by Dan Freed, or the official statement of the problem due to Pierre Deligne.

Update: The authors have withdrawn their claim to have disproven the Hodge conjecture, acknowledging problems with their argument beginning in section 5 of the paper.

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36 Responses to A Counterexample to the Hodge Conjecture?

  1. Pingback: Ars Mathematica » Blog Archive » Hodge Conjecture False?

  2. anon_hodge says:

    Nobody posted this breaking news to slashdot yet.

  3. mustang says:

    ya like slashdot care.

  4. Harry Walton says:

    Hello,

    A plea for help:

    For a long time now I’ve tried to view the videos on the Clay Millenium Prize website.

    I consistently get an error message from Real Player – the same thing happens with the Hodge Conjecture link you’ve provided here.

    I watch plenty of other video web feeds with no problem.

    I’ve emailed the Clay site – but no response.

    Is this a known problem or is it just a problem for me?

  5. nonblogger says:

    These two authors got a paper a few years ago in Ann. Math. where they disproved a fairly old conjecture in Dynamical Systems.
    http://www.arxiv.org/abs/math.DS/9907095

    So at least their new paper does deserve to be studied carefully.

  6. I don’t see exactly why it would be so shocking. A lot of these conjectures seem to be held in far too high a regard. It seems a bit paradoxical that something someones was “unable” to prove should be named after them.

    It’s like the Riemann Hypothesis. There could be some sort of fractal set of transcendental complex numbers with zeros somewhere in the (0,1) strip and we might never know. So if someone finally proves the danm thing false, why the surprise? The fact that you were unable to prove it true should be taken as a hint.

  7. D. Eppstein says:

    Harry: I am also unable to view the Freed video. RealPlayer tells me that it can’t access the video data at the address given in the ram file.

  8. Johan Richter says:

    From the rules of the Clay Institute:

    “In the case of the P versus NP problem and the Navier-Stokes problem, the SAB will consider the award of the Millennium Prize for deciding the question in either direction. In the case of the other problems if a counterexample is proposed, the SAB will consider this counterexample after publication and the same two-year waiting period as for a proposed solution will apply. If, in the opinion of the SAB, the counterexample effectively resolves the problem then the SAB may recommend the award of the Prize. If the counterexample shows that the original problem survives after reformulation or elimination of some special case, then the SAB may recommend that a small prize be awarded to the author. The money for this prize will not be taken from the Millennium Prize Problem fund, but from other CMI funds. ”

    Would the present paper effectively resolve the problem if correct?

  9. Davis says:

    If the counterexample shows that the original problem survives after reformulation…

    The counterexample comes from taking certain products of varieties; as such, I wouldn’t be surprised if this allowed for some sort of reformulation excluding such examples. That’s assuming their argument is correct, of course.

  10. Thomas Mulligan says:

    If their proof is sound, it’s a good year for mathematics: first Poincare, now Hodge. I agree with “ObsessiveMathsFreak” about the undue confidence people put in their assumed truth of these conjectures; history has demonstrated (Hilbert’s Fourteenth, Euler’s conjecture, etc.) that assumed elegance should not be used as warrant for believing a conjecture to be true. Furthermore, since the Hodge conjecture is so deeply buried in abstract, complicated, specialized mathematics, it’s hard to believe our intuitions would be of any utility at all. . . .

  11. So if someone finally proves the danm thing false, why the surprise? The fact that you were unable to prove it true should be taken as a hint.

    Well, there’s an empirical argument: in all the areas I’m familiar with, there have been far more conjectures that stayed open for years and were finally proved, than conjectures that stayed open for years and were finally disproved. Of course the latter often get a disproportionate amount of attention.

    Incidentally, does anyone know if there’s a reasonably elementary statement that’s equivalent to the Hodge conjecture? I read the links from this post and a few other articles, and I still don’t understand what’s being asked.

  12. jb says:

    I know many experts were not at all convinced the HC had to be true, unlike with, say, the Riemann Hypothesis.

    As for equivalent elementary statements, I’m not aware of any.

  13. Kea says:

    Peter

    How many Millenium problems are there left?

  14. Yatima says:

    I would guess 7:

    http://www.claymath.org/millennium/

    Remember…remember… how many positive and negative proofs there have been for P = NP ?

    (Peter – Great Book by the way; unfortunately my mathematical ability has suffered somewhat by years in ICT)

  15. Walt says:

    Kea: 5, assuming this holds up.

    Yatima: I assume you mean 6 (it seems pretty likely now that Poincare is settled).

    Scott: I don’t think there is a more elementary formulation. Algebraic geometry has been around long enough that it’s become incredibly technical.

    ObsessiveMathsFreak: The Hodge conjecture is unusual in that there is so little supporting evidence. The other conjectures, such as the Riemann hypothesis, have more supporting evidence. For example, it’s numerically easy to compute zeroes of the zeta function, and the 2 billion that have been found all lie along the critical line.

