Torelli theorem

Remarks

One proof, due to Andreotti, recovers $C$ from the geometry of $\Theta$. Let $C$ be a curve that is not hyperelliptic. The Gauss map sends a smooth point of $\Theta$ to its tangent hyperplane, and its branch locus is the dual of the canonical model of $C$. Projective duality then gives back $C$. Over $\C$, let $M_g$ be the moduli of curves of genus $g$ and $A_g$ the moduli of principally polarized abelian varieties of dimension $g$. The theorem says that the Torelli map $M_g \to A_g$ is injective on points. For example, two elliptic curves are isomorphic exactly when their Jacobians, which are the curves themselves, are isomorphic. The polarization cannot be dropped in general: there are non-isomorphic curves whose Jacobians are isomorphic as unpolarized abelian varieties.

Going deeper

For a student

The theorem reduces questions about curves to questions about principally polarized abelian varieties, which are described by period matrices. Over $\C$, the period matrix of $C$ in the Siegel upper half space determines $C$. The image of $M_g$ in $A_g$ is the Jacobian locus, and describing it is the Schottky problem. There are Torelli theorems for other varieties, such as the global Torelli theorem for K3 surfaces. In them the curve is replaced by the variety and the Jacobian by a Hodge structure.