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Complex Differential Form
In mathematics, a complex differential form is a differential form on a manifold (usually a complex manifold) which is permitted to have complex coefficients. Complex forms have broad applications in differential geometry. On complex manifolds, they are fundamental and serve as the basis for much of algebraic geometry, Kähler geometry, and Hodge theory. Over non-complex manifolds, they also play a role in the study of almost complex structures, the theory of spinors, and CR structures. Typically, complex forms are considered because of some desirable decomposition that the forms admit. On a complex manifold, for instance, any complex ''k''-form can be decomposed uniquely into a sum of so-called (''p'', ''q'')-forms: roughly, wedges of ''p'' differentials of the holomorphic coordinates with ''q'' differentials of their complex conjugates. The ensemble of (''p'', ''q'')-forms becomes the primitive object of study, and determines a finer geometrical structure on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tensor
In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tensors. There are many types of tensors, including scalars and vectors (which are the simplest tensors), dual vectors, multilinear maps between vector spaces, and even some operations such as the dot product. Tensors are defined independent of any basis, although they are often referred to by their components in a basis related to a particular coordinate system. Tensors have become important in physics because they provide a concise mathematical framework for formulating and solving physics problems in areas such as mechanics ( stress, elasticity, fluid mechanics, moment of inertia, ...), electrodynamics (electromagnetic tensor, Maxwell tensor, permittivity, magnetic susceptibility, ...), general relativity (stress–energy tensor, c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Frölicher Spectral Sequence
In mathematics, the Frölicher spectral sequence (often misspelled as Fröhlicher) is a tool in the theory of complex manifolds, for expressing the potential failure of the results of cohomology theory that are valid in general only for Kähler manifolds. It was introduced by . A spectral sequence is set up, the degeneration of which would give the results of Hodge theory and Dolbeault's theorem In mathematics, in particular in algebraic geometry and differential geometry, Dolbeault cohomology (named after Pierre Dolbeault) is an analog of de Rham cohomology for complex manifolds. Let ''M'' be a complex manifold. Then the Dolbeault cohomo .... See also * Hodge–de Rham spectral sequence References * {{DEFAULTSORT:Frohlicher Spectral Sequence Complex manifolds Spectral sequences ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dolbeault Complex
In mathematics, in particular in algebraic geometry and differential geometry, Dolbeault cohomology (named after Pierre Dolbeault) is an analog of de Rham cohomology for complex manifolds. Let ''M'' be a complex manifold. Then the Dolbeault cohomology groups H^(M, \Complex) depend on a pair of integers ''p'' and ''q'' and are realized as a subquotient of the space of complex differential forms of degree (''p'',''q''). Construction of the cohomology groups Let Ω''p'',''q'' be the vector bundle of complex differential forms of degree (''p'',''q''). In the article on complex forms, the Dolbeault operator is defined as a differential operator on smooth sections :\bar:\Omega^\to\Omega^ Since :\bar^2=0 this operator has some associated cohomology. Specifically, define the cohomology to be the quotient space :H^(M,\Complex)=\frac . Dolbeault cohomology of vector bundles If ''E'' is a holomorphic vector bundle on a complex manifold ''X'', then one can define likewise ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sheaf (mathematics)
In mathematics, a sheaf is a tool for systematically tracking data (such as sets, abelian groups, rings) attached to the open sets of a topological space and defined locally with regard to them. For example, for each open set, the data could be the ring of continuous functions defined on that open set. Such data is well behaved in that it can be restricted to smaller open sets, and also the data assigned to an open set is equivalent to all collections of compatible data assigned to collections of smaller open sets covering the original open set (intuitively, every piece of data is the sum of its parts). The field of mathematics that studies sheaves is called sheaf theory. Sheaves are understood conceptually as general and abstract objects. Their correct definition is rather technical. They are specifically defined as sheaves of sets or as sheaves of rings, for example, depending on the type of data assigned to the open sets. There are also maps (or morphisms) from o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
De Rham Cohomology
In mathematics, de Rham cohomology (named after Georges de Rham) is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly adapted to computation and the concrete representation of cohomology classes. It is a cohomology theory based on the existence of differential forms with prescribed properties. On any smooth manifold, every exact form is closed, but the converse may fail to hold. Roughly speaking, this failure is related to the possible existence of "holes" in the manifold, and the de Rham cohomology groups comprise a set of topological invariants of smooth manifolds that precisely quantify this relationship. Definition The de Rham complex is the cochain complex of differential forms on some smooth manifold , with the exterior derivative as the differential: :0 \to \Omega^0(M)\ \stackrel\ \Omega^1(M)\ \stackrel\ \Omega^2(M)\ \stackrel\ \Omega^3(M) \t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ddbar Lemma
