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Deligne Cohomology
In mathematics, Deligne cohomology is the hypercohomology of the Deligne complex of a complex manifold. It was introduced by Pierre Deligne in unpublished work in about 1972 as a cohomology theory for algebraic varieties that includes both ordinary cohomology and intermediate Jacobians. For introductory accounts of Deligne cohomology see , , and . Definition The analytic Deligne complex Z(''p'')D, an on a complex analytic manifold ''X'' is0\rightarrow \mathbf Z(p)\rightarrow \Omega^0_X\rightarrow \Omega^1_X\rightarrow\cdots\rightarrow \Omega_X^ \rightarrow 0 \rightarrow \dotswhere Z(''p'') = (2π i)''p''Z. Depending on the context, \Omega^*_X is either the complex of smooth (i.e., ''C''∞) differential forms or of holomorphic forms, respectively. The Deligne cohomology is the ''q''-th hypercohomology of the Deligne complex. An alternative definition of this complex is given as the homotopy limit of the diagram\begin & & \mathbb \\ & & \downarrow \\ \Omega_X^ & \to & \Omega_ ... [...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] |
Special Values Of L-functions
In mathematics, the study of special values of L-functions is a subfield of number theory devoted to generalising formulae such as the Leibniz formula for pi, namely :1 \,-\, \frac \,+\, \frac \,-\, \frac \,+\, \frac \,-\, \cdots \;=\; \frac,\! by the recognition that expression on the left-hand side is also ''L''(1) where ''L''(''s'') is the Dirichlet L-function for the Gaussian field. This formula is a special case of the analytic class number formula, and in those terms reads that the Gaussian field has class number 1, and also contains four roots of unity, so accounting for the factor ¼. Conjectures There are two families of conjectures, formulated for general classes of ''L''-functions (the very general setting being for ''L''-functions ''L''(''s'') associated to Chow motives over number fields), the division into two reflecting the questions of: how to replace π in the Leibniz formula by some other "transcendental" number (whether or not it is yet possible fo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sheaf Theory
In mathematics, a sheaf is a tool for systematically tracking data (such as sets, abelian groups, rings) attached to the open sets of a topological space and defined locally with regard to them. For example, for each open set, the data could be the ring of continuous functions defined on that open set. Such data is well behaved in that it can be restricted to smaller open sets, and also the data assigned to an open set is equivalent to all collections of compatible data assigned to collections of smaller open sets covering the original open set (intuitively, every piece of data is the sum of its parts). The field of mathematics that studies sheaves is called sheaf theory. Sheaves are understood conceptually as general and abstract objects. Their correct definition is rather technical. They are specifically defined as sheaves of sets or as sheaves of rings, for example, depending on the type of data assigned to the open sets. There are also maps (or morphisms) from one ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inventiones Mathematicae
''Inventiones Mathematicae'' is a mathematical journal published monthly by Springer Science+Business Media. It was established in 1966 and is regarded as one of the most prestigious mathematics journals in the world. The current managing editors are Camillo De Lellis (Institute for Advanced Study, Princeton) and Jean-Benoît Bost (University of Paris-Sud Paris-Sud University (French: ''Université Paris-Sud''), also known as University of Paris — XI (or as Université d'Orsay before 1971), was a French research university distributed among several campuses in the southern suburbs of Paris, in ...). Abstracting and indexing The journal is abstracted and indexed in: References External links *{{Official website, https://www.springer.com/journal/222 Mathematics journals Publications established in 1966 English-language journals Springer Science+Business Media academic journals Monthly journals ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Academic Press
