Conjectures
Hodge Conjecture
HOJ kon-JEK-cher (rhymes with lodge; named for William Vallance Douglas Hodge)
Geometry
Citation Formats
General Reference
APA Style
BibTeX
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 Cross-Tradition 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.
The Atiyah-Hirzebruch theorem constrains the Hodge conjecture, showing it cannot hold at the level of integral cohomology classes.
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.
Posed By
Sources
1. Clay Mathematics Institute
Clay Mathematics Institutehttps://www.claymath.org/wp-content/uploads/2022/06/hodge.pdfQuote, https://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.
1. Clay Mathematics Institute
Clay Mathematics InstituteAssociated With: Michael Atiyah, Section 2, Remark (iv), p. 2Quote, Associated With: Michael Atiyah, Section 2, Remark (iv), p. 2
the Atiyah and Hirzebruch theorem [2] that the Hodge conjecture cannot hold integrally
1. Clay Mathematics Institute
Clay Mathematics InstituteAssociated With: Alexander Grothendieck, Section 2, Remark (vi), p. 2Quote, 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].
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.
Hodge Conjecture (Wikipedia)
Wikimedia FoundationIn Branch: Geometry, Opening paragraphQuote, In Branch: Geometry, Opening paragraph
the Hodge conjecture is a major unsolved problem in algebraic geometry and complex geometry
Hodge Conjecture (Wikipedia)
Wikimedia FoundationIn Branch: Algebra, Opening paragraphQuote, In Branch: Algebra, Opening paragraph
a major unsolved problem in algebraic geometry and complex geometry
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
Reader Challenges (0 open reader challenges)
No disputes yet. Spotted an error or a better source? Open the first one.
Sign in to dispute this or suggest a correction.
View At A Past Year
The atlas records no dated fact of its own for this entry, so there is no other year to choose.