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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 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 differential form, 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 (linear algebra), quotient space :H^(M,\Complex)=\frac . Dolbeault cohomology of vector bundles If ''E'' is a holomorphic vector bundle on a c ... [...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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polydisc
In the theory of functions of several complex variables, a branch of mathematics, a polydisc is a Cartesian product of discs. More specifically, if we denote by D(z,r) the open disc of center ''z'' and radius ''r'' in the complex plane, then an open polydisc is a set of the form :D(z_1,r_1) \times \dots \times D(z_n,r_n). It can be equivalently written as :\. One should not confuse the polydisc with the open ball in Cn, which is defined as :\. Here, the norm is the Euclidean distance in Cn. When n > 1, open balls and open polydiscs are ''not'' biholomorphically equivalent, that is, there is no biholomorphic mapping between the two. This was proven by Poincaré in 1907 by showing that their automorphism groups have different dimensions as Lie groups.Poincare, H,Les fonctions analytiques de deux variables et la r?epresentation conforme, Rend. Circ. Mat. Palermo23 (1907), 185-220 When n=2 the term ''bidisc'' is sometimes used. A polydisc is an example of logarithmicall ... [...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] |
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 duality co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fubini–Study Metric
In mathematics, the Fubini–Study metric is a Kähler metric on projective Hilbert space, that is, on a complex projective space CP''n'' endowed with a Hermitian form. This metric was originally described in 1904 and 1905 by Guido Fubini and Eduard Study. A Hermitian form in (the vector space) C''n''+1 defines a unitary subgroup U(''n''+1) in GL(''n''+1,C). A Fubini–Study metric is determined up to homothety (overall scaling) by invariance under such a U(''n''+1) action; thus it is homogeneous. Equipped with a Fubini–Study metric, CP''n'' is a symmetric space. The particular normalization on the metric depends on the application. In Riemannian geometry, one uses a normalization so that the Fubini–Study metric simply relates to the standard metric on the (2''n''+1)-sphere. In algebraic geometry, one uses a normalization making CP''n'' a Hodge manifold. Construction The Fubini–Study metric arises naturally in the quotient space construction of complex projective space. ... [...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 view: ... [...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] |
Complex Projective Space
In mathematics, complex projective space is the projective space with respect to the field of complex numbers. By analogy, whereas the points of a real projective space label the lines through the origin of a real Euclidean space, the points of a complex projective space label the ''complex'' lines through the origin of a complex Euclidean space (see below for an intuitive account). Formally, a complex projective space is the space of complex lines through the origin of an (''n''+1)-dimensional complex vector space. The space is denoted variously as P(C''n''+1), P''n''(C) or CP''n''. When , the complex projective space CP1 is the Riemann sphere, and when , CP2 is the complex projective plane (see there for a more elementary discussion). Complex projective space was first introduced by as an instance of what was then known as the "geometry of position", a notion originally due to Lazare Carnot, a kind of synthetic geometry that included other projective geometries as well. Sub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fine Sheaf
In mathematics, injective sheaves of abelian groups are used to construct the resolutions needed to define sheaf cohomology (and other derived functors, such as sheaf Ext). There is a further group of related concepts applied to sheaves: flabby (''flasque'' in French), fine, soft (''mou'' in French), acyclic. In the history of the subject they were introduced before the 1957 " Tohoku paper" of Alexander Grothendieck, which showed that the abelian category notion of ''injective object'' sufficed to found the theory. The other classes of sheaves are historically older notions. The abstract framework for defining cohomology and derived functors does not need them. However, in most concrete situations, resolutions by acyclic sheaves are often easier to construct. Acyclic sheaves therefore serve for computational purposes, for example the Leray spectral sequence. Injective sheaves An injective sheaf \mathcal is a sheaf that is an injective object of the category of abelian sheaves; ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inventiones Mathematicae
''Inventiones Mathematicae'' is a mathematical journal published monthly by Springer Science+Business Media. It was established in 1966 and is regarded as one of the most prestigious mathematics journals in the world. The current managing editors are Camillo De Lellis (Institute for Advanced Study, Princeton) and Jean-Benoît Bost (University of Paris-Sud Paris-Sud University (French: ''Université Paris-Sud''), also known as University of Paris — XI (or as Université d'Orsay before 1971), was a French research university distributed among several campuses in the southern suburbs of Paris, in ...). Abstracting and indexing The journal is abstracted and indexed in: References External links *{{Official website, https://www.springer.com/journal/222 Mathematics journals Publications established in 1966 English-language journals Springer Science+Business Media academic journals Monthly journals ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logarithmic Form
In contexts including complex manifolds and algebraic geometry, a logarithmic differential form is a meromorphic differential form with poles of a certain kind. The concept was introduced by Deligne. Let ''X'' be a complex manifold, ''D'' ⊂ ''X'' a divisor, and ω a holomorphic ''p''-form on ''X''−''D''. If ω and ''d''ω have a pole of order at most one along ''D'', then ω is said to have a logarithmic pole along ''D''. ω is also known as a logarithmic ''p''-form. The logarithmic ''p''-forms make up a subsheaf of the meromorphic ''p''-forms on ''X'' with a pole along ''D'', denoted :\Omega^p_X(\log D). In the theory of Riemann surfaces, one encounters logarithmic one-forms which have the local expression :\omega = \frac =\left(\frac + \frac\right)dz for some meromorphic function (resp. rational function) f(z) = z^mg(z) , where ''g'' is holomorphic and non-vanishing at 0, and ''m'' is the order of ''f'' at ''0''. That is, for some open covering, there are local represent ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
