Griffiths' conjecture: does every ample vector bundle carry a Griffiths-positive metric?

False in rank two. Theorem 1.1 exhibits an ample rank-two holomorphic vector bundle on the abelian surface $C \times C$, $C : y^2 = x^3 - x$, with no smooth Griffiths-semipositive Hermitian metric, so a fortiori no Griffiths-positive one. The obstruction is established first on a split bundle, shown to persist under small deformations, and then transferred to an ample bundle through an irreducible moduli space of stable sheaves using openness of ampleness. Scope, from the paper's Remark 2.6: the obstruction concerns smooth Hermitian metrics, and the argument does not apply to general strongly pseudoconvex complex Finsler squared norms, so the Finsler characterisation of ampleness is not refuted.

False in rank two. Theorem 1.1 exhibits an ample rank-two holomorphic vector bundle on the abelian surface $C \times C$, $C : y^2 = x^3 - x$, with no smooth Griffiths-semipositive Hermitian metric, so a fortiori no Griffiths-positive one. The obstruction is established first on a split bundle, shown to persist under small deformations, and then transferred to an ample bundle through an irreducible moduli space of stable sheaves using openness of ampleness. Scope, from the paper's Remark 2.6: the obstruction concerns smooth Hermitian metrics, and the argument does not apply to general strongly pseudoconvex complex Finsler squared norms, so the Finsler characterisation of ampleness is not refuted.