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Stein Manifold
In mathematics, in the theory of several complex variables and complex manifolds, a Stein manifold is a complex submanifold of the vector space of ''n'' complex dimensions. They were introduced by and named after . A Stein space is similar to a Stein manifold but is allowed to have singularities. Stein spaces are the analogues of affine varieties or affine schemes in algebraic geometry. Definition Suppose X is a complex manifold of complex dimension n and let \mathcal O(X) denote the ring of holomorphic functions on X. We call X a Stein manifold if the following conditions hold: * X is holomorphically convex, i.e. for every compact subset K \subset X, the so-called ''holomorphically convex hull'', ::\bar K = \left \, :is also a ''compact'' subset of X. * X is holomorphically separable, i.e. if x \neq y are two points in X, then there exists f \in \mathcal O(X) such that f(x) \neq f(y). Non-compact Riemann surfaces are Stein manifolds Let ''X'' be a connected, non-compact Riema ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Function Of Several Complex Variables
The theory of functions of several complex variables is the branch of mathematics dealing with complex-valued functions. The name of the field dealing with the properties of function of several complex variables is called several complex variables (and analytic space), that has become a common name for that whole field of study and Mathematics Subject Classification has, as a top-level heading. A function f:(z_1,z_2, \ldots, z_n) \rightarrow f(z_1,z_2, \ldots, z_n) is -tuples of complex numbers, classically studied on the complex coordinate space \Complex^n. As in complex analysis of functions of one variable, which is the case , the functions studied are ''holomorphic'' or ''complex analytic'' so that, locally, they are power series in the variables . Equivalently, they are locally uniform limits of polynomials; or locally square-integrable solutions to the -dimensional Cauchy–Riemann equations. For one complex variable, every domainThat is an open connected subset. (D \subs ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cartan's Theorems A And B
In mathematics, Cartan's theorems A and B are two results proved by Henri Cartan around 1951, concerning a coherent sheaf on a Stein manifold . They are significant both as applied to several complex variables, and in the general development of sheaf cohomology. Theorem B is stated in cohomological terms (a formulation that