Jacobian variety
Encyclopedia
In mathematics
, the Jacobian variety J(C) of a non-singular algebraic curve
C of genus
g is the moduli space
of degree 0 line bundle
s. It is the connected component of the identity in the Picard group of C, hence an abelian variety
.
, of dimension
g, and hence, over the complex numbers, it is a complex torus
. If p is a point of C, then the curve C can be mapped to a subvariety
of J with the given point p mapping to the identity of J, and C generates J as a group
.
V/L, where V is the dual of the vector space
of all global holomorphic differentials on C and L is the lattice
of all elements of V of the form
where γ is a closed path
in C.
The Jacobian of a curve over an arbitrary field was constructed by as part of his proof of the Riemann hypothesis for curves over a finite field.
The Abel-Jacobi theorem states that the torus thus built is a variety, the classical Jacobian of a curve, that indeed parametrizes the degree 0 line bundles, that is, it can be identified with its Picard variety of degree 0 divisors modulo linear equivalence.
The Schottky problem
asks which principally polarized abelian varieties are the Jacobians of curves.
The Picard variety, the Albanese variety
, and intermediate Jacobian
s are generalizations of the Jacobian for higher dimensional varieties. For varieties of higher dimension the construction of the Jacobian variety as a quotient of the space of holomorphic 1-forms generalizes to give the Albanese variety
, but in general this need not be isomorphic to the Picard variety.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the Jacobian variety J(C) of a non-singular algebraic curve
Algebraic curve
In algebraic geometry, an algebraic curve is an algebraic variety of dimension one. The theory of these curves in general was quite fully developed in the nineteenth century, after many particular examples had been considered, starting with circles and other conic sections.- Plane algebraic curves...
C of genus
Genus (mathematics)
In mathematics, genus has a few different, but closely related, meanings:-Orientable surface:The genus of a connected, orientable surface is an integer representing the maximum number of cuttings along non-intersecting closed simple curves without rendering the resultant manifold disconnected. It...
g is the moduli space
Moduli space
In algebraic geometry, a moduli space is a geometric space whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects...
of degree 0 line bundle
Line bundle
In mathematics, a line bundle expresses the concept of a line that varies from point to point of a space. For example a curve in the plane having a tangent line at each point determines a varying line: the tangent bundle is a way of organising these...
s. It is the connected component of the identity in the Picard group of C, hence an abelian variety
Abelian variety
In mathematics, particularly in algebraic geometry, complex analysis and number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular functions...
.
Introduction
The Jacobian variety is named after Carl Gustav Jacobi, who proved the complete version Abel-Jacobi theorem, making the injectivity statement of Niels Abel into an isomorphism. It is a principally polarized abelian varietyAbelian variety
In mathematics, particularly in algebraic geometry, complex analysis and number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular functions...
, of dimension
Dimension
In physics and mathematics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it...
g, and hence, over the complex numbers, it is a complex torus
Complex torus
In mathematics, a complex torus is a particular kind of complex manifold M whose underlying smooth manifold is a torus in the usual sense...
. If p is a point of C, then the curve C can be mapped to a subvariety
Subvariety
In botanical nomenclature, a subvariety is a taxonomic rank below that of variety but above that of form : it is an infraspecific taxon. Its name consists of three parts: a genus name, a specific epithet and an infraspecific epithet. To indicate the rank, the abbreviation "subvar." should be put...
of J with the given point p mapping to the identity of J, and C generates J as a group
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
.
Construction over for complex curves
Over the complex numbers, the Jacobian variety can be realized as the quotient spaceQuotient space
In topology and related areas of mathematics, a quotient space is, intuitively speaking, the result of identifying or "gluing together" certain points of a given space. The points to be identified are specified by an equivalence relation...
V/L, where V is the dual of the vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
of all global holomorphic differentials on C and L is the lattice
Lattice
Lattice may refer to:In art and design:* Latticework an ornamental criss-crossed framework, an arrangement of crossing laths or other thin strips of material* Lattice In engineering:* A lattice shape truss structure...
of all elements of V of the form
where γ is a closed path
Path (topology)
In mathematics, a path in a topological space X is a continuous map f from the unit interval I = [0,1] to XThe initial point of the path is f and the terminal point is f. One often speaks of a "path from x to y" where x and y are the initial and terminal points of the path...
in C.
The Jacobian of a curve over an arbitrary field was constructed by as part of his proof of the Riemann hypothesis for curves over a finite field.
The Abel-Jacobi theorem states that the torus thus built is a variety, the classical Jacobian of a curve, that indeed parametrizes the degree 0 line bundles, that is, it can be identified with its Picard variety of degree 0 divisors modulo linear equivalence.
Further notions
Torelli's theorem states that a complex curve is determined by its Jacobian (with its polarization).The Schottky problem
Schottky problem
In mathematics, the Schottky problem, named after Friedrich Schottky, is a classical question of algebraic geometry, asking for a characterisation of Jacobian varieties amongst abelian varieties.-Geometric formulation:...
asks which principally polarized abelian varieties are the Jacobians of curves.
The Picard variety, the Albanese variety
Albanese variety
In mathematics, the Albanese variety A, named for Giacomo Albanese, is a generalization of the Jacobian variety of a curve, and is the abelian variety generated by a variety V. In other words there is a morphism from the variety V to its Albanese variety A, such that any morphism from V to an...
, and intermediate Jacobian
Intermediate Jacobian
In mathematics, the intermediate Jacobian of a compact Kähler manifold or Hodge structure is a complex torus that is a common generalization of the Jacobian variety of a curve and the Picard variety and the Albanese variety. It is obtained by putting a complex structure on the torus Hn/Hn for n odd...
s are generalizations of the Jacobian for higher dimensional varieties. For varieties of higher dimension the construction of the Jacobian variety as a quotient of the space of holomorphic 1-forms generalizes to give the Albanese variety
Albanese variety
In mathematics, the Albanese variety A, named for Giacomo Albanese, is a generalization of the Jacobian variety of a curve, and is the abelian variety generated by a variety V. In other words there is a morphism from the variety V to its Albanese variety A, such that any morphism from V to an...
, but in general this need not be isomorphic to the Picard variety.