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  "body": "\n## Summary\n\nA complex torus of dimension $g$ is the quotient of $\\C^g$ by a [[lattice|WN9V7WSJKF]] of rank $2g$.\nIt is a compact connected complex manifold and a commutative complex Lie group.\nEvery complex [[abelian variety|R7W03TXRS9]] is a complex torus, but for $g \\ge 2$ most complex tori are not algebraic.\n\n## Explanations\n\n### L1 human\n\nTake the plane $\\C$ and the grid of points $m + n i$ with integers $m$ and $n$.\nGlue together any two points that differ by a grid point.\nEach square of the grid is folded up into the surface of a doughnut, the quotient $\\C / (\\Z + \\Z i)$.\nA complex torus does the same in $\\C^g$, with a grid that has $2g$ independent directions.\nAddition in $\\C^g$ passes to the quotient, so the points of a complex torus form a group.\n",
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