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Torus Action
In algebraic geometry, a torus action on an algebraic variety is a group action of an algebraic torus on the variety. A variety equipped with an action of a torus ''T'' is called a ''T''-variety. In differential geometry, one considers an action of a real or complex torus on a manifold (or an orbifold). A normal algebraic variety with a torus acting on it in such a way that there is a dense orbit is called a toric variety (for example, orbit closures that are normal are toric varieties). Linear action of a torus A linear action of a torus can be simultaneously diagonalized, after extending the base field if necessary: if a torus ''T'' is acting on a finite-dimensional vector space ''V'', then there is a direct sum decomposition: :V = \bigoplus_ V_ where *\chi: T \to \mathbb_m is a group homomorphism, a character of ''T''. *V_ = \, ''T''-invariant subspace called the weight subspace of weight \chi. The decomposition exists because the linear action determines (and is determined ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Variety
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. Modern definitions generalize this concept in several different ways, while attempting to preserve the geometric intuition behind the original definition. Conventions regarding the definition of an algebraic variety differ slightly. For example, some definitions require an algebraic variety to be irreducible, which means that it is not the union of two smaller sets that are closed in the Zariski topology. Under this definition, non-irreducible algebraic varieties are called algebraic sets. Other conventions do not require irreducibility. The fundamental theorem of algebra establishes a link between algebra and geometry by showing that a monic polynomial (an algebraic object) in one variable with complex number coefficients is determined ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cellular Decomposition
In geometric topology, a cellular decomposition ''G'' of a manifold ''M'' is a decomposition of ''M'' as the disjoint union of cells (spaces homeomorphic to ''n''-balls ''Bn''). The quotient space ''M''/''G'' has points that correspond to the cells of the decomposition. There is a natural map from ''M'' to ''M''/''G'', which is given the quotient topology. A fundamental question is whether ''M'' is homeomorphic to ''M''/''G''. Bing's dogbone space is an example with ''M'' (equal to R3) not homeomorphic to ''M''/''G''. Definition Cellular decomposition of X is an open cover \mathcal with a function \text:\mathcal\to \mathbb for which: * Cells are disjoint: for any distinct e,e'\in\mathcal, e\cap e' = \varnothing. * No set gets mapped to a negative number: \text^(\) = \varnothing. * Cells look like balls: For any n\in\mathbb N_0 and for any e\in \text^(n) there exists a continuous map \phi:B^n\to X that is an isomorphism \textB^n\cong e and also \phi(\partial B^n) \subseteq \cup \t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Groups
In mathematics, an algebraic group is an algebraic variety endowed with a group structure which is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory. Many groups of geometric transformations are algebraic groups; for example, orthogonal groups, general linear groups, projective groups, Euclidean groups, etc. Many matrix groups are also algebraic. Other algebraic groups occur naturally in algebraic geometry, such as elliptic curves and Jacobian varieties. An important class of algebraic groups is given by the affine algebraic groups, those whose underlying algebraic variety is an affine variety; they are exactly the algebraic subgroups of the general linear group, and are therefore also called ''linear algebraic groups''. Another class is formed by the abelian varieties, which are the algebraic groups whose underlying variety is a projective variety. Chevalley's structure theorem s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical problems about these sets of zeros. The fundamental objects of study in algebraic geometry are algebraic varieties, which are geometric manifestations of solutions of systems of polynomial equations. Examples of the most studied classes of algebraic varieties are: plane algebraic curves, which include lines, circles, parabolas, ellipses, hyperbolas, cubic curves like elliptic curves, and quartic curves like lemniscates and Cassini ovals. A point of the plane belongs to an algebraic curve if its coordinates satisfy a given polynomial equation. Basic questions involve the study of the points of special interest like the singular points, the inflection points and the points at infinity. More advanced questions involve the topology of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monomial Ideal
In abstract algebra, a monomial ideal is an ideal generated by monomials in a multivariate polynomial ring over a field. A toric ideal is an ideal generated by differences of monomials (provided the ideal is a prime ideal). An affine or projective algebraic variety defined by a toric ideal or a homogeneous toric ideal is an affine or projective toric variety, possibly non-normal. Definitions and Properties Let \mathbb be a field and R = \mathbb /math> be the polynomial ring over \mathbb with ''n'' variables x = x_1, x_2, \dotsc, x_n. A monomial in R is a product x^ = x_1^ x_2^ \cdots x_n^ for an ''n''-tuple \alpha = (\alpha_1, \alpha_2, \dotsc, \alpha_n) \in \mathbb^n of nonnegative integers. The following three conditions are equivalent for an ideal I \subseteq R: # I is generated by monomials, # If f = \sum_ c_ x^ \in I , then x^ \in I , provided that c_ is nonzero. # I is torus fixed, i.e, given (c_1, c_2, \dotsc, c_n) \in (\mathbb^*)^ , then I is fixed under the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equivariant Cohomology