  16. Andrew says:

    Actually, their paper doesn’t look correct. They’re talking about something well-known called the Kuga-Satake-Deligne correspondence, which is implied by the Hodge conjecture. The entire paper is poorly written, doesn’t seem to even try to prove the theorems they state, it looks like they completely misinterpreted the actual statement of the correspondence. All of the correct parts of the paper are lifted almost verbatim from van Geemen’s paper “Kuga-Satake Varieties and the Hodge Conjecture.” One of them is a computer scientist, so it looks like some amateurs are just trying to have a good time proving some famous conjectures.

  17. werdna says:

    Who is the computer scientist, and how do you know that? Theoretical computer science can be highly mathematical, and some CS people are excellent mathematicians. In fact, the P=NP problem came from CS.

  18. not a Hodge Conjecture expert says:

    Scott,

    The Hodge Conjecture is widely considered to be the most difficult millennium problem to explain to the general public, but a watered down (probably inaccurate, definitely less general) elementary statement, understandable to most mathematicians and fancy theoretical physicists, might go something like this:

    On a complex algebraic variety, every homology class that could reasonably contain a subvariety does contain a subvariety.

    The “could reasonably contain” part mostly refers to an obvious obstruction: The homology class of a complex subvariety must be Poincare dual to a differential form of type (p,p).

    I’m not sure why the Hodge Conjecture is rarely dumbed down in this way. I think it’s because algebraic geometers don’t realize that no one else understands algebraic geometry.

  19. Davis says:

    werdna:

    some CS people are excellent mathematicians…

    True, but very few CS people are good algebraic geometers. Algebraic geometry is a notoriously difficult field to jump into, and has only a handful of intersections with CS stuff (that I’m aware of).

    not a Hodge Conjecture expert:

    I think it’s because algebraic geometers don’t realize that no one else understands algebraic geometry.

    I think algebraic geometers just sort of gave up on developing explanations accessible to folks outside the field (which is unfortunate, in my opinion). These days it can be challenging to explain cutting-edge AG to other algebraic geometers, nevermind non-AGers.

  20. On a complex algebraic variety, every homology class that could reasonably contain a subvariety does contain a subvariety.

    So, to translate into my doofus computer-scientist terms: We have a solution set, S, of some system of polynomial equations over the complex numbers. We’re interested in what sorts of subsets T of S can also arise as the solution set of a system of polynomial equations. In particular, what are the possible ways that T can embed into S topologically? How can T wrap around the holes of S, and so on? The Hodge conjecture basically says that T can embed into S in “every way that it reasonably could.”

    Is that completely off-base? (I don’t doubt that the conjecture is more general, but I’d be happy to understand any nontrivial special case…)

  21. Walt says:

    It’s sort of dual to what you describe. Let’s say S has a hole. The Hodge conjecture answers the question, when is there a T that wraps around that hole?

  22. Lolo says:

    That’s strange to me what non-algebraic geometers think about algebraic geometry : Hodge conjecture is from far one of the easiest conjecture in algebraic geometry to state (and, apart from its deepness, I think this is one of the reasons why it was chosed for the millenium problems, so that non-algebraic geometers can understand its statement). It is very “concrete” compared to other conjectures in algebraic geometry and can be understood by any graduate strudent in algebraic geometry begining his PHD.

    By “concrete” I mean : it deals with smooth projective algebraic varieties over C and their cohomology. Thus you can think of them as smooth complex manifolds and for the cohomology you can think of it as singular/De Rham cohomology. There’s no étale/cristalline/syntomic/motivic cohomology in this statement and it does not need the theory of schemes or algebraic stacks or stuff like that to be stated since you deal with smooth projective varieties over C, you can use your differential geometry intuition.

  23. Walt says:

    Lolo: What you say is false, unless you mean by “starting his Ph.D.” someone who’s already had two years of graduate work, including lots of differential geometry. I actually find schemes easier to understand than complex analytic varieties, but that’s probably just me.

  24. not a Hodge Conjecture expert says:

    Lolo, with apologies, I take back my unnecessarily snarky remark about algebraic geometers. According to the official Clay statement, the Hodge Conjecture states:

    “On a projective non-singular algebraic variety over C, any Hodge class is a rational linear combination of classes cl(Z) of algebraic cycles,”

    where a Hodge class is a rational homology class of type (p,p), and an algebraic cycle is just an algebraic subvariety.

    This description is actually quite simple and really not that fancy. So I will make a different snarky remark: Algebraic geometers have intimidated the rest of us so much that we immediately assume that we can’t possibly know what they’re talking about, even when we can. ;)

    More seriously, I guess the problem is that a lot of people know what (projective) varieties (over C) are, and a lot of people know what homology is, but many of these people don’t know enough complex geometry to understand the (p,p) part, I guess?

