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Dolbeault Operator
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 the ... [...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 a fin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...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 associated with 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; those components form an array, which can be thought of as a high-dimensional matrix. 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, quantum mechanics, fluid mechanics, moment of inertia, ...), electrodynamics ( electromagnetic ten ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sheaf (mathematics)
In mathematics, a sheaf (: sheaves) 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 are well-behaved in that they can be restricted to smaller open sets, and also the data assigned to an open set are equivalent to all collections of compatible data assigned to collections of smaller open sets covering the original open set (intuitively, every datum is the sum of its constituent data). The field of mathematics that studies sheaves is called sheaf theory. Sheaves are understood conceptually as general and abstract objects. Their precise 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 ... [...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 Closed and exact differential forms, exact form is closed, but the converse may fail to hold. Roughly speaking, this failure is related to the possible existence of Hole#In mathematics, "holes" in the manifold, and the de Rham cohomology groups comprise a set of Topological invariant, 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)\ ... [...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 space, 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 mani ... [...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 of vi ... [...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 p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Poincaré Lemma
In mathematics, the Poincaré lemma gives a sufficient condition for a closed differential form to be exact (while an exact form is necessarily closed). Precisely, it states that every closed ''p''-form on an open ball in R''n'' is exact for ''p'' with . The lemma was introduced by Henri Poincaré in 1886. Informal Discussion Especially in calculus, the Poincaré lemma also says that every closed 1-form on a simply connected open subset in \mathbb^n is exact. In simpler terms, it means that if a differential form is closed in a region that can be shrunk to a point, then it can be written as the derivative of another form; i.e. if dα = 0 on a simplely connected region, we can always find α = dβ; therefore we have d(dβ) = 0, expressed simply as d2 = 0. This concept is used in mathematical physics, particularly in the context of electromagnetism and differential geometry, where it relates to the fact that the boundary of a boundary is always empty, i.e. if you have a surface (a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Star Domain
In geometry, a 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, or 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 , y\right~:=~ \left\ ~=~ x + (y - x) , 1 where z , 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 _0, s\right\subseteq S. A set S is and is called a if there exists some point s_0 \in S such that S ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
