{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $X$ be a compact [[Riemann surface|5K0N8QMV6Q]] of genus $g \\ge 1$.\nLet $\\omega_1, \\ldots, \\omega_g$ be a basis of the space of holomorphic [[Abelian differentials|P014C13K5H]] on $X$.\nLet $\\Lambda \\subset \\C^g$ be the set of vectors $\\left( \\int_\\gamma \\omega_1, \\ldots, \\int_\\gamma \\omega_g \\right)$, where $\\gamma$ runs over the closed loops in $X$.\nThen $\\Lambda$ is a lattice of rank $2g$ in $\\C^g$, and $J(X) = \\C^g / \\Lambda$ is the Jacobian variety of $X$.\nFix a base point $P_0 \\in X$.\nThe *Abel–Jacobi map* is\n$$A \\colon X \\to J(X), \\qquad A(P) = \\left( \\int_{P_0}^{P} \\omega_1, \\ldots, \\int_{P_0}^{P} \\omega_g \\right) \\bmod \\Lambda,$$\nwhere each integral is taken along the same path from $P_0$ to $P$.\nIt extends to divisors by $A\\left( \\sum_i n_i P_i \\right) = \\sum_i n_i A(P_i)$.\n\n## Remarks\n\nThe map is well defined: two paths from $P_0$ to $P$ differ by a closed loop, which changes the vector of integrals by an element of $\\Lambda$.\nOn divisors of degree $0$ the result does not depend on $P_0$.\n\nExample: let $X = \\C / \\Lambda'$ be a complex torus with $g = 1$ and $\\omega_1 = dz$.\nThe integral from $P_0$ to $P$ is $P - P_0$ modulo the periods of $dz$, which form $\\Lambda'$.\nSo $A(P) = P - P_0$ is an isomorphism from $X$ onto $J(X) = \\C / \\Lambda'$.\n\nNon-example: the map $P \\mapsto \\left( \\int_{P_0}^{P} \\omega_i \\right)_i$ into $\\C^g$ itself, without the quotient by $\\Lambda$, is not well defined on $X$.\nGoing once around a loop with a nonzero period changes its value.\n\nAbel's theorem: a divisor $D$ of degree $0$ is the divisor of a meromorphic function if and only if $A(D) = 0$.\nJacobi inversion: the induced map from effective divisors of degree $g$ to $J(X)$ is surjective.\nFor $g \\ge 1$ the map $A \\colon X \\to J(X)$ is injective, and for $g \\ge 2$ it is a holomorphic embedding.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "defines": "HE8F7KBF42",
    "kind": "definition",
    "msc": [
      "14H40",
      "30F30"
    ],
    "name": "Abel–Jacobi map",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Abel%E2%80%93Jacobi_map"
      }
    ],
    "tag": "FMQEG0PVYB",
    "uses": [
      "5K0N8QMV6Q",
      "P014C13K5H"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/FMQEG0PVYB/",
  "links": [
    {
      "field": "defines",
      "target": "HE8F7KBF42"
    },
    {
      "field": "uses",
      "target": "5K0N8QMV6Q"
    },
    {
      "field": "uses",
      "target": "P014C13K5H"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/FMQEG0PVYB.md",
  "msc": [
    "14H40",
    "30F30"
  ],
  "name": "Abel–Jacobi map",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/FMQ/FMQEG0PVYB.md",
  "status": null,
  "tag": "FMQEG0PVYB",
  "uri": "m2://FMQEG0PVYB",
  "url": "https://math2.tamnd.com/wiki/abel-jacobi-map-definition/",
  "writing": {
    "tag": "FMQEG0PVYB",
    "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
      }
    ]
  }
}