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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 of a smooth projective algebraic variety V is an abelian variety \operatorname(V) together with a morphism V\to \operatorname(V) such that any morphism from V to an abelian variety factors uniquely through this morphism. For complex manifolds, defined the Albanese variety in a similar way, as a morphism from V to a complex torus \operatorname(V) such that any morphism to a complex torus factors uniquely through this map. (The complex torus \operatorname(V) need not be algebraic in this case.) Properties For compact space, compact Kähler manifolds the dimension of the Albanese variety is the Hodge theory, 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 of differential forms, any holomorphic 1-form on V is a pullba ... [...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] |
Annals Of Mathematics
The ''Annals of Mathematics'' is a mathematical journal published every two months by Princeton University and the Institute for Advanced Study. History The journal was established as ''The Analyst'' in 1874 and with Joel E. Hendricks as the founding editor-in-chief. It was "intended to afford a medium for the presentation and analysis of any and all questions of interest or importance in pure and applied Mathematics, embracing especially all new and interesting discoveries in theoretical and practical astronomy, mechanical philosophy, and engineering". It was published in Des Moines, Iowa, and was the earliest American mathematics journal to be published continuously for more than a year or two. This incarnation of the journal ceased publication after its tenth year, in 1883, giving as an explanation Hendricks' declining health, but Hendricks made arrangements to have it taken over by new management, and it was continued from March 1884 as the ''Annals of Mathematics''. T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Annales Scientifiques De L'École Normale Supérieure
''Annales Scientifiques de l'École Normale Supérieure'' is a French scientific journal of mathematics published by the Société Mathématique de France. It was established in 1864 by the French chemist Louis Pasteur and published articles in mathematics, physics, chemistry, biology, and geology. In 1900, it became a purely mathematical journal. It is published with help of the Centre national de la recherche scientifique. Its web site is hosted by the mathematics department of the École Normale Supérieure École or Ecole may refer to: * an elementary school in the French educational stages normally followed by Secondary education in France, secondary education establishments (collège and lycée) * École (river), a tributary of the Seine flowing i .... External links * Archive(1864–2013) Mathematics journals Publications established in 1864 Multidisciplinary scientific journals Société Mathématique de France academic journals {{math-journal-stub English-F ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Motivic Albanese
In music, a motif () or motive is a short musical idea, a salient recurring figure, musical fragment or succession of notes that has some special importance in or is characteristic of a composition. The motif is the smallest structural unit possessing thematic identity. History The defines a motif as a "melodic, rhythmic, or harmonic cell", whereas the 1958 maintains that it may contain one or more cells, though it remains the smallest analyzable element or phrase within a subject. It is commonly regarded as the shortest subdivision of a theme or phrase that still maintains its identity as a musical idea. "The smallest structural unit possessing thematic identity". Grove and Larousse also agree that the motif may have harmonic, melodic and/or rhythmic aspects, Grove adding that it "is most often thought of in melodic terms, and it is this aspect of the motif that is connoted by the term 'figure'." A harmonic motif is a series of chords defined in the abstract, tha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Albanese Scheme
Albanese is an Italian surname. It means "Albanian", in reference to the Arbëreshë people (Italo-Albanians) of southern Italy or someone of Albanian origin. The surname is common in southern Italy but more rare elsewhere in the country. "Albanése, -i : dall'etnico Albanése o, nel Sud, 'appartenente alle colonie albanesi' (in Abruzzo, Puglie, Campania, Calabria e Sicilia)." Notable people with the surname include: * Albano Albanese (1921–2010), Italian hurdler and high jumper * Alessandro Albanese (born 2000), Belgian professional footballer * Antonio Albanese (1937–2013), Italian fencer *Anthony Albanese (born 1963), Australian politician and current prime minister of Australia * Antonio Albanese (born 1964), Italian comedian, actor, director, and writer * Catherine L. Albanese (born 1940), American religious studies scholar, professor, lecturer, and author * Charles Albanese (1937–1995), American serial killer * Diego Albanese (born 1973), Argentine rugby union player * D ... [...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 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 i^, while the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Invertible Sheaves
