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Mixed Hodge Structure
In algebraic geometry, a mixed Hodge structure is an algebraic structure containing information about the cohomology of general algebraic varieties. It is a generalization of a Hodge structure, which is used to study smooth projective varieties. In mixed Hodge theory, where the decomposition of a cohomology group H^k(X) may have subspaces of different weights, i.e. as a direct sum of Hodge structures :H^k(X) = \bigoplus_i (H_i, F_i^\bullet) where each of the Hodge structures have weight k_i. One of the early hints that such structures should exist comes from the long exact sequence \dots \to H^(Y) \to H^i_c(U) \to H^i(X) \to \dotsassociated to a pair of smooth projective varieties Y \subset X . This sequence suggests that the cohomology groups H^i_c(U) (for U = X - Y ) should have differing weights coming from both H^(Y) and H^i(X) . Motivation Originally, Hodge structures were introduced as a tool for keeping track of abstract Hodge decompositions on the cohomology groups o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Geometry
Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; the modern approach generalizes this in a few different aspects. The fundamental objects of study in algebraic geometry are algebraic variety, algebraic varieties, which are geometric manifestations of solution set, solutions of systems of polynomial equations. Examples of the most studied classes of algebraic varieties are line (geometry), lines, circles, parabolas, ellipses, hyperbolas, cubic curves like elliptic curves, and quartic curves like lemniscate of Bernoulli, lemniscates and Cassini ovals. These are plane algebraic curves. A point of the plane lies on an algebraic curve if its coordinates satisfy a given polynomial equation. Basic questions involve the study of points of special interest like singular point of a curve, singular p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graded Ring
In mathematics, in particular abstract algebra, a graded ring is a ring such that the underlying additive group is a direct sum of abelian groups R_i such that . The index set is usually the set of nonnegative integers or the set of integers, but can be any monoid. The direct sum decomposition is usually referred to as gradation or grading. A graded module is defined similarly (see below for the precise definition). It generalizes graded vector spaces. A graded module that is also a graded ring is called a graded algebra. A graded ring could also be viewed as a graded -algebra. The associativity is not important (in fact not used at all) in the definition of a graded ring; hence, the notion applies to non-associative algebras as well; e.g., one can consider a graded Lie algebra. First properties Generally, the index set of a graded ring is assumed to be the set of nonnegative integers, unless otherwise explicitly specified. This is the case in this article. A graded ring is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Motive (algebraic Geometry)
In algebraic geometry, motives (or sometimes motifs, following French usage) is a theory proposed by Alexander Grothendieck in the 1960s to unify the vast array of similarly behaved cohomology theories such as singular cohomology, de Rham cohomology, etale cohomology, and crystalline cohomology. Philosophically, a "motif" is the "cohomology essence" of a variety. In the formulation of Grothendieck for smooth projective varieties, a motive is a triple (X, p, m), where X is a smooth projective variety, p: X \vdash X is an idempotent correspondence, and ''m'' an integer; however, such a triple contains almost no information outside the context of Grothendieck's category of pure motives, where a morphism from (X, p, m) to (Y, q, n) is given by a correspondence of degree n-m. A more object-focused approach is taken by Pierre Deligne in ''Le Groupe Fondamental de la Droite Projective Moins Trois Points''. In that article, a motive is a "system of realisations" – that is, a tuple ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gysin Sequence
In the field of mathematics known as algebraic topology, the Gysin sequence is a long exact sequence which relates the cohomology classes of the base space, the fiber and the total space of a Fiber bundle#Sphere bundles, sphere bundle. The Gysin sequence is a useful tool for calculating the cohomology rings given the Euler class of the sphere bundle and vice versa. It was introduced by , and is generalized by the Serre spectral sequence. Definition Consider a fiber-oriented sphere bundle with total space ''E'', base space ''M'', fiber ''S''''k'' and projection map \pi: S^k \hookrightarrow E \stackrel M. Any such bundle defines a degree ''k'' + 1 cohomology class ''e'' called the Euler class of the bundle. De Rham cohomology Discussion of the sequence is clearest with de Rham cohomology. There cohomology classes are represented by differential forms, so that ''e'' can be represented by a (''k'' + 1)-form. The projection map \pi induces a map in cohomology H^ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quartic Surface
In mathematics, especially in algebraic geometry, a quartic surface is a surface defined by an equation of degree 4. More specifically there are two closely related types of quartic surface: affine and projective. An ''affine'' quartic surface is the solution set of an equation of the form :f(x,y,z)=0\ where is a polynomial of degree 4, such as . This is a surface in affine space . On the other hand, a projective quartic surface is a surface in projective space of the same form, but now is a ''homogeneous'' polynomial of 4 variables of degree 4, so for example . If the base field is or the surface is said to be '' real'' or ''complex'' respectively. One must be careful to distinguish between algebraic Riemann surfaces, which are in fact quartic curves over , and quartic surfaces over . For instance, the Klein quartic is a ''real'' surface given as a quartic curve over . If on the other hand the base field is finite, then it is said to be an ''arithmetic quartic surfac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Verdier Duality
