{"slug":"hodge-conjecture","title":"Hodge conjecture","question":"Is every Hodge class on a projective variety a combination of algebraic cycles?","field":"mathematics","sinceYear":1950,"status":"OPEN","source":{"name":"Clay Mathematics Institute","list":"Millennium Prize Problems","url":"https://www.claymath.org/millennium/hodge-conjecture/"},"checkedOn":"2026-09-29","publicRecordCount":0,"claimCount":10,"summary":"Asks whether certain cohomology classes on smooth complex projective varieties can always be built from algebraic subvarieties.","prize":{"sponsor":"Clay Mathematics Institute","amount":1000000,"currency":"USD","rulesUrl":"https://www.claymath.org/millennium-problems/rules/"},"statusHistory":[{"status":"OPEN","source":{"name":"Clay Mathematics Institute","list":"Millennium Prize Problems","url":"https://www.claymath.org/millennium/hodge-conjecture/"},"sourceDate":null,"checkedOn":"2026-09-29"}],"records":[],"seoTitle":"Hodge conjecture: what is known and what is open — Papex","metaDescription":"Is every Hodge class on a projective variety a combination of algebraic cycles? Known cases, abelian varieties, recent claims and open sub-questions.","page":{"whyOpen":"A smooth complex projective variety is a smooth space cut out by polynomial equations in complex projective space. Its cohomology with rational coefficients records its topology, and every closed subvariety of complex codimension p defines a class in degree 2p [4]. Hodge theory splits each cohomology group, taken with complex coefficients, into pieces of type (p, q). The class of a subvariety always lies in the (p, p) piece, and rational classes of type (p, p) are called Hodge classes [4]. An algebraic cycle is a formal combination of subvarieties with rational coefficients [5]. The conjecture states that every Hodge class is the class of an algebraic cycle [4].\n\nFew cases are known for every variety: classes of degree 2, by the Lefschetz theorem on (1,1)-classes, and, as a consequence, classes of curves [5]. In Claire Voisin's 2016 survey, \"Not much is known about the Hodge conjecture\" beyond this and a theorem of Eduardo Cattani, Pierre Deligne and Aroldo Kaplan [5]. The Clay Mathematics Institute notes that the conjecture is known for varieties of dimension less than four, and unknown in dimension four [10]. Both restrictions in the statement, rational coefficients and projective varieties, are needed (see below).\n\nA resolution would be a proof for every smooth complex projective variety in every degree, or a Hodge class shown not to be algebraic. For the Clay Mathematics Institute's prize, a solution must appear in a qualifying outlet, at least two years must pass after publication, and it must gain general acceptance in the mathematics community [10]. Deligne names the obstacle: \"Progress is blocked by a lack of methods to construct interesting algebraic cycles\" [4]. Voisin wrote in 2026 that \"beyond the formal definition, there is no good understanding of what is a Hodge class from the viewpoint of algebraic geometry\" [8].","known":"**Formulation and corrections (1950–1969).** W. V. D. Hodge posed the conjecture in a paper for the 1950 International Congress of Mathematicians, published in 1952 [4]. With integer coefficients it is false, as Michael Atiyah and Friedrich Hirzebruch showed in 1962 [1, 4], and János Kollár later found hypersurfaces on which the integral form fails [5]. Alexander Grothendieck observed in 1969 that a more general conjecture of Hodge was \"trivially false\", and corrected it [4]. The conjecture also fails for compact Kähler manifolds: some complex tori carry Hodge classes that come from no analytic subvariety [4, 5], and in 2002 Voisin found tori on which some do not even come from Chern classes of coherent sheaves [5].\n\n**Hodge classes behave as if algebraic (1995).** Cattani, Deligne and Kaplan proved that in a family of smooth projective varieties, the set of parameters where a given integral class stays of type (p, p) is algebraic [3]. The conjecture predicts exactly this, and Voisin calls the theorem the best known evidence for it [5]. The conjecture also predicts that, for a family defined over the algebraic numbers, this set is defined over the algebraic numbers too; that is not known [4, 5].