The Hodge conjecture asserts that, for smooth projective complex algebraic varieties , every Hodge cycle is a rational linear combination of cohomology
classes of algebraic cycles .
The separate claim for products of projective complex K3 surfaces (OpenAI 2026f) was withdrawn on Oct. 7, 2026. A sign error invalidated
a cancellation argument in a paper on Weil classes
and the construction used by the dependent Kuga-Satake and K3-product manuscripts
(OpenAI 2026d). This withdraws the claimed proofs and does
not give a counterexample to the Hodge conjecture.
An unconditional family of positive cases is supplied by Floccari and Fu (2026). They proved the Hodge conjecture for every projective algebraic
variety deformation equivalent to a generalized
Kummer variety of complex dimension prime , and for every Cartesian
product of finitely many copies of that variety. For even Lefschetz standard conjecture
for the variety.
See also Abelian Variety ,
Algebraic Cycle ,
Algebraic Variety ,
Diagonal
Complete Intersection ,
Generalized
Hodge Conjecture ,
Generalized Kummer
Variety ,
Hodge Cycle ,
K3
Surface ,
Kuga-Satake Correspondence ,
Lefschetz Standard Conjecture ,
Millennium Prize Problems ,
Milne's
Rationality Conjecture ,
Special Subvariety ,
Weil Class
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References Clay Mathematics Institute. "Hodge Conjecture." Accessed 11 Sep 2026. https://www.claymath.org/millennium/hodge-conjecture/ . Deligne,
P. "The Hodge Conjecture." https://www.claymath.org/wp-content/uploads/2022/06/hodge.pdf . Floccari,
S. and Fu, L. "On the Hodge Conjecture for Hyper-Kähler Varieties of Generalized
Kummer Type of Dimension Divisible by Four." 22 Oct 2026. https://arxiv.org/abs/2610.31740 . Grothendieck,
A. "Hodge's General Conjecture Is False for Trivial Reasons." Topology 8 ,
299-303, 1969. 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. OpenAI.
"History." Oct. 7, 2026d. https://github.com/openai/math/blob/main/history.md . OpenAI.
"The Rational Hodge Conjecture for Products of K3 Surfaces." Oct. 4,
2026f. https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-rational-Hodge-conjecture-for-products-of-K3-surfaces-October-4-2026/hodge-conjecture-products-k3.pdf . Referenced
on Wolfram|Alpha Hodge Conjecture
Cite this as:
Weisstein, Eric W. "Hodge Conjecture."
From MathWorld https://mathworld.wolfram.com/HodgeConjecture.html
Subject classifications More... Less...
, where 
and 
is prime, and for every Cartesian
product of finitely many copies of that variety. For even 
, Floccari and Fu (2026) proved the same conclusion assuming
the Lefschetz standard conjecture
for the variety.
