On the Hodge conjecture

[This is a guest post by Burt Totaro. This blog post was initially written in a different file format and converted using AI. — T.]

These are strange times for mathematicians. It now seems possible that AI companies will burn through vast resources in order to prove some new fact about the Hodge conjecture. I’d like to discuss the current status of the Hodge conjecture, in order to think about the value to the mathematical community of exploring such hard problems. Thanks to Terry Tao for suggesting this guest post.

The Hodge conjecture crystallizes a big mystery: the relation between topology and algebraic geometry, or (what amounts to the same) between real and complex geometry. In short, real submanifolds are flexible, whereas complex submanifolds are quite rigid, and it is a challenge to relate the two. The setting is a “smooth complex projective variety” , a complex manifold defined by algebraic equations. We want to understand the possible shapes of complex algebraic subspaces of . (By a famous result of Wei-Liang Chow (1949), every complex analytic submanifold of is in fact algebraic.) One could ask whether every real submanifold of can be moved continuously to a complex submanifold. That is far too optimistic. Still, to a first approximation, the Hodge conjecture predicts which real submanifolds can be moved continuously to a complex submanifold, in terms of whether certain integrals are zero. (More precisely: every Hodge class in the rational cohomology of should be the class of an algebraic cycle.) The conjecture goes back to William Hodge (1950).

Now suppose in some counterfactual world that the day after the conjecture was made some inhuman oracle told us that the answer was “yes” and the conjecture was considered “solved.” We can see, by examining our real world, the vast body of mathematical knowledge that would have been lost. In our world, what we want from a great conjecture is a challenge that inspires all kinds of other discoveries, even if the original question remains open. The Hodge conjecture has been that kind of conjecture for algebraic geometers.

A key piece of evidence for the Hodge conjecture is the “Lefschetz -theorem“, which says that the Hodge conjecture is true for algebraic cycles of codimension 1 (that is, of complex dimension , if has complex dimension ), and also for cycles of dimension 1. Solomon Lefschetz’s work was quite early (around 1924), well before the Hodge conjecture was stated in general. Interestingly, some of the most important proofs by both Lefschetz and Hodge have gaps, from our current perspective. They had tremendous insight, but the full justification of their results required hard analysis by many mathematicians, notably Kunihiko Kodaira (around 1950).

By some measures, one could say that progress on the Hodge conjecture has been limited. In view of Lefschetz’s results, the first open case is for codimension-2 cycles on a variety of complex dimension 4, and that case (for arbitrary varieties of dimension 4) still seems far out of reach. But the Hodge conjecture has been enormously successful for inspiring new mathematical theories, such as Phillip Griffiths’s “Variations of Hodge structure” (starting in the 1960s) and Pierre Deligne’s “absolute Hodge cycles“. Broadly speaking, the Hodge conjecture suggests that the structure of algebraic cycles should be controlled by Hodge theory, in other words by integrals of algebraic functions. This hope has led to vast numbers of proved insights about the structure of algebraic cycles. (For example, Griffiths disproved Grothendieck’s conjecture that algebraic and homological equivalence were the same, which was a surprise; but he used Hodge theory in order to do it.) In recent decades, Claire Voisin has been a leader in using Hodge theory in new ways to get information about algebraic cycles.

An exciting recent development is Eyal Markman’s 2025 proof of the Hodge conjecture for abelian varieties of dimension 4 and 5. (Abelian varieties, the tori with a complex algebraic structure, are very special compared to all algebraic varieties; but they are a particularly important class in many ways, for example in number theory.) This is a monumental piece of work that builds on many earlier developments. André Weil (1977) identified a class of abelian varieties, those of Weil type, for which there are “unexpected” Hodge classes that do not appear on most other abelian varieties. In low dimensions such as 4, Ben Moonen and Yuri Zarhin (1995) showed that the Hodge conjecture for all abelian varieties would follow from the special case of abelian varieties of Weil type. Spencer Bloch (1972), extended by Ragnar-Olaf Buchweitz and Hubert Flenner (2003), defined a property called “semi-regularity” which, when it holds, allows proving the Hodge conjecture for a continuous family of varieties when it holds for one variety in the family.

For a long time, however, semi-regularity seemed far too strong a condition ever to apply to hard cases of the Hodge conjecture. Markman found, with great ingenuity, how to construct enough examples of semi-regular sheaves to prove the Hodge conjecture for abelian varieties in dimensions 4 and 5. His ideas use several big theories that have been developed in recent decades, notably about derived equivalences between algebraic varieties and about hyperkähler manifolds. In the end, Markman actually needed an extension of Buchweitz–Flenner’s semi-regularity theorem to “twisted sheaves”, which was supplied by Jonathan Pridham (2024).

At this point, I am confident that with enough effort, it will be possible to push Markman’s ideas further. But an AI-generated proof of some fact about the Hodge conjecture will not be satisfying by itself. The great power of mathematical ideas comes from a continual negotiation among people, as I’ve tried to indicate. A new paper should be taking part in a coherent conversation, not just piling up facts.