\n\n**Abelian varieties (1982–2025).** An abelian variety is a complex torus that can be embedded in projective space [8]. In notes published in 1982, Deligne proved that every Hodge class on an abelian variety is \"absolute Hodge\": it stays a Hodge class when any field automorphism of the complex numbers is applied to the defining equations, as classes of algebraic cycles do [2, 5]. His proof reduces to abelian varieties of CM type [2]. André Weil identified a two-dimensional space of Hodge classes, now called Weil classes, on abelian varieties of dimension 2n with a suitable action of an imaginary quadratic field (\"Weil type\") [7]. In a 2025 preprint, Eyal Markman proved that Weil classes are algebraic on every abelian sixfold of Weil type with discriminant −1, deduced the same for every abelian fourfold of Weil type, and noted that the conjecture for abelian fourfolds follows [7]; Voisin called this \"a big recent progress\" [8]. For abelian varieties of CM type, a 2020 note by James Milne shows, following Deligne and Yves André, that the Hodge classes can be expressed in terms of divisor classes and \"split Weil classes\" [6].\n\n**Recent claims.** On 6 October 2026 OpenAI opened the public repository github.com/openai/math of \"mathematical manuscripts and supporting proof artifacts produced by an internal OpenAI model\", each cited there as an \"OpenAI Math Release preprint\" [9]. Its manuscript \"The rational Hodge conjecture for CM abelian varieties\" (dated 30 September 2026; a 6 October edition updates citations) claims the conjecture, with rational coefficients and in every codimension, for complex abelian varieties with complex multiplication: a special case, not the full conjecture [9]. It defines these as abelian varieties whose endomorphism algebra contains a commutative semisimple subalgebra of twice the variety's dimension, and also claims consequences, through theorems of Milne, for the Tate conjecture for abelian varieties over finite fields [9]. The README lists the result among the \"exceptions\" to the fixed procedure used for most results, without saying how it was produced; on 14 October 2026 the repository's Lean formalisation catalogue listed no formal proof of it [9]. The repository's history records, under 7 October 2026, the withdrawal of three other Hodge-related manuscripts (on split abelian eightfolds, Kuga–Satake correspondences for K3 surfaces and products of K3 surfaces) after \"a sign error\"; the notices say this \"does not assert that the mathematical statement is false\", and the CM manuscript was not withdrawn [9]. No published response by a named specialist to the CM manuscript had been found on 14 October 2026, when the Clay Mathematics Institute still listed the conjecture as unsolved [10].","approaches":"**Weil classes on abelian varieties.** Reduce the Hodge classes of abelian varieties to special classes and construct cycles for those; on abelian varieties of CM type, Hodge classes can be written through divisor classes and split Weil classes [6]. *Strongest evidence:* Markman's algebraicity of Weil classes on sixfolds of discriminant −1 and on all fourfolds of Weil type, which gives the conjecture for abelian fourfolds [7, 8]. *Main difficulty:* Voisin counts Weil classes on abelian varieties with complex multiplication among the \"more mysterious classes\" [5], and Markman's argument uses semiregular sheaves, which she thinks very unlikely to settle the variational form of the conjecture in general [8].\n\n**Hodge classes in families.** Follow a class as the variety varies. *Strongest evidence:* the algebraicity of the locus where a class stays Hodge [3]; Voisin's survey shows that the variational Hodge conjecture (a class that is algebraic on one member of a family and stays Hodge is algebraic on every member) follows from the Lefschetz standard conjecture [5]. *Main difficulty:* the arithmetic counterpart, that this locus is defined over a number field when the family is, is in Voisin's words \"completely open\", and the Lefschetz standard conjecture is itself unproved in general [5].\n\n**Absolute Hodge classes.** Voisin splits the conjecture into two independent parts: every Hodge class is absolute Hodge, and every absolute Hodge class is algebraic [5]. *Strongest evidence:* the first part is proved for abelian varieties by Deligne and, through the Kuga–Satake construction, for powers of hyper-Kähler varieties [2, 5]. *Main difficulty:* proving the first part leaves the second open [5]; Deligne notes that such substitutes do not allow reduction modulo a prime [4], and the second part includes the Weil classes above [5].\n\n**Analytic and inductive methods.** Build cycles from holomorphic objects, or argue by induction on dimension through pencils of hyperplane sections, as in proofs of the degree-2 case [5]. *Strongest evidence:* the degree-2 case, which has both kinds of proof [5]. *Main difficulty:* the Kähler counterexamples mean any proof must use algebraicity [5, 8]; Deligne explains that the induction method of Phillip Griffiths needs all of the intermediate Jacobian of the hyperplane sections to come from algebraic cycles, and that known examples have made it \"generally inapplicable\" [4]; and an approach of Maurizio Cornalba and Griffiths through vector bundles \"has alas not been successful\" [8].","subQuestions":[{"number":1,"text":"**Other Weil sixfolds.** Markman's theorem covers abelian sixfolds of Weil type with discriminant −1 [7]. Can his secant-sheaf construction, or another, prove the same for other discriminants?"},{"number":2,"text":"**A class every variety has.** If the degree-2k cohomology of X has dimension b, its top exterior power gives a Hodge class on the product of b copies of X, absolute Hodge but not known to be algebraic [5, 8]. Is it algebraic for smooth hypersurfaces, where Voisin notes it is at least \"motivated\" in André's sense [5]?"},{"number":3,"text":"**A rigid counterexample.** Is there a variety not defined over a number field with a Hodge class that stops being Hodge under every non-trivial deformation? Such a pair would disprove the arithmetic counterpart of the Cattani–Deligne–Kaplan theorem, and with it the Hodge conjecture [5, 8]."},{"number":4,"text":"**Limits of semi-regularity.** Is there a family in which a class is algebraic on one member and stays Hodge, but no multiple of it is represented by a semiregular cycle or sheaf [8]?"},{"number":5,"text":"**Absolute Hodge beyond abelian varieties.** The property is proved for abelian varieties and for powers of hyper-Kähler varieties [2, 5]. Can it be proved for the Hodge classes of another natural class of varieties?"}],"references":[{"number":1,"citation":"Atiyah, M. F. & Hirzebruch, F. (1962). Analytic cycles on complex manifolds. *Topology* 1(1), 25–45.","doi":"10.1016/0040-9383(62)90094-0","url":"https://doi.org/10.1016/0040-9383(62)90094-0","kind":"journal","verifiedOn":"2026-10-14","contentChecked":"descriptions of the result in reference 4 (remark iv) and reference 5 (Remark 1)"},{"number":2,"citation":"Deligne, P. (1982). Hodge cycles on abelian varieties (notes by J. S. Milne). In Deligne, P., Milne, J. S., Ogus, A. & Shih, K.-y., *Hodge Cycles, Motives, and Shimura Varieties*, Lecture Notes in Mathematics 900, Springer, 9–100.","doi":"10.1007/978-3-540-38955-2_3","url":"https://doi.org/10.1007/978-3-540-38955-2_3","kind":"book","verifiedOn":"2026-10-14","contentChecked":"introduction of J. S. Milne's online edition (https://www.jmilne.org/math/Documents/Deligne82.pdf), which states the main theorem and the reduction to abelian varieties of CM type, and the statement in reference 5"},{"number":3,"citation":"Cattani, E., Deligne, P. & Kaplan, A. (1995). On the locus of Hodge classes. *Journal of the American Mathematical Society* 8(2), 483–506.","doi":"10.1090/S0894-0347-1995-1273413-2","url":"https://doi.org/10.1090/S0894-0347-1995-1273413-2","kind":"journal","verifiedOn":"2026-10-14","contentChecked":"abstract (Crossref) and Theorem 6 in reference 5"},{"number":4,"citation":"Deligne, P. The Hodge conjecture. Official problem description, Clay Mathematics Institute (prepared for the Millennium Prize; its most recent reference is from 2000).","doi":null,"url":"https://www.claymath.org/wp-content/uploads/2022/06/hodge.pdf","kind":"web","verifiedOn":"2026-10-14","contentChecked":null},{"number":5,"citation":"Voisin, C. (2016). The Hodge conjecture. In *Open Problems in Mathematics*, Springer, 521–543.","doi":"10.1007/978-3-319-32162-2_17","url":"https://doi.org/10.1007/978-3-319-32162-2_17","kind":"book","verifiedOn":"2026-10-14","contentChecked":"full text (author's PDF, https://webusers.imj-prg.fr/~claire.voisin/Articlesweb/voisinhodge.pdf)"},{"number":6,"citation":"Milne, J. S. (2020). Hodge classes on abelian varieties. arXiv:2010.08857 (version 2, 11 November 2020; a short note whose arXiv comment reads \"No new results, but the proofs are shorter\").","doi":null,"url":"https://arxiv.org/abs/2010.08857","kind":"preprint","verifiedOn":"2026-10-14","contentChecked":"abstract (arXiv)"},{"number":7,"citation":"Markman, E. (2025). Cycles on abelian 2n-folds of Weil type from secant sheaves on abelian n-folds. arXiv:2502.03415 (version 2, 8 June 2025; preprint, no journal reference listed on 2026-10-14).","doi":null,"url":"https://arxiv.org/abs/2502.03415","kind":"preprint","verifiedOn":"2026-10-14","contentChecked":"abstract (arXiv)"},{"number":8,"citation":"Voisin, C. (2026). The status of the Hodge conjecture. Guest post on Terence Tao's blog *What's new*, 12 September 2026.","doi":null,"url":"https://terrytao.wordpress.com/2026/09/12/the-status-of-the-hodge-conjecture/","kind":"web","verifiedOn":"2026-10-14","contentChecked":"full post"},{"number":9,"citation":"OpenAI (2026). The rational Hodge conjecture for CM abelian varieties. OpenAI Math Release preprint, editions dated 30 September and 6 October 2026, in the repository github.com/openai/math (opened 6 October 2026), with the repository's README.md, history.md, withdrawal notices and Lean formalisation catalogue (lean/formalization.yaml).","doi":null,"url":"https://github.com/openai/math/blob/main/preprints/The-rational-Hodge-conjecture-for-CM-abelian-varieties-October-6-2026/paper.pdf","kind":"preprint","verifiedOn":"2026-10-14","contentChecked":"abstract and introduction of the 6 October edition; README.md; history.md; the three withdrawal notices; lean/formalization.yaml and lean/docs, which list no formalisation for this result"},{"number":10,"citation":"Clay Mathematics Institute. Hodge Conjecture (problem page, status \"Unsolved\") and Rules for the Millennium Prize Problems.","doi":null,"url":"https://www.claymath.org/millennium/hodge-conjecture/","kind":"web","verifiedOn":"2026-10-14","contentChecked":null}],"reviewedOn":"2026-10-14","version":1},"claims":[{"id":"a9bd3d9f-848b-4185-8839-1c5be3ab274a","kind":"partial","state":"current","title":"Algebraicity of Weil classes on abelian sixfolds of Weil type for ℚ(√−3) with discriminant −2","quote":"We prove that the Weil classes of every polarized abelian sixfold of Weil type for K = ℚ(√−3) with Hermitian discriminant −2 are algebraic.","claimant":"Benjamin Severn","aiInvolvement":"Text declaration: Primarily AI-generated. Human-understanding declaration: Some parts of this contribution, but not all, have been checked and understood by a human. Additional provenance: This paper was produced in a research programme directed by the author, in which AI systems did most of the mathematical derivation, the computer-algebra code and the writing.","modelName":"Claude Opus 5.5, Claude Sonnet 5.5; Codex (GPT-5.6 Sol, GPT-6 Astra, GPT-5.6 Luna)","venue":"Hexagon","sourceUrl":"https://hexagonmath.org/abs/2610.00246v3","claimedOn":"2026-10-22","formalSystem":null,"formalProofUrl":null,"writeupUrl":null,"verifiers":null,"note":null,"checkedOn":"2026-10-27","updates":[]},{"id":"38f48c42-5d5c-497a-b43f-394508c7bfdd","kind":"partial","state":"current","title":"Weil classes on every abelian sixfold of