A counterexample to Griffiths’ positivity conjecture
OpenAI research collection · 1 manuscript · Source checked 2026-10-19
OpenAI's result summary
Constructs ample rank-two bundles on \mathbb P^1×\mathbb P^1 with no smooth Hermitian metric of strictly Griffiths-positive curvature, disproving Griffiths' positivity conjecture already on the quadric surface.
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MANUSCRIPT 01
Ample rank-two bundles on the quadric surface without Griffiths-positive metrics
We give counterexamples to Griffiths' conjecture in rank two on the quadric surface \mathbb P^1×\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…
Full source abstract
We give counterexamples to Griffiths' conjecture in rank two on the quadric surface \mathbb P^1×\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.
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