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No all-powers or all-dimensional statement is included.","result":"proved","report_ids":["hodge-conjecture:hodge-broe-tate-fourfolds:report:v1"],"independent_review":"pending"},{"claim":{"id":"claim-hodge-openai-cm-proposal","version":2},"relation":"special_case","relationship_note":"Provider proposal for a specified family in arbitrary dimension; not rational Hodge for every smooth projective variety.","result":"proposed","report_ids":["hodge-conjecture:hodge-openai-cm-proposal:report:v1"],"independent_review":"pending"},{"claim":{"id":"claim-hodge-openai-milne-proposal","version":2},"relation":"related_variant","relationship_note":"A provider proposal about rationality of pairings at one reduction place. Rationality alone is weaker than producing an algebraic representative for every complex Hodge class.","result":"proposed","report_ids":["hodge-conjecture:hodge-openai-milne-proposal:report:v1"],"independent_review":"pending"}],"connections":[{"id":"hodge-connection-integral","source":{"id":"claim-hodge-rational-target","version":2},"target":{"id":"claim-integral-hodge-counterexamples","version":3},"relation":"related_to","assertion":null,"explanation":"Integral coefficients ask for a stronger assertion and admit counterexamples, whereas the canonical question permits rational combinations.","limitation":"An integral obstruction can disappear after tensoring with Q and does not refute rational Hodge.","source_entry_ids":["hodge-conjecture:integral-hodge-counterexamples:v1","hodge-conjecture:hodge-rational-target:v1"],"independent_review":"pending"},{"id":"hodge-connection-kahler","source":{"id":"claim-hodge-rational-target","version":2},"target":{"id":"claim-hodge-kahler-extension","version":3},"relation":"related_to","assertion":null,"explanation":"Dropping projectivity changes the class of spaces and yields a false extension.","limitation":"A proof of rational Hodge must use the smooth projective setting; these counterexamples have a different domain.","source_entry_ids":["hodge-conjecture:hodge-kahler-boundary:v1","hodge-conjecture:hodge-rational-target:v1"],"independent_review":"pending"},{"id":"hodge-connection-fourfolds","source":{"id":"claim-hodge-abelian-fourfolds","version":2},"target":{"id":"claim-hodge-weil-fourfold-powers","version":3},"relation":"related_to","assertion":null,"explanation":"The all-abelian-fourfold theorem and the all-powers theorem intersect at a single Weil fourfold with discriminant 1.","limitation":"One covers every abelian fourfold but not all its powers; the other covers all powers of a smaller family.","source_entry_ids":["hodge-conjecture:hodge-abelian-fourfolds:v1","hodge-conjecture:hodge-weil-fourfold-powers:v1"],"independent_review":"pending"},{"id":"hodge-connection-tate","source":{"id":"claim-hodge-abelian-fourfolds","version":2},"target":{"id":"claim-hodge-broe-tate-fourfolds","version":2},"relation":"analogue_of","assertion":null,"explanation":"Hodge asks for Betti Hodge classes over C; Tate asks for Galois-invariant ℓ-adic classes over finite fields. Broe’s proof uses Markman’s geometric input with further specialization arguments.","limitation":"This relationship is not a direct implication from the fourfold corollary alone, or a new case of canonical Hodge.","source_entry_ids":["hodge-conjecture:hodge-abelian-fourfolds:v1","hodge-conjecture:hodge-broe-tate-fourfolds:v1"],"independent_review":"pending"},{"id":"hodge-connection-projectors","source":{"id":"claim-hodge-rational-target","version":2},"target":{"id":"claim-hodge-kunneth-consequence","version":2},"relation":"related_to","assertion":null,"explanation":"Applying the global conjecture to X×X would algebraize its diagonal projectors.","limitation":"The linked Claim proves the implication under Hodge on the product; these projectors are not constructed unconditionally.","source_entry_ids":["hodge-conjecture:hodge-rational-target:v1","hodge-conjecture:hodge-projector-consequence:v1"],"independent_review":"pending"}],"milestones":[{"id":"hodge-milestone-origin","date":{"value":"1952"},"date_basis":"publication","category":"foundation","title":"Hodge’s ICM conjecture and its corrected scope","summary":"The published proceedings introduce the broader cycle-algebraicity question. The modern record uses rational coefficients and smooth projective varieties.","limitation":"1952 is the proceedings publication year, not a precise formulation date or an established theorem.","claims":[{"id":"claim-hodge-rational-target","version":2}],"source_entry_ids":["hodge-conjecture:hodge-origin-history:v1"],"independent_review":"pending"},{"id":"hodge-milestone-divisors","date":{"value":"1953"},"date_basis":"publication","category":"special_case","title":"The divisor-class criterion in the cited exposition","summary":"The Kodaira–Spencer account cited by Deligne records the Lefschetz (1,1) theorem, the codimension-one foundation.","limitation":"This is the cited exposition’s date, not a claim of first discovery. Higher codimension does not follow merely by multiplying divisors.","claims":[{"id":"claim-hodge-divisor-classes","version":2}],"source_entry_ids":["hodge-conjecture:hodge-divisor-history:v1","hodge-conjecture:hodge-divisor-classes:v1"],"independent_review":"pending"},{"id":"hodge-milestone-integral","date":{"value":"1962"},"date_basis":"publication","category":"correction","title":"Integral Hodge counterexamples fix the coefficient boundary","summary":"Atiyah–Hirzebruch’s published counterexamples invalidate the integral extension.","limitation":"They