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Projective Varieties
In algebraic geometry, a projective variety is an algebraic variety that is a closed subvariety of a projective space. That is, it is the zero-locus in \mathbb^n of some finite family of homogeneous polynomials that generate a prime ideal, the defining ideal of the variety. A projective variety is a projective curve if its dimension is one; it is a projective surface if its dimension is two; it is a projective hypersurface if its dimension is one less than the dimension of the containing projective space; in this case it is the set of zeros of a single homogeneous polynomial. If ''X'' is a projective variety defined by a homogeneous prime ideal ''I'', then the quotient ring :k _0, \ldots, x_nI is called the homogeneous coordinate ring of ''X''. Basic invariants of ''X'' such as the degree and the dimension can be read off the Hilbert polynomial of this graded ring. Projective varieties arise in many ways. They are complete, which roughly can be expressed by saying that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Addition On Cubic (clean Version)
Addition (usually signified by the plus symbol, +) is one of the four basic operations of arithmetic, the other three being subtraction, multiplication, and division. The addition of two whole numbers results in the total or '' sum'' of those values combined. For example, the adjacent image shows two columns of apples, one with three apples and the other with two apples, totaling to five apples. This observation is expressed as , which is read as "three plus two equals five". Besides counting items, addition can also be defined and executed without referring to concrete objects, using abstractions called numbers instead, such as integers, real numbers, and complex numbers. Addition belongs to arithmetic, a branch of mathematics. In algebra, another area of mathematics, addition can also be performed on abstract objects such as vectors, matrices, subspaces, and subgroups. Addition has several important properties. It is commutative, meaning that the order of the numb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Serre Duality
In algebraic geometry, a branch of mathematics, Serre duality is a duality for the coherent sheaf cohomology of algebraic varieties, proved by Jean-Pierre Serre. The basic version applies to vector bundles on a smooth projective variety, but Alexander Grothendieck found wide generalizations, for example to singular varieties. On an ''n''-dimensional variety, the theorem says that a cohomology group H^i is the dual space of another one, H^. Serre duality is the analog for coherent sheaf cohomology of Poincaré duality in topology, with the canonical line bundle replacing the orientation sheaf. The Serre duality theorem is also true in complex geometry more generally, for compact complex manifolds that are not necessarily projective complex algebraic varieties. In this setting, the Serre duality theorem is an application of Hodge theory for Dolbeault cohomology, and may be seen as a result in the theory of elliptic operators. These two different interpretations of Serre du ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hodge Theory
In mathematics, Hodge theory, named after W. V. D. Hodge, is a method for studying the cohomology groups of a smooth manifold ''M'' using partial differential equations. The key observation is that, given a Riemannian metric on ''M'', every cohomology class has a canonical representative, a differential form that vanishes under the Laplacian operator of the metric. Such forms are called harmonic. The theory was developed by Hodge in the 1930s to study algebraic geometry, and it built on the work of Georges de Rham on de Rham cohomology. It has major applications in two settings—Riemannian manifolds and Kähler manifolds. Hodge's primary motivation, the study of complex projective varieties, is encompassed by the latter case. Hodge theory has become an important tool in algebraic geometry, particularly through its connection to the study of algebraic cycles. While Hodge theory is intrinsically dependent upon the real and complex numbers, it can be applied to questions ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coherent Sheaf
In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a class of sheaves closely linked to the geometric properties of the underlying space. The definition of coherent sheaves is made with reference to a sheaf of rings that codifies this geometric information. Coherent sheaves can be seen as a generalization of vector bundles. Unlike vector bundles, they form an abelian category, and so they are closed under operations such as taking kernels, images, and cokernels. The quasi-coherent sheaves are a generalization of coherent sheaves and include the locally free sheaves of infinite rank. Coherent sheaf cohomology is a powerful technique, in particular for studying the sections of a given coherent sheaf. Definitions A quasi-coherent sheaf on a ringed space (X, \mathcal O_X) is a sheaf \mathcal F of \mathcal O_X- modules that has a local presentation, that is, every point in X has an open neighborhood U in which there ... [...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] |
Algebraic Geometry And Analytic Geometry
