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.

Exact FrontierDelta

Prior state unknown→disproved

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Occurred: Sep 22, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: unreviewed. Publication: preprint. AI contribution: ai-assisted. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

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

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Registry verification: unreviewed · preprint · candidate

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VibeMathed
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Yun-Heng Du
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Song-Yan Xie
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VibeMathed record: Griffiths' conjecture: does every ample vector bundle carry a Griffiths-positive metric?

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No independent mathematician has checked this yet. Checked here on 27 September 2026 against arXiv:2609.26504v1, twenty pages, posted 22 September. Theorem 1.1 is read as stated: there is an ample holomorphic vector bundle $E$ of rank two on $X = C \times C$ with $C : y^2 = x^3 - x$ admitting no smooth Griffiths-semipositive Hermitian metric, hence no Griffiths-positive one. The route was read in outline: a curvature obstruction on a split bundle $L_1 \oplus L_2$ with a strict Levi-form bound, persistence under small deformations, then nonsplit extensions placed with an ample bundle in one irreducible moduli space of stable sheaves, where openness of ampleness supplies an ample member inside the obstruction neighbourhood. The paper's own Remark 2.6 bounds the claim: the operator argument applies to Hermitian squared norms and not to general strongly pseudoconvex complex Finsler norms, so the Finsler characterisation of ampleness is untouched. The mathematics was not checked line by line here, and no referee has seen it. The AI Use Disclosure is quoted in full in the AI role field; it names no model and no vendor, which is the thinnest disclosure in this batch and is recorded as such.