do not invalidate the canonical rational conjecture.","claims":[{"id":"claim-integral-hodge-counterexamples","version":3}],"source_entry_ids":["hodge-conjecture:integral-hodge-counterexamples:v1"],"independent_review":"pending"},{"id":"hodge-milestone-transfer","date":{"value":"2005-12-16"},"date_basis":"preprint","category":"conditional_result","title":"Transfer through motivated varieties","summary":"Arapura’s selected final author revision gives a precise transfer from Hodge on all powers of X to all powers of Y.","limitation":"Y must be motivated by X in the stated sense, and Hodge on all powers of X remains a premise.","claims":[{"id":"claim-hodge-motivated-transfer","version":2}],"source_entry_ids":["hodge-conjecture:hodge-motivated-transfer:v1"],"independent_review":"pending"},{"id":"hodge-milestone-markman","date":{"value":"2025-06-08"},"date_basis":"preprint","category":"special_case","title":"All complex abelian fourfolds","summary":"Markman’s selected v2 constructs Weil classes and obtains rational Hodge for abelian fourfolds, including arbitrary discriminants.","limitation":"General smooth projective fourfolds in codimension two remain outside this result.","claims":[{"id":"claim-hodge-abelian-fourfolds","version":2}],"source_entry_ids":["hodge-conjecture:hodge-abelian-fourfolds:v1"],"independent_review":"pending"},{"id":"hodge-milestone-powers","date":{"value":"2026-06"},"date_basis":"publication","category":"special_case","title":"A published geometric proof for selected powers","summary":"Floccari–Fu’s journal publication proves rational Hodge for every power of a Weil fourfold with discriminant 1; the cited author v2 fixes theorem text.","limitation":"The discriminant and Weil-type hypotheses are essential. This is a publication event, not an assertion that the theorem first appeared in June.","claims":[{"id":"claim-hodge-weil-fourfold-powers","version":3}],"source_entry_ids":["hodge-conjecture:hodge-weil-journal:v1","hodge-conjecture:hodge-weil-fourfold-powers:v1"],"independent_review":"pending"},{"id":"hodge-milestone-tate","date":{"value":"2026-08-19"},"date_basis":"preprint","category":"related_result","title":"A finite-field Tate theorem for abelian fourfolds","summary":"Broe’s preprint proves the ℓ-adic cycle-class surjectivity theorem and deduces Tate in all codimensions for abelian fourfolds.","limitation":"A different-field analogue; it is excluded from direct Hodge progress and does not prove Tate for all dimensions or powers.","claims":[{"id":"claim-hodge-broe-tate-fourfolds","version":2}],"source_entry_ids":["hodge-conjecture:hodge-broe-tate-fourfolds:v1"],"independent_review":"pending"},{"id":"hodge-milestone-openai-cm","date":{"value":"2026-10-06"},"date_basis":"preprint","category":"proposal","title":"OpenAI’s proposed all-dimensional CM case","summary":"The provider manuscript claims rational Hodge for all CM abelian varieties, their finite products and powers.","limitation":"It stays proposed pending independent assessment. Arbitrary non-CM and non-abelian varieties remain outside the claimed scope.","claims":[{"id":"claim-hodge-openai-cm-proposal","version":2}],"source_entry_ids":["hodge-conjecture:hodge-openai-cm:v1"],"independent_review":"pending"},{"id":"hodge-milestone-openai-milne","date":{"value":"2026-10-07"},"date_basis":"preprint","category":"proposal","title":"A proposed rationality theorem after good reduction","summary":"A separate provider manuscript claims rationality and realization independence for specialized Hodge/divisor pairings.","limitation":"The weaker pairing statement is distinct from constructing all complex algebraic cycles and remains unassessed here.","claims":[{"id":"claim-hodge-openai-milne-proposal","version":2}],"source_entry_ids":["hodge-conjecture:hodge-openai-milne:v1"],"independent_review":"pending"},{"id":"hodge-milestone-withdrawal","date":{"value":"2026-10-08"},"date_basis":"assessment","category":"correction","title":"A recent CM-fourfold manuscript is withdrawn","summary":"Ningyi Li’s arXiv v2 records withdrawal after reassessing the contribution of the September manuscript.","limitation":"The withdrawal adds no established theorem, and does not refute the independently cited Markman or Floccari–Fu results.","claims":[],"source_entry_ids":["hodge-conjecture:hodge-withdrawal-2026:v1"],"independent_review":"pending"}],"progress":[{"claim_id":"claim-hodge-divisor-classes","aspect":"Algebraicity by codimension","summary":"All divisor Hodge classes, in any dimension.","remaining":"Higher codimension is not covered by this theorem.","overlaps_with":[],"overlap_note":"This covers the divisor case. It overlaps the low-dimensional theorem and every other indexed family in codimension one.","category":"special_case"},{"claim_id":"claim-hodge-dimension-at-most-three","aspect":"Algebraicity by dimension","summary":"All rational Hodge classes on curves, surfaces and threefolds.","remaining":"General fourfolds and higher-dimensional varieties are not covered.","overlaps_with":["claim-hodge-divisor-classes"],"overlap_note":"Includes the divisor case. For threefolds the curve-class case follows from hard Lefschetz and the divisor case; arbitrary fourfolds are excluded.","category":"special_case"},{"claim_id":"claim-hodge-abelian-fourfolds","aspect":"Algebraicity for families","summary":"Every abelian fourfold, including arbitrary Weil discriminants.","remaining":"Nonabelian fourfolds and arbitrary higher-dimensional abelian varieties remain outside this result.","overlaps_with":["claim-hodge-divisor-classes"],"overlap_note":"Corollary 