A counterexample to Griffiths’ positivity conjecture

1 · The essentials

The manuscript challenges a proposed link between two kinds of positivity in geometry. One concerns the overall structure of an object; the other asks whether it can be equipped with a smooth way of measuring lengths that has positive curvature. The construction takes place on a surface made by pairing two projective lines. Starting with one explicit object, the manuscript builds a family indexed by positive whole numbers. Every member has the first kind of positivity, but all members with sufficiently large indices lack the second. This gives counterexamples to Griffiths’ positivity conjecture in the setting described. The main result has a Lean formalization, although the supplied summary does not specify its detailed scope.

A picture to help (an imperfect image, not to be taken literally)

As an imperfect image, imagine a family of structures that all pass an overall design test but, beyond some point in the family, cannot be fitted with a measuring system that passes a separate local test everywhere. Passing the first test does not guarantee that the second requirement can be met.

Why it matters

The result separates ampleness from the existence of a smooth strictly Griffiths-positive Hermitian metric. According to the supplied summary, this failure already occurs for rank-two bundles on the quadric surface, rather than requiring a different setting.

2 · Going further

The precise question is whether an ample vector bundle must admit a smooth Hermitian metric with strictly Griffiths-positive curvature. The manuscript gives counterexamples in rank two on the quadric surface $P^{1} \times P^{1}$ . Its conclusion concerns the nonexistence of any metric satisfying those requirements, not merely the failure of a particular choice of metric.

The construction begins with an explicit bundle $G$ . The abstract describes taking its coordinatewise power pullbacks and then tensoring them with $O (1, 1)$ . These bundles are ample for every positive power, while for all sufficiently large powers they admit no smooth strictly Griffiths-positive Hermitian metric. The contrast between these two conclusions is the mechanism of the counterexample; the supplied abstract does not describe the arguments establishing ampleness or obstructing the metrics.

The repository describes this as a counterexample to Griffiths’ positivity conjecture already on the quadric surface. Only one manuscript is supplied, so no relationship with companion manuscripts can be reported. The main result has a Lean formalization, but the provided material does not identify which intermediate statements or additional claims are included in it.

Useful notions

Projective lines and their product
Rank-two vector bundles
Pullbacks and tensor products of bundles
Ampleness
Hermitian metrics and curvature

What was known before / what is new

Before

The source identifies Griffiths’ positivity conjecture as the claim being challenged: ampleness should lead to the existence of a smooth strictly Griffiths-positive Hermitian metric. It supplies no account of earlier partial results or of the conjecture’s history.

New

The manuscript constructs an explicit bundle $G$ whose coordinatewise power pullbacks, tensored with $O (1, 1)$ , are ample for every positive power. For all sufficiently large powers, these bundles have no smooth strictly Griffiths-positive Hermitian metric. This supplies the stated rank-two counterexamples on $P^{1} \times P^{1}$ .

Limits and assumptions

The stated construction concerns rank-two bundles on $P^{1} \times P^{1}$ and specifically uses coordinatewise power pullbacks followed by tensoring with $O (1, 1)$ . Ampleness is asserted for every positive power, but nonexistence of the specified metrics is asserted only for sufficiently large powers; no numerical threshold is supplied. The metric requirement is both smoothness and strict Griffiths positivity, so the statement should not be broadened to other metric conditions. The main result has a Lean formalization, whose detailed coverage is not given here. The source does not specify further open questions or the behavior at smaller powers.

Mini glossary

Quadric surface: The surface in this construction is $P^{1} \times P^{1}$ , the product of two projective lines.
Ampleness: Ampleness is the bundle positivity condition that the manuscript asserts for every positive power in its construction.
Strict Griffiths positivity: Strict Griffiths positivity is the curvature condition required of the smooth Hermitian metrics ruled out for sufficiently large powers.
Coordinatewise power pullback: This is the operation described in the abstract for producing new bundles from $G$ using power maps in each coordinate.
Lean formalization: A Lean formalization is a machine-checked mathematical verification; the main result has one, but its detailed scope is not supplied.

Keywords : Griffiths’ positivity conjecture · Ample vector bundles · Quadric surface

AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.

3 · Technical details and sources

Claim made in the manuscript (source text)

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

Verified by Lean: the Lean software checked the part of the reasoning described below, sometimes only part of the manuscript. This does not mean independent mathematicians have validated the result.

Scope of the Lean formalization (source text)

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lean/docs/050.md

Manuscripts in this family

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

24/09/2026

Verified by Lean

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

PDF GitHub folder Lean formalization