the Abel-Jacobi map
/ AH-bel yah-KOH-bee /
A Riemann surface of positive genus is a curvy, hole-filled thing with no obvious group structure. The Abel-Jacobi map performs a small miracle: it maps the surface into a flat torus — the Jacobian — in a way that linearizes its geometry, turning the tangled question of which divisors are linearly equivalent into simple addition in a group. It is the bridge from the curve to its 'linear shadow.'
Precisely, fix a base point p_0 on a compact surface X of genus g, and a basis omega_1, ..., omega_g of holomorphic 1-forms. The Abel-Jacobi map sends a point p to the vector of integrals u(p) = (integral from p_0 to p of omega_1, ..., integral from p_0 to p of omega_g), taken in the Jacobian J(X) = C^g / Lambda, where Lambda is the period lattice generated by integrating the omega_i over a basis of loops. The quotient by Lambda is exactly what makes the answer independent of the integration path (different paths differ by a loop, i.e. by a lattice vector). The map extends additively to divisors: a degree-0 divisor D = sum n_p [p] goes to sum n_p u(p) in J(X).
Why it matters: it converts divisor questions into group theory. Abel's theorem says a degree-0 divisor is principal (i.e. linearly equivalent to 0) IF AND ONLY IF its Abel-Jacobi image is 0 in the Jacobian — so the Jacobian exactly measures the failure of 'same degree implies linearly equivalent.' Jacobi's inversion theorem says the map from the g-fold symmetric product to the Jacobian is surjective, so every point of J(X) is hit by an effective divisor of degree g. Together they identify Pic^0(X) with J(X) as groups. Honesty notes: the map depends on choices (base point p_0 shifts it by a translation; basis of forms changes coordinates), but the induced map on degree-0 divisor CLASSES is canonical. For g = 1 the Abel-Jacobi map is an isomorphism of the surface onto its own Jacobian, which is the group law on an elliptic curve; for g >= 2 the image is a curve sitting inside the higher-dimensional torus, NOT all of it.
For an elliptic curve (g = 1) with one holomorphic form dz, the Abel-Jacobi map p -> integral of dz from p_0 to p is literally the identification of the torus C/L with its own Jacobian. Abel's theorem then becomes the elliptic group law: three points p, q, r sum to zero in the group exactly when [p] + [q] + [r] - 3[p_0] is principal, i.e. when they are the three intersections with a line.
For g = 1 the Abel-Jacobi map identifies the curve with its Jacobian — and Abel's theorem is the elliptic group law.
The point map depends on the base point and form-basis, but the induced map on degree-0 divisor CLASSES is canonical (it equals Pic^0 -> J). For g >= 2 the image is a curve in the g-dimensional Jacobian, not the whole torus; only for g = 1 is it onto.