HEXAGONbeta

Weak positivity of Schur forms under decomposable curvature

A mix of human-written and AI-generated textHuman understanding: some partsmath.FA — Functional Analysismath.CV — Complex Variablesmath.DG — Differential Geometryquant-ph — Quantum Physics

Contributed by Suvrit Sra ↗

Version 1 / Nov 02, 2026 / CC BY 4.0

Abstract

Griffiths's 1969 conjecture predicts positivity of Schur forms for Griffiths-positive vector bundles. Finski reduced its weak form to a sign question for the double mixed discriminant Φn(H)\Phi_n(H) of a positive linear map. Du's rank-nine counterexample shows that Φn(H)≥0\Phi_n(H)\ge0 fails in general. We prove Φn(H)≥0\Phi_n(H)\ge0 for every decomposable map HH, a sum of a completely positive and a completely copositive map, using a quaternionic Gaussian determinant representation. Combined with Finski's incidence formula, this inequality gives weak nonnegativity of every Schur form under decomposable curvature, without splitting assumptions. Beyond the decomposable cone, nonnegativity of Φk(HV,U)\Phi_k(H_{V,U}) on every square compression is equivalent to Φn(H+S)≥0\Phi_n(H+S)\ge0 for every entanglement-breaking map SS. This class contains explicit indecomposable maps, but is nonconvex in every size n≥9n\ge9. We prove the nonconvexity by expressing Du's map as an average of four members, and the obstruction persists when all compression inequalities are strict.

Provenance statement

The results in this paper were developed through extensive use of an AI assistant GPT-6 Astra (Pro). The choice of the questions worth answering was fully made by the author, as was the selection of the subset of results worth retaining. This choice of material evolved over many months, including up to the recent disproof of Φ_9 ≥ 0. None of the claims were obtained zero-shot: they emerged through sustained back-and-forth discussion, including failed attempts, counterexamples, audits, and repairs. The writing, polishing, and presentation were done with GPT-6 as well as Claude's (Fable 5.1 and Opus 5.5) assistance. The main text has not yet been vetted line by line by the author.