1.6.1, using algebraicity of Weil classes and prior classification results. This does not prove the conjecture for arbitrary fourfolds or for all powers of every abelian fourfold.","category":"special_case"},{"claim_id":"claim-hodge-weil-fourfold-powers","aspect":"Algebraicity for families","summary":"Every power of a Weil-type fourfold with discriminant 1.","remaining":"Does not assert the all-powers statement for arbitrary discriminants or arbitrary abelian varieties.","overlaps_with":["claim-hodge-divisor-classes","claim-hodge-abelian-fourfolds"],"overlap_note":"For m=1 this overlaps Markman’s fourfold corollary; m>1 covers selected varieties of dimension 4m. The discriminant-1 hypothesis is retained for the all-powers assertion.","category":"special_case"},{"claim_id":"claim-hodge-motivated-transfer","aspect":"Conditional reductions","summary":"Hodge algebraicity transfers from all powers of a motivating variety.","remaining":"The motivating relation and the all-powers hypothesis are not automatically available for an arbitrary variety.","overlaps_with":["claim-hodge-weil-fourfold-powers"],"overlap_note":"Transfers established cases rather than resolving the unrestricted conjecture. Choosing a base with all-powers Hodge algebraicity turns the implication into an unconditional case.","category":"conditional_result"},{"claim_id":"claim-hodge-kunneth-consequence","aspect":"Algebraic correspondences","summary":"Algebraic representatives for the Künneth components of the diagonal, if Hodge holds on X×X.","remaining":"Algebraicity of these particular classes remains open in general; the implication proves no new unconditional case.","overlaps_with":[],"overlap_note":"A consequence under an open premise, not an additional resolved part of that premise.","category":"conditional_implication"},{"claim_id":"claim-hodge-inverse-lefschetz-consequence","aspect":"Standard-conjecture consequences","summary":"Inverse Lefschetz isomorphisms would be induced by cycles, under Hodge on the product.","remaining":"The premise must cover X×X in the specified codimension. General cycle construction is still missing.","overlaps_with":[],"overlap_note":"A consequence under an open premise, not an additional resolved part of that premise.","category":"conditional_implication"}],"activities":[{"id":"hodge-withdrawal-2026","title":"Recent source update · withdrawn preprint","kind":"source_correction","source_entry_ids":["hodge-conjecture:hodge-withdrawal-2026:v1"],"summary":"A September 2026 manuscript claiming results on powers of CM fourfolds was withdrawn by its author on 8 October 2026 after reassessing its contribution.","limitation":"The withdrawal is not a refutation of rational Hodge or of the other cited results. This manuscript contributes no additional proved progress in this record.","related_claim_ids":["claim-hodge-abelian-fourfolds","claim-hodge-weil-fourfold-powers"]},{"id":"hodge-openai-proposals","title":"Company AI proposals: CM algebraicity and specialized rationality","kind":"research_direction","source_entry_ids":["hodge-conjecture:hodge-openai-cm:v1","hodge-conjecture:hodge-openai-milne:v1"],"summary":"The October release proposes two distinct mathematical scopes: CM Hodge classes and rationality of pairings after good reduction.","limitation":"The original manuscripts are pinned and linked. Their dependencies, independent review and any formal correspondence remain unassessed; neither is promoted to established progress.","related_claim_ids":["claim-hodge-openai-cm-proposal","claim-hodge-openai-milne-proposal"]}],"curation":{"checked_on":"2026-10-10","selection":"Selected classical cases, the 2025 abelian-fourfold result, a February 2026 revised proof for selected powers, one conditional transfer theorem and an integral-variant boundary. Includes selected source-dated history, exact Claim connections, interpretations, approaches and grounded conditional impacts.","search_scope":"Clay descriptions; author arXiv revisions and theorem statements; recent Hodge/abelian-fourfold literature; public Google DeepMind, Axiom, Harmonic and mathematical formalization search. Search cutoff: 10 October 2026. This selection is not exhaustive. Rechecked the June 2026 Floccari–Fu journal record, Broe v1 (August 2026), September status/direction literature, and October OpenAI CM-Hodge and Milne manuscripts at a pinned release commit.","formal_status":"No complete Lean proof corresponding to these selected Hodge Claims has been established by this curation. PDF proofs are external mathematical arguments, not OpenMath Verification Results.","ai_status":"October OpenAI CM-Hodge and Milne rationality manuscripts are source-reported AI proposals with exact scopes, not selected proved progress. No independently accepted general Hodge proof or locally executed corresponding Lean proof is established by this curation.","independent_review":"pending"},"schema_version":"research-record-v3","claims":[{"statement":"For a smooth projective complex variety X, every rational class in H²(X,Q) of type (1,1) is a rational combination of divisor classes.","id":"claim-hodge-divisor-classes","slug":"hodge-divisor-classes","version":2,"kind":"theorem","title":"Divisor classes · Lefschetz (1,1)","assumptions":["X is smooth and projective over C.","The class has cohomological degree 2 and Hodge type (1,1)."],"scope":"Codimension one in any dimension; rational coefficients.","source_entry_ids":["hodge-conjecture:hodge-divisor-classes:v1"],"formal_declarations":[]},{"statement":"The rational Hodge conjecture holds for every smooth projective