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List Of Things Named After W. V. D. Hodge
These are things named after W. V. D. Hodge, a Scottish mathematician. {{incomplete-list, date=May 2013 * Hodge algebra * Hodge–Arakelov theory * Hodge bundle * Hodge conjecture * Hodge cycle * Hodge–de Rham spectral sequence * Hodge diamond * Hodge duality * Hodge filtration * Hodge index theorem * Hodge group * Hodge star operator * Hodge structure ** Mixed Hodge structure * Hodge–Tate module * Hodge theory * Mixed Hodge module * Hodge–Arakelov theory * p-adic Hodge theory In mathematics, ''p''-adic Hodge theory is a theory that provides a way to classify and study ''p''-adic Galois representations of characteristic 0 local fields with residual characteristic ''p'' (such as Q''p''). The theory has its beginnings i ... Hodge ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hodge Algebra
In mathematics, a Hodge algebra or algebra with straightening law is a commutative algebra that is a free module over some ring ''R'', together with a given basis similar to the basis of standard monomials of the coordinate ring of a Grassmannian In mathematics, the Grassmannian is a space that parameterizes all -dimensional linear subspaces of the -dimensional vector space . For example, the Grassmannian is the space of lines through the origin in , so it is the same as the projective .... Hodge algebras were introduced by , who named them after W. V. D. Hodge. References * Commutative algebra {{abstract-algebra-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mumford–Tate Group
In algebraic geometry, the Mumford–Tate group (or Hodge group) ''MT''(''F'') constructed from a Hodge structure ''F'' is a certain algebraic group ''G''. When ''F'' is given by a rational representation of an algebraic torus, the definition of ''G'' is as the Zariski closure of the image in the representation of the circle group, over the rational numbers. introduced Mumford–Tate groups over the complex numbers under the name of Hodge groups. introduced the ''p''-adic analogue of Mumford's construction for Hodge–Tate modules, using the work of on p-divisible groups, and named them Mumford–Tate groups. Formulation The algebraic torus ''T'' used to describe Hodge structures has a concrete matrix representation, as the 2×2 invertible matrices of the shape that is given by the action of ''a''+''bi'' on the basis of the complex numbers C over R: :\begin a & b \\ -b & a \end. The circle group inside this group of matrices is the unitary group ''U''(1). Hodg ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mixed Hodge Module
In mathematics, mixed Hodge modules are the culmination of Hodge theory, mixed Hodge structures, intersection cohomology, and the decomposition theorem yielding a coherent framework for discussing variations of degenerating mixed Hodge structures through the six functor formalism. Essentially, these objects are a pair of a filtered D-module (M, F^\bullet) together with a perverse sheaf \mathcal such that the functor from the Riemann–Hilbert correspondence sends (M, F^\bullet) to \mathcal. This makes it possible to construct a Hodge structure on intersection cohomology, one of the key problems when the subject was discovered. This was solved by Morihiko Saito who found a way to use the filtration on a coherent D-module as an analogue of the Hodge filtration for a Hodge structure. This made it possible to give a Hodge structure on an intersection cohomology sheaf, the simple objects in the Abelian category of perverse sheaves. Abstract structure Before going into the nitty gritty ... [...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 in nu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hodge–Tate Module
In mathematics, a Hodge–Tate module is an analogue of a Hodge structure over p-adic fields. introduced and named Hodge–Tate structures using the results of on p-divisible groups. Definition Suppose that ''G'' is the absolute Galois group of a ''p''-adic field ''K''. Then ''G'' has a canonical cyclotomic character χ given by its action on the ''p''th power roots of unity. Let ''C'' be the completion of the algebraic closure of ''K''. Then a finite-dimensional vector space over ''C'' with a semi-linear action of the Galois group ''G'' is said to be of Hodge–Tate type if it is generated by the eigenvectors of integral powers of χ. See also *p-adic Hodge theory *Mumford–Tate group In algebraic geometry, the Mumford–Tate group (or Hodge group) ''MT''(''F'') constructed from a Hodge structure ''F'' is a certain algebraic group ''G''. When ''F'' is given by a rational representation of an algebraic torus, the definition of ' ... References * * * Algebraic geometry ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 of a pair of smooth projective varieties Y \subset X . The cohomology groups H^i_c(U) (for U = X - Y ) should have differing weights coming from both H^i(X) and H^(Y) . Motivation Originally, Hodge structures were introduced as a tool for keeping track of abstract Hodge decompositions on the cohomology groups of smooth projective algebraic varieties. These structures gave geometers new tools for s ... [...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] |