Tools used

OpenAI
GPT-6 Astra (Pro)
Anthropic
Claude Fable

References

  1. Alimzhan Amanov and Damir Yeliussizov. Tensor slice rank and Cayley 's first hyperdeterminant. Linear Algebra and its Applications , 656: 0 224--246, 2023. https://doi.org/10.1016/j.laa.2022.09.027. Numbered locators refer to https://arxiv.org/abs/2107.08864v1.DOI
  2. Alexander Barvinok. Polynomial time algorithms to approximate permanents and mixed discriminants within a simply exponential factor. Random Structures & Algorithms , 14 0 (1): 0 29--61, 1999. Author copy: https://deepblue.lib.umich.edu/handle/2027.42/35110.link
  3. Raoul Bott and Shiing-Shen Chern. Hermitian vector bundles and the equidistribution of the zeroes of their holomorphic sections. Acta Mathematica , 114: 0 71--112, 1965. https://doi.org/10.1007/BF02391818.DOI
  4. Heinz-Peter Breuer. Optimal entanglement criterion for mixed quantum states. Physical Review Letters , 97: 0 080501, 2006. https://doi.org/10.1103/PhysRevLett.97.080501; https://arxiv.org/abs/quant-ph/0605036.DOI
  5. Man-Duen Choi. Completely positive linear maps on complex matrices. Linear Algebra and its Applications , 10 0 (3): 0 285--290, 1975 a . https://doi.org/10.1016/0024-3795(75)90075-0.DOI
  6. Man-Duen Choi. Positive semidefinite biquadratic forms. Linear Algebra and its Applications , 12 0 (2): 0 95--100, 1975 b . https://doi.org/10.1016/0024-3795(75)90058-0.DOI
  7. Dariusz Chru ś ci ń ski and Andrzej Kossakowski. A class of positive atomic maps. Journal of Physics A: Mathematical and Theoretical , 41: 0 215201, 2008. https://doi.org/10.1088/1751-8113/41/21/215201; https://arxiv.org/abs/0711.4483.DOI
  8. Nir Cohen and Stefano De Leo. The quaternionic determinant. Electronic Journal of Linear Algebra , 7: 0 100--111, 2000. https://arxiv.org/abs/math-ph/9907015v2.link
  9. Yun-Heng Du. Griffiths positivity does not imply positivity of the top Chern form, 2026. arXiv:2610.05931v1, 5 October 2026. https://arxiv.org/abs/2610.05931v1.arXiv
  10. Yun-Heng Du and Song-Yan Xie. Ample vector bundles without Griffiths -semipositive metrics, 2026. arXiv:2609.26504v1, 22 September 2026. https://arxiv.org/abs/2609.26504v1.arXiv
Show all 31 referencesShow fewer references
  1. G. P. Egorychev. The solution of van der Waerden 's problem for permanents. Advances in Mathematics , 42 0 (3): 0 299--305, 1981. https://doi.org/10.1016/0001-8708(81)90044-X.DOI
  2. D. I. Falikman. Proof of the van der Waerden conjecture regarding the permanent of a doubly stochastic matrix. Mathematical Notes , 29 0 (6): 0 475--479, 1981. Russian original: Matematicheskie Zametki 29 (1981), 931--938. https://doi.org/10.1007/BF01163285.DOI
  3. Siarhei Finski. On characteristic forms of positive vector bundles, mixed discriminants and pushforward identities. Journal of the London Mathematical Society , 106 0 (2): 0 1539--1579, 2022. https://doi.org/10.1112/jlms.12605. Numbered locators in this paper refer to https://arxiv.org/abs/2009.13107v2.DOI
  4. Peter J. Forrester. Dualities for characteristic polynomial averages of complex symmetric and self dual non- Hermitian random matrices, 2024. arXiv:2411.07356v1. https://arxiv.org/abs/2411.07356v1.arXiv
  5. William Fulton and Robert Lazarsfeld. Positive polynomials for ample vector bundles. Annals of Mathematics , 118 0 (1): 0 35--60, 1983. https://doi.org/10.2307/2006953.DOI
  6. Mark Girard, Seung-Hyeok Kye, and Erling Størmer. Convex cones in mapping spaces between matrix algebras. Linear Algebra and its Applications , 608: 0 248--269, 2021. https://doi.org/10.1016/j.laa.2020.09.008; https://arxiv.org/abs/2002.09614.DOI
  7. Phillip A. Griffiths. Hermitian differential geometry, Chern classes, and positive vector bundles. In Donald C. Spencer and Shokichi Iyanaga, editors, Global Analysis (Papers in Honor of K. Kodaira) , pages 185--251. University of Tokyo Press, Tokyo, 1969. Author copy: https://publications.ias.edu/sites/default/files/hermdiffgeometrychern.pdf.link
  8. Leonid Gurvits. Quantum matching theory, 2002. arXiv:quant-ph/0201022v2. https://arxiv.org/abs/quant-ph/0201022v2.arXiv
  9. Leonid Gurvits. Classical complexity and quantum entanglement. Journal of Computer and System Sciences , 69 0 (3): 0 448--484, 2004. https://doi.org/10.1016/j.jcss.2004.06.003. Numbered locators refer to the preprint Classical deterministic complexity of Edmonds' problem and quantum entanglement , arXiv:quant-ph/0303055v1, https://arxiv.org/abs/quant-ph/0303055v1.arXivDOI
  10. William Hall. Constructions of indecomposable positive maps based on a new criterion for indecomposability, 2006. arXiv:quant-ph/0607035. https://arxiv.org/abs/quant-ph/0607035.arXiv
  11. Michael Horodecki, Peter W. Shor, and Mary Beth Ruskai. Entanglement breaking channels. Reviews in Mathematical Physics , 15 0 (6): 0 629--641, 2003. https://arxiv.org/abs/quant-ph/0302031v2.link
  12. Hong-Jong Kim and Seung-Hyeok Kye. Indecomposable extreme positive linear maps in matrix algebras. Bulletin of the London Mathematical Society , 26 0 (6): 0 575--581, 1994. https://doi.org/10.1112/blms/26.6.575.DOI
  13. Anish Kulkarni, Kohei Kawabata, and Shinsei Ryu. Non-linear sigma models for non- Hermitian random matrices in symmetry classes AI†\mathrm{AI}^{\dagger} and AII†\mathrm{AII}^{\dagger} . Journal of Physics A: Mathematical and Theoretical , 58: 0 225202, 2025. https://doi.org/10.1088/1751-8121/adc95f; https://arxiv.org/abs/2410.24043.DOI
  14. Seung-Hyeok Kye. Positive linear maps between matrix algebras which fix diagonals. Linear Algebra and its Applications , 216: 0 239--256, 1995. https://doi.org/10.1016/0024-3795(93)00140-U.DOI
  15. Ping Li. Nonnegative Hermitian vector bundles and Chern numbers. Mathematische Annalen , 380: 0 21--41, 2021. https://doi.org/10.1007/s00208-020-01974-x. Numbered locators refer to https://arxiv.org/abs/1702.01701v6.DOI
  16. Jean-Gabriel Luque and Jean-Yves Thibon. Hankel hyperdeterminants and Selberg integrals. Journal of Physics A: Mathematical and General , 36 0 (19): 0 5267--5292, 2003. https://doi.org/10.1088/0305-4470/36/19/306; https://arxiv.org/abs/math-ph/0211044.DOI
  17. Marvin Marcus. The permanent analogue of the Hadamard determinant theorem. Bulletin of the American Mathematical Society , 69 0 (4): 0 494--496, 1963. https://doi.org/10.1090/S0002-9904-1963-10975-1.DOI
  18. Harry Tamvakis. The connection between representation theory and Schubert calculus. L'Enseignement Mathématique , 50: 0 267--286, 2004. Author manuscript: https://math.umd.edu/~harryt/papers/reptoschub.pdf; https://arxiv.org/abs/math/0306414.link
  19. Xueyuan Wan. Positivity of Schur forms for strongly decomposably positive vector bundles. Forum of Mathematics, Sigma , 12: 0 e9, 2024. https://doi.org/10.1017/fms.2023.125.DOI
  20. Xueyuan Wan. Positivity of the third Chern form for Griffiths positive vector bundles, 2026. arXiv:2601.10424v2, 20 January 2026. https://arxiv.org/abs/2601.10424v2.arXiv
  21. Dan Wang, Wei Chen, Sei Zhen Khong, and Li Qiu. On the phases of a complex matrix. Linear Algebra and its Applications , 593: 0 152--179, 2020. https://doi.org/10.1016/j.laa.2020.01.035; https://arxiv.org/abs/1904.07211v2.DOI

Version history

  1. v1Submitted by Suvrit SraInitial depositCurrentOct 31, 2026