Tom Graber recently asked me why people believe the Hodge conjecture, given the sparse evidence for its truth. I didn’t have time then to answer fully (I was giving a talk), but it’s a question that deserves a full answer. So I’ve sketched below what I feel are the reasons for believing the Hodge conjecture.
The Hodge conjecture is perhaps the most famous problem in algebraic geometry. But progress on the Hodge conjecture is slow, and a lot of algebraic geometry goes in different directions from the Hodge conjecture. Why should we believe the Hodge conjecture? How important will it be to solve the problem?
The Hodge conjecture is about the relation between topology and algebraic geometry. The cohomology with complex coefficients of a smooth complex projective variety splits as a direct sum of linear subspaces, the Hodge decomposition H i(X,C) = Σj = 0i H j,i-j(X). The cohomology class of a complex subvariety of codimension p lies in the middle piece H p,p(X) of H 2p(X,C). The Hodge conjecture asserts that any element of H 2p(X,Q) which lies in the middle piece of the Hodge decomposition is the class of an algebraic cycle, meaning a Q‑linear combination of complex subvarieties.
The main evidence for the Hodge conjecture is the Lefschetz (1,1)-theorem, which implies the Hodge conjecture for codimension-1 cycles. Together with the hard Lefschetz theorem, this also implies the Hodge conjecture of cycles of dimension 1. These results are part of algebraic geometers’ good understanding of line bundles and codimension-one subvarieties.
Not much is known about the Hodge conjecture in other cases, starting with 2‑cycles on 4‑folds. For example, it holds for uniruled 4‑folds (Conte-Murre, 1978). That includes 4‑fold hypersurfaces of degree at most 5, but the Hodge conjecture remains unknown for smooth 4‑fold hypersurfaces of degree at least 6. Why should we believe the conjecture?
One reason to believe the Hodge conjecture is that it suggests a close relation between Hodge theory and algebraic cycles, and this hope has led to a long series of discoveries about algebraic cycles. For example, Griffiths used Hodge theory to show that homological and algebraic equivalence for algebraic cycles can be different. (That is, an algebraic cycle with rational coefficients can represent zero in cohomology without being connected to zero through a continuous family of algebraic cycles.) Mumford used Hodge theory to show that the Chow group of zero-cycles modulo rational equivalence can be infinite-dimensional. There are many more discoveries in the same spirit, many of them summarized in Voisin’s book Hodge Theory and Complex Algebraic Geometry.
Another reason for hope about the Hodge conjecture is that it is part of a wide family of conjectures about algebraic cycles. These conjectures add conviction to each other, and some of them have been proved, or checked for satisfying families of examples.
The closest analog is the Tate conjecture, which describes the image of algebraic cycles in etale cohomology for a smooth projective variety over a finitely generated field, as the space of cohomology classes fixed by the Galois group. The Tate conjecture is not known even for codimension-1 cycles. But Tate proved the Tate conjecture for codimension-1 cycles on abelian varieties over finite fields. Faltings proved the Tate conjecture for codimension-1 cycles on abelian varieties over number fields by a deep argument, part of his proof of the Mordell conjecture. An important piece of evidence for the Hodge conjecture is Deligne’s theorem that Hodge cycles on abelian varieties are “absolute Hodge”, meaning that they satisfy the arithmetic properties (Galois invariance) that algebraic cycles would satisfy. This means that the Hodge and Tate conjectures for abelian varieties are closely related.
The Tate conjecture belongs to a broad family of conjectures about algebraic cycles in an arithmetic context. These include the Birch–Swinnerton-Dyer conjecture, on the arithmetic of elliptic curves, and a vast generalization, the Bloch–Kato conjecture on special values of zeta functions. One relation among these conjectures is that the Birch–Swinnerton-Dyer conjecture for elliptic curves over global fields of positive characteristic is equivalent to the Tate conjecture for elliptic surfaces, by Tate. Some of the main advances in number theory over the past 30 years, by Kolyvagin and others, have proved the Birch–Swinnerton-Dyer conjecture for elliptic curves over the rationals of analytic rank at most 1.
The Hodge conjecture belongs to several other families of conjectures. There is Bloch’s conjecture that the Hodge theory of an algebraic surface should determine whether the Chow group of zero cycle is finite-dimensional. There is the Beilinson–Lichtenbaum conjecture, recently proved by Voevodsky and Rost, which asserts that certain motivic cohomology groups with finite coefficients map isomorphically to etale cohomology.