complex variety of complex dimension at most three.","id":"claim-hodge-dimension-at-most-three","slug":"hodge-dimension-at-most-three","version":3,"kind":"theorem","title":"All codimensions in dimension ≤ 3","assumptions":["X is smooth and projective over C.","Complex dimension of X is at most 3.","Hodge classes and cycle combinations have rational coefficients."],"scope":"Every codimension, restricted to complex dimension ≤ 3.","source_entry_ids":["hodge-conjecture:hodge-dimension-at-most-three:v2"],"formal_declarations":[]},{"statement":"The rational Hodge conjecture holds for every complex abelian variety of dimension four.","id":"claim-hodge-abelian-fourfolds","slug":"hodge-abelian-fourfolds","version":2,"kind":"theorem","title":"Abelian fourfolds · Markman (2025)","assumptions":["A is a complex abelian variety.","Complex dimension of A is 4.","Rational coefficients in cohomology and cycles."],"scope":"All codimensions on abelian fourfolds. No restriction to Weil discriminant 1 is needed for this corollary.","source_entry_ids":["hodge-conjecture:hodge-abelian-fourfolds:v1"],"formal_declarations":[]},{"statement":"If A is a complex abelian fourfold of Weil type with discriminant 1, the rational Hodge conjecture holds for A^m for every positive integer m.","id":"claim-hodge-weil-fourfold-powers","slug":"hodge-weil-fourfold-powers","version":3,"kind":"theorem","title":"All powers of Weil fourfolds with discriminant 1","assumptions":["A is a complex abelian fourfold of Weil type: an imaginary quadratic field acts with signature (2,2).","A has a compatible polarization of discriminant 1, in the paper’s Weil-type convention.","m is any positive integer; coefficients are rational."],"scope":"All codimensions on powers of this specified Weil-type family.","source_entry_ids":["hodge-conjecture:hodge-weil-fourfold-powers:v1","hodge-conjecture:hodge-weil-journal:v1"],"formal_declarations":[]},{"statement":"If smooth projective complex varieties X and Y satisfy that Y is motivated by X in Arapura’s sense, and the rational Hodge conjecture holds on every power of X, it holds on every power of Y.","id":"claim-hodge-motivated-transfer","slug":"hodge-motivated-transfer","version":2,"kind":"theorem","title":"Conditional transfer through motivated varieties","assumptions":["X and Y are smooth projective over C.","Y is motivated by X in the category of André motives used in the paper, not merely weakly motivated.","Assume the rational Hodge conjecture for every positive power of X."],"scope":"A proved implication; its Hodge hypothesis must be supplied for the chosen X.","source_entry_ids":["hodge-conjecture:hodge-motivated-transfer:v1"],"formal_declarations":[]},{"statement":"The integral analogue of the Hodge conjecture is false: there exist integral Hodge classes on smooth projective complex varieties not represented by integral algebraic cycles.","id":"claim-integral-hodge-counterexamples","slug":"integral-hodge-counterexamples","version":3,"kind":"theorem","title":"Integral Hodge variant · counterexamples","assumptions":["Integral, rather than rational, coefficients.","Smooth projective complex varieties; the assertion is existential."],"scope":"Atiyah–Hirzebruch counterexamples to the integral variant; not a counterexample to rational Hodge.","source_entry_ids":["hodge-conjecture:integral-hodge-counterexamples:v1"],"formal_declarations":[]},{"statement":"For every smooth projective variety X over C and 0 ≤ p ≤ dim X, the rational cycle-class map CH^p(X)⊗Q → H^{2p}(X,Q) ∩ H^{p,p}(X) is surjective.","id":"claim-hodge-rational-target","slug":"hodge-rational-target","version":2,"kind":"conjecture","title":"rational Hodge holds in all codimensions","assumptions":["Smooth, projective and defined over C.","Rational coefficients; the intersection is taken after complexification."],"scope":"All varieties and codimensions. A representative may be a rational combination of cycles, without positivity or uniqueness.","source_entry_ids":["hodge-conjecture:hodge-rational-target:v1"],"formal_declarations":[]},{"statement":"If the rational Hodge conjecture holds on X×X for a smooth projective complex variety X of dimension d, each rational Künneth component of [Δ_X] in H^{2d}(X×X,Q) is represented by a rational codimension-d algebraic cycle.","id":"claim-hodge-kunneth-consequence","slug":"hodge-kunneth-consequence","version":2,"kind":"theorem","title":"Hodge would make cohomological projectors algebraic","assumptions":["X smooth projective over C.","Hodge conjecture on X×X, in codimension d."],"scope":"A proved conditional implication using the Hodge type of diagonal components, not an unconditional construction of those cycles.","source_entry_ids":["hodge-conjecture:hodge-projector-consequence:v1"],"formal_declarations":[]},{"statement":"For smooth projective X/C of dimension d, an ample hyperplane class η, and 0 ≤ p ≤ d, assume rational Hodge on X×X in codimension d−p. Then the inverse of η^p: H^{d−p}(X,Q) → H^{d+p}(X,Q) is induced by a rational algebraic correspondence of codimension d−p on X×X.","id":"claim-hodge-inverse-lefschetz-consequence","slug":"hodge-inverse-lefschetz-consequence","version":2,"kind":"theorem","title":"Hodge would algebraize inverse Lefschetz maps","assumptions":["X smooth projective over C and η an ample hyperplane class.","Hodge conjecture on X×X in the specified codimension."],"scope":"A proved implication for the Lefschetz standard-conjecture classes in characteristic zero; no assertion in arbitrary positive characteristic follows.","source_entry_ids":["hodge-conjecture:hodge-lefschetz-consequence:v1"],"formal_declarations":[]},{"statement":"The