In complex geometry, the \partial \bar \partial lemma (pronounced ddbar lemma) is a mathematical lemma about the de Rham cohomology class of a complex differential form. The \partial \bar \partial-lemma is a result of Hodge theory and the Kähler identities on a compact Kähler manifold. Sometimes it is also known as the dd^c-lemma, due to the use of a related operator d^c = -\frac(\partial - \bar \partial), with the relation between the two operators being i\partial \bar \partial = dd^c and so \alpha = dd^c \beta. Statement The \partial \bar \partial lemma asserts that if (X,\omega) is a compact Kähler manifold and \alpha \in \Omega^(X) is a complex differential form of bidegree (p,q) (with p,q\ge 1) whose class alpha\in H_^(X,\mathbb) is zero in de Rham cohomology, then there exists a form \beta\in \Omega^(X) of bidegree (p-1,q-1) such that \alpha = i\partial \bar \partial \beta, where \partial and \bar \partial are the Dolbeault operators of the complex manifold X. d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kähler Manifold
In mathematics and especially differential geometry, a Kähler manifold is a manifold with three mutually compatible structures: a complex structure, a Riemannian structure, and a symplectic structure. The concept was first studied by Jan Arnoldus Schouten and David van Dantzig in 1930, and then introduced by Erich Kähler in 1933. The terminology has been fixed by André Weil. Kähler geometry refers to the study of Kähler manifolds, their geometry and topology, as well as the study of structures and constructions that can be performed on Kähler manifolds, such as the existence of special connections like Hermitian Yang–Mills connections, or special metrics such as Kähler–Einstein metrics. Every smooth complex projective variety is a Kähler manifold. Hodge theory is a central part of algebraic geometry, proved using Kähler metrics. Definitions Since Kähler manifolds are equipped with several compatible structures, they can be described from different points ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Local Ddbar Lemma
In complex geometry, the \partial \bar \partial lemma (pronounced ddbar lemma) is a mathematical lemma about the de Rham cohomology class of a complex differential form. The \partial \bar \partial-lemma is a result of Hodge theory and the Kähler identities on a compact Kähler manifold. Sometimes it is also known as the dd^c-lemma, due to the use of a related operator d^c = -\frac(\partial - \bar \partial), with the relation between the two operators being i\partial \bar \partial = dd^c and so \alpha = dd^c \beta. Statement The \partial \bar \partial lemma asserts that if (X,\omega) is a compact Kähler manifold and \alpha \in \Omega^(X) is a complex differential form of bidegree (p,q) (with p,q\ge 1) whose class alpha\in H_^(X,\mathbb) is zero in de Rham cohomology, then there exists a form \beta\in \Omega^(X) of bidegree (p-1,q-1) such that \alpha = i\partial \bar \partial \beta, where \partial and \bar \partial are the Dolbeault operators of the complex manifold X. ddbar po ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Star Domain
In geometry, a Set (mathematics), set S in the Euclidean space \R^n is called a star domain (or star-convex set, star-shaped set or radially convex set) if there exists an s_0 \in S such that for all s \in S, the line segment from s_0 to s lies in S. This definition is immediately generalizable to any Real number, real, or Complex number, complex, vector space. Intuitively, if one thinks of S as a region surrounded by a wall, S is a star domain if one can find a vantage point s_0 in S from which any point s in S is within line-of-sight. A similar, but distinct, concept is that of a radial set. Definition Given two points x and y in a vector space X (such as Euclidean space \R^n), the convex hull of \ is called the and it is denoted by \left[x, y\right] ~:=~ \left\ ~=~ x + (y - x) [0, 1], where z [0, 1] := \ for every vector z. A subset S of a vector space X is said to be s_0 \in S if for every s \in S, the closed interval \left[s_0, s\right] \subseteq S. A set S is and i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dolbeault Cohomology
In mathematics, in particular in algebraic geometry and differential geometry, Dolbeault cohomology (named after Pierre Dolbeault) is an analog of de Rham cohomology for complex manifolds. Let ''M'' be a complex manifold. Then the Dolbeault cohomology groups H^(M, \Complex) depend on a pair of integers ''p'' and ''q'' and are realized as a subquotient of the space of complex differential forms of degree (''p'',''q''). Construction of the cohomology groups Let Ω''p'',''q'' be the vector bundle of complex differential forms of degree (''p'',''q''). In the article on complex forms, the Dolbeault operator is defined as a differential operator on smooth sections :\bar:\Omega^\to\Omega^ Since :\bar^2=0 this operator has some associated cohomology. Specifically, define the cohomology to be the quotient space :H^(M,\Complex)=\frac . Dolbeault cohomology of vector bundles If ''E'' is a holomorphic vector bundle on a complex manifold ''X'', then one can define likewi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
