{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "ESBGJHPEDT"
    },
    {
      "field": "defines",
      "target": "HWEED6YTNN"
    },
    {
      "field": "specializes",
      "target": "WJ8TRE4F1K"
    }
  ],
  "body": "\n## Summary\n\nGriffiths positivity is a notion of positive curvature for a holomorphic Hermitian vector bundle of any rank.\nIt asks that the curvature of the [[Chern connection|7RC9CN2Q49]] be positive when evaluated on one tangent vector and one fibre vector at a time.\nFor line bundles it is the same as being a [[positive line bundle|J45KWFHENP]].\nGriffiths positive bundles are ample, and Griffiths asked whether the converse holds; this question is open in general.\n\n## Explanations\n\n### L1 human\n\nFor a line bundle, curvature is a single $2$-form, and positivity has one clear meaning.\nFor a bundle of higher rank, curvature is a matrix of $2$-forms, and there are several reasonable ways to call it positive.\nGriffiths positivity is the weakest common one: for each direction on the base and each vector in the fibre, a certain number built from the curvature must be positive.\nA direct sum of copies of a positive line bundle, with the sum metric, is Griffiths positive.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "32L05",
      "53C55"
    ],
    "name": "Griffiths positivity",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Ample_line_bundle"
      }
    ],
    "tag": "FVCDGMNSAN"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/FVCDGMNSAN/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/FVCDGMNSAN.md",
  "msc": [
    "32L05",
    "53C55"
  ],
  "name": "Griffiths positivity",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/FVC/FVCDGMNSAN.md",
  "status": null,
  "tag": "FVCDGMNSAN",
  "uri": "m2://FVCDGMNSAN",
  "url": "https://tamnd.github.io/math2/wiki/griffiths-positivity/",
  "writing": {
    "tag": "FVCDGMNSAN",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}