Hodge Star Operator
In mathematics, the Hodge star operator or Hodge star is a linear map defined on the exterior algebra of a finite-dimensional oriented vector space endowed with a nondegenerate symmetric bilinear form. Applying the operator to an element of the algebra produces the Hodge dual of the element. This map was introduced by W. V. D. Hodge. For example, in an oriented 3-dimensional Euclidean space, an oriented plane can be represented by the exterior product of two basis vectors, and its Hodge dual is the normal vector given by their cross product; conversely, any vector is dual to the oriented plane perpendicular to it, endowed with a suitable bivector. Generalizing this to an -dimensional vector space, the Hodge star is a one-to-one mapping of -vectors to -vectors; the dimensions of these spaces are the binomial coefficients \tbinom nk = \tbinom. The naturalness of the star operator means it can play a role in differential geometry, when applied to the cotangent bundle of a ps ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hodge Index Theorem
In mathematics, the Hodge index theorem for an algebraic surface ''V'' determines the signature of the intersection pairing on the algebraic curves ''C'' on ''V''. It says, roughly speaking, that the space spanned by such curves (up to linear equivalence) has a one-dimensional subspace on which it is positive definite (not uniquely determined), and decomposes as a direct sum of some such one-dimensional subspace, and a complementary subspace on which it is negative definite. In a more formal statement, specify that ''V'' is a non-singular projective surface, and let ''H'' be the divisor class on ''V'' of a hyperplane section of ''V'' in a given projective embedding. Then the intersection :H \cdot H = d\ where ''d'' is the degree of ''V'' (in that embedding). Let ''D'' be the vector space of rational divisor classes on ''V'', up to algebraic equivalence. The dimension of ''D'' is finite and is usually denoted by ρ(''V''). The Hodge index theorem says that the subspace spanned ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hodge–Arakelov Theory
In mathematics, Hodge–Arakelov theory of elliptic curves is an analogue of classical and p-adic Hodge theory for elliptic curves carried out in the framework of Arakelov theory. It was introduced by . It bears the name of two mathematicians, Suren Arakelov and W. V. D. Hodge. The main comparison in his theory remains unpublished as of 2019. Mochizuki's main comparison theorem in Hodge–Arakelov theory states (roughly) that the space of polynomial functions of degree less than ''d'' on the universal extension of a smooth elliptic curve in characteristic 0 is naturally isomorphic (via restriction) to the ''d''2-dimensional space of functions on the ''d''- torsion points. It is called a 'comparison theorem' as it is an analogue for Arakelov theory of comparison theorems in cohomology relating de Rham cohomology to singular cohomology of complex varieties or étale cohomology of ''p''-adic varieties. In and he pointed out that arithmetic Kodaira–Spencer map and Gauss–Manin c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hodge Filtration
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 = \bigo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hodge Duality
In mathematics, the Hodge star operator or Hodge star is a linear map defined on the exterior algebra of a finite-dimensional oriented vector space endowed with a nondegenerate symmetric bilinear form. Applying the operator to an element of the algebra produces the Hodge dual of the element. This map was introduced by W. V. D. Hodge. For example, in an oriented 3-dimensional Euclidean space, an oriented plane can be represented by the exterior product of two basis vectors, and its Hodge dual is the normal vector given by their cross product; conversely, any vector is dual to the oriented plane perpendicular to it, endowed with a suitable bivector. Generalizing this to an -dimensional vector space, the Hodge star is a one-to-one mapping of -vectors to -vectors; the dimensions of these spaces are the binomial coefficients \tbinom nk = \tbinom. The naturalness of the star operator means it can play a role in differential geometry, when applied to the cotangent bundle of a pseud ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |