Generalized Jacobian

In algebraic geometry

In mathematics, there are several notions of generalized Jacobians, which are algebraic groups or complex manifolds that are in some sense analogous to the Jacobian variety of an algebraic curve, or related to the Albanese variety and Picard variety that generalize it to higher-dimensional objects. They all carry a commutative group law.

One kind is an algebraic group, typically an extension of an abelian variety by an affine algebraic group. This was studied in particular by Maxwell Rosenlicht, and can be used to study ramified coverings of a curve, with abelian Galois group.

There are two other important definitions as complex manifolds; each of these is a complex torus, defined by Hodge theory data. The definition by André Weil is an abelian variety, while the definition by Phillip Griffiths, the intermediate Jacobian, is not, but varies holomorphically.

In numerical analysis

Alternatively, generalized Jacobian may refer to a surrogate matrix that can be used in lieu of the conventional Jacobian matrix within a modified Newton's method for solving non-linear equations in the case of a non-differentiable function. This method is particularly useful for solving non-linear complementarity problems.