In mathematics, equivariant cohomology (or ''Borel cohomology'') is a cohomology theory from algebraic topology which applies to topological spaces with a ''group action''. It can be viewed as a common generalization of group cohomology and an ordinary cohomology theory. Specifically, the equivariant cohomology ring of a space X with action of a topological group G is defined as the ordinary cohomology ring with coefficient ring \Lambda of the homotopy quotient EG \times_G X: :H_G^*(X; \Lambda) = H^*(EG \times_G X; \Lambda). If G is the trivial group, this is the ordinary cohomology ring of X, whereas if X is contractible, it reduces to the cohomology ring of the classifying space BG (that is, the group cohomology of G when ''G'' is finite.) If ''G'' acts freely on ''X'', then the canonical map EG \times_G X \to X/G is a homotopy equivalence and so one gets: H_G^*(X; \Lambda) = H^*(X/G; \Lambda). Definitions It is also possible to define the equivariant cohomology H_G^*(X;A) of X ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
GKM Variety
In algebraic geometry, a GKM variety is a complex algebraic variety equipped with a torus action that meets certain conditions. The concept was introduced by Mark Goresky, Robert Kottwitz, and Robert MacPherson in 1998. The torus action of a GKM variety must be ''skeletal'': both the set of fixed points of the action, and the number of one-dimensional orbits of the action, must be finite. In addition, the action must be ''equivariantly formal'', a condition that can be phrased in terms of the torus' rational cohomology. See also *equivariant cohomology In mathematics, equivariant cohomology (or ''Borel cohomology'') is a cohomology theory from algebraic topology which applies to topological spaces with a ''group action''. It can be viewed as a common generalization of group cohomology and an ordi ... References {{algebraic-geometry-stub Algebraic geometry ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sumihiro's Theorem
In algebraic geometry, Sumihiro's theorem, introduced by , states that a normal algebraic variety with an action of a torus can be covered Cover or covers may refer to: Packaging * Another name for a lid * Cover (philately), generic term for envelope or package * Album cover, the front of the packaging * Book cover or magazine cover ** Book design ** Back cover copy, part of co ... by torus-invariant affine open subsets. The "normality" in the hypothesis cannot be relaxed. The hypothesis that the group acting on the variety is a torus can also not be relaxed. Notes References *. External links * Theorems in algebraic geometry {{algebraic-geometry-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Morse Theory
In mathematics, specifically in differential topology, Morse theory enables one to analyze the topology of a manifold by studying differentiable functions on that manifold. According to the basic insights of Marston Morse, a typical differentiable function on a manifold will reflect the topology quite directly. Morse theory allows one to find CW structures and handle decompositions on manifolds and to obtain substantial information about their homology. Before Morse, Arthur Cayley and James Clerk Maxwell had developed some of the ideas of Morse theory in the context of topography. Morse originally applied his theory to geodesics ( critical points of the energy functional on the space of paths). These techniques were used in Raoul Bott's proof of his periodicity theorem. The analogue of Morse theory for complex manifolds is Picard–Lefschetz theory. Basic concepts To illustrate, consider a mountainous landscape surface M (more generally, a manifold). If f is the function M ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebra Automorphism
In mathematics, an algebra homomorphism is a homomorphism between two associative algebras. More precisely, if and are algebras over a field (or commutative ring) , it is a function F\colon A\to B such that for all in and in , * F(kx) = kF(x) * F(x + y) = F(x) + F(y) * F(xy) = F(x) F(y) The first two conditions say that is a ''K''-linear map (or ''K''-module homomorphism if ''K'' is a commutative ring), and the last condition says that is a (non-unital) ring homomorphism. If admits an inverse homomorphism, or equivalently if it is bijective, is said to be an isomorphism between and . Unital algebra homomorphisms If ''A'' and ''B'' are two unital algebras, then an algebra homomorphism F:A\rightarrow B is said to be ''unital'' if it maps the unity of ''A'' to the unity of ''B''. Often the words "algebra homomorphism" are actually used to mean "unital algebra homomorphism", in which case non-unital algebra homomorphisms are excluded. A unital algebra homomorphism is a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group-scheme Action
In algebraic geometry, an action of a group scheme is a generalization of a group action to a group scheme. Precisely, given a group ''S''-scheme ''G'', a left action of ''G'' on an ''S''-scheme ''X'' is an ''S''-morphism :\sigma: G \times_S X \to X such that * (associativity) \sigma \circ (1_G \times \sigma) = \sigma \circ (m \times 1_X), where m: G \times_S G \to G is the group law, * (unitality) \sigma \circ (e \times 1_X) = 1_X, where e: S \to G is the identity section of ''G''. A right action of ''G'' on ''X'' is defined analogously. A scheme equipped with a left or right action of a group scheme ''G'' is called a ''G''-scheme. An equivariant morphism between ''G''-schemes is a morphism of schemes that intertwines the respective ''G''-actions. More generally, one can also consider (at least some special case of) an action of a group functor: viewing ''G'' as a functor, an action is given as a natural transformation satisfying the conditions analogous to the above.In details, gi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