Weil type for ℚ(√−3)","quote":"We prove that the Weil classes of every polarized complex abelian sixfold of Weil type for K = ℚ(√−3), of every Hermitian discriminant, are algebraic.","claimant":"Benjamin Severn","aiInvolvement":"Text declaration: Primarily AI-generated. Human-understanding declaration: Some parts of this contribution, but not all, have been checked and understood by a human. Additional provenance: This paper was produced in a research programme directed by the author, in which AI systems did most of the mathematical derivation, the computer-algebra code and the writing.","modelName":"Claude Opus 5.5, Claude Sonnet 5.5; Codex (GPT-6 Astra, GPT-5.6 Sol, GPT-5.6 Luna)","venue":"Hexagon","sourceUrl":"https://hexagonmath.org/abs/2610.00387v2","claimedOn":"2026-10-22","formalSystem":null,"formalProofUrl":null,"writeupUrl":null,"verifiers":null,"note":null,"checkedOn":"2026-10-27","updates":[]},{"id":"ac7646a8-5317-4e3f-8185-54c3635bdbbc","kind":"partial","state":"current","title":"The Hodge Conjecture for CM abelian varieties","quote":"Following the recent work [Ope26], we give a proof of the Hodge conjecture for CM abelian varieties.","claimant":"Pol van Hoften, Jack Sempliner","aiInvolvement":"Text declaration: Primarily human-written. Human-understanding declaration: All parts of this contribution have been checked and understood by a human. Additional provenance: This article is entirely human written. That said, we made extensive use of ChatGPT Pro during the preparation of this argument.","modelName":"ChatGPT 6.1","venue":"Hexagon","sourceUrl":"https://hexagonmath.org/abs/2610.00259v2","claimedOn":"2026-10-18","formalSystem":null,"formalProofUrl":null,"writeupUrl":null,"verifiers":null,"note":null,"checkedOn":"2026-10-27","updates":[]},{"id":"0dfa516e-d6d2-409a-9eea-039129e4230b","kind":"partial","state":"current","title":"Abelian covers, Gale correspondences, and the Hodge conjecture for powers","quote":"We prove the rational Hodge conjecture on every self-power of the Jacobian at each tensor Hodge-generic point of the full marked variation of a connected abelian cover of curves. This holds in every base genus and for every compatible branching pattern.","claimant":"OpenAI","aiInvolvement":"This repository contains mathematical manuscripts and supporting proof artifacts produced by an internal OpenAI model.","modelName":"an unreleased internal OpenAI model","venue":"openai/math repository","sourceUrl":"https://github.com/openai/math/blob/main/preprints/Abelian-covers-Gale-correspondences-and-the-Hodge-conjecture-for-powers-October-6-2026/paper.pdf","claimedOn":"2026-10-06","formalSystem":null,"formalProofUrl":null,"writeupUrl":null,"verifiers":null,"note":null,"checkedOn":"2026-10-14","updates":[]},{"id":"059af417-3a47-4cfe-812e-584dfa0616b7","kind":"partial","state":"current","title":"Algebraic Kuga–Satake correspondences and Hodge conjectures on a K3 quadratic locus","quote":"We prove the rational Hodge and generalized Hodge conjectures in every cohomological degree of every self-power of a projective complex K3 surface whose transcendental quadratic space, with its cup-product form, admits a rational isometric embedding in U_Q^{⊕2} ⊥ ⟨−1⟩^4. This includes every ample P-polarized K3 surface, including all Picard jumps, for P = U ⊕ D8(−1) ⊕ D4(−1).","claimant":"OpenAI","aiInvolvement":"This repository contains mathematical manuscripts and supporting proof artifacts produced by an internal OpenAI model.","modelName":"an unreleased internal OpenAI model","venue":"openai/math repository","sourceUrl":"https://github.com/openai/math/blob/main/preprints/Algebraic-Kuga-Satake-correspondences-and-Hodge-conjectures-on-a-K3-quadratic-locus-September-30-2026/paper.pdf","claimedOn":"2026-10-06","formalSystem":null,"formalProofUrl":null,"writeupUrl":null,"verifiers":null,"note":null,"checkedOn":"2026-10-14","updates":[]},{"id":"387c3bcc-3e47-4c24-81cd-6555cc20094a","kind":"partial","state":"current","title":"The rational Hodge conjecture for CM abelian varieties","quote":"We prove the rational Hodge conjecture for