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

Griffiths proved in 1969 that a holomorphic vector bundle carrying a smooth Hermitian metric whose Chern curvature is Griffiths-positive is ample, and conjectured the converse: every ample holomorphic vector bundle on a compact complex manifold should admit a smooth Griffiths-positive Hermitian metric. It holds for line bundles and, by Murakami's 2026 analytic proof, on compact Riemann surfaces, and Demailly's nonlinear system and direct-image methods have been aimed at it for two decades. Does the converse hold in rank at least two?

Result
Disproved(see note)
Status
Candidate (review pending)
AI contribution
AI-assisted
Method
Construction
Field
Complex geometry; positivity of vector bundles
Posed by
Phillip A. Griffiths, Hermitian differential geometry, Chern classes, and positive vector bundles (1969)
Year posed
1969
Years open
57y
Solved
2026-09-22
Model
Not disclosed
Vendor
Not disclosed
Collaborators
Yun-Heng Du, Song-Yan Xie
Verification
Unreviewed
Publication
Preprint
Significance
55 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

False in rank two. Theorem 1.1 exhibits an ample rank-two holomorphic vector bundle on the abelian surface C×CC \times C, C:y2=x3−xC : 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.

What the AI did

According to the authors' AI Use Disclosure, the research direction and initial mathematical framework were developed by the authors. Generative AI was then used interactively during the mathematical exploration, including the development, testing, and refinement of parts of the construction and proof. The authors independently verified the final arguments and take full responsibility for the paper. The disclosure does not identify the model or specify which individual components of the construction or proof involved AI.

Verification

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 EE of rank two on X=C×CX = C \times C with C:y2=x3−xC : 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 L1⊕L2L_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.

Sources

Submitted by Holomorphic on

Changelog2 changes

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