This web of conjectures mutually support each other. Mathematicians continually make progress on one or the other of them. Trying to prove them has led to a vast amount of progress in number theory, algebra, and algebraic geometry. For me, this is the best reason to believe the Hodge conjecture.
Geometry Bulletin Board 
Thanks alot for the post.
I have tried several times to approach those beliefs and have always found it really vexing. Between the overloading of proper names (Tate, Hodge…) and the mixing of number theory and topology/geometry, it’s really tough to make oneself a clear picture of what is known, what is not, and what is believed (and why).
So your post is very highly welcome, and leaves me yearning for more. Or actually maybe not so much. It seems that it may be an illusion, if I had a short exhaustive roadmap to motivic conjectures, I think I would only be fooling myself that I understand things before actually looking at all the details.
Still, a couple of questions:
When you write “for elliptic surfaces, by Tate”, you mean Tate proved his conjecture for elliptic surfaces? Is it known now for all surfaces now? Threefolds? Abelian varieties? I guess there is a good reference for this, what is it?
It seems to me that abelian varieties are a crucial component of our understanding of the Hodge and Tate conjectures but I cannot see further, the details of this. Do the Hodge and Tate conjectures (together perhaps) for abelian varieties (at least conjecturally) imply the general case?
I would like to be able to comment more on this. I will think. Thanks again.
Tate and Milne proved the equivalence of two problems, the Tate conjecture for elliptic surfaces over finite fields and the Birch-Swinnerton-Dyer conjecture for elliptic curves over global fields of positive characteristic. Both problems remain open. See for example Ulmer’s notes on elliptic curves over function fields.
In particular, the Tate conjecture is not known for surfaces over finite fields. As I mentioned, Tate proved the Tate conjecture for codimension-1 cycles on abelian varieties over finite fields, but it is not known for cycles of arbitrary dimension on abelian varieties over finite fields. See for example Milne’s notes on the Tate conjecture.
There is no reason to think that the Hodge or Tate conjecture for abelian varieties should imply the general case. Rather, these problems should be much easier for abelian varieties than
for varieties in general.
Among the good books in this area are Voisin’s Hodge Theory and Complex Algebraic Geometry and Andre’s Une introduction aux motifs.
Regarding your remark on the Tate conjecture for abelian varieties not implying it for all varieties. Here is a comment of Milne:
Actually it seems the Tate conjecture for abelian varieties does imply it in general. Kahn remarks in his Handbook of K-theory survey that this together with Beilinson’s conjecture (that rational and numerical equivalence coincide over a finite field, taking cycles with Q-coefficients), implies that Voevodsky’s triangulated category of mixed motives over a finite field k (in the Nisnevich topology) with rational coefficient (and perhaps also over an arbitrary scheme of finite type over a field of positive characteristic) is the derived category of pure motives over k, with morphisms tensored with Q. So the canonical t-structure on D(PureMotives_Q) gives a motivic t-structure and mixed motives with Q-coefficients over k are actually pure. He also remarks that taking finer topologies than the Nisnevich should give a t-structure directly for DM, but I am not sure he thinks this would also be equivalent to D(PureMotives) without tensoring with Q. Probably not in fact, looking into Milne’s article, remark 2.50, it seems this is related to the Bloch-Kato or Quillen-Lichtenbaum conjectures -I’d have to think about it more.
I think Milne proved all that in his “Motives over finite fields” article in the first volume of Seattle’s Motives conference. I looked at the article a little and it looks awesome, but I have not understood much so far, so if anybody has related references that could help. I will try later, looking at Milne’s website too.
No, the Tate conjecture for abelian varieties over a finite field F_q is not known to imply it for arbitrary smooth projective varieties over F_q. The point is that, by the Weil conjectures, the eigenvalues of Frobenius on the cohomology of any variety over F_q are equal to eigenvalues of Frobenius on some abelian variety over F_q. That suggests that every motive over F_q should be a summand of the motive of an abelian variety. But in order to prove that, you would need the Tate conjecture on arbitrary varieties, not just on abelian varieties. (You need the Tate conjecture to go from an isomorphism between the cohomology of two varieties as Galois modules to an algebraic correspondence between them.)
Thanks alot, and I am very sorry for my mistakes. Your reply is really great and I think (hope) it is definitely worth it that you spend some time correcting such mistakes.
Just in case others read this, to be clear, Milne in his motives article does assume the Tate conjecture for all smooth projective varieties over the base finite field. So does Kahn in his articles.
It’s a great post. Can you say more on this? Like why believe motives?