complex abelian varieties with complex multiplication: every rational Hodge class on such a variety is a rational linear combination of algebraic cycle classes. As consequences, we obtain the generalized Hodge conjecture for CM abelian varieties, the Tate conjecture for abelian varieties over finite fields, and the Hodge standard conjecture for abelian varieties in arbitrary characteristic.","claimant":"OpenAI","aiInvolvement":"This repository contains mathematical manuscripts and supporting proof artifacts produced by an internal OpenAI model. … Exceptions to this fixed procedure include work on a zero-free region for the Riemann zeta function and proof of the Hodge Conjecture for CM abelian varieties.","modelName":"an unreleased internal OpenAI model","venue":"openai/math repository","sourceUrl":"https://github.com/openai/math/blob/main/preprints/The-rational-Hodge-conjecture-for-CM-abelian-varieties-October-6-2026/paper.pdf","claimedOn":"2026-10-06","formalSystem":null,"formalProofUrl":null,"writeupUrl":null,"verifiers":null,"note":null,"checkedOn":"2026-10-14","updates":[]},{"id":"1686ac70-da92-4c2a-8c77-0cf5af60e340","kind":"partial","state":"current","title":"Weil classes and Hodge classes on abelian powers","quote":"We prove the rational Hodge conjecture in every codimension on every self-power of a complex abelian sixfold with an imaginary-quadratic action and a compatible polarization whose rational homological Hermitian form is hyperbolic of signature (3,3). We also prove it in every codimension on every self-power of a complex abelian variety of dimension at most five admitting an imaginary-quadratic action.","claimant":"OpenAI","aiInvolvement":"This repository contains mathematical manuscripts and supporting proof artifacts produced by an internal OpenAI model.","modelName":"an unreleased internal OpenAI model","venue":"openai/math repository","sourceUrl":"https://github.com/openai/math/blob/main/preprints/Weil-classes-and-Hodge-classes-on-abelian-powers-October-6-2026/paper.pdf","claimedOn":"2026-10-06","formalSystem":null,"formalProofUrl":null,"writeupUrl":null,"verifiers":null,"note":null,"checkedOn":"2026-10-14","updates":[]},{"id":"d4102f2d-fb3d-4ae2-9e93-27c7b97d36e0","kind":"partial","state":"withdrawn","title":"Algebraicity of Kuga–Satake Correspondences for K3 Surfaces","quote":"We prove that the Kuga–Satake correspondence is algebraic for every smooth projective complex K3 surface. More precisely, the prescribed embedding of its transcendental cohomology into the cohomology of its Kuga–Satake abelian variety is induced by a rational algebraic cycle, with the fixed normalization and full even-Clifford target.","claimant":"OpenAI","aiInvolvement":"This repository contains mathematical manuscripts and supporting proof artifacts produced by an internal OpenAI model.","modelName":"an unreleased internal OpenAI model","venue":"openai/math repository","sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Algebraicity-of-Kuga-Satake-Correspondences-for-K3-Surfaces-October-3-2026/manuscript.pdf","claimedOn":"2026-10-06","formalSystem":null,"formalProofUrl":null,"writeupUrl":null,"verifiers":null,"note":null,"checkedOn":"2026-10-14","updates":[{"kind":"withdrawal","on":"2026-10-06","by":"OpenAI","text":"The proof relies on an adaptation of the flawed stabilization-trace construction in Algebraicity of Weil classes on split abelian eightfolds. The sign error in that construction prevents the required signed-double-point cancellation. The resulting gap leaves this paper's claimed algebraicity theorem unproved. This withdrawal concerns the proof; it does not assert that the mathematical statement is false.","url":"https://github.com/openai/math/blob/main/preprints/Algebraicity-of-Kuga-Satake-Correspondences-for-K3-Surfaces-October-3-2026/README.md"},{"kind":"response","on":"2026-10-22","by":"Yankı Lekili, Nick Sheridan","text":"The documents [OAI26a, OAI26b, OAI26c] claimed to prove cases of the Hodge conjecture via homological mirror symmetry, before being withdrawn by OpenAI. We discuss the source of the