In mathematics, Verdier duality is a cohomological duality in algebraic topology that generalizes Poincaré duality for manifolds. Verdier duality was introduced in 1965 by as an analog for locally compact topological spaces of Alexander Grothendieck's theory of Poincaré duality in étale cohomology for schemes in algebraic geometry. It is thus (together with the said étale theory and for example Grothendieck's coherent duality) one instance of Grothendieck's six operations formalism. Verdier duality generalises the classical Poincaré duality of manifolds in two directions: it applies to continuous maps from one space to another (reducing to the classical case for the unique map from a manifold to a one-point space), and it applies to spaces that fail to be manifolds due to the presence of singularities. It is commonly encountered when studying constructible or perverse sheaves. Verdier duality Verdier duality states that (subject to suitable finiteness conditions disc ... [...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] |
Constructible Sheaf
In mathematics, a constructible sheaf is a sheaf of abelian groups over some topological space ''X'', such that ''X'' is the union of a finite number of locally closed subsets on each of which the sheaf is a locally constant sheaf. It has its origins in algebraic geometry, where in étale cohomology constructible sheaves are defined in a similar way . For the derived category of constructible sheaves, see a section in ℓ-adic sheaf. The finiteness theorem in étale cohomology states that the higher direct images of a constructible sheaf are constructible. Definition of étale constructible sheaves on a scheme ''X'' Here we use the definition of constructible étale sheaves from the book by Freitag and Kiehl referenced below. In what follows in this subsection, all sheaves \mathcal on schemes X are étale sheaves unless otherwise noted. A sheaf \mathcal is called constructible if X can be written as a finite union of locally closed subschemes i_Y:Y \to X such that for each subsc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Distinguished Triangle
In mathematics, a triangulated category is a category with the additional structure of a "translation functor" and a class of "exact triangles". Prominent examples are the derived category of an abelian category, as well as the stable homotopy category. The exact triangles generalize the short exact sequences in an abelian category, as well as fiber sequences and cofiber sequences in topology. Much of homological algebra is clarified and extended by the language of triangulated categories, an important example being the theory of sheaf cohomology. In the 1960s, a typical use of triangulated categories was to extend properties of sheaves on a space ''X'' to complexes of sheaves, viewed as objects of the derived category of sheaves on ''X''. More recently, triangulated categories have become objects of interest in their own right. Many equivalences between triangulated categories of different origins have been proved or conjectured. For example, the homological mirror symmetry co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bézout's Theorem
In algebraic geometry, Bézout's theorem is a statement concerning the number of common zeros of polynomials in indeterminates. In its original form the theorem states that ''in general'' the number of common zeros equals the product of the degrees of the polynomials. It is named after Étienne Bézout. In some elementary texts, Bézout's theorem refers only to the case of two variables, and asserts that, if two plane algebraic curves of degrees d_1 and d_2 have no component in common, they have d_1d_2 intersection points, counted with their multiplicity, and including points at infinity and points with complex coordinates. In its modern formulation, the theorem states that, if is the number of common points over an algebraically closed field of projective hypersurfaces defined by homogeneous polynomials in indeterminates, then is either infinite, or equals the product of the degrees of the polynomials. Moreover, the finite case occurs almost always. In the case of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logarithmic Form
In algebraic geometry and the theory of complex manifolds, a logarithmic differential form is a differential form with poles of a certain kind. The concept was introduced by Pierre Deligne. In short, logarithmic differentials have the mildest possible singularities needed in order to give information about an open submanifold (the complement of the divisor of poles). (This idea is made precise by several versions of de Rham's theorem discussed below.) Let ''X'' be a complex manifold, ''D'' ⊂ ''X'' a reduced divisor (a sum of distinct codimension-1 complex subspaces), and ω a holomorphic ''p''-form on ''X''−''D''. If both ω and ''d''ω have a pole of order at most 1 along ''D'', then ω is said to have a logarithmic pole along ''D''. ω is also known as a logarithmic ''p''-form. The ''p''-forms with log poles along ''D'' form a subsheaf of the meromorphic ''p''-forms on ''X'', denoted :\Omega^p_X(\log D). The name comes from the fact that in complex analysis, d(\log z)=dz/z; ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complete Intersection
In mathematics, an algebraic variety ''V'' in projective space is a complete intersection if the ideal of ''V'' is generated by exactly ''codim V'' elements. That is, if ''V'' has dimension ''m'' and lies in projective space ''P''''n'', there should exist ''n'' − ''m'' homogeneous polynomials: :F_i(X_0,\cdots,X_n), 1\leq i\leq n - m, in the homogeneous coordinates ''X''''j'', which generate all other homogeneous polynomials that vanish on ''V''. Geometrically, each ''F''''i'' defines a hypersurface; the intersection of these hypersurfaces should be ''V''. The intersection of hypersurfaces will always have dimension at least ''m'', assuming that the field of scalars is an algebraically closed field such as the complex numbers. The question is essentially, can we get the dimension down to ''m'', with no extra points in the intersection? This condition is fairly hard to check as soon as the codimension . When then ''V'' is automatically a hypersurface and there is nothin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
