Griffiths' conjecture (Griffiths 1969) asserted that every ample holomorphic vector bundle over a compact complex manifold admits a smooth Hermitian metric that is positive in the sense of Griffiths. This positivity means that the bundle curvature of the Chern connection is positive on every nonzero decomposable tangent-fiber vector.
Du and Xie (2026) reported a counterexample of bundle rank 2 over the abelian variety 
of complex dimension
2, where 
is the elliptic curve 
. The vector bundle
is ample but admits no smooth Griffiths-semipositive Hermitian
metric, and therefore no Griffiths-positive Hermitian
metric. The result does not refute the more general characterization of ample
vector bundles by strongly pseudoconvex complex
Finsler metrics.
The authors report that generative AI was used interactively to explore and refine parts of the argument, which they then independently verified. As of Sep. 27, 2026, independent specialist review had not been reported.