  25. Lolo says:

    I agree on your last remark : the difficulty is to define what means (p,p) in the conjecture.

    But I agree with you on your remark before : for me it’s easier now to understand scheme theory….but it took me let’s say something like 4 years before I can say that; to understand the definition of a scheme and basic properties takes you a few months, but to be able to work with and manipulate them as if you had ever been living with took me a long time, which was not the case for differential manifolds where just after you’ve seen the definition your intuition is avaible (just because you’ve passed the preceding 4 years to work on differential calculus on open subsets of R and after R^n as an undergraduate, if you had done the same work
    in commutative algebra and functors theory then scheme theory would be immediatly accessible to your intuition).

  26. Algebraic Geometry Joe says:

    I’ve also taken a look at this paper, and I can confirm that it is garbage. It looks like the Hodge conjecture still stands. The paper is full of mistakes and the authors don’t really know what they’re writing about. It’s a shame that this made it onto the arxiv.

  27. Danny says:

    I agree that it looks like the authors don’t know what they’re talking about, but nothing in the paper is really new except the computer calculation and a statement about incompatible cup product structures. Otherwise it’s a poorly written survey of other people’s work. It’s well known that the Hodge conjecture implies the Kuga-Satake-Deligne correspondence, and the relationship between this and the clifford algebra stuff in the paper. I don’t think that the authors don’t know much Hodge theory, but maybe these computer scientists just found something numerical implied by the Hodge conjecture and wrote a program to find a counterexample.

  28. Bromskloss says:

    Yep, I too have problems with the video. I use Media Player Classic.

  29. Speculator says:

    “Algebraic Geometry Joe” and “Danny”:

    Could you please indicate what did you found wrong in the preprint?

    Since you state that “it is full of mistakes” this shouldn’t be difficult.

    For your information Kim and Roush are not “computer scientists”. They did disprove a few years ago Williams conjecture in Symbolic Dynamics, the main outstanding problem in the field (published in Annals of Math in 1999). Their publication list exhibits a long track of “problem solving”.

    Maybe it is just that Algebraic Geometers are not that good at finding counterexamples?

  30. Algebraic Geometry Joe says:

    I should have been more precise in my critisism. Certainly, it is possible that the Hodge conjecture has been disproved. However, the paper is badly written – there are many examples of this. Finding problems in the logic of the paper is very difficult when it makes little sense except to those who wrote it.

    Here’s one possible problem, though: Proposition 5.1 assumes the existance of a small deformation of a k3 surface with a transcendental lattice of constant rank. I was under the impression that this is impossible. For example math.AG/0011258.

    Having said this, whoever is clever enough to disprove the Hodge conjecture is also clever enough to write a paper that I find confusing. It’ll be interesting to see what consensus appears.

  31. Speculator says:

    This sounds different from “…it is garabage…full of mistakes…”.

    Don’t worry, there are such deformations….just take the constant one…

    Notice also that A=>B is still true if A is an empty condition…

  32. Clark says:

    I have to agree with “Algebraic Geometry Joe.” It’s at best very poorly written. Section 5 in particular (on which everything depends) doesn’t make much sense to me.

    But perhaps a more irritating problem with the paper is that in the proofs of the main results, the authors refer to propositions not contained within the paper, e.g., to “7.1,” despite the fact that their paper only has 6 sections. There are at least 3 examples of this.

    Certainly a more legible account would be helpful…

  33. Davis says:

    Certainly a more legible account would be helpful…

    Agreed. I tried to read it last night, and it’s not clearly written at all. And the authors really need to learn to use LaTeX’s theorem and proof environments.

  34. DA says:

    Hi Peter, You might remember me, we were Postdocs together
    on West Coast years ago, I was the only algebraic geometer at the
    time….

    I just thought I’d jump in to this discussion with a few comments. It would accurate to say Hodge suggested (rather than conjectured) various things in his 1950 ICM talk. Some of these thing are known to be false (e.g. a counter example to the Hodge conjecture for integer coefficients was found early on by Atiyah-Hirzebruch, later Grothendieck found a counterexample to the general Hodge conjecture in its original formulation).

    So I wouldn’t be altogether suprised if the Hodge conjecture in its present form were to fail also. But, unfortunately, this preprint seems pretty unclear in various places. For example, as people have pointed out here, prop 5.1 looks suspect. The statement itself is ambiguous, and the proof seems bogus. I don’t want to dismiss this paper
    outright (they may be on to something), but this kind of thing doesn’t inspire a lot of confidence.

  35. woit says:

    Hi DA,

    Of course I remember you, thanks a lot for the comment.

    I’ve also heard privately from other experts the same evaluation: prop. 5.1 may or may not be true, but the proof given doesn’t work.

  36. hack says:

    What? They withdrew their paper just because one of the arguments is wrong? Culture shock.

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