Conjecture
Hodge Conjecture
HOJ kon-JEK-cher (rhymes with lodge; named for William Vallance Douglas Hodge)
Geometry
One of the seven Clay Mathematics Institute Millennium Prize Problems, in algebraic geometry. William Vallance Douglas Hodge posed it at the 1950 International Congress of Mathematicians in Cambridge, Massachusetts, published in the proceedings in 1952. It asks whether every Hodge class, a special kind of cohomology class on a smooth complex projective algebraic variety, is really built out of the classes of actual algebraic subvarieties. The codimension one case is proved true, by the Lefschetz theorem on (1,1) classes as reproved cohomologically by Kunihiko Kodaira and Donald Spencer in 1953; Michael Atiyah and Friedrich Hirzebruch showed in 1962 that the natural stronger integral version of the conjecture is false, which is why the conjecture is stated with rational rather than integral coefficients; and Alexander Grothendieck showed in 1969 that a further generalization Hodge had also proposed is false for trivial reasons and gave a corrected version. The conjecture remains open in general.
Facts
StatementOn a smooth complex projective algebraic variety, every Hodge class is a rational linear combination of the cohomology classes of algebraic cycles. 1 Proposed Year Prize StatusOne of the seven Clay Mathematics Institute Millennium Prize Problems, named in 2000; a correct proof carries a one million dollar award. 1 Progress Toward ResolutionProved in codimension one (the Lefschetz (1,1) theorem, reproved cohomologically by Kodaira and Spencer, 1953). Known false if the algebraic cycle classes are required to combine with integer rather than rational coefficients (Atiyah and Hirzebruch, 1962). A further, stronger conjecture Hodge also proposed was shown false for trivial reasons by Grothendieck in 1969, who supplied a corrected version. Several concrete cases, including the diagonal cycle's Kunneth components, remain explicitly open. 1 Classification
Resolution Status Prize Status
Prize Status (category)Millennium Prize Problem 1 Connections
Associated With
Grothendieck showed that a further conjecture Hodge proposed alongside the Hodge conjecture proper was false for trivial reasons, and gave a corrected version of it.
Source Clay Mathematics Institute
The Atiyah-Hirzebruch theorem constrains the Hodge conjecture, showing it cannot hold at the level of integral cohomology classes.
Source Clay Mathematics Institute
In Branch
Precise field is algebraic geometry, the study of algebraic cycles on complex varieties; the atlas's taxonomy has no dedicated algebraic-geometry branch, so both its nearest neighbors are recorded.
Additional Source Hodge Conjecture (Wikipedia)Opening paragraph
Additional Source Hodge Conjecture (Wikipedia)Opening paragraph
Open Questions
Source Clay Mathematics Institute
Posed By
Source Wolfram MathWorld
Sources
1. Clay Mathematics Institute
Clay Mathematics Institutehttps://www.claymath.org/wp-content/uploads/2022/06/hodge.pdf
On a projective non-singular algebraic variety over C, any Hodge class is a rational linear combination of classes cl(Z) of algebraic cycles.
Associated With: Michael Atiyah, Section 2, Remark (iv), p. 2
the Atiyah and Hirzebruch theorem [2] that the Hodge conjecture cannot hold integrally
Associated With: Alexander Grothendieck, Section 2, Remark (vi), p. 2
Grothendieck observed that this further conjecture is trivially false, and gave a corrected version of it in [5].
View the Source Hodge Conjecture (Wikipedia)
Wikimedia FoundationLead section, resolution status
In mathematics, the Hodge conjecture is a major unsolved problem in algebraic geometry and complex geometry that relates the algebraic topology of a non-singular complex algebraic variety to its subvarieties.
In Branch: Geometry, Opening paragraph
the Hodge conjecture is a major unsolved problem in algebraic geometry and complex geometry
In Branch: Algebra, Opening paragraph
a major unsolved problem in algebraic geometry and complex geometry
View the Source Wolfram MathWorld
Wolfram Research, Inc.Posed By: William Vallance Douglas Hodge, References section, Hodge W.V.D. 1952 entryQuote, Posed By: William Vallance Douglas Hodge, References section, Hodge W.V.D. 1952 entry
Hodge, W. V. D. The Topological Invariants of Algebraic Varieties. Proc. Internat. Congress Math., Cambridge, Mass., 1950, Vol. 1. Providence, RI: Amer. Math. Soc., pp. 182-192, 1952.
Open Questions (1 open question)
Is every Hodge class on a smooth complex projective algebraic variety really a rational combination of the cohomology classes of actual algebraic subvarieties?
Proved only in codimension one, by the Lefschetz theorem on (1,1) classes. Higher codimension cases, including named examples such as the diagonal cycle's Kunneth components, remain open, and the natural integral version of the conjecture is known to be false.
What would resolve this A general proof for every codimension, or an explicit Hodge class shown not to be a combination of algebraic cycle classes.
Algebraic geometryClay Mathematics Institute
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