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CONDITIONAL
Asymptotics of local height pairing without Hodge conjecture
The Hodge conjecture is no longer required for the asymptotic expansion of the Archimedean local height pairing between algebraic cycles.
arxiv:2610.30574 · math.AG · 2026-10
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ACCEPT
Rigidity, flexibility, and a sign error
The OpenAI Hodge conjecture proof attempts fail because a theorem they rely on always produces a degree-2 self-intersection point, making the Lagrangian obstructed.
arxiv:2610.30976 · math.SG · 2026-10
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CONDITIONAL
Lengths of Hodge characters in Fermat varieties
The Hodge conjecture is now proved for every Fermat variety of degree below 65 (except 44, 51, 52) in all dimensions, via algorithmic length reduction.
arxiv:2609.27301 · math.AG · 2026-09
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ACCEPT
The Chern character of a coherent sheaf on a smooth projective hypersurface
An explicit formula for the Chern character of a coherent sheaf on a hypersurface in terms of matrix factorizations is proved and applied to the Hodge conjecture.
arxiv:2609.12759 · math.AG · 2026-09
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ACCEPT
The Tate conjecture for abelian fivefolds over finite fields
Under the assumption of the Tate conjecture for abelian fourfolds over finite fields, the Tate conjecture is proved for abelian fivefolds in every codimension.
arxiv:2609.06265 · math.NT · 2026-09
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CONDITIONAL
A holomorphic (2, 2)-Theorem for Abelian varieties of CM-type
A partial Hodge conjecture statement: on complex tori with algebraic normalized period matrix, including CM abelian varieties, the analytic Milnor K2 dlog map surjects onto rational (2,2) Hodge classes.
arxiv:2609.00269 · math.AG · 2026-08
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ACCEPT
A note on the noncommutative Hodge conjecture for graded matrix factorizations
For homogeneous polynomials that split into binary forms, the rational noncommutative Hodge conjecture holds for the graded matrix factorization category, yielding explicit algebraic generators for all rational Hodge classes.
arxiv:2609.03784 · math.AG · 2026-09
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CONDITIONAL
Algebraic cycles and Gushel-Mukai fivefolds
Gushel-Mukai fivefolds admit a multiplicative Chow-Kunneth decomposition, and a tautological subring of the Chow ring of their powers injects into cohomology.
arxiv:2009.11061 · math.AG · 2020-09
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REJECT
Exceptional Symmetry as a Source of Algebraic Cycles: Non-Constructive Methods for the Hodge Conjecture for a Special Class of Calabi-Yau 5-Folds
The paper's claimed proof of the (1,1) Hodge conjecture for CY 5-folds with h^{1,1}=56 rests on an unproved self-cited criterion, and its key dimension bound contradicts the paper's own isomorphism.
arxiv:2506.17754 · math.GM · 2025-06
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ACCEPT
On the integral Tate conjecture for abelian varieties
If the integral Hodge conjecture fails for a complex abelian variety, then the integral Tate conjecture fails for the corresponding variety over some finite extension of its field of definition.
arxiv:2509.06707 · math.AG · 2025-09
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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
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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
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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
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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
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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
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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
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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