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Vector Bundles On Algebraic Curves
In mathematics, vector bundles on algebraic curves may be studied as holomorphic vector bundles on compact Riemann surfaces, which is the classical approach, or as locally free sheaves on algebraic curves ''C'' in a more general, algebraic setting (which can for example admit singular points). Some foundational results on classification were known in the 1950s. The result of , that holomorphic vector bundles on the Riemann sphere are sums of line bundles, is now often called the Birkhoff–Grothendieck theorem, since it is implicit in much earlier work of on the Riemann–Hilbert problem. gave the classification of vector bundles on elliptic curves. The Riemann–Roch theorem for vector bundles was proved by , before the 'vector bundle' concept had really any official status. Although, associated ruled surfaces were classical objects. See Hirzebruch–Riemann–Roch theorem for his result. He was seeking a generalization of the Jacobian variety, by passing from holomorphic lin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hirzebruch–Riemann–Roch Theorem
In mathematics, the Hirzebruch–Riemann–Roch theorem, named after Friedrich Hirzebruch, Bernhard Riemann, and Gustav Roch, is Hirzebruch's 1954 result generalizing the classical Riemann–Roch theorem on Riemann surfaces to all complex algebraic varieties of higher dimensions. The result paved the way for the Grothendieck–Hirzebruch–Riemann–Roch theorem proved about three years later. Statement of Hirzebruch–Riemann–Roch theorem The Hirzebruch–Riemann–Roch theorem applies to any holomorphic vector bundle ''E'' on a compact complex manifold ''X'', to calculate the holomorphic Euler characteristic of ''E'' in sheaf cohomology, namely the alternating sum : \chi(X,E) = \sum_^ (-1)^ \dim_ H^(X,E) of the dimensions as complex vector spaces, where ''n'' is the complex dimension of ''X''. Hirzebruch's theorem states that χ(''X'', ''E'') is computable in terms of the Chern classes ''ck''(''E'') of ''E'', and the Todd classes \operatorname_(X) of the holomorphi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal Für Die Reine Und Angewandte Mathematik
''Crelle's Journal'', or just ''Crelle'', is the common name for a mathematics journal, the ''Journal für die reine und angewandte Mathematik'' (in English: ''Journal for Pure and Applied Mathematics''). History The journal was founded by August Leopold Crelle (Berlin) in 1826 and edited by him until his death in 1855. It was one of the first major mathematical journals that was not a proceedings of an academy. It has published many notable papers, including works of Niels Henrik Abel, Georg Cantor, Gotthold Eisenstein, Carl Friedrich Gauss and Otto Hesse. It was edited by Carl Wilhelm Borchardt from 1856 to 1880, during which time it was known as ''Borchardt's Journal''. The current editor-in-chief is Daniel Huybrechts (Rheinische Friedrich-Wilhelms-Universität Bonn). Past editors * 1826–1856: August Leopold Crelle * 1856–1880: Carl Wilhelm Borchardt * 1881–1888: Leopold Kronecker, Karl Weierstrass Karl Theodor Wilhelm Weierstrass (; ; 31 October 1815 â ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
American Journal Of Mathematics
The ''American Journal of Mathematics'' is a bimonthly mathematics journal published by the Johns Hopkins University Press. History The ''American Journal of Mathematics'' is the oldest continuously published mathematical journal in the United States, established in 1878 at the Johns Hopkins University by James Joseph Sylvester, an English-born mathematician who also served as the journal's editor-in-chief from its inception through early 1884. Initially W. E. Story was associate editor in charge; he was replaced by Thomas Craig (mathematician), Thomas Craig in 1880. For volume 7 Simon Newcomb became chief editor with Craig managing until 1894. Then with volume 16 it was "Edited by Thomas Craig with the Co-operation of Simon Newcomb" until 1898. Other notable mathematicians who have served as editors or editorial associates of the journal include Frank Morley, Oscar Zariski, Lars Ahlfors, Hermann Weyl, Wei-Liang Chow, S. S. Chern, André Weil, Harish-Chandra, Jean Dieudonné, Hen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Transactions Of The American Mathematical Society
The ''Transactions of the American Mathematical Society'' is a monthly peer-reviewed scientific journal of pure and applied mathematics published by the American Mathematical Society. It was established in 1900. As a requirement, all articles must be more than 15 printed pages. Its ISSN number is 0002-9947. See also * ''Bulletin of the American Mathematical Society'' * ''Journal of the American Mathematical Society'' * '' Memoirs of the American Mathematical Society'' * '' Notices of the American Mathematical Society'' * ''Proceedings of the American Mathematical Society'' References External links * ''Transactions of the American Mathematical Society''on JSTOR JSTOR ( ; short for ''Journal Storage'') is a digital library of academic journals, books, and primary sources founded in 1994. Originally containing digitized back issues of academic journals, it now encompasses books and other primary source ... American Mathematical Society academic journals Mathematics jo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hitchin System
