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Jacobian Varieties
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 bundles. It is the connected component of the identity in the Picard group of ''C'', hence an abelian variety. Introduction The Jacobian variety is named after Carl Gustav Jacobi, who proved the complete version of the Abel–Jacobi theorem, making the injectivity statement of Niels Abel into an isomorphism. It is a principally polarized 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. Construction for complex curves Over the complex numbers, the Jacobian variety can be realized as the quotient space ''V''/''L'', where ''V'' is the dual of the vector space of all global holomorphic differentials on ''C'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Path (topology)
In mathematics, a path in a topological space X is a continuous function from the closed unit interval , 1/math> into X. Paths play an important role in the fields of topology and mathematical analysis. For example, a topological space for which there exists a path connecting any two points is said to be path-connected. Any space may be broken up into path-connected components. The set of path-connected components of a space X is often denoted \pi_0(X). One can also define paths and loops in pointed spaces, which are important in homotopy theory. If X is a topological space with basepoint x_0, then a path in X is one whose initial point is x_0. Likewise, a loop in X is one that is based at x_0. Definition A ''curve'' in a topological space X is a continuous function f : J \to X from a non-empty and non-degenerate interval J \subseteq \R. A in X is a curve f : , b\to X whose domain , b/math> is a compact non-degenerate interval (meaning a is homeomorphic to , 1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
1807
Events January–March * January 7 – The United Kingdom of Great Britain and Ireland issues an Order in Council prohibiting British ships from trading with France or its allies. * January 20 – The Sierra Leone Company, faced with bankruptcy because of the outlawing of the slave trade in British colonies, petitions the British government for purchase and transfer of its property to the Crown; Parliament approves the transfer on July 29, and it takes effect on January 1, 1808. * February 3 – Napoleonic Wars and Anglo-Spanish War – Battle of Montevideo: The British Army captures Montevideo from the Spanish Empire, as part of the British invasions of the Río de la Plata. * February 7 – Napoleon leads the forces of the French Empire in an invasion of the Russian Empire, and begins fighting at the Battle of Eylau against Russian and Prussian forces. * February 8 – Battle of Eylau: Napoleon fights a hard but inconclusive battle against the Russians under Benni ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 manifold, complex structure on the torus H^n(M,\R)/H^n(M,\Z) for ''n'' odd. There are several different natural ways to put a complex structure on this torus, giving several different sorts of intermediate Jacobians, including one due to and one due to . The ones constructed by Weil have natural polarizations if ''M'' is projective, and so are abelian varieties, while the ones constructed by Griffiths behave well under holomorphic deformations. A complex structure on a real vector space is given by an automorphism ''I'' with square -1. The complex structures on H^n(M,\R) are defined using the Hodge decomposition : H^(M,) \otimes = H^(M)\oplus\cdots\oplus H^(M). On H^ the Weil complex structure I_W is multiplication by ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Honda–Tate Theorem
In mathematics, the Honda–Tate theorem classifies abelian varieties over finite fields up to isogeny. It states that the isogeny classes of simple abelian varieties over a finite field of order ''q'' correspond to algebraic integers all of whose conjugates (given by eigenvalues of the Frobenius endomorphism on the first cohomology group or Tate module In mathematics, a Tate module of an abelian group, named for John Tate, is a module constructed from an abelian group ''A''. Often, this construction is made in the following situation: ''G'' is a commutative group scheme over a field ''K'', ''K ...) have absolute value . showed that the map taking an isogeny class to the eigenvalues of the Frobenius is injective, and showed that this map is surjective, and therefore a bijection. References * * * Theorems in algebraic geometry {{abstract-algebra-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hodge Structure
In mathematics, a Hodge structure, named after W. V. D. Hodge, is an algebraic structure at the level of linear algebra, similar to the one that Hodge theory gives to the cohomology groups of a smooth and compact Kähler manifold. Hodge structures have been generalized for all complex varieties (even if they are singular and non-complete) in the form of mixed Hodge structures, defined by Pierre Deligne (1970). A variation of Hodge structure is a family of Hodge structures parameterized by a manifold, first studied by Phillip Griffiths (1968). All these concepts were further generalized to mixed Hodge modules over complex varieties by Morihiko Saito (1989). Hodge structures Definition of Hodge structures A pure Hodge structure of integer weight ''n'' consists of an abelian group H_ and a decomposition of its complexification ''H'' into a direct sum of complex subspaces H^, where p+q=n, with the property that the complex conjugate of H^ is H^: :H := H_\otimes_ \Complex = \bigop ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Period Matrix
