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.