 Given forms and closed loops we can integrate, and we define 2"g" vectors

It follows from the Riemann bilinear relations that the formula_6 generate a nondegenerate lattice formula_7 (that is, they are a real basis for formula_8), and the Jacobian is defined by

The Abel–Jacobi map is then defined as follows.We pick some base point formula_10 and, nearly mimicking the definition of formula_11 define the map

Although this is seemingly dependent on a path from formula_13 to formula_14 any two such paths define a closed loop in formula_15 and, therefore, an element of formula_16 so integration over it gives an element of formula_17 Thus the difference is erased in the passage to the quotient by formula_7.Changing base-point formula_13 does change the map, but only by a translation of the torus.