Open problem, Geometry, Algebra, posed 1950

Hodge conjecture

OpenMillennium

Let $X$ be a smooth projective algebraic variety over $\mathbb{C}$ . Every rational Hodge class — a class in $H^{2p}(X,\mathbb{Q}) \cap H^{p,p}(X)$ coming from the Hodge decomposition of complex cohomology — is a $\mathbb{Q}$ -linear combination of the cohomology classes of algebraic subvarieties of $X$ of codimension $p$ .

Research frontier as of 2026

As of 2026 the Hodge conjecture is open in general. The codimension-1 case is fully settled by the Lefschetz $(1,1)$ theorem (1924), and codimension $\dim X$ is trivial by Poincaré duality. For abelian varieties — long a key testing ground because of their rich endomorphism structure — the conjecture is now known in every dimension up to 5: Tate handled powers of a single elliptic curve, Tankeev and Ribet handled simple abelian varieties of prime dimension, Deligne proved it for varieties of CM type via his theory of absolute Hodge cycles, and in 2025 Markman completed dimensions 4 and 5 by constructing algebraic cycles for the previously mysterious Weil classes using secant sheaves. Beyond dimension 5, and for general (non-abelian) varieties beyond low dimension, no comparable technique is known, and the integral refinement is already known to fail (Atiyah–Hirzebruch, 1962), so any proof must produce actual algebraic cycles with rational, not just topological, coefficients.

Best known results

The Lefschetz $(1,1)$ theorem settles codimension 1 for every smooth projective variety (1924).
The conjecture is fully proved for abelian varieties of dimension up to 5 (Tate; Tankeev–Ribet; Deligne; Markman 2025, arXiv:2502.03415).
Atiyah and Hirzebruch (1962) pin down exactly why the naive integral version fails, via torsion obstructions — clarifying what the correct (rational) statement must be.

Tools and where they stop

Tool	Achieved	Where it stops
Exponential sheaf sequence / Lefschetz $(1,1)$ theorem	Proves every codimension-1 Hodge class is algebraic, for any smooth projective (even compact Kähler) variety, by relating line bundles to Chern classes.	The argument is fundamentally about line bundles and does not generalize to algebraic cycles of higher codimension.
Absolute Hodge cycles (Deligne)	Shows Hodge classes on abelian varieties of CM type are algebraic, and underlies most of the known cases for abelian varieties.	Relies on the extra symmetry of complex multiplication or prime dimension; a generic abelian variety, let alone a general algebraic variety, has no such structure to exploit.
Secant sheaves on Weil-type abelian varieties (Markman)	Constructs explicit algebraic cycles representing the previously inaccessible Weil classes, completing dimensions 4 and 5 (2025).	Tied to the specific geometry of abelian varieties; no analogous construction is known for Weil classes in dimension 6 and above, or for non-abelian varieties.

Open questions

Does Markman's secant-sheaf construction extend to Weil classes on abelian varieties of dimension 6 and higher?
Given that the integral Hodge conjecture fails, is there any structural reason to expect the rational Hodge conjecture to fail for some exotic non-abelian variety, or is a uniform proof plausible?

References

William V. D. Hodge (1952). The topological invariants of algebraic varieties
Michael F. Atiyah, Friedrich Hirzebruch (1962). Analytic cycles on complex manifolds · DOI:10.1016/0040-9383(62)90094-0
Eyal Markman (2025). Cycles on abelian 2n-folds of Weil type from secant sheaves on abelian n-folds · arXiv:2502.03415 [preprint, not peer-reviewed]
Claire Voisin (2002). Hodge Theory and Complex Algebraic Geometry I