In mathematics, in the field of algebraic geometry, the period mapping relates families of Kähler manifolds to families of Hodge structures. Ehresmann's theorem Let be a holomorphic submersive morphism. For a point ''b'' of ''B'', we denote the fiber of ''f'' over ''b'' by ''X''''b''. Fix a point 0 in ''B''. Ehresmann's theorem guarantees that there is a small open neighborhood ''U'' around 0 in which ''f'' becomes a fiber bundle. That is, is diffeomorphic to . In particular, the composite map :X_b \hookrightarrow f^(U) \cong X_0 \times U \twoheadrightarrow X_0 is a diffeomorphism. This diffeomorphism is not unique because it depends on the choice of trivialization. The trivialization is constructed from smooth paths in ''U'', and it can be shown that the homotopy class of the diffeomorphism depends only on the choice of a homotopy class of paths from ''b'' to 0. In particular, if ''U'' is contractible, there is a well-defined diffeomorphism up to homotopy. The diffe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 manifold, complex structure on the torus H^n(M,\R)/H^n(M,\Z) for ''n'' odd. There are several different natural ways to put a complex structure on this torus, giving several different sorts of intermediate Jacobians, including one due to and one due to . The ones constructed by Weil have natural polarizations if ''M'' is projective, and so are abelian varieties, while the ones constructed by Griffiths behave well under holomorphic deformations. A complex structure on a real vector space is given by an automorphism ''I'' with square -1. The complex structures on H^n(M,\R) are defined using the Hodge decomposition : H^(M,) \otimes = H^(M)\oplus\cdots\oplus H^(M). On H^ the Weil complex structure I_W is multiplication by ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generalized Jacobian
In algebraic geometry a generalized Jacobian is a commutative algebraic group associated to a curve with a divisor, generalizing the Jacobian variety of a complete curve. They were introduced by Maxwell Rosenlicht in 1954, and can be used to study ramified coverings of a curve, with abelian Galois group. Generalized Jacobians of a curve are extensions of the Jacobian of the curve by a commutative affine algebraic group, giving nontrivial examples of Chevalley's structure theorem. Definition Suppose ''C'' is a complete nonsingular curve, ''m'' an effective divisor on ''C'', ''S'' is the support of ''m'', and ''P'' is a fixed base point on ''C'' not in ''S''. The generalized Jacobian ''J''''m'' is a commutative algebraic group with a rational map ''f'' from ''C'' to ''J''''m'' such that: *''f'' takes ''P'' to the identity of ''J''''m''. *''f'' is regular outside ''S''. *''f''(''D'') = 0 whenever ''D'' is the divisor of a rational function ''g'' on ''C'' such that ''g''≡1 mod ''m''. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Albanese Variety
In mathematics, the Albanese variety A(V), named for Giacomo Albanese, is a generalization of the Jacobian variety of a curve. Precise statement The Albanese variety is the abelian variety A generated by a variety V taking a given point of V to the identity of A. In other words, there is a morphism from the variety V to its Albanese variety \operatorname(V), such that any morphism from V to an abelian variety (taking the given point to the identity) factors uniquely through \operatorname(V). For complex manifolds, defined the Albanese variety in a similar way, as a morphism from V to a torus \operatorname(V) such that any morphism to a torus factors uniquely through this map. (It is an analytic variety in this case; it need not be algebraic.) Properties For compact Kähler manifolds the dimension of the Albanese variety is the Hodge number h^, the dimension of the space of differentials of the first kind on V, which for surfaces is called the irregularity of a surface. In terms ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 More precisely, one should consider algebraic curves C of a given genus g, and their Jacobians \operatorname(C). There is a moduli space \mathcal_g of such curves, and a moduli space of abelian varieties, \mathcal_g, of dimension g, which are ''principally polarized''. There is a morphism\operatorname: \mathcal_g \to \mathcal_gwhich on points (geometric points, to be more accurate) takes isomorphism class /math> to operatorname(C)/math>. The content of Torelli's theorem is that \operatorname is injective (again, on points). The Schottky problem asks for a description of the image of \operatorname, denoted \mathcal_g = \operatorname(\mathcal_g). The dimension of \mathcal_g is 3g - 3, for g \geq 2, while the dimension of ''\mathcal_g'' is ''g''(''g'' + 1)/2. This me ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
