Math 2.0

Algebraic geometry Cohomology Conjecture

Hodge conjecture

Builds on Projective varietySmooth morphismSingular cohomologyHodge decompositionHodge structureAlgebraic cycle

Statement

Let XX be a smooth projective variety over ℂ\C, and let p≥0p \ge 0. Let Hp,p(X)H^{p,p}(X) be the (p,p)(p, p) part of the Hodge decomposition of H2p(X,ℂ)H^{2p}(X, \C). Here H2p(X,ℚ)H^{2p}(X, \Q) is singular cohomology, viewed inside H2p(X,ℂ)H^{2p}(X, \C). Let Hdgp(X)=H2p(X,ℚ)∩Hp,p(X)\mathrm{Hdg}^p(X) = H^{2p}(X, \Q) \cap H^{p,p}(X) be the group of rational Hodge classes. Then every class in Hdgp(X)\mathrm{Hdg}^p(X) is a ℚ\Q-linear combination of classes of closed subvarieties of XX of codimension pp. In other words, it is the class of an algebraic cycle with rational coefficients.

Remarks

The conjecture is open. It is one of the Millennium Prize Problems of the Clay Mathematics Institute. The class of a subvariety of codimension pp always lies in Hdgp(X)\mathrm{Hdg}^p(X), so the conjecture asks for the converse.

The case p=1p = 1 is true: it is the Lefschetz theorem on (1,1)(1,1)-classes, which follows from the exponential sequence. It shows that every class in H2(X,ℤ)H^{2}(X, \Z) whose image lies in H1,1H^{1,1} is the first Chern class of a line bundle. By the hard Lefschetz theorem, the case p=1p = 1 implies the case p=n−1p = n - 1, where n=dim⁡Xn = \dim X. So the conjecture holds for varieties of dimension at most 33.

The version with integer coefficients is false: Atiyah and Hirzebruch found torsion classes that are not algebraic, and Kollár found non-torsion counterexamples. The analogous statement for compact Kähler manifolds, with analytic cycles or Chern classes of coherent sheaves, is false by an example of Voisin.

Reading path

To understand this page from the ground up, read these 15 pages in order. The foundations come first.

Show the path as a graph
Algebraic cycle, concept, no evidence levelAlgebraic cycleconceptDifferential form, concept, no evidence levelDifferential formconceptHodge star operator, concept, no evidence levelHodge star operatorconceptHodge structure, concept, no evidence levelHodge structureconceptOrientation of a manifold, concept, no evidence levelOrientation of a mani…conceptProjective variety, concept, no evidence levelProjective varietyconceptRiemannian manifold, concept, no evidence levelRiemannian manifoldconceptSheaf cohomology, concept, no evidence levelSheaf cohomologyconceptSingular cohomology, concept, no evidence levelSingular cohomologyconceptSmooth morphism, concept, no evidence levelSmooth morphismconceptde Rham cohomology, concept, no evidence levelde Rham cohomologyconceptExterior derivative, concept, no evidence levelExterior derivativeconceptHodge theorem, statement, no evidence levelHodge theoremstatementHodge decomposition, statement, no evidence levelHodge decompositionstatementHodge conjecture, conjecture, no evidence levelHodge conjectureconjecture
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Algebraic cycle concept
  2. Differential form concept
  3. Hodge star operator concept
  4. Hodge structure concept
  5. Orientation of a manifold concept
  6. Projective variety concept
  7. Riemannian manifold concept
  8. Sheaf cohomology concept
  9. Singular cohomology concept
  10. Smooth morphism concept
  11. de Rham cohomology concept
  12. Exterior derivative concept
  13. Hodge theorem statement
  14. Hodge decomposition statement
  15. Hodge conjecture conjecture

Further reading

  1. Web https://en.wikipedia.org/wiki/Hodge_conjecture
About this page
TagXAEEDWV3PM
URIm2://XAEEDWV3PM
Kindconjecture
WritingW0 stub
MSC14C30, 14C25
StatusThis copy of the site has no build data. Run m2 status XAEEDWV3PM.
AboutAlgebraic cycle, Hodge structure
UsesProjective variety, Smooth morphism, Singular cohomology, Hodge decomposition, Hodge structure, Algebraic cycle
Same as
Changed2 November 2026

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