the Jacobian variety
/ yah-KOH-bee-an /
Attached to every compact Riemann surface is a second, very different geometric object: not a curvy curve but a flat, smooth, higher-dimensional torus that carries a group law. This is the Jacobian variety. It packages all the surface's degree-0 divisor classes into one coherent abelian group with geometry, and it is the home where the Abel-Jacobi map lives. Where the curve is rigid and complicated, its Jacobian is supple and linear.
Precisely, for a compact Riemann surface X of genus g, the Jacobian is the complex torus J(X) = C^g / Lambda, where C^g is the dual of the space of holomorphic 1-forms (g-dimensional) and Lambda is the period lattice — the rank-2g lattice obtained by integrating a basis of 1-forms over a basis of H_1(X, Z). As a real manifold it is a 2g-dimensional torus; as a complex manifold it is a g-dimensional complex torus; and crucially it is an ABELIAN VARIETY: it is projective (it embeds in projective space, via theta functions) and its group operation is holomorphic. The natural isomorphism J(X) = Pic^0(X) identifies it with the group of degree-0 divisor classes under linear equivalence.
Why it matters: the Jacobian linearizes the curve. Abel's theorem identifies it with degree-0 divisor classes; the Abel-Jacobi map embeds the curve into it; the Riemann theta function lives on it and its zero locus (the theta divisor) reconstructs the curve (the Torelli theorem: the Jacobian with its principal polarization determines X). It is the prototype of an abelian variety and the meeting point of curves with the theory of abelian varieties and theta functions. Honesty caveat: NOT every g-dimensional complex torus is a Jacobian — Jacobians of curves form a special (Schottky) locus inside the moduli of all principally polarized abelian varieties, and characterizing which tori are Jacobians (the Schottky problem) is hard. Also a 'complex torus' need not be projective at all; the Jacobian is special precisely because its period lattice satisfies the Riemann bilinear relations, making it an abelian variety.
For an elliptic curve (g = 1), the Jacobian is the curve itself: J(X) = C/L, a 1-dimensional complex torus, and the identification J(X) = Pic^0(X) IS the elliptic group law. For a genus-2 curve, the Jacobian is a 2-dimensional abelian variety (a 4-real-dimensional torus), and the curve embeds into it as a genuine curve via Abel-Jacobi — its theta divisor.
For g = 1 the Jacobian is the elliptic curve itself; for g = 2 it is a 2-dimensional abelian variety containing the curve.
Not every g-dimensional complex torus is a Jacobian, and not even every principally polarized abelian variety is — the Jacobians form the Schottky locus, and deciding which tori are Jacobians is the open Schottky problem. The Jacobian is abelian (projective) only because its periods satisfy the Riemann relations.