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The Millennium Museum·Algebraic geometry

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The Hodge Conjecture

On a non-singular complex projective variety, is every rational Hodge class a rational combination of classes of algebraic cycles?

Open. A Lean statement is being drafted in Formal Conjectures.

The exhibit

The problem, three ways

Curious

Some shapes in geometry are carved out by polynomial equations; others are known only through their holes and symmetries. The Hodge conjecture says that a particular large family of those topological features always comes from genuine algebraic shapes, tying topology to algebra.

Undergraduate

For a smooth complex projective variety X, the classes in H^(2k)(X, ℚ) that are also of Hodge type (k, k) should be exactly the rational combinations of classes of algebraic subvarieties of codimension k. The case k = 1 is the Lefschetz (1,1) theorem.

Specialist

Known cases include divisors, and curves by hard Lefschetz. The integral version is false (Atiyah and Hirzebruch), and the analogue for compact Kähler manifolds fails (Voisin). Cattani, Deligne and Kaplan proved that Hodge loci are algebraic, as the conjecture predicts.

What would count

  • A proof or counterexample to the conjecture with rational coefficients for smooth complex projective varieties, as in Pierre Deligne's Clay description.
  • Clay's conditions: publication in a qualifying outlet, two years, general acceptance, then a decision by its board.

Fidelity traps

  • With integer coefficients the statement is known to be false, so an integral counterexample says nothing.
  • Compact Kähler manifolds that are not projective are outside the conjecture.
  • A formal statement depends on Hodge theory that Mathlib is still building; one is being drafted in Formal Conjectures.

In the register

Formal record: unclassified

No Lean statement is held yet.

No AI contribution is recorded against this problem.

The full record →

Who is attacking it

  • Formal Conjectures: Drafting Lean statements for Hodge, Birch and Swinnerton-Dyer, and Yang–Mills. Source
  • OpenAI: Says it began large-scale work on all the open Millennium problems on 1 September 2026. Source

The history

8 events
  1. 1920s
  2. 1924

    progress

    The Lefschetz (1,1) theorem

    Solomon Lefschetz proves the case of divisors, decades before the conjecture is stated.

    Source

  3. 1950s
  4. 1950

    posed

    Hodge at the ICM

    W. V. D. Hodge presents the conjecture at the International Congress of Mathematicians in Cambridge, Massachusetts.

    Source

  5. 1960s
  6. 1962

    barrier

    The integral version fails

    Atiyah and Hirzebruch show that the statement with integer coefficients is false.

    Source

  7. 1990s
  8. 1995

    progress

    Hodge loci are algebraic

    Cattani, Deligne and Kaplan prove that Hodge loci are algebraic, as the conjecture predicts.

    Source

  9. 2000s
  10. 2000-05-24

    prize

    A Millennium Prize Problem

    The Clay Mathematics Institute names the conjecture one of seven carrying a $1,000,000 prize, with an official description by Pierre Deligne.

    Source

  11. 2002

    barrier

    The Kähler analogue fails

    Claire Voisin shows that the natural generalisation to compact Kähler manifolds is false.

    Source

  12. 2026
  13. 2026-09

    formal

    A Lean statement in draft

    Formal Conjectures begins drafting Lean statements for the Hodge conjecture, Birch and Swinnerton-Dyer, and Yang–Mills.

    Source

  14. 2026-09-01

    AI

    OpenAI turns to the Millennium problems

    OpenAI says it began large-scale work on all the open Millennium problems.

    Source

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