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)

Manuscript

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

September 24, 2026 · source · Lean formalization available · cite as \cite{OAI:ample-rank-two-bundles-on-the-quadric-surface-without-griffiths-positive-metrics-September-24-2026}

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.

Result 050 · Algebraic and complex geometry. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.