Griffiths' positivity conjecture
Griffiths' positivity conjecture asks whether every ample holomorphic vector bundle on a smooth complex projective variety admits a smooth Hermitian metric with strictly Griffiths-positive Chern curvature, that is, curvature positive in every pair of nonzero tangent and fiber directions. Such a metric always implies ampleness; the conjecture asks for the converse (Griffiths 1969, Problem (0.9)). A manuscript dated 24 September 2026 in OpenAI's public mathematics collection (github.com/openai/math) claims counterexamples in rank two on the quadric surface. A counterexample on an abelian surface was posted by other authors two days earlier. This page summarizes the published claims and lists related public work. Statements are the respective authors' claims; the publication of a paper, code or a formal proof is not presented here as independent verification.
Background
- As recalled in the manuscript's introduction: for line bundles the algebraic and curvature notions agree (Umemura), and on compact Riemann surfaces the metric equivalence holds (Umemura; Campana–Flenner), so a surface is the first possible base dimension for a counterexample. Positive results recover strong curvature after modifying the bundle: Berndtsson proved Nakano positivity of E ⊗ det E for ample E, and Liu, Sun and Yang treated determinant-twisted and large symmetric powers. Analytic approaches include Demailly's Hermitian–Yang–Mills-type system and later work of Pingali and Murakami. Kobayashi characterized ampleness through strongly pseudoconvex complex Finsler metrics, which need not be Hermitian.
The original result
- Claim. On S = P1 × P1 with P = O(1, 1), let fm raise the homogeneous coordinates in each factor to their m-th powers. There is an explicit algebraic rank-two bundle G on S such that Em = fm*G ⊗ P is ample for every positive integer m, and there is an integer m0 such that Em admits no smooth Hermitian metric of strictly Griffiths-positive curvature for every m ≥ m0 (Theorem 1.1). The manuscript states that the ampleness has a quantitative very-ampleness certificate and that m0 is obtained by compactness and is ineffective. Source: the manuscript Ample rank-two bundles on the quadric surface without Griffiths-positive metrics, dated 24 September 2026 (repository commit
1581d4c). - Formalization, as described by the repository. The scope note lean/docs/050.md states that the formalized counterexample covers the bundles Em above (ample for every positive power, without such a metric for all sufficiently large powers), and that the separate very-ampleness result for all admissible even exponents is not included. The comparator statement is QuadricBundles.lean.
- Relation to other work, as stated in the manuscript. The introduction cites the preprint of Du and Xie posted on 22 September 2026 (below) for a rank-two counterexample on an abelian surface, and describes its own contribution as a construction on the rational quadric with a specified power-map family, an ampleness certificate valid for all m and a general limiting statement.
Related work by others
| Date | Work | Relation | Public claim | Public status (checked 15 October 2026) |
|---|---|---|---|---|
| 2026-09-22 | Yun-Heng Du and Song-Yan Xie, arXiv:2609.26504 (v1), Ample vector bundles without Griffiths-semipositive metrics | Earlier counterexample (cited by the manuscript) | A rank-two ample holomorphic vector bundle on an abelian surface (the manuscript specifies C × C with C the elliptic curve y2 = x3 − x) admitting no smooth Griffiths-semipositive Hermitian metric, giving a counterexample to the Griffiths conjecture. The abstract describes the proof as combining a curvature obstruction that persists under deformations with openness of ampleness in an irreducible moduli space of stable sheaves. | arXiv preprint, version 1. |
| 2026-09-22 | Yun-Heng Du and Song-Yan Xie, arXiv:2609.26528 (v1), A first-jet characterization of Griffiths positivity | Companion paper (cited by the manuscript) | A first-jet characterization of Griffiths positivity in terms of an adjoint-Levi obstruction, as described in the manuscript. | arXiv preprint, version 1. |
| 2026-09-29 | Kefeng Liu and Xueyuan Wan, arXiv:2609.37309 (v1), Stable ample vector bundles without semipositive Kobayashi curvature | Related (cited by the manuscript) | As described in the manuscript: ample bundles on every Hirzebruch surface, including P1 × P1, that are slope-stable for suitable polarizations and admit no smooth strongly pseudoconvex Finsler metric of semipositive Kobayashi curvature, a condition imposed on the bundle itself. | arXiv preprint, version 1. |
Notes
- Dates are those shown by arXiv and by the manuscript. The list covers sources found up to the date above; it makes no priority or completeness claim.