{
  "id": "griffiths-positivity-on-the-quadric",
  "title": "Griffiths positivity on the quadric",
  "description": "Griffiths asked whether every ample vector bundle carries a Hermitian metric with positive curvature in the sense of Griffiths. The answer is no, already in rank two on P¹ × P¹: an explicit bundle glued along the graphs of three projective transformations gives a family of ample bundles of which all but finitely many carry no such metric. The proof: symmetric-power filtrations for ampleness; for the metrics, a plurisubharmonic quadratic limit of dual norms along power maps, and an invariant measure that two transformations of order three with a hyperbolic product cannot share. Proved in full in the programme, with no named hypothesis. The course is an exposition of OpenAI's proof (2026).",
  "units": [
    {
      "id": "hermitian-metrics-and-griffiths-positivity",
      "title": "Hermitian metrics and Griffiths positivity",
      "reader": "hermitian-metrics-and-griffiths-positivity.html",
      "source": "src/hermitian-metrics-and-griffiths-positivity.md",
      "source_sha256": "08d84e8e41fb28537d13415813878cc1d20e0834abd81a497d608fd18b92e50b",
      "status": "author_self_checked"
    },
    {
      "id": "a-bundle-glued-along-three-graphs",
      "title": "A bundle glued along three graphs",
      "reader": "a-bundle-glued-along-three-graphs.html",
      "source": "src/a-bundle-glued-along-three-graphs.md",
      "source_sha256": "9bcde5f62de801299ed7d44be6a3e163c3309887d8b473323f85519316eb9040",
      "status": "author_self_checked"
    },
    {
      "id": "ampleness-of-the-power-family",
      "title": "Ampleness of the power family",
      "reader": "ampleness-of-the-power-family.html",
      "source": "src/ampleness-of-the-power-family.md",
      "source_sha256": "7da9ae45f7921cd2c013cff242bc135102a9c9fc27b5ed49bdf11bc687752409",
      "status": "author_self_checked"
    },
    {
      "id": "a-quadratic-limit-of-dual-norms",
      "title": "A quadratic limit of dual norms",
      "reader": "a-quadratic-limit-of-dual-norms.html",
      "source": "src/a-quadratic-limit-of-dual-norms.md",
      "source_sha256": "7257948420efdfe696058b60f158d9d166e4cdb2e29cb54762569510544f2c1d",
      "status": "author_self_checked"
    },
    {
      "id": "an-invariant-measure-and-the-proof",
      "title": "An invariant measure and the proof",
      "reader": "an-invariant-measure-and-the-proof.html",
      "source": "src/an-invariant-measure-and-the-proof.md",
      "source_sha256": "219b7c1122388c890129908ade6fcbf66b7bed933dacab9738ce4e1e8a53f991",
      "status": "author_self_checked"
    }
  ],
  "prerequisites": [
    "course Distributions, kernels and analytic singularities, lesson Compactness and complex-line positivity: plurisubharmonic functions, the Levi criterion, radial averages, decreasing limits",
    "courses Sheaves and schemes, Morphisms of schemes and Cohomology of quasi-coherent sheaves: projective bundles of quotients, Veronese and Segre embeddings, ampleness, Cech cohomology, cohomology of projective space",
    "course Foundations of von Neumann algebras, lesson on weak topologies: Banach-Alaoglu and weak compactness in Hilbert space",
    "the lessons prove every other fact they use; the historical context in Section 5 of the first lesson is quoted"
  ],
  "status": "author_self_checked",
  "proof_providers": "providers.json",
  "provenance": "provenance.json",
  "licence": "CC0-1.0"
}