Well, some versions of the category of motives can be defined now
(Grothendieck motives, Chow motives, and so on). Many mathematicians find motives useful. (I have to admit that I haven’t used the language of motives much myself.) They provide a language for describing the relation between different algebraic varieties. I think the question about “believing motives” refers to whether a given category of motives has all the properties that we expect. For example, the category of Grothendieck motives would have especially good properties if Grothendieck’s standard conjectures hold. These conjectures would follow from the Hodge conjecture, so I certainly believe that they are true. But you can use the language of motives to analyze particular varieties without knowing these conjectures in general.
One reference on these ideas is Yves Andre’s book Une introduction aux motifs (SMF, 2004). I also like the 2-volume conference proceedings Motives (AMS, 1994), which includes a lot of expository papers.
Can you say more specifically, what compelling evidence is there for the Hodge conjecture? I am afraid that some mathematicians might not see a web of related conjectures as a reason to “believe” the Hodge conjecture.
Fair enough, Jason. Part of what I’m talking about is the psychology of mathematical research. It’s useful to believe the web of conjectures about algebraic cycles, as a way to encourage yourself to prove parts of these conjectures. If you take the opposite point of view, that “nobody knows anything about algebraic cycles”, then you may be discouraged from working in the area at all. And that would be a shame. There have been enough spectacular developments in the area that I think there is plenty of hope for further advances on the big conjectures. (Some of my favorites are Tate’s proof of the Tate conjecture for divisors on abelian varieties over finite fields, Faltings’s proof of the same statement over number fields, and the recent advances on the Tate conjecture for K3 surfaces over finite fields by Maulik and Madapusi Pera which I expect to yield a complete proof soon.)
Pingback: Fifteenth Linkfest
Here is one possible take on “why believe in the Hodge conjecture” and “why believe in motives”. Take the Tannakian viewpoint. The category of Grothendieck (pure) motives over a field k is conjectured to be Tannakian, yielding a motivic Galois group(oid). Specifically, take k = C. The standard conjecture necessary to have this is that homological and numerical equivalences agree on all smooth projective C-varieties (I’ll take homological equivalence with respect to Betti cohomology). This is an open question, but it is known for motives which are twisted direct summands of motives of abelian varieties, because Lieberman proved this standard conjecture for abelian varieties over C. So, if we restrict ourselves to this rigid subcategory, we do have a well-defined motivic Galois group Mot(ab), which is a proreductive algebraic group over Q.
Now, if you take the Tannakian category of pure, polarisable Q-Hodge structures, you get another proreductive algebraic group MT; when you restrict to the Tannakian subcategory generated by pol. Hodge structures of type (1,0) + (0,1), you get a quotient MT(ab), and the functor which to a variety associates its Hodge cohomology gives you a morphism
(*) MT(ab)\to Mot(ab).
The Hodge conjecture, restricted to complex abelian varieties, is equivalent to saying that this morphism is an isomorphism. (In fact, the Hodge conjecture would imply that MT\to Mot would be epi or faithfully flat granting the existence of the “big” motivic Galois group Mot, hence that (*) is also epi; but on the other hand, any pol. Hodge structure of type (1,0) + (0,1) comes from an abelian variety, and this provides “isomorphism”.)
Now, one can examine (*) “case by case” and look at morphisms
(*A) MT(A)\to Mot(A)
for abelian varieties A, where MT(A) is the Mumford-Tate group of A and Mot(A) is the quotient of Mot(ab) corresponding to the Tannakian subcategory of motives generated by the motive of A. This is now a homomorphism of (finitely generated) reductive groups, which is mono because both groups sit in GL(H^1(A))\times G_m. of course, (*) is equivalent to (*A) for all A. In many cases, one succeeds in proving the full Hodge conjecture for A and all its powers by just looking at the structure of MT(A), and eliminating cases. For example, products of elliptic curves, and a number of isolated examples of simple abelian varieties A.
On the other hand, it is very hard to put these examples together and there is no general argument to say that if (*A) and (*B) are true, then (*A\times B) is true. (The converse is OK.) So, imagine that the Hodge conjecture is true for certain A’s, and false for certain others. This would create such a messy picture that I find it compelling enough empirically!
I agree. Some people prefer to look for counterexamples to the Hodge conjecture, which is fine. For me, the study of algebraic cycles gains fascination from the possibility of a grand unifying story. There are plenty of other problems, in algebraic geometry or other subjects, where things just seem complicated, with no big pattern. For me, believing in the Hodge conjecture is like believing in progress.