open problem

The Hodge Conjecture

Whether certain de Rham cohomology classes on projective varieties are algebraic. A Clay Millennium Prize problem.

7 machine-reviewed papers whose extracted claims name this problem. Matching is literal, over the claim graph, so every listing traces to a sentence in a review.

This is the review hub: papers that name the problem, with the machine's verdict on each. The claims shelf lists the individual claims themselves, formal (Lean) or stated.

Share X Bluesky LinkedIn Reddit HN

ACCEPT Cycles on abelian 2n-folds of Weil type from secant sheaves on abelian n-folds
A new proof shows Weil classes on abelian sixfolds of discriminant -1 are algebraic, which implies the Hodge conjecture for abelian fourfolds.

arxiv:2502.03415 · math.AG · 2025-02
UNVERDICTED Varieties with representable CH₀-group and a question of Colliot-Th\'{e}l\`{e}ne
Voisin builds a variety with representable CH_0 and no universal 0-cycle, answering Colliot-Thélène.

arxiv:2508.02331 · math.AG · 2025-08
UNVERDICTED A Conditional Reduction of the Rational Hodge Conjecture for Threefolds and Deformation-Theoretic Verifications in Several Families
Under a proposed Hypothesis BB, every codimension-2 Hodge class on a smooth projective threefold is a specialization of complete-intersection curves, which would imply the rational Hodge conjecture for such threefolds.

arxiv:2508.08321 · math.AG · 2025-08
UNVERDICTED The semiregularity theorem for equivariant noncommutative varieties
Semiregularity extends to equivariant noncommutative varieties, implying geometric origin of many invariant categories and aiding Markman's Hodge work on abelian fourfolds.

arxiv:2604.00511 · math.AG · 2026-04
UNVERDICTED Refined index obstructions for Brauer classes on an abelian variety
Refined index obstructions for Brauer classes on abelian varieties are shown to be stricter than prior versions and produce additional counterexamples to the integral Hodge conjecture.

arxiv:2605.26407 · math.AG · 2026-05
UNVERDICTED Algebraic cycles on Gushel-Mukai varieties
Proves generalized Hodge, Mumford-Tate and Tate conjectures for GM varieties, computes most Chow groups, and establishes motive isomorphisms for partners and duals.

arxiv:2207.01118 · math.AG · 2022-07
UNVERDICTED The Lichtenbaum-Quillen dimension of complex varieties
Authors define Lichtenbaum-Quillen dimension of complex varieties from K-theory stabilization and apply it to rationality obstructions and new cases of the integral Hodge conjecture.

arxiv:2312.06930 · math.AG · 2023-12