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Weil classes on every abelian sixfold of Weil type for Q(−3)\mathbb{Q}(\sqrt{-3})

Primarily AI-generated textHuman understanding: some partsmath.AG — Algebraic Geometrymath.GR — Group Theorymath.NT — Number Theory

Contributed by Benjamin Severn

Version 2 / Oct 20, 2026 / CC BY 4.0

Abstract

We prove that the Weil classes of every polarized complex abelian sixfold of Weil type for K=Q(−3)K=\mathbb{Q}(\sqrt{-3}), of every Hermitian discriminant, are algebraic. The same holds for every abelian eightfold of Weil type for KK of signature (4,4)(4,4) with split discriminant, and every power of such a sixfold whose special Mumford--Tate group is SUHSU_H satisfies the Hodge conjecture. The proof applies the transfer theorem of the companion preprint to the Shephard--Todd group G34G_{34} with twenty reflections as local monodromies, and passes from eightfolds to sixfolds by a compensation argument. Two general tools are proved on the way. If every local monodromy of a transfer datum is conjugate to its inverse, the lattice condition of the transfer theorem follows from Brauer's induction theorem. Dominance of the period map follows from exact data at a maximal degeneration, by the nilpotent orbit theorem. We also give an explicit higher-dimensional form of Schoen's compensation argument, which shows that one decided cell of 2n2n-folds, n≥4n\ge4, decides every cell of sixfolds over the same imaginary quadratic field. No outside unrefereed manuscript is used. This version, with its computational certificates, is archived on Zenodo: https://doi.org/10.5281/zenodo.23422649

Provenance statement

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. Claude models (Anthropic) derived the arguments, wrote the GAP and SageMath certificates, and drafted and revised the text. Each step was checked by an independent adversarial referee from a different model family (OpenAI GPT-5.6 and GPT-6 models). It began as part of the transfer-theorem manuscript, which went through four rounds of adversarial review by ChatGPT (OpenAI); after the split, this paper went through two further rounds on drafts v0.2 and v0.3 (verdicts: accept with revisions, then accept with minor revisions), and every objection raised was answered in the text or the certificates. This is version 0.5. The author set the research direction, chose which results to pursue, relayed the outside reviews and decided each revision. The boundary certificate for the G34 datum (the tuples, the lattice and dominance data, an independent SageMath reproduction, and a GAP chain check) consists of exact computations in GAP 4.14.0 and SageMath 10.7, supplied with READMEs and SHA-256 hashes of every file. The same version is archived on Zenodo, https://doi.org/10.5281/zenodo.23422649, with the certificates. The reviews were source-level: no human expert other than the author has checked the paper. No outside unrefereed manuscript is used.

Tools used

Anthropic
Claude OpusVersion 5.5
Anthropic
Claude SonnetVersion 5.5
OpenAI
CodexVersion GPT-6 Astra
OpenAI
CodexVersion GPT-5.6 Sol
OpenAI
CodexVersion GPT-5.6 Luna
The GAP Group
GAPVersion 4.14.0
SageMath
SageMathVersion 10.7

References

  1. S. Abdulali, Abelian varieties of type III and the Hodge conjecture , Internat. J. Math. 10 (1999), 667--675.
  2. V. Alexeev, R. Donagi, G. Farkas, E. Izadi and A. Ortega, The uniformization of the moduli space of principally polarized abelian 6-folds , J. Reine Angew. Math. 761 (2020), 163--217.
  3. C. Birkenhake and H. Lange, Complex Abelian Varieties , second edition, Grundlehren 302, Springer, 2004.
  4. P. Brosnan, Discriminants of Hermitian forms, maximal degenerations and Kontsevich's tropical approach to the Hodge conjecture , arXiv:2609.14169, 2026.arXiv
  5. E. Cattani, A. Kaplan and W. Schmid, Degeneration of Hodge structures , Ann. of Math. 123 (1986), 457--535.
  6. C. H. Clemens, Degeneration of Kähler manifolds , Duke Math. J. 44 (1977), 215--290.
  7. M. Dettweiler and S. Wewers, Variation of local systems and parabolic cohomology , Israel J. Math. 156 (2006), 157--185.
  8. M. Dettweiler and S. Wewers, Variation of parabolic cohomology and Poincaré duality , arXiv:math/0411119.arXiv
  9. P. Frediani, A. Ghigi and M. Penegini, Shimura varieties in the Torelli locus via Galois coverings , Int. Math. Res. Not. IMRN 2015, 10595--10623.
  10. M. Fried and H. Völklein, The inverse Galois problem and rational points on moduli spaces , Math. Ann. 290 (1991), 771--800.
Show all 35 referencesShow fewer references
  1. J. Harris and D. Mumford, On the Kodaira dimension of the moduli space of curves , Invent. Math. 67 (1982), 23--86.
  2. A. Hatcher, Algebraic Topology , Cambridge University Press, 2002.
  3. R. Howe, Remarks on classical invariant theory , Trans. Amer. Math. Soc. 313 (1989), 539--570.
  4. F. Knudsen, The projectivity of the moduli space of stable curves II: the stacks Mg,nM_{g,n} , Math. Scand. 52 (1983), 161--199.
  5. K. Koike, Algebraicity of some Weil Hodge classes , Canad. Math. Bull. 47 (2004), 566--572.
  6. H. Kraft and C. Procesi, Classical Invariant Theory: A Primer , preliminary version, July 1996, University of Basel.
  7. E. Markman, The monodromy of generalized Kummer varieties and algebraic cycles on their intermediate Jacobians , arXiv:1805.11574.arXiv
  8. E. Markman, Cycles on abelian 2n2n-folds of Weil type from secant sheaves on abelian nn-folds , arXiv:2502.03415.arXiv
  9. E. Markman, Secant sheaves and Weil classes on abelian varieties , arXiv:2509.23403.arXiv
  10. J. S. Milne, Lefschetz classes on abelian varieties , Duke Math. J. 96 (1999), 639--675.
  11. A. Mostaed, McMullen's curve, the Weil locus, and the Hodge conjecture for abelian sixfolds , arXiv:2603.20268, 2026.arXiv
  12. W. Schmid, Variation of Hodge structure: the singularities of the period mapping , Invent. Math. 22 (1973), 211--319.
  13. C. Schoen, Hodge classes on self-products of a variety with an automorphism , Compositio Math. 65 (1988), 3--32.
  14. C. Schoen, Addendum to: Hodge classes on self-products of a variety with an automorphism , Compositio Math. 114 (1998), 329--336.
  15. P. Schapira, An Introduction to Categories and Sheaves , lecture notes, version of 1 October 2023.
  16. J.-P. Serre, Linear Representations of Finite Groups , Graduate Texts in Mathematics 42, Springer, 1977.
  17. B. Severn, Algebraicity of Weil classes on abelian sixfolds of Weil type for $\Q(\sqrt{-3})$ with discriminant −2-2 , preprint, version 0.10, HexagonMath hexagon:2610.00246v1, 2026.hexagon
  18. G. C. Shephard and J. A. Todd, Finite unitary reflection groups , Canad. J. Math. 6 (1954), 274--304.
  19. B. van Geemen, An introduction to the Hodge conjecture for abelian varieties , in: Algebraic Cycles and Hodge Theory (Torino, 1993), Lecture Notes in Math. 1594, Springer, 1994, 233--252.
  20. B. van Geemen and A. Verra, Quaternionic Pryms and Hodge classes , Topology 42 (2003), 35--53.
  21. J. Voight, Quaternion Algebras , Graduate Texts in Mathematics 288, Springer, 2021.
  22. C. Voisin, La conjecture de Hodge pour les variétés abéliennes de dimension au plus 5 [d'après Markman] , Séminaire Bourbaki, exposé 1248, 2026.
  23. C. Voisin, Hodge Theory and Complex Algebraic Geometry II , Cambridge Studies in Advanced Mathematics 77, Cambridge University Press, 2003.
  24. A. Weil, Abelian varieties and the Hodge ring , in: Collected Papers III, Springer, 1980, 421--429.
  25. H. Weyl, The Classical Groups: Their Invariants and Representations , second edition, Princeton University Press, 1946.

Version history

  1. v1Submitted by Benjamin SevernInitial depositSupersededOct 19, 2026
  2. v2Submitted by Benjamin SevernSubmitted as a new versionCurrentOct 22, 2026