{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $X$ be a [[complex manifold|JSGWAN2RJV]] of dimension $n$ and let $E \\to X$ be a [[holomorphic vector bundle|TKA958Z4KZ]] of rank $r$ with a [[Hermitian metric|Q6X6W2N29G]] $h$.\nAt a point $x$, write the curvature $\\Theta$ of the [[Chern connection|7RC9CN2Q49]] in local coordinates and a frame orthonormal at $x$ as\n$$i \\Theta(x) = i \\sum_{j,k=1}^{n} \\sum_{\\lambda,\\mu=1}^{r} c_{jk\\lambda\\mu}\\, dz_j \\wedge d\\bar z_k \\otimes e_\\lambda^* \\otimes e_\\mu.$$\nThe bundle $(E, h)$ is *Nakano positive* if at every point $x \\in X$\n$$\\sum_{j,k,\\lambda,\\mu} c_{jk\\lambda\\mu}\\, \\tau_{j\\lambda} \\bar\\tau_{k\\mu} \u003e 0$$\nfor every nonzero complex $n \\times r$ matrix $\\tau = (\\tau_{j\\lambda})$.\nSuch a matrix is the same as a tensor $\\tau \\in T_x X \\otimes E_x$.\n\n## Remarks\n\nTaking $\\tau_{j\\lambda} = \\xi_j v_\\lambda$ recovers the condition of [[Griffiths positivity|FVCDGMNSAN]], so Nakano positive bundles are Griffiths positive.\nFor $r = 1$ or $n = 1$ every tensor $\\tau$ is of this form, and the two notions agree.\n\nExample: $E = L^{\\oplus r}$, where $L$ is a positive line bundle with curvature $i \\sum a_{jk} dz_j \\wedge d\\bar z_k$, with the direct sum metric.\nThen $c_{jk\\lambda\\mu} = a_{jk} \\delta_{\\lambda\\mu}$, and the sum is $\\sum_\\lambda \\sum_{j,k} a_{jk} \\tau_{j\\lambda} \\bar\\tau_{k\\lambda}$.\nEach inner sum is $\\ge 0$ because $(a_{jk})$ is positive definite, and some inner sum is $\u003e 0$ because $\\tau \\ne 0$.\n\nNon-example: a trivial bundle $X \\times \\C^r$ with the standard metric has $\\Theta = 0$, so the sum is $0$ for every $\\tau$.\n\nNakano's vanishing theorem: if $X$ is compact Kähler and $E$ is Nakano positive, then $H^q(X, K_X \\otimes E) = 0$ for $q \\ge 1$, where $K_X$ is the canonical bundle.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "WJ8TRE4F1K",
    "kind": "definition",
    "msc": [
      "32L05",
      "32L20",
      "53C55"
    ],
    "name": "Nakano positivity",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Kodaira_vanishing_theorem"
      }
    ],
    "tag": "ESBGJHPEDT",
    "uses": [
      "TKA958Z4KZ",
      "Q6X6W2N29G",
      "7RC9CN2Q49",
      "FVCDGMNSAN",
      "JSGWAN2RJV"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/ESBGJHPEDT/",
  "links": [
    {
      "field": "defines",
      "target": "WJ8TRE4F1K"
    },
    {
      "field": "uses",
      "target": "TKA958Z4KZ"
    },
    {
      "field": "uses",
      "target": "Q6X6W2N29G"
    },
    {
      "field": "uses",
      "target": "7RC9CN2Q49"
    },
    {
      "field": "uses",
      "target": "FVCDGMNSAN"
    },
    {
      "field": "uses",
      "target": "JSGWAN2RJV"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/ESBGJHPEDT.md",
  "msc": [
    "32L05",
    "32L20",
    "53C55"
  ],
  "name": "Nakano positivity",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/ESB/ESBGJHPEDT.md",
  "status": null,
  "tag": "ESBGJHPEDT",
  "uri": "m2://ESBGJHPEDT",
  "url": "https://tamnd.github.io/math2/wiki/nakano-positivity-definition/",
  "writing": {
    "tag": "ESBGJHPEDT",
    "kind": "definition",
    "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
      }
    ]
  }
}