assertion that every rational (p,p) class on every compact Kähler manifold is a rational combination of analytic cycle classes is false; complex-torus counterexamples are recorded in the cited description.","id":"claim-hodge-kahler-extension","slug":"hodge-kahler-extension","version":3,"kind":"theorem","title":"The unrestricted compact Kähler extension is false","assumptions":["Compact Kähler complex manifolds, without the projectivity assumption.","Analytic cycle classes and rational coefficients."],"scope":"A refuted different-domain extension; it is not a counterexample among smooth projective varieties.","source_entry_ids":["hodge-conjecture:hodge-kahler-boundary:v1"],"formal_declarations":[]},{"statement":"For every abelian fourfold A over a finite field k = F_q and prime ℓ ≠ char(k), CH²(A)⊗Q_ℓ → H⁴_et(A_kbar,Q_ℓ(2))^{Gal(kbar/k)} is surjective. Combined with the divisor theorem and hard Lefschetz, Tate holds for A in all codimensions.","id":"claim-hodge-broe-tate-fourfolds","slug":"hodge-broe-tate-fourfolds","version":2,"kind":"theorem","title":"Tate is proved for abelian fourfolds over finite fields","assumptions":["A is an abelian variety of dimension four over a finite field.","ℓ differs from the field characteristic; cycles defined over k; invariants under its absolute Galois group."],"scope":"A cited preprint theorem about ℓ-adic classes over finite fields. No all-powers or all-dimensional statement is included.","source_entry_ids":["hodge-conjecture:hodge-broe-tate-fourfolds:v1"],"formal_declarations":[]},{"statement":"For every complex CM abelian variety A and every p ≥ 0, CH^p(A)⊗Q → H^{2p}(A,Q) ∩ H^{p,p}(A) is surjective. The provider includes all finite products and powers of CM abelian varieties.","id":"claim-hodge-openai-cm-proposal","slug":"hodge-openai-cm-proposal","version":2,"kind":"theorem","title":"Proposed Hodge result for all CM abelian varieties","assumptions":["Complex abelian variety of CM type: End(A)⊗Q contains a commutative semisimple Q-algebra of dimension 2 dim A.","Rational cycle classes; every codimension."],"scope":"Provider proposal for a specified family in arbitrary dimension; not rational Hodge for every smooth projective variety.","source_entry_ids":["hodge-conjecture:hodge-openai-cm:v1"],"formal_declarations":[]},{"statement":"Let A/Qbar have dimension d and good reduction A₀/Fpbar at a fixed p-adic place, and let γ be a rational Hodge class of degree 2r. For every list of d−r Cartier divisors on A₀, the trace of the specialized γ times their classes is a common q ∈ Q in every ℓ-adic realization (ℓ ≠ p) and in the specified de Rham–crystalline realization.","id":"claim-hodge-openai-milne-proposal","slug":"hodge-openai-milne-proposal","version":2,"kind":"theorem","title":"Proposed rationality of specialized Hodge pairings","assumptions":["A over Qbar, embedded in C, with good reduction after finite descent and extension at the chosen place.","0 ≤ r ≤ d; top-Tate-twist trace normalized to degree of a zero-cycle.","The crystalline comparison and divisor-product conventions of Theorem 1.1."],"scope":"A provider proposal about rationality of pairings at one reduction place. Rationality alone is weaker than producing an algebraic representative for every complex Hodge class.","source_entry_ids":["hodge-conjecture:hodge-openai-milne:v1"],"formal_declarations":[]}],"problem":{"resolution_scope":{"resolution":"unresolved","extent":"general_case","summary":"Divisors, varieties of dimension at most three, abelian fourfolds and selected powers are covered by cited theorems; conditional transfer results are also indexed.","report_ids":["hodge-conjecture:report:0:v2"],"progress_claim_ids":["claim-hodge-divisor-classes","claim-hodge-dimension-at-most-three","claim-hodge-abelian-fourfolds","claim-hodge-weil-fourfold-powers","claim-hodge-motivated-transfer"],"remaining":"Arbitrary smooth projective varieties remain open, including general fourfolds in codimension two. Abelian fourfolds are a solved family, not all fourfolds."},"statement_kind":"conjecture","problem_id":"problem-hodge-conjecture","slug":"hodge-conjecture","title":"Hodge conjecture","question":"On a smooth complex projective variety, is every rational Hodge class a rational linear combination of classes of algebraic cycles?","scope":"Rational Hodge classes on smooth projective varieties over C, in every codimension. Integral coefficients, nonprojective spaces, the generalized Hodge conjecture and the Tate conjecture are distinct variants.","aliases":["Hodge conjecture"],"fields":["Algebraic geometry"],"msc":["14C30"],"topics":["Scoped progress","Conditional results"],"background":"specialist","scale":"major conjecture","depth":"source-linked","source_entry_ids":["hodge-conjecture:entry:0:v1","hodge-conjecture:entry:0:v2","hodge-conjecture:hodge-divisor-classes:v1","hodge-conjecture:hodge-dimension-at-most-three:v2","hodge-conjecture:hodge-abelian-fourfolds:v1","hodge-conjecture:hodge-weil-fourfold-powers:v1","hodge-conjecture:hodge-motivated-transfer:v1","hodge-conjecture:integral-hodge-counterexamples:v1","hodge-conjecture:hodge-withdrawal-2026:v1","hodge-conjecture:hodge-rational-target:v1","hodge-conjecture:hodge-projector-consequence:v1","hodge-conjecture:hodge-lefschetz-consequence:v1","hodge-conjecture:hodge-kahler-boundary:v1","hodge-conjecture:hodge-broe-tate-fourfolds:v1","hodge-conjecture:hodge-weil-journal:v1","hodge-conjecture:hodge-openai-cm:v1","hodge-conjecture:hodge-openai-milne:v1","hodge-conjecture:hodge-origin-history:v1","hodge-conjecture:hodge-divisor-history:v1","hodge-conjecture:hodge-cycle-obstruction:v1"],"status_report_ids":["hodge-conjecture:report:0:v1","hodge-conjecture:report:0:v2"],"editorial":null,"literature":[{"title":"Clay problem page and official description","url":"https://www.claymath.org/millennium/hodge-conjecture/","note":"Clay reports the general question unsolved, observed 10 October 2026. This current-state observation is separate from the dated theorem and artifact citations below."