Hodge Conjecture
Is every Hodge class on a smooth projective variety a combination of algebraic cycles? Proved only in special cases, such as abelian fourfolds in 2025.
Open in the interactive tree →The Hodge conjecture says that certain cohomology classes on a smooth complex projective variety, the Hodge classes, are always rational combinations of algebraic cycles, the shapes cut out by polynomial equations. For cycles of codimension one this is the Lefschetz (1,1) theorem; apart from that and a few sporadic results, little was known until recently. Clay prize: one million US dollars; Pierre Deligne wrote the official problem description.
As of October 2026
October 2026: unsolved. Eyal Markman proved the conjecture for abelian fourfolds in 2025 (arXiv 2502.03415, after his earlier proof for the discriminant-1 Weil case via hyper-Kähler varieties of generalized Kummer type); organisers of a June 2026 workshop in France called it arguably the most significant advance since the conjecture was formulated. Floccari and Fu (April 2025) and van Geemen and Rapagnetta (20 July 2026) posted further preprints with new proofs for discriminant 1. On 10 September 2026 OpenAI told the New York Times it had made substantial progress on an unnamed Millennium problem; it has not said which problem, and nothing has been verified.
What is missing
- A method that works for all varieties, not only abelian ones with extra structure such as Weil type
- Any general construction of algebraic cycles from Hodge classes; the 2025 proofs rely on hyper-Kähler geometry special to those varieties
- Proofs beyond codimension one for other families such as K3 products and Calabi-Yau varieties
- Independent verification and, ideally, machine-checked versions of the new abelian-fourfold proofs
Becomes possible once solved
- A way to tell from topology and analysis which shapes polynomial equations can cut out
- Progress on the arithmetic counterpart, the Tate conjecture
- Stronger bridges between algebraic geometry, topology and number theory
Open steps
- Higher-dimensional abelian varieties Low AI leverageExtend the 2025 abelian-fourfold results to Weil-type abelian varieties of higher dimension, where Hodge classes are known only in sporadic cases.
- Lean proof of the fourfold case Medium AI leverageFormalize Markman's proof and the independent discriminant-1 proofs in Lean, including deformation of twisted sheaves on hyper-Kähler varieties.
- Other families: K3, Calabi-Yau Low AI leverageProve the conjecture for further families, such as K3 products or hyper-Kähler varieties of known types, using moduli and deformation arguments like those used for Weil fourfolds.
- Search for a counterexample Medium AI leverageTest the conjecture on explicit varieties with unusual Hodge structure, looking for a Hodge class that provably cannot be algebraic; none is known.
Where AI could help
Low AI leverage. A proof needs new geometric ideas; AI can formalize and check long arguments and search examples, but no confirmed AI result on Hodge exists.
- Formalize long algebraic-geometry proofs, such as Markman's, in Lean
- Test candidate cycles on explicit varieties with computer algebra
- Survey which special cases still lack proofs
- Check long arguments and flag gaps for human experts
Prerequisites
- Geometry (Euclid)~300 BC
- Calculus (Analysis)1684
- Number Theory & Group Theory1801
- Riemannian Geometry1854
- Topology1895The Hodge conjecture links algebraic geometry to the topology of cohomology classes