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  "body": "\n## Statement\n\nLet $k$ be an [[algebraically closed field|0PF6TM8TWT]].\nLet $C$ and $C'$ be smooth projective connected [[curves|XJM2MT4S3Y]] over $k$.\nLet $(J(C), \\Theta)$ and $(J(C'), \\Theta')$ be their [[Jacobian varieties|ZQ84M9D9TC]] with the principal polarizations given by their theta divisors.\nSuppose that an isomorphism of [[abelian varieties|R7W03TXRS9]] $J(C) \\to J(C')$ carries the polarization of $\\Theta$ to that of $\\Theta'$.\nThen $C$ and $C'$ are isomorphic.\n\n## Remarks\n\nOne proof, due to Andreotti, recovers $C$ from the geometry of $\\Theta$.\nLet $C$ be a curve that is not hyperelliptic.\nThe Gauss map sends a smooth point of $\\Theta$ to its tangent hyperplane, and its branch locus is the dual of the canonical model of $C$.\nProjective duality then gives back $C$.\nOver $\\C$, let $M_g$ be the moduli of curves of genus $g$ and $A_g$ the moduli of principally polarized abelian varieties of dimension $g$.\nThe theorem says that the Torelli map $M_g \\to A_g$ is injective on points.\nFor example, two elliptic curves are isomorphic exactly when their Jacobians, which are the curves themselves, are isomorphic.\nThe polarization cannot be dropped in general: there are non-isomorphic curves whose Jacobians are isomorphic as unpolarized abelian varieties.\n\n## Explanations\n\n### L1 human\n\nEvery curve has a Jacobian, an abelian variety built from the curve, together with a special divisor $\\Theta$ on it.\nThe theorem says that the Jacobian with this divisor remembers the curve completely.\nSo no information is lost when one passes from a curve to its polarized Jacobian.\n\n### L2 human\n\nThe theorem reduces questions about curves to questions about principally polarized abelian varieties, which are described by period matrices.\nOver $\\C$, the period matrix of $C$ in the Siegel upper half space determines $C$.\nThe image of $M_g$ in $A_g$ is the Jacobian locus, and describing it is the Schottky problem.\nThere are Torelli theorems for other varieties, such as the global Torelli theorem for K3 surfaces.\nIn them the curve is replaced by the variety and the Jacobian by a Hodge structure.\n",
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