}],"formalizations":[],"related":[],"tasks":[],"insights":[{"id":"hodge-interpretation-cycles","title":"A class is a cohomological shadow of a cycle","summary":"The conjecture asks whether rational classes satisfying the Hodge condition come from algebraic subvarieties. It concerns equality of cohomology classes, rather than unique cycles or effective positive representatives.","review":"pending","kind":"interpretation","source_entry_ids":["hodge-conjecture:hodge-rational-target:v1"],"claims":[{"id":"claim-hodge-rational-target","version":2}],"limitation":"It neither identifies all cycles up to rational equivalence nor promises an algorithm that finds equations for a representative."},{"id":"hodge-interpretation-scope","title":"Dimension, codimension and powers are different axes","summary":"Divisors are covered in every dimension; all codimensions are covered in dimension ≤3. An abelian fourfold is a special family, and a power of a fourfold usually has larger dimension.","review":"pending","kind":"interpretation","source_entry_ids":["hodge-conjecture:hodge-divisor-classes:v1","hodge-conjecture:hodge-dimension-at-most-three:v2","hodge-conjecture:hodge-abelian-fourfolds:v1"],"claims":[{"id":"claim-hodge-divisor-classes","version":2},{"id":"claim-hodge-dimension-at-most-three","version":3},{"id":"claim-hodge-abelian-fourfolds","version":2}],"limitation":"These scopes overlap and cannot be added as disjoint fractions. Arbitrary fourfolds remain open in codimension two."},{"id":"hodge-approach-geometry","title":"Construct cycles in controlled geometric families","summary":"Markman uses secant sheaves and Weil-type geometry; Floccari–Fu relates selected abelian fourfolds to singular OG6 geometry. These provide actual algebraic representatives within their families.","review":"pending","kind":"approach","source_entry_ids":["hodge-conjecture:hodge-abelian-fourfolds:v1","hodge-conjecture:hodge-weil-fourfold-powers:v1"],"claims":[{"id":"claim-hodge-abelian-fourfolds","version":2},{"id":"claim-hodge-weil-fourfold-powers","version":3}],"limitation":"The constructions have precise family hypotheses. Neither proves the conjecture for every smooth projective variety."},{"id":"hodge-approach-transfer","title":"Reduce to a motivating variety and its powers","summary":"Arapura’s transfer theorem can propagate known cases once a suitable motivated relationship and all-powers base theorem are supplied.","review":"pending","kind":"approach","source_entry_ids":["hodge-conjecture:hodge-motivated-transfer:v1"],"claims":[{"id":"claim-hodge-motivated-transfer","version":2}],"limitation":"Checking one variety X is weaker than Hodge on every X^m. Replacing “motivated” by a weaker relationship changes the theorem."},{"id":"hodge-approach-obstruction","title":"Detecting a Hodge class does not construct its cycle","summary":"Deligne explains the difficulty of passing from cohomological conditions and families to actual algebraic cycles. An intermediate-Jacobian induction needs more than Hodge for hyperplane sections.","review":"pending","kind":"approach","source_entry_ids":["hodge-conjecture:hodge-cycle-obstruction:v1","hodge-conjecture:hodge-kahler-boundary:v1"],"claims":[{"id":"claim-hodge-kahler-extension","version":3}],"limitation":"The historical exposition describes obstructions, not a complete current catalogue of approaches; analytic arguments must retain projectivity."},{"id":"hodge-impact-projectors","title":"Realize cohomological decompositions by correspondences","summary":"Hodge on every product X×X would give algebraic representatives for its diagonal’s Künneth components. These correspondences are key inputs to decomposing cohomological motives.","review":"pending","kind":"impact","source_entry_ids":["hodge-conjecture:hodge-projector-consequence:v1"],"claims":[{"id":"claim-hodge-kunneth-consequence","version":2}],"scenario":{"id":"claim-hodge-rational-target","version":2},"limitation":"The selected implication concerns cohomological classes; it does not assert mutually orthogonal Chow idempotents modulo rational equivalence."},{"id":"hodge-impact-lefschetz","title":"Give the inverse Lefschetz operators geometric representatives","summary":"A general proof would algebraize the inverse hard-Lefschetz maps on smooth projective complex varieties, supplying the indicated Lefschetz standard-conjecture classes.","review":"pending","kind":"impact","source_entry_ids":["hodge-conjecture:hodge-lefschetz-consequence:v1"],"claims":[{"id":"claim-hodge-inverse-lefschetz-consequence","version":2}],"scenario":{"id":"claim-hodge-rational-target","version":2},"limitation":"Hard Lefschetz already proves the cohomological isomorphism. The new consequence is algebraicity of its inverse correspondence, with no positive-characteristic transfer asserted."