In mathematics, the Hitchin integrable system is an integrable system depending on the choice of a complex reductive group and a compact Riemann surface, introduced by Nigel Hitchin in 1987. It lies on the crossroads of algebraic geometry, the theory of Lie algebras and integrable system theory. It also plays an important role in the geometric Langlands correspondence over the field of complex numbers through conformal field theory. A genus zero analogue of the Hitchin system, the Garnier system, was discovered by René Garnier somewhat earlier as a certain limit of the Schlesinger equations, and Garnier solved his system by defining spectral curves. (The Garnier system is the classical limit of the Gaudin model. In turn, the Schlesinger equations are the classical limit of the Knizhnik–Zamolodchikov equations). Almost all integrable systems of classical mechanics can be obtained as particular cases of the Hitchin system or their common generalization defined by Bottacin a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometric Invariant Theory
In mathematics, geometric invariant theory (or GIT) is a method for constructing quotients by group actions in algebraic geometry, used to construct moduli spaces. It was developed by David Mumford in 1965, using ideas from the paper in classical invariant theory. Geometric invariant theory studies an action of a group on an algebraic variety (or scheme) and provides techniques for forming the 'quotient' of by as a scheme with reasonable properties. One motivation was to construct moduli spaces in algebraic geometry as quotients of schemes parametrizing marked objects. In the 1970s and 1980s the theory developed interactions with symplectic geometry and equivariant topology, and was used to construct moduli spaces of objects in differential geometry, such as instantons and monopoles. Background Invariant theory is concerned with a group action of a group on an algebraic variety (or a scheme) . Classical invariant theory addresses the situation when is a vect ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Moduli Space
In mathematics, in particular algebraic geometry, a moduli space is a geometric space (usually a scheme (mathematics), scheme or an algebraic stack) whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects. Such spaces frequently arise as solutions to classification problems: If one can show that a collection of interesting objects (e.g., the smooth algebraic curves of a fixed genus (topology), genus) can be given the structure of a geometric space, then one can parametrize such objects by introducing coordinates on the resulting space. In this context, the term "modulus" is used synonymously with "parameter"; moduli spaces were first understood as spaces of parameters rather than as spaces of objects. A variant of moduli spaces is formal moduli. Bernhard Riemann first used the term "moduli" in 1857. Motivation Moduli spaces are spaces of solutions of geometric classification problems. That is, the points of a moduli space corr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Holomorphic Line Bundle
In mathematics, a holomorphic vector bundle is a complex vector bundle over a complex manifold such that the total space is a complex manifold and the projection map is holomorphic. Fundamental examples are the holomorphic tangent bundle of a complex manifold, and its dual, the holomorphic cotangent bundle. A holomorphic line bundle is a rank one holomorphic vector bundle. By Serre's GAGA, the category of holomorphic vector bundles on a smooth complex projective variety ''X'' (viewed as a complex manifold) is equivalent to the category of algebraic vector bundles (i.e., locally free sheaves of finite rank) on ''X''. Definition through trivialization Specifically, one requires that the trivialization maps :\phi_U : \pi^(U) \to U \times \mathbf^k are biholomorphic maps. This is equivalent to requiring that the transition functions :t_ : U\cap V \to \mathrm_k(\mathbf) are holomorphic maps. The holomorphic structure on the tangent bundle of a complex manifold is guarant ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jacobian Variety
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 differenti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ruled Surface
In geometry, a Differential geometry of surfaces, surface in 3-dimensional Euclidean space is ruled (also called a scroll) if through every Point (geometry), point of , there is a straight line that lies on . Examples include the plane (mathematics), plane, the lateral surface of a cylinder (geometry), cylinder or cone (geometry), cone, a conical surface with ellipse, elliptical directrix (rational normal scroll), directrix, the right conoid, the helicoid, and the tangent developable of a smooth curve in space. A ruled surface can be described as the set of points swept by a moving straight line. For example, a cone is formed by keeping one point of a line fixed whilst moving another point along a circle. A surface is doubly ruled if through every one of its points there are two distinct lines that lie on the surface. The hyperbolic paraboloid and the hyperboloid of one sheet are doubly ruled surfaces. The plane is the only surface which contains at least three distinct lines thr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Holomorphic Vector Bundle
In mathematics, a holomorphic vector bundle is a complex vector bundle over a complex manifold such that the total space is a complex manifold and the projection map is holomorphic. Fundamental examples are the holomorphic tangent bundle of a complex manifold, and its dual, the holomorphic cotangent bundle. A holomorphic line bundle is a rank one holomorphic vector bundle. By Serre's GAGA, the category of holomorphic vector bundles on a smooth complex projective variety ''X'' (viewed as a complex manifold) is equivalent to the category of algebraic vector bundles (i.e., locally free sheaves of finite rank) on ''X''. Definition through trivialization Specifically, one requires that the trivialization maps :\phi_U : \pi^(U) \to U \times \mathbf^k are biholomorphic maps. This is equivalent to requiring that the transition functions :t_ : U\cap V \to \mathrm_k(\mathbf) are holomorphic maps. The holomorphic structure on the tangent bundle of a complex manifold is guaran ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
