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Riemann–Roch-type Theorem
In algebraic geometry, there are various generalizations of the Riemann–Roch theorem; among the most famous is the Grothendieck–Riemann–Roch theorem, which is further generalized by the formulation due to Fulton et al. Formulation due to Baum, Fulton and MacPherson Let G_* and A_* be functors on the category ''C'' of schemes separated and locally of finite type over the base field ''k'' with proper morphisms such that *G_*(X) is the Grothendieck group of coherent sheaves on ''X'', *A_*(X) is the rational Chow group of ''X'', *for each proper morphism ''f'', G_*(f), A_*(f) are the direct images (or push-forwards) along ''f''. Also, if f: X \to Y is a (global) local complete intersection morphism; i.e., it factors as a closed regular embedding X \hookrightarrow P into a smooth scheme ''P'' followed by a smooth morphism P \to Y, then let :T_f = _X- _/math> be the class in the Grothendieck group of vector bundles on ''X''; it is independent of the factorization and is ca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Riemann–Roch Theorem
The Riemann–Roch theorem is an important theorem in mathematics, specifically in complex analysis and algebraic geometry, for the computation of the dimension of the space of meromorphic functions with prescribed zeros and allowed poles. It relates the complex analysis of a connected compact Riemann surface with the surface's purely topological genus ''g'', in a way that can be carried over into purely algebraic settings. Initially proved as Riemann's inequality by , the theorem reached its definitive form for Riemann surfaces after work of Riemann's short-lived student . It was later generalized to algebraic curves, to higher-dimensional varieties and beyond. Preliminary notions A Riemann surface X is a topological space that is locally homeomorphic to an open subset of \Complex, the set of complex numbers. In addition, the transition maps between these open subsets are required to be holomorphic. The latter condition allows one to transfer the notions and methods of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Chern Class
In mathematics, in particular in algebraic topology, differential geometry and algebraic geometry, the Chern classes are characteristic classes associated with complex vector bundles. They have since found applications in physics, Calabi–Yau manifolds, string theory, Chern–Simons theory, knot theory, Gromov–Witten invariants, topological quantum field theory, the Chern theorem etc. Chern classes were introduced by . Geometric approach Basic idea and motivation Chern classes are characteristic classes. They are topological invariants associated with vector bundles on a smooth manifold. The question of whether two ostensibly different vector bundles are the same can be quite hard to answer. The Chern classes provide a simple test: if the Chern classes of a pair of vector bundles do not agree, then the vector bundles are different. The converse, however, is not true. In topology, differential geometry, and algebraic geometry, it is often important to count ho ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in Berlin, it expanded internationally in the 1960s, and through mergers in the 1990s and a sale to venture capitalists it fused with Wolters Kluwer and eventually became part of Springer Nature in 2015. Springer has major offices in Berlin, Heidelberg, Dordrecht, and New York City. History Julius Springer founded Springer-Verlag in Berlin in 1842 and his son Ferdinand Springer grew it from a small firm of 4 employees into Germany's then second largest academic publisher with 65 staff in 1872.Chronology ". Springer Science+Business Media. In 1964, Springer expanded its business internationall ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Kawasaki's Riemann–Roch Formula
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In differential geometry, Kawasaki's Riemann–Roch formula, introduced by Tetsuro Kawasaki, is the Riemann–Roch formula for orbifolds. It can compute the Euler characteristic of an orbifold. Kawasaki's original proof made a use of the equivariant index theorem. Today, the formula is known to follow from the Riemann–Roch formula for quotient stacks. References *Tetsuro Kawasaki. The Riemann-Roch theorem for complex V-manifolds. Osaka J. Math., 16(1):151–159, 1979 Theorems in differential geometry Theorems in algebraic geometry See also *Riemann–Roch-type theorem In algebraic geometry, there are various generalizations of the Riemann–Roch theorem; among the most famous is the Grothendieck–Riemann–Roch theorem, which is further generalized by the formulation due to Fulton et al. Formulation due to Bau ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Virtual Fundamental Class
In mathematics, specifically enumerative geometry, the virtual fundamental class \text_ of a space X is a replacement of the classical fundamental class \in A^*(X) in its chow ring which has better behavior with respect to the enumerative problems being considered. In this way, there exists a cycle with can be used for answering specific enumerative problems, such as the number of degree d rational curves on a quintic threefold. For example, in Gromov–Witten theory, the Kontsevich moduli spaces\overline_(X,\beta)for X a scheme and \beta a class in A_1(X), their behavior can be wild at the boundary, such aspg 503 having higher-dimensional components at the boundary than on the main space. One such example is in the moduli space\overline_(\mathbb^2,1 for H the class of a line in \mathbb^2. The non-compact "smooth" component is empty, but the boundary contains maps of curvesf:C \to \mathbb^2whose components consist of one degree 3 curve which contracts to a point. There is a virt ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Quotient Stack
In algebraic geometry, a quotient stack is a stack (mathematics), stack that parametrizes equivariant objects. Geometrically, it generalizes a quotient of a scheme or a variety by a group: a quotient variety, say, would be a coarse approximation of a quotient stack. The notion is of fundamental importance in the study of stacks: a stack that arises in nature is often either a quotient stack itself or admits a stratification by quotient stacks (e.g., a Deligne–Mumford stack.) A quotient stack is also used to construct other stacks like classifying stacks. Definition A quotient stack is defined as follows. Let ''G'' be an affine smooth group scheme over a scheme ''S'' and ''X'' an ''S''-scheme on which ''G'' group-scheme action, acts. Let the quotient stack [X/G] be the fibered category, category over the category of ''S''-schemes: *an object over ''T'' is a torsor (algebraic geometry), principal ''G''-bundle P\to T together with equivariant map P\to X; *an arrow from P\to T to P' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Kawasaki's Riemann–Roch
