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  "body": "\n## Definition\n\nLet $g \\ge 1$ be an integer.\nA *lattice* in $\\C^g$ is a subgroup $\\Lambda \\subseteq \\C^g$ generated by $2g$ vectors that are linearly independent over $\\R$.\nA *complex torus* of dimension $g$ is the [[quotient group|0KYX2N69J0]] $X = \\C^g / \\Lambda$ for a [[lattice|WN9V7WSJKF]] $\\Lambda$ in $\\C^g$.\nIt carries the complex manifold structure for which the projection $\\C^g \\to X$ is a local biholomorphism.\n\n## Remarks\n\nExample: $\\Lambda = \\Z + \\Z \\tau$ with $\\tau \\in \\C$ and $\\operatorname{Im} \\tau \u003e 0$.\nThe numbers $1$ and $\\tau$ are linearly independent over $\\R$, so $\\Lambda$ is a lattice in $\\C$, and $\\C / \\Lambda$ is a complex torus of dimension $1$.\nEvery complex torus of dimension $1$ is the set of complex points of an elliptic curve.\n\nNon-example: $\\Z \\subseteq \\C$ is a discrete subgroup, but it is generated by only one vector, not two.\nThe quotient $\\C / \\Z$ is not compact; the map $z \\mapsto e^{2 \\pi i z}$ identifies it with $\\C^{*}$.\n\nLet $X = \\C^g / \\Lambda$ be a complex torus.\nThen $X$ comes from an [[abelian variety|R7W03TXRS9]] if and only if $\\C^g$ has a positive definite Hermitian form $H$ with $\\operatorname{Im} H(\\Lambda \\times \\Lambda) \\subseteq \\Z$.\nSuch a form is called a Riemann form or a polarization.\n",
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