Result 050, Algebraic and complex geometry

A counterexample to Griffiths’ positivity conjecture

Constructs ample rank-two bundles on $\mathbb P^1\times\mathbb P^1$ with no smooth Hermitian metric of strictly Griffiths-positive curvature, disproving Griffiths' positivity conjecture already on the quadric surface.

Lean formalization Disproof or counterexample

The bigger picture

Why it matters

Algebraic geometry and curvature offer different ways to describe a vector bundle as positive. This unreviewed manuscript reports that algebraic ampleness need not yield strictly positive Griffiths curvature, contradicting a conjectured bridge between the two.

What changes?

The construction starts with an explicit rank-two bundle G on the quadric surface, the product of two complex projective lines. Such a bundle assigns a two-dimensional complex vector space to each point. Pullbacks by coordinatewise power maps, tensored with the line bundle of degree one in each factor, are ample, an algebraic positivity condition, for every positive integer power. For all sufficiently large powers, they reportedly admit no smooth Hermitian metric with strictly Griffiths-positive curvature.

What does that help mathematicians do?

Strict Griffiths positivity asks for positive curvature in every nonzero tangent direction and every nonzero fiber direction. The claimed counterexamples would show that ampleness alone cannot guarantee such a metric, even for rank two on a surface. Researchers therefore could not infer this curvature positivity from ampleness alone; arguments needing it would require additional hypotheses or a separate metric construction.

Are there practical applications?

The immediate value is foundational: clarifying the limits of translating algebraic positivity into curvature-based methods in complex geometry. The power-pullback family also provides concrete test cases for proposed criteria that might distinguish ample bundles admitting Griffiths-positive metrics from those that do not.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

Ample rank-two bundles on the quadric surface without Griffiths-positive metrics

September 24, 2026 18 pages Main result formalized in Lean

We give counterexamples to Griffiths' conjecture in rank two on the quadric surface $\mathbb P^1\times\mathbb P^1$ . We construct an explicit bundle G whose coordinatewise power pullbacks, tensored with $\mathcal O(1,1)$ , are ample for every positive power, but admit no smooth strictly Griffiths-positive Hermitian metric for all sufficiently large powers.

PDF LaTeX source

Cite (BibTeX)

@misc{OAI:ample-rank-two-bundles-on-the-quadric-surface-without-griffiths-positive-metrics-September-24-2026,
  author = {{OpenAI}},
  title = {{Ample rank-two bundles on the quadric surface without Griffiths-positive metrics}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/ample-rank-two-bundles-on-the-quadric-surface-without-griffiths-positive-metrics-September-24-2026/paper.pdf}{OAI:ample-rank-two-bundles-on-the-quadric-surface-without-griffiths-positive-metrics-September-24-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/050.md.

A counterexample to Griffiths’ positivity conjecture

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Ample rank-two bundles on the quadric surface without Griffiths-positive metrics

Scope

Griffiths' positivity conjecture predicts that every ample holomorphic vector bundle on a smooth complex projective variety admits a smooth Hermitian metric with strictly Griffiths-positive curvature. The formalized counterexample starts with a rank-two algebraic bundle $G$ on $\mathbb P^1\times\mathbb P^1$ . Its coordinatewise power pullbacks, tensored with $\mathcal O(1,1)$ , are ample for every positive power but admit no such metric for all sufficiently large powers.

The separate very-ampleness result for all admissible even exponents is not included.

Comparator links

Result	Comparator statement
Ample bundles without Griffiths-positive metrics	QuadricBundles.lean

At a glance

Subject: Algebraic and complex geometry
Manuscripts: 1
Pages: 18
Dated: Sep 24, 2026
Lean formalization: Yes
Reasoning summary: No

Named problems addressed

Griffiths' positivity conjectureDisproved

Indexed by Emergent Mind with an LLM from the text above; check the manuscripts for the precise statements.

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By similarity of summaries and abstracts.

On the map

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