}],"coverage_gaps":["Selected results, not an exhaustive literature survey or a percentage of the conjecture solved.","Source-attributed theorem statements; independent mathematical and relationship review is pending.","External papers, programs and Lean placeholders have no OpenMath execution or independent reproduction attached.","AI involvement is recorded only where the cited source reports it; absence from this selection does not establish absence of work.","Timeline dates refer to cited research events at their supported precision. Overlapping results and conditional consequences do not define a percentage solved."],"checked_on":"2026-10-12"},"sources":[{"id":"clay-millennium@observed-2026-10-01","source_id":"clay-millennium","title":"Clay · Millennium Problems","owner":"Clay Mathematics Institute","url":"https://www.claymath.org/millennium-problems/","subject_scope":"Seven Millennium Prize Problems; five selected here","revision":"observed-2026-10-01","content_sha256":null,"update_method":"Manual checked import; new observations require staff preview and publication","identifier_policy":"Canonical source URL plus explicit item/declaration locator; no fuzzy identity merge","observed_on":"2026-10-01","source_updated_on":null,"rights":"metadata-link","license":"Not established; links and OpenMath paraphrases only","terms_url":"https://www.claymath.org/millennium-problems/","attribution":"Clay Mathematics Institute","rights_note":"Store metadata and original OpenMath paraphrases; follow the original source for authoritative wording.","check_interval_days":30},{"id":"clay-millennium@observed-2026-10-10","source_id":"clay-millennium","title":"Clay · Millennium Problems","owner":"Clay Mathematics Institute","url":"https://www.claymath.org/millennium-problems/","subject_scope":"Seven Millennium Prize Problems; 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full text stays upstream.","check_interval_days":30}],"entries":[{"id":"hodge-conjecture:entry:0:v1","source_version_id":"clay-millennium@observed-2026-10-01","item_id":"https://www.claymath.org/millennium/hodge-conjecture/","url":"https://www.claymath.org/millennium/hodge-conjecture/","wording":"On a smooth complex projective variety, is every rational Hodge class a rational linear combination of classes of algebraic cycles?","wording_kind":"paraphrase","locator":"https://www.claymath.org/millennium/hodge-conjecture/","supersedes":null},{"id":"hodge-conjecture:entry:0:v2","source_version_id":"clay-millennium@observed-2026-10-10","item_id":"https://www.claymath.org/millennium/hodge-conjecture/","url":"https://www.claymath.org/millennium/hodge-conjecture/","wording":"On a smooth complex projective variety, is every rational Hodge class a rational linear combination of classes of algebraic cycles?","wording_kind":"paraphrase","locator":"https://www.claymath.org/millennium/hodge-conjecture/","supersedes":"hodge-conjecture:entry:0:v1"},{"id":"hodge-conjecture:hodge-divisor-classes:v1","source_version_id":"clay-deligne-hodge@official-description","item_id":"Section 2(iii), PDF page 2","url":"https://www.claymath.org/wp-content/uploads/2022/06/hodge.pdf","wording":"For a smooth projective complex variety X, every rational class in H²(X,Q) of type (1,1) is a rational combination of divisor classes.","wording_kind":"paraphrase","locator":"Section 2(iii), PDF page 2","supersedes":null},{"id":"hodge-conjecture:hodge-dimension-at-most-three:v2","source_version_id":"clay-millennium@observed-2026-10-10","item_id":"Clay Hodge page, overview: dimension less than four","url":"https://www.claymath.org/millennium/hodge-conjecture/","wording":"The rational Hodge conjecture holds for every smooth projective complex variety of complex dimension at most three.","wording_kind":"paraphrase","locator":"Clay Hodge page, overview: dimension less than four","supersedes":null},{"id":"hodge-conjecture:hodge-abelian-fourfolds:v1","source_version_id":"arxiv-2502.03415@arxiv-v2","item_id":"Corollary 1.6.1, PDF page 9; 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The discriminant-1 hypothesis is retained for the all-powers assertion.","result":"proved","report_ids":["hodge-conjecture:hodge-weil-fourfold-powers:v1:report","hodge-conjecture:hodge-weil-journal:report:v1"],"independent_review":"pending"},{"statement":"If smooth projective complex varieties X and Y satisfy that Y is motivated by X in Arapura’s sense, and the rational Hodge conjecture holds on every power of X, it holds on every power of Y.","id":"claim-hodge-motivated-transfer","slug":"hodge-motivated-transfer","version":2,"kind":"theorem","title":"Conditional transfer through motivated varieties","assumptions":["X and Y are smooth projective over C.","Y is motivated by X in the category of André motives used in the paper, not merely weakly motivated.","Assume the rational Hodge conjecture for every positive power of X."],"scope":"A proved implication; its Hodge hypothesis must be supplied for the chosen X.","source_entry_ids":["hodge-conjecture:hodge-motivated-transfer:v1"],"formal_declarations":[],"relation":"conditional_result","relationship_note":"Transfers established cases rather than resolving the unrestricted conjecture. Choosing a base with all-powers Hodge algebraicity turns the implication into an unconditional case.","result":"proved","report_ids":["hodge-conjecture:hodge-motivated-transfer:v1:report"],"independent_review":"pending"}],"entries":[{"id":"hodge-conjecture:hodge-weil-fourfold-powers:v1","source_version_id":"arxiv-2504.13607@arxiv-v2","item_id":"Theorem 1.2 and its proof, PDF pages 2 and 16","url":"https://arxiv.org/abs/2504.13607v2","wording":"If A is a complex abelian fourfold of Weil type with discriminant 1, the rational Hodge conjecture holds for A^m for every positive integer m.","wording_kind":"paraphrase","locator":"Theorem 1.2 and its proof, PDF pages 2 and 