of the generalized Jacobian making it a complex Lie group. The analytic subgroup underlying the generalized Jacobian can be described as follows. (This does not always determine the algebraic structure as two non-isomorphic commutative algebraic groups may be isomorphic as analytic groups.) Suppose that C is a curve with an effective divisor m with support S. There is a natural map from the homology group H1(C − S) to the dual Ω(−m)* of the complex vector space Ω(−m) (1-forms with poles on m) induced by the integral of a 1-form over a 1-cycle. The analytic generalized Jacobian is then the quotient group Ω(−m)*/H1(C − S). References Algebraic groups Algebraic curves