Academic Press (AP) is an academic book publisher founded in 1941. It was acquired by Harcourt, Brace & World in 1969. Reed Elsevier bought Harcourt in 2000, and Academic Press is now an imprint of Elsevier. Academic Press publishes reference books, serials and online products in the subject areas of: * Communications engineering * Economics * Environmental science * Finance * Food science and nutrition * Geophysics * Life sciences * Mathematics and statistics * Neuroscience * Physical sciences * Psychology Well-known products include the ''Methods in Enzymology'' series and encyclopedias such as ''The International Encyclopedia of Public Health'' and the ''Encyclopedia of Neuroscience''. See also * Akademische Verlagsgesellschaft (AVG) — the German predecessor, founded in 1906 by Leo Jolowicz (1868–1940), the father of Walter Jolowicz Walter may refer to: People * Walter (name), both a surname and a given name * Little Walter, American blues harmonica player Marion Wa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
1706
In the Swedish calendar it was a common year starting on Monday, one day ahead of the Julian and ten days behind the Gregorian calendar. Events January–March * January 26 – War of Spanish Succession: Bavarian uprising of 1705–06, The uprising by Bavarians against the occupation of the Electorate of Bavaria by Austrian Habsburgs, Austrian troops ends after 75 days, and ends the plans of Maximilian II Emanuel, Elector of Bavaria, Maximilian, the Elector of Bavaria, to bring Bavaria under the rule of the House of Wittelsbach. * January 26 – Great Northern War: The Battle of Grodno (1706), Battle of Grodno begins when a coalition of 34,000 Swedish and Polish troops besieges the then-Lithuanian city in the winter time and clashes with 41,000 Russian and Saxon troops. After almost three months of fighting that lasts to April 10, Sweden takes control of the city, which is now located in Belarus. * February 6 – The city of Albuquerque, New Mexico, is in ... [...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] |
Motivic Cohomology
Motivic cohomology is an invariant of algebraic varieties and of more general schemes. It is a type of cohomology related to motives and includes the Chow ring of algebraic cycles as a special case. Some of the deepest problems in algebraic geometry and number theory are attempts to understand motivic cohomology. Motivic homology and cohomology Let ''X'' be a scheme of finite type over a field ''k''. A key goal of algebraic geometry is to compute the Chow groups of ''X'', because they give strong information about all subvarieties of ''X''. The Chow groups of ''X'' have some of the formal properties of Borel–Moore homology in topology, but some things are missing. For example, for a closed subscheme ''Z'' of ''X'', there is an exact sequence of Chow groups, the localization sequence :CH_i(Z) \rightarrow CH_i(X) \rightarrow CH_i(X-Z) \rightarrow 0, whereas in topology this would be part of a long exact sequence. This problem was resolved by generalizing Chow groups to a bigrad ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bundle Gerbe
In mathematics, a bundle gerbe is a geometrical model of certain 1-gerbes with connection, or equivalently of a 2-class in Deligne cohomology. Topology U(1)-principal bundles over a space M (see circle bundle) are geometrical realizations of 1-classes in Deligne cohomology which consist of 1-form connections and 2-form curvatures. The topology of a U(1) bundle is classified by its Chern class, which is an element of H^2(M, \mathbb), the second integral cohomology of M. Gerbes, or more precisely 1-gerbes, are abstract descriptions of Deligne 2-classes, which each define an element of H^3(M, \mathbb), the third integral cohomology of ''M''. As a cohomology class in Deligne cohomology Recall for a smooth manifold M the p-th Deligne cohomology groups are defined by the hypercohomology of the complex \mathbb(q)_D^\infty = \underline(q) \to \mathcal_^0 \xrightarrow \mathcal_^1 \xrightarrow \cdots \xrightarrow \mathcal_^ called the weight q Deligne complex, where \mathcal^k_ is the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symmetric Spectrum
In algebraic topology, a symmetric spectrum ''X'' is a spectrum of pointed simplicial sets that comes with an action of the symmetric group \Sigma_n on X_n such that the composition of structure maps :S^1 \wedge \dots \wedge S^1 \wedge X_n \to S^1 \wedge \dots \wedge S^1 \wedge X_ \to \dots \to S^1 \wedge X_ \to X_ is equivariant with respect to \Sigma_p \times \Sigma_n. A morphism between symmetric spectra is a morphism of spectra that is equivariant with respect to the actions of symmetric groups. The technical advantage of the category \mathcalp^\Sigma of symmetric spectra is that it has a closed symmetric monoidal structure (with respect to smash product). It is also a simplicial model category. A symmetric ring spectrum is a monoid in \mathcalp^\Sigma; if the monoid is commutative, it's a commutative ring spectrum In the mathematical field of algebraic topology, a commutative ring spectrum, roughly equivalent to a E_\infty-ring spectrum, is a commutative monoid in a goodsymmet ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