16","supersedes":null},{"id":"hodge-conjecture:hodge-motivated-transfer:v1","source_version_id":"arxiv-math-0501348@arxiv-v3","item_id":"Lemma 4.2, PDF pages 9–10","url":"https://arxiv.org/abs/math/0501348v3","wording":"If smooth projective complex varieties X and Y satisfy that Y is motivated by X in Arapura’s sense, and the rational Hodge conjecture holds on every power of X, it holds on every power of Y.","wording_kind":"paraphrase","locator":"Lemma 4.2, PDF pages 9–10","supersedes":null},{"id":"hodge-conjecture:hodge-weil-journal:v1","source_version_id":"arxiv-2504.13607@journal-2026-103876","item_id":"Journal metadata: volume 210 (June 2026), article 103876; abstract","url":"https://doi.org/10.1016/j.matpur.2026.103876","wording":"The publisher records the published Floccari–Fu proof for Weil fourfolds of discriminant 1 and all their powers.","wording_kind":"paraphrase","locator":"Journal metadata: volume 210 (June 2026), article 103876; abstract","supersedes":null}],"sources":[{"work":{"authors":["Salvatore Floccari","Lie Fu"],"kind":"research_paper","publication":"accepted","publication_note":"arXiv v2 states final version, to appear in Journal de Mathématiques Pures et Appliquées; acceptance is source-reported.","ai_role":"not_reported","ai_note":"The cited work does not report AI involvement; this is not an assertion that no AI was used.","artifacts":[{"kind":"pdf","title":"Proof PDF · v2","url":"https://arxiv.org/pdf/2504.13607v2","version":"arXiv:2504.13607v2","assessment":"paper_proof","note":"External evidence linked; no OpenMath execution or independent reproduction is asserted."}]},"id":"arxiv-2504.13607@arxiv-v2","source_id":"arxiv-2504.13607","title":"Floccari–Fu · Weil fourfolds and powers","owner":"Salvatore Floccari; Lie Fu","url":"https://arxiv.org/abs/2504.13607v2","subject_scope":"Floccari–Fu · Weil fourfolds and powers","revision":"arxiv-v2","content_sha256":null,"update_method":"Manual source and revision check; recheck upstream before a new publication.","identifier_policy":"Stable bibliographic identifier and explicit revision; theorem locators belong to source entries.","observed_on":"2026-10-10","source_updated_on":"2026-02-10","rights":"metadata-link","license":"Links and original OpenMath paraphrases only; upstream license governs external files.","terms_url":"https://arxiv.org/abs/2504.13607v2","attribution":"Salvatore Floccari; Lie Fu","rights_note":"No upstream full text or code is republished by this record.","check_interval_days":90},{"work":{"authors":["Donu Arapura"],"kind":"research_paper","publication":"published","publication_note":"Published in Advances in Mathematics 207 (2006), 762–781; author version cited.","ai_role":"not_reported","ai_note":"The cited work does not report AI involvement; this is not an assertion that no AI was used.","artifacts":[{"kind":"pdf","title":"Author proof PDF · v3","url":"https://arxiv.org/pdf/math/0501348v3","version":"arXiv:math/0501348v3","assessment":"paper_proof","note":"External evidence linked; no OpenMath execution or independent reproduction is asserted."}]},"id":"arxiv-math-0501348@arxiv-v3","source_id":"arxiv-math-0501348","title":"Arapura · motivation for Hodge cycles","owner":"Donu Arapura","url":"https://arxiv.org/abs/math/0501348v3","subject_scope":"Arapura · motivation for Hodge cycles","revision":"arxiv-v3","content_sha256":null,"update_method":"Manual source and revision check; recheck upstream before a new publication.","identifier_policy":"Stable bibliographic identifier and explicit revision; theorem locators belong to source entries.","observed_on":"2026-10-10","source_updated_on":"2005-12-16","rights":"metadata-link","license":"Links and original OpenMath paraphrases only; upstream license governs external files.","terms_url":"https://arxiv.org/abs/math/0501348v3","attribution":"Donu Arapura","rights_note":"No upstream full text or code is republished by this record.","check_interval_days":90},{"work":{"authors":["Salvatore Floccari","Lie Fu"],"kind":"research_paper","publication":"published","publication_note":"Published in Journal de Mathématiques Pures et Appliquées 210 (June 2026), article 103876, DOI 10.1016/j.matpur.2026.103876. The author v2 remains separately pinned for theorem text.","ai_role":"not_reported","ai_note":"The cited work does not report AI involvement; this is not an assertion that no AI was used.","artifacts":[{"kind":"pdf","title":"Proof PDF · v2","url":"https://arxiv.org/pdf/2504.13607v2","version":"arXiv:2504.13607v2","assessment":"paper_proof","note":"External evidence linked; no OpenMath execution or independent reproduction is asserted."}],"published":{"value":"2026-06"}},"id":"arxiv-2504.13607@journal-2026-103876","source_id":"arxiv-2504.13607","title":"Floccari–Fu · Weil fourfolds and powers","owner":"Salvatore Floccari; Lie Fu","url":"https://doi.org/10.1016/j.matpur.2026.103876","subject_scope":"Floccari–Fu · Weil fourfolds and powers","revision":"journal-2026-103876","content_sha256":null,"update_method":"Manual source and revision check; recheck upstream before a new publication.","identifier_policy":"Stable bibliographic identifier and explicit revision; theorem locators belong to source entries.","observed_on":"2026-10-10","source_updated_on":"2026-02-10","rights":"metadata-link","license":"Links and original OpenMath paraphrases only; upstream license governs external files.","terms_url":"https://arxiv.org/abs/2504.13607v2","attribution":"Salvatore Floccari; Lie Fu","rights_note":"No upstream full text or code is republished by this record.","check_interval_days":90}]}}}