In mathematics, an invertible sheaf is a sheaf on a ringed space that has an inverse with respect to tensor product of sheaves of modules. It is the equivalent in algebraic geometry of the topological notion of a line bundle. Due to their interactions with Cartier divisors, they play a central role in the study of algebraic varieties. Definition Let (''X'', ''O''''X'') be a ringed space. Isomorphism classes of sheaves of ''O''''X''-modules form a monoid under the operation of tensor product of ''O''''X''-modules. The identity element for this operation is ''O''''X'' itself. Invertible sheaves are the invertible elements of this monoid. Specifically, if ''L'' is a sheaf of ''O''''X''-modules, then ''L'' is called invertible if it satisfies any of the following equivalent conditions:Stacks Project, tag 01CR * There exists a sheaf ''M'' such that L \otimes_ M \cong \mathcal_X. * The natural homomorphism L \otimes_ L^\vee \to \mathcal_X is an isomorphism, where L^\vee denotes the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Picard Scheme
In mathematics, the Picard group of a ringed space ''X'', denoted by Pic(''X''), is the group of isomorphism classes of invertible sheaves (or line bundles) on ''X'', with the group operation being tensor product. This construction is a global version of the construction of the divisor class group, or ideal class group, and is much used in algebraic geometry and the theory of complex manifolds. Alternatively, the Picard group can be defined as the sheaf cohomology group :H^1 (X, \mathcal_X^).\, For integral schemes the Picard group is isomorphic to the class group of Cartier divisors. For complex manifolds the exponential sheaf sequence gives basic information on the Picard group. The name is in honour of Émile Picard's theories, in particular of divisors on algebraic surfaces. Examples * The Picard group of the spectrum of a Dedekind domain is its ''ideal class group''. * The invertible sheaves on projective space P''n''(''k'') for ''k'' a field, are the twisting sheaves ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Connected Space
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union (set theory), union of two or more disjoint set, disjoint Empty set, non-empty open (topology), open subsets. Connectedness is one of the principal topological properties that distinguish topological spaces. A subset of a topological space X is a if it is a connected space when viewed as a Subspace topology, subspace of X. Some related but stronger conditions are #Path connectedness, path connected, Simply connected space, simply connected, and N-connected space, n-connected. Another related notion is Locally connected space, locally connected, which neither implies nor follows from connectedness. Formal definition A topological space X is said to be if it is the union of two disjoint non-empty open sets. Otherwise, X is said to be connected. A subset of a topological space is said to be connected if it is connected under its subspace topology. So ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Duality Theory Of Abelian Varieties
In mathematics, a dual abelian variety can be defined from an abelian variety ''A'', defined over a field ''k''. A 1-dimensional abelian variety is an elliptic curve, and every elliptic curve is isomorphic to its dual, but this fails for higher-dimensional abelian varieties, so the concept of dual becomes more interesting in higher dimensions. Definition Let ''A'' be an abelian variety over a field ''k''. We define \operatorname^0 (A) \subset \operatorname (A) to be the subgroup of the Picard group consisting of line bundles ''L'' such that m^*L \cong p^*L \otimes q^*L, where m, p, q are the multiplication and projection maps A \times_k A \to A respectively. An element of \operatorname^0(A) is called a degree 0 line bundle on ''A''. To ''A'' one then associates a dual abelian variety ''A''v (over the same field), which is the solution to the following moduli problem. A family of degree 0 line bundles parametrized by a ''k''-variety ''T'' is defined to be a line bundle ''L'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Borel–Moore Homology
In topology, Borel−Moore homology or homology with closed support is a homology theory for locally compact spaces, introduced by Armand Borel and John Moore in 1960. For reasonable compact spaces, Borel−Moore homology coincides with the usual singular homology. For non-compact spaces, each theory has its own advantages. In particular, a closed oriented submanifold defines a class in Borel–Moore homology, but not in ordinary homology unless the submanifold is compact. Note: Borel equivariant cohomology is an invariant of spaces with an action of a group ''G''; it is defined as H^*_G(X) = H^*((EG \times X)/G). That is not related to the subject of this article. Definition There are several ways to define Borel−Moore homology. They all coincide for reasonable spaces such as manifolds and locally finite CW complexes. Definition via sheaf cohomology For any locally compact space ''X'', Borel–Moore homology with integral coefficients is defined as the cohomology of the dual ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
