Griffiths positivity versus ampleness

Prove that every ample holomorphic vector bundle can be endowed with a Griffiths-positive Hermitian metric, thereby establishing the equivalence between ampleness and Griffiths positivity for vector bundles.

Background

Griffiths positivity is a differential-geometric positivity notion for Hermitian holomorphic vector bundles, while ampleness is its algebro-geometric counterpart. The two notions are equivalent for line bundles and for vector bundles over curves, but the equivalence for general ample vector bundles remains unresolved. The paper notes that constructing a Griffiths-positive metric on a general ample vector bundle is difficult, although a strategy due to Demailly has been proposed.

References

It is well-known that Griffiths positivity implies ampleness, and the converse was conjectured to also hold true in [Gr69], i.e., an ample vector bundle can be endowed with a Griffiths positive metric.

— Chern numbers on positive vector bundles and combinatorics  (2501.08833 - Li, 15 Jan 2025) in Section 1, Introduction

Griffiths conjectured that $E$ is ample if and only if it admits a smooth Hermitian metric with Griffiths-positive Chern curvature. The implication from curvature positivity to ampleness is classical, while the converse remains open in higher dimension.

— Higgs-Demailly System and Positivity of Higgs Bundles  (2609.00556 - Zhang, 1 Sep 2026) in Section 1, Introduction