Kawasaki disease is a syndrome of unknown cause that results in a fever and mainly affects children under 5 years of age. It is a form of vasculitis, where blood vessels become inflamed throughout the body. The fever typically lasts for more than five days and is not affected by usual medications. Other common symptoms include large lymph nodes in the neck, a rash in the genital area, lips, palms, or soles of the feet, and red eyes. Within three weeks of the onset, the skin from the hands and feet may peel, after which recovery typically occurs. In some children, coronary artery aneurysms form in the heart. While the specific cause is unknown, it is thought to result from an excessive immune system response to an infection in children who are genetically predisposed. It does not spread between people. Diagnosis is usually based on a person's signs and symptoms. Other tests such as an ultrasound of the heart and blood tests may support the diagnosis. Diagnosis must ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Equivariant Index Theorem
In differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ..., the equivariant index theorem, of which there are several variants, computes the (graded) trace of an element of a compact Lie group acting in given setting in terms of the integral over the fixed points of the element. If the element is neutral, then the theorem reduces to the usual index theorem. The classical formula such as the Atiyah–Bott formula is a special case of the theorem. Statement Let \pi: E \to M be a clifford module bundle. Assume a compact Lie group ''G'' acts on both ''E'' and ''M'' so that \pi is equivariant bundle, equivariant. Let ''E'' be given a connection that is compatible with the action of ''G''. Finally, let ''D'' be a Dirac operator on ''E'' associated to the given data. In ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Localized Chern Class
In algebraic geometry, a localized Chern class is a variant of a Chern class, that is defined for a chain complex of vector bundles as opposed to a single vector bundle. It was originally introduced in Fulton's ''intersection theory'', as an algebraic counterpart of the similar construction in algebraic topology. The notion is used in particular in the Riemann–Roch-type theorem. S. Bloch later generalized the notion in the context of arithmetic schemes (schemes over a Dedekind domain) for the purpose of giving #Bloch's conductor formula that computes the non-constancy of Euler characteristic of a degenerating family of algebraic varieties (in the mixed characteristic case). Definitions Let ''Y'' be a pure-dimensional regular scheme of finite type over a field or discrete valuation ring and ''X'' a closed subscheme. Let E_ denote a complex of vector bundles on ''Y'' :0 = E_ \to E_n \to \dots \to E_m \to E_ = 0 that is exact on Y - X. The localized Chern class of this complex ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Todd Class
In mathematics, the Todd class is a certain construction now considered a part of the theory in algebraic topology of characteristic classes. The Todd class of a vector bundle can be defined by means of the theory of Chern classes, and is encountered where Chern classes exist — most notably in differential topology, the theory of complex manifolds and algebraic geometry. In rough terms, a Todd class acts like a reciprocal of a Chern class, or stands in relation to it as a conormal bundle does to a normal bundle. The Todd class plays a fundamental role in generalising the classical Riemann–Roch theorem to higher dimensions, in the Hirzebruch–Riemann–Roch theorem and the Grothendieck–Hirzebruch–Riemann–Roch theorem. History It is named for J. A. Todd, who introduced a special case of the concept in algebraic geometry in 1937, before the Chern classes were defined. The geometric idea involved is sometimes called the Todd-Eger class. The general defini ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Grothendieck–Riemann–Roch Theorem
In mathematics, specifically in algebraic geometry, the Grothendieck–Riemann–Roch theorem is a far-reaching result on coherent cohomology. It is a generalisation of the Hirzebruch–Riemann–Roch theorem, about complex manifolds, which is itself a generalisation of the classical Riemann–Roch theorem for line bundles on compact Riemann surfaces. Riemann–Roch type theorems relate Euler characteristics of the cohomology of a vector bundle with their topological degrees, or more generally their characteristic classes in (co)homology or algebraic analogues thereof. The classical Riemann–Roch theorem does this for curves and line bundles, whereas the Hirzebruch–Riemann–Roch theorem generalises this to vector bundles over manifolds. The Grothendieck–Riemann–Roch theorem sets both theorems in a relative situation of a morphism between two manifolds (or more general schemes) and changes the theorem from a statement about a single bundle, to one applying to chain com ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Natural Transformation
In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved. Hence, a natural transformation can be considered to be a "morphism of functors". Informally, the notion of a natural transformation states that a particular map between functors can be done consistently over an entire category. Indeed, this intuition can be formalized to define so-called functor categories. Natural transformations are, after categories and functors, one of the most fundamental notions of category theory and consequently appear in the majority of its applications. Definition If F and G are functors between the categories C and D , then a natural transformation \eta from F to G is a family of morphisms that satisfies two requirements. # The natural transformation must associate, to every object X in C, a morphism \eta_X ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |