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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.

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Sep 24, 2026·paper.pdfLean

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.

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