In mathematics, algebraic geometry and analytic geometry are two closely related subjects. While algebraic geometry studies algebraic varieties, analytic geometry deals with complex manifolds and the more general analytic spaces defined locally by the vanishing of analytic functions of several complex variables. The deep relation between these subjects has numerous applications in which algebraic techniques are applied to analytic spaces and analytic techniques to algebraic varieties. Main statement Let X be a projective complex algebraic variety. Because X is a complex variety, its set of complex points X(\C) can be given the structure of a compact complex analytic space. This analytic space is denoted X^\mathrm. Similarly, if \mathcal is a sheaf on X, then there is a corresponding sheaf \mathcal^\text on X^\mathrm. This association of an analytic object to an algebraic one is a functor. The prototypical theorem relating X and X^\mathrm says that for any two coherent s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a + bi, where and are real numbers. Because no real number satisfies the above equation, was called an imaginary number by René Descartes. For the complex number is called the , and is called the . The set of complex numbers is denoted by either of the symbols \mathbb C or . Despite the historical nomenclature, "imaginary" complex numbers have a mathematical existence as firm as that of the real numbers, and they are fundamental tools in the scientific description of the natural world. Complex numbers allow solutions to all polynomial equations, even those that have no solutions in real numbers. More precisely, the fundamental theorem of algebra asserts that every non-constant polynomial equation with real or complex coefficie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chow Variety
In mathematics, particularly in the field of algebraic geometry, a Chow variety is an algebraic variety whose points correspond to effective algebraic cycles of fixed dimension and degree on a given projective space. More precisely, the Chow variety \operatorname(k,d,n) is the fine moduli variety parametrizing all effective algebraic cycles of dimension k-1 and degree d in \mathbb^. The Chow variety \operatorname(k,d,n) may be constructed via a Chow embedding into a sufficiently large projective space. This is a direct generalization of the construction of a Grassmannian variety via the Plücker embedding, as Grassmannians are the d=1 case of Chow varieties. Chow varieties are distinct from Chow groups, which are the abelian group of all algebraic cycles on a variety (not necessarily projective space) up to rational equivalence. Both are named for Wei-Liang Chow (周煒良), a pioneer in the study of algebraic cycles. Background on algebraic cycles If X is a closed subvarie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Teichmüller Space
In mathematics, the Teichmüller space T(S) of a (real) topological (or differential) surface S is a space that parametrizes complex structures on S up to the action of homeomorphisms that are isotopic to the identity homeomorphism. Teichmüller spaces are named after Oswald Teichmüller. Each point in a Teichmüller space T(S) may be regarded as an isomorphism class of "marked" Riemann surfaces, where a "marking" is an isotopy class of homeomorphisms from S to itself. It can be viewed as a moduli space for marked hyperbolic structure on the surface, and this endows it with a natural topology for which it is homeomorphic to a ball of dimension 6g-6 for a surface of genus g \ge 2. In this way Teichmüller space can be viewed as the universal covering orbifold of the Riemann moduli space. The Teichmüller space has a canonical complex manifold structure and a wealth of natural metrics. The study of geometric features of these various structures is an active body of researc ... [...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] |
Grassmannian
In mathematics, the Grassmannian \mathbf_k(V) (named in honour of Hermann Grassmann) is a differentiable manifold that parameterizes the set of all k-dimension (vector space), dimensional linear subspaces of an n-dimensional vector space V over a field (mathematics), field K that has a differentiable structure. For example, the Grassmannian \mathbf_1(V) is the space of lines through the origin in V, so it is the same as the projective space \mathbf(V) of one dimension lower than V. When V is a real number, real or complex number, complex vector space, Grassmannians are compact space, compact smooth manifolds, of dimension k(n-k). In general they have the structure of a nonsingular projective algebraic variety. The earliest work on a non-trivial Grassmannian is due to Julius Plücker, who studied the set of projective lines in real projective 3-space, which is equivalent to \mathbf_2(\mathbf^4), parameterizing them by what are now called Plücker coordinates. (See below.) Herma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hilbert Scheme
In algebraic geometry, a branch of mathematics, a Hilbert scheme is a scheme that is the parameter space for the closed subschemes of some projective space (or a more general projective scheme), refining the Chow variety. The Hilbert scheme is a disjoint union of projective subschemes corresponding to Hilbert polynomials. The basic theory of Hilbert schemes was developed by . Hironaka's example shows that non-projective varieties need not have Hilbert schemes. Hilbert scheme of projective space The Hilbert scheme \mathbf(n) of \mathbb^n classifies closed subschemes of projective space in the following sense: For any locally Noetherian scheme , the set of -valued points :\operatorname(S, \mathbf(n)) of the Hilbert scheme is naturally isomorphic to the set of closed subschemes of \mathbb^n \times S that are flat over . The closed subschemes of \mathbb^n \times S that are flat over can informally be thought of as the families of subschemes of projective space parameterized b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
