View source for 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 ([https://github.com/openai/math 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'' = '''P'''<sup>1</sup> × '''P'''<sup>1</sup> with ''P'' = ''O''(1, 1), let ''f''<sub>''m''</sub> 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 ''E''<sub>''m''</sub> = ''f''<sub>''m''</sub><sup>*</sup>''G'' ⊗ ''P'' is ample for every positive integer ''m'', and there is an integer ''m''<sub>0</sub> such that ''E''<sub>''m''</sub> admits no smooth Hermitian metric of strictly Griffiths-positive curvature for every ''m'' ≥ ''m''<sub>0</sub> (Theorem 1.1). The manuscript states that the ampleness has a quantitative very-ampleness certificate and that ''m''<sub>0</sub> is obtained by compactness and is ineffective. Source: the manuscript [https://github.com/openai/math/tree/1581d4cbdd69a0faf34b2089614fe2a814316a6a/preprints/ample-rank-two-bundles-on-the-quadric-surface-without-griffiths-positive-metrics-September-24-2026 Ample rank-two bundles on the quadric surface without Griffiths-positive metrics], dated 24 September 2026 (repository commit <code>1581d4c</code>).
* '''Formalization, as described by the repository.''' The scope note [https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/050.md lean/docs/050.md] states that the formalized counterexample covers the bundles ''E''<sub>''m''</sub> 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 [https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/ComparatorChallenges/QuadricBundles.lean 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 ==
{| class="wikitable"
! Date !! Work !! Relation !! Public claim !! Public status (checked 15 October 2026)
|-
| 2026-09-22 || Yun-Heng Du and Song-Yan Xie, [https://arxiv.org/abs/2609.26504 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 ''y''<sup>2</sup> = ''x''<sup>3</sup> − ''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, [https://arxiv.org/abs/2609.26528 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, [https://arxiv.org/abs/2609.37309 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 '''P'''<sup>1</sup> × '''P'''<sup>1</sup>, 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.

[[Category:OpenAI results]]