error (but do not analyse the rest of the argument).","url":"https://arxiv.org/abs/2610.30976"}]},{"id":"b734f790-5731-40e3-8edc-5ae752cd3acc","kind":"partial","state":"withdrawn","title":"Algebraicity of Weil classes on split abelian eightfolds","quote":"The Weil classes are a particularly concrete test of the Hodge conjecture: they form a rational two-dimensional space defined by an imaginary quadratic action, while their algebraic representatives generally do not come from products of divisor classes. This paper establishes their algebraicity in the split eight-dimensional case.","claimant":"OpenAI","aiInvolvement":"This repository contains mathematical manuscripts and supporting proof artifacts produced by an internal OpenAI model.","modelName":"an unreleased internal OpenAI model","venue":"openai/math repository","sourceUrl":"https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Algebraicity-of-Weil-classes-on-split-abelian-eightfolds-September-18-2026/paper.pdf","claimedOn":"2026-10-06","formalSystem":null,"formalProofUrl":null,"writeupUrl":null,"verifiers":null,"note":null,"checkedOn":"2026-10-14","updates":[{"kind":"withdrawal","on":"2026-10-06","by":"OpenAI","text":"The proof contains a sign error in the stabilization-trace argument used to obtain negative double points. … The Eliashberg-Murphy cancellation theorem invoked at this step requires zero signed double-point count, so its hypothesis is not met. The subsequent oriented-surgery and embedded-brane construction is therefore unsupported, and the paper does not establish its claimed algebraicity theorem. This withdrawal concerns the proof; it does not assert that the mathematical statement is false.","url":"https://github.com/openai/math/blob/main/preprints/Algebraicity-of-Weil-classes-on-split-abelian-eightfolds-September-18-2026/README.md"},{"kind":"response","on":"2026-10-22","by":"Yankı Lekili, Nick Sheridan","text":"The documents [OAI26a, OAI26b, OAI26c] claimed to prove cases of the Hodge conjecture via homological mirror symmetry, before being withdrawn by OpenAI. We discuss the source of the error (but do not analyse the rest of the argument).","url":"https://arxiv.org/abs/2610.30976"}]},{"id":"b8c039d9-5405-4f37-8b62-22a3ea88a440","kind":"partial","state":"withdrawn","title":"The rational Hodge conjecture for products of K3 surfaces","quote":"We prove the rational Hodge conjecture for every finite product of projective complex K3 surfaces: every rational Hodge class is algebraic. The factors may be distinct or repeated, with no restrictions on their Picard numbers, periods, or endomorphism fields.","claimant":"OpenAI","aiInvolvement":"This repository contains mathematical manuscripts and supporting proof artifacts produced by an internal OpenAI model.","modelName":"an unreleased internal OpenAI model","venue":"openai/math repository","sourceUrl":"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","claimedOn":"2026-10-06","formalSystem":null,"formalProofUrl":null,"writeupUrl":null,"verifiers":null,"note":null,"checkedOn":"2026-10-14","updates":[{"kind":"withdrawal","on":"2026-10-06","by":"OpenAI","text":"The proof relies on an adaptation of the flawed stabilization-trace construction in Algebraicity of Weil classes on split abelian eightfolds. The sign error in that construction prevents the required signed-double-point cancellation. The resulting gap leaves this paper's claimed rational Hodge theorem for products of projective K3 surfaces unproved. This withdrawal concerns the proof; it does not assert that the mathematical statement is false.","url":"https://github.com/openai/math/blob/main/preprints/The-rational-Hodge-conjecture-for-products-of-K3-surfaces-October-4-2026/README.md"},{"kind":"response","on":"2026-10-22","by":"Yankı Lekili, Nick Sheridan","text":"The documents [OAI26a, OAI26b, OAI26c] claimed to prove cases of the Hodge conjecture via homological mirror symmetry, before being withdrawn by OpenAI. We discuss the source of the error (but do not analyse the rest of the argument).","url":"https://arxiv.org/abs/2610.30976"}]}]}