CATALOGUE 050
A counterexample to Griffiths’ positivity conjecture
The manuscript constructs a rank-two bundle on $\mathbf P^1\times\mathbf P^1$ whose family $E_m=f_m^*G\otimes\mathcal O(1,1)$ is ample for every positive integer $m$, while admitting no smooth strictly Griffiths-positive Hermitian metric for all sufficiently large $m$. This guide distinguishes the manuscript's claims from the scope of editorial checking.
1. Paper in this catalogue
These summaries give the principal scope; consult the cited original results for the complete hypotheses and statements.
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| # | Manuscript and overview | Principal claim | Role and available overview |
|---|---|---|---|
| 01 | Ample rank-two bundles on the quadric surface without Griffiths-positive metrics →Short name: Ample rank-two bundles on the quadric surfaceSeptember 24, 2026 · 18 pagesOriginal paper (GitHub) ↗ | For an explicit rank-two $G$, the bundle $E_m=f_m^*G\otimes\mathcal O_S(1,1)$ is ample for every $m\ge1$. There is an $m_0$ such that, for every $m\ge m_0$, it has no smooth strictly Griffiths-positive Hermitian metric. Theorem 1.1 · p. 2 ↗ | Connects a symmetric-power filtration from two quotients to an invariant-measure obstruction from the graph identities. Read the proof overview → |
2. Entry into the proof
The individual article starts with the explicit construction of $G$ along three graphs. It then follows ampleness through two quotient sequences and a measure obstruction to a quadratic psh limit on the dual bundle. The graph identities connect the two arguments. The contradicted conjecture concerns a smooth strictly Griffiths-positive Hermitian metric on the ample bundle itself; neither exclusion of semipositive metrics nor an effective threshold is asserted.
There is one paper in this catalogue. Direct external inputs are collected in the article: removable singularities and decreasing limits for psh functions, Čech comparison, Künneth, and projective-bundle constructions. Comparisons with other counterexample manuscripts are not treated as proof dependencies.
3. Sources and checking scope
The official subject is Algebraic and complex geometry and the official catalogue number is 050. The internal article ID is separate from that number.
OpenAI Research Catalog · Algebraic and complex geometry, 050 · p. 7
The source is the September 24, 2026 version, 18 pages. Checking covers the main theorem, the proof route in §§2–6, and statements and uses of direct external inputs. It does not constitute independent verification of complete proofs, recursive verification of external literature, or expert peer review. The individual article records the detailed scope.