{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $X$ be a compact connected [[Riemann surface|5K0N8QMV6Q]] of [[genus|NJ787GZS1N]] $g \\ge 1$.\nLet $\\Omega(X)$ be the complex vector space of holomorphic [[Abelian differentials|P014C13K5H]] on $X$, which has dimension $g$, and let $\\Omega(X)^*$ be its dual space.\nEach class $\\gamma \\in H_1(X, \\Z)$ gives the linear form $\\omega \\mapsto \\int_\\gamma \\omega$ on $\\Omega(X)$, and this defines a map $H_1(X, \\Z) \\to \\Omega(X)^*$ whose image $\\Lambda$ is a [[lattice|WN9V7WSJKF]] of rank $2g$.\nThe *Jacobian variety* of $X$ is the [[complex torus|Y7MA6KBE9M]]\n$$J(X) = \\Omega(X)^* / \\Lambda.$$\n\n## Remarks\n\nChoosing a basis $\\omega_1, \\ldots, \\omega_g$ of $\\Omega(X)$ identifies $J(X)$ with $\\C^g / \\Lambda$, where $\\Lambda$ is spanned by the period vectors $\\left( \\int_\\gamma \\omega_1, \\ldots, \\int_\\gamma \\omega_g \\right)$.\nThe Riemann bilinear relations give $J(X)$ a principal polarization, so it is an abelian variety.\nBy the Abel–Jacobi theorem, $J(X)$ is isomorphic to the group $\\operatorname{Pic}^0(X)$ of divisor classes of degree $0$; this description defines the Jacobian of a smooth projective curve over any field.\nExample: for $X = \\C / \\Lambda_0$ with $g = 1$ and $\\omega_1 = dz$, the periods of $dz$ are the elements of $\\Lambda_0$, so $J(X) \\cong \\C / \\Lambda_0 \\cong X$.\nNon-example: a general complex torus $\\C^2 / \\Lambda$ of dimension $2$ admits no nonconstant meromorphic functions, so it is not an abelian variety and is not the Jacobian of any curve.\nTorelli's theorem says that $X$ is determined by $J(X)$ together with its principal polarization.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "ZQ84M9D9TC",
    "kind": "definition",
    "msc": [
      "14H40",
      "14K30"
    ],
    "name": "Jacobian variety",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Jacobian_variety"
      }
    ],
    "tag": "9GS8XQMZED",
    "uses": [
      "5K0N8QMV6Q",
      "P014C13K5H",
      "Y7MA6KBE9M",
      "WN9V7WSJKF",
      "NJ787GZS1N"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/9GS8XQMZED/",
  "links": [
    {
      "field": "defines",
      "target": "ZQ84M9D9TC"
    },
    {
      "field": "uses",
      "target": "5K0N8QMV6Q"
    },
    {
      "field": "uses",
      "target": "P014C13K5H"
    },
    {
      "field": "uses",
      "target": "Y7MA6KBE9M"
    },
    {
      "field": "uses",
      "target": "WN9V7WSJKF"
    },
    {
      "field": "uses",
      "target": "NJ787GZS1N"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/9GS8XQMZED.md",
  "msc": [
    "14H40",
    "14K30"
  ],
  "name": "Jacobian variety",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/9GS/9GS8XQMZED.md",
  "status": null,
  "tag": "9GS8XQMZED",
  "uri": "m2://9GS8XQMZED",
  "url": "https://tamnd.github.io/math2/wiki/jacobian-variety-definition/",
  "writing": {
    "tag": "9GS8XQMZED",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}