André Weil and the Hodge Conjecture | Articles

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André Weil and the Hodge Conjecture
Jiahao Cao, Munich
It is said that OpenAI may very possibly have constructed a counterexample to the Hodge conjecture.André Weil was one of the earliest mathematicians to doubt the Hodge conjecture and to attempt to construct counterexamples, and perhaps also one of those who came closest to doing so successfully.
Weil actually had little direct interaction with Hodge himself. The earliest record seems to be from 1947, when Weil mentioned in a letter to H. Cartan that he was studying Hodge’s book and trying to apply it to algebraic varieties.
Weil strongly recognized the importance of Hodge theory, but at the same time sharply criticized its notation as looking like “a horrible salad of tensors.”
Ten years later, Weil wrote Introduction à l’étude des variétés kählériennes, rewriting and organizing the Kähler–Hodge techniques in a cleaner language, while stating that this was done to “facilitate the reader’s understanding.”
The notation and organization of Hodge theory in its modern form were to a large extent fixed in Weil’s book. Following Weil’s passing, the book was singled out in his obituary as the first modern exposition of Hodge theory.
Then came the famous first postwar Fields Medal. Hodge was among the members of the committee who most strongly supported awarding it to Weil, warning that otherwise they “might be shirking our duty.” His efforts ultimately did not prevail. Perhaps to spare future committees from facing quite the same dilemma, the Fields Medal would later acquire its famous age limit of forty.
At the International Congress of Mathematicians where the prizes were awarded, Weil gave a plenary lecture presenting a unified perspective on number theory and algebraic geometry, summarizing and anticipating much of the development of algebraic geometry over the following decades. Five days later, Hodge presented the original version of the Hodge conjecture in another lecture.

The early Hodge conjecture looked like a natural extrapolation of Lefschetz’s (1,1)-theorem, so there were not many reasons to doubt it.
Lefschetz had already solved the case (p=1), and degree-2 Hodge classes all come from divisors. Therefore, if the entire Hodge ring of a variety were generated by degree-2 Hodge classes, the Hodge conjecture would follow automatically.
In the 1960s, Mumford and Tate tried precisely to pursue this route. Unfortunately, Mumford found a counterexample that cut off this approach, namely the so-called exceptional Hodge classes. The example is a CM-type abelian fourfold (A) possessing Hodge ((2,2)) classes that cannot be written as products of divisors.
What happened next is known from a letter Tate wrote to Serre.
In 1965, Tate told Weil about this counterexample. Weil quickly realized that this example was essentially only a special case of a 4-dimensional family of examples.
He then considered a class of special abelian varieties with an additional imaginary quadratic field symmetry. This symmetry divides the complex directions evenly into two groups. Taking the product of these directions over the imaginary quadratic field gives a two-dimensional rational cohomology subspace. The fact that the two groups have equal size ensures that this space is of ((n,n))-type, and therefore it is a two-dimensional Hodge class space: what later became known as the Weil classes.
These Weil classes arise automatically from symmetry and linear-algebraic structure, without any need to know in advance of a corresponding algebraic subvariety. Weil therefore began to doubt the conjecture: if these classes can be generated automatically in this way by symmetry, why must they necessarily correspond to genuine algebraic cycles? If even one Weil class could be shown not to correspond to any algebraic cycle, the Hodge conjecture would be disproved. In this way, a counterexample to one possible proof strategy was generalized by Weil into a possible counterexample to the conjecture itself.
Weil therefore said to Tate:

As you and Mumford seem to believe Hodge’s conjecture, it is now up to you to exhibit algebraic cycles corresponding to these abnormal classes. As I incline to disbelieve it, I shall rather attempt to show that there is no such cycle.

Weil then began trying to construct a counterexample. Of course, his efforts did not succeed; otherwise the conjecture would not later have become one of the Millennium Prize Problems.
This was much like Weil’s work on the Mordell/Riemann conjectures, and even on the Fermat and Langlands conjectures. Faced with these great conjectures, he could either make an initial breakthrough that no one had achieved for a century, or immediately recognize the importance of a conjecture, or of a particular route toward its solution, for the future development of mathematics, and thereby help direct the course of the subject as a whole. Although in the end he was usually not the final solver of these problems, Weil may be precisely the kind of mathematician most needed in the AI era.

Later still, according to Benedict Gross’s recollection, at a conference celebrating Ahlfors’s seventieth birthday about ten years later, Weil was still publicly discussing, in a rather provocative way, how one might attack the Hodge conjecture.
Gross, who was still at an early stage of his career, sat in the audience trying to modify Weil’s ideas in order to prove a period identity related to the Chowla–Selberg formula, and soon turned the argument into a paper.
At the same conference, Serre introduced him to Deligne, and Gross then explained this result obtained by adapting Weil’s ideas.
Deligne was astonished and asked Gross: “Have you proved the Hodge conjecture?”
Gross cheerfully replied that if that were really the case, he would certainly be pleased. He was still looking for a PhD thesis topic, and proving the Hodge conjecture should presumably be enough for a thesis.
A few weeks later, Deligne sent Gross a three-page manuscript. That manuscript eventually developed into the famous result that all Hodge cycles on abelian varieties are absolutely Hodge.

When Milne discussed this history, he remarked that in relation to Weil’s doubts, abelian varieties should originally have counted as one of the easier cases of the Hodge conjecture.
Yet after fifty years, the mathematical community still could not prove that these Weil Hodge classes are algebraic. This obviously did a great deal to weaken mathematicians’ confidence in the Hodge conjecture……
As for what came later in the history of the Hodge conjecture—Grothendieck, Deligne, Griffiths, and the rest—that will have to be added another time.
Finally, regarding the possibility that OpenAI may have constructed a counterexample to the Hodge conjecture, Weil’s own words may be especially appropriate:

“La question que pose la “conjecture de Hodge” est bien naturelle… Par malheur, en d´epit du mot de “conjecture”, il n’y a, que je sache, pas l’ombre d’une raison d’y croire; on rendrait service aux g´eom`etres si l’on pouvait trancher la question au moyen d’un contre-exemple.”
“The question posed by the ‘Hodge conjecture’ is a very natural one… Unfortunately, despite the word ‘conjecture’, as far as I know there is not the slightest reason to believe it; one would be doing geometers a service if the question could be settled by means of a counterexample.”

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André Weil was among the earliest mathematicians to express doubt regarding the Hodge conjecture and to attempt the construction of counterexamples. His relationship with Hodge was complex; although he recognized the importance of Hodge theory, he sharply criticized its notation as appearing disorganized, describing it as a "horrible salad of tensors." Ten years later, Weil revised and organized Kähler-Hodge techniques in his book Introduction à l’étude des variétés kählériennes, aiming to facilitate the understanding of the subject. Following his death, this book was recognized as a modern exposition of Hodge theory.

Weil participated in the formal recognition of Hodge's work, as he supported the awarding of the Fields Medal to Hodge, though this effort did not ultimately succeed. At the International Congress of Mathematicians, Weil presented a unified perspective bridging number theory and algebraic geometry, anticipating much of the subsequent development in algebraic geometry. The initial formulation of the Hodge conjecture seemed intuitive, following naturally from Lefschetz’s (1,1)-theorem, as degree-2 Hodge classes are known to correspond to divisors.

The pursuit of the conjecture faced obstacles when Mumford found exceptional Hodge classes, such as those on a CM-type abelian fourfold, which could not be expressed as products of divisors, halting this line of inquiry. In 1965, Tate presented Weil with such a counterexample, prompting Weil to consider a class of special abelian varieties with additional imaginary quadratic field symmetry. By taking the product of complex directions based on this symmetry, Weil identified a two-dimensional rational cohomology subspace, which he termed Weil classes. These classes automatically arose from symmetry and linear-algebraic structure, leading Weil to question whether they necessarily corresponded to genuine algebraic cycles. Weil suggested that if these classes could be generated automatically by symmetry, then finding a counterexample would be the most effective way to settle the conjecture.

Weil’s approach reflected a broader strategy when facing profound conjectures like the Mordell/Riemann or Langlands conjectures: either achieve a breakthrough, recognize the conjecture's importance, or devise a specific route toward its solution. Despite his efforts to construct a counterexample, Weil ultimately did not succeed. Later work by Gross, Serre, and Deligne resulted in the proof that all Hodge cycles on abelian varieties are absolutely Hodge. However, the persistent lack of a general proof that these Weil Hodge classes are algebraic has continued to weaken confidence in the conjecture among mathematicians. Weil’s own statement encapsulates his view on the matter: that if the conjecture is truly as natural as it seems, a counterexample would serve to help geometers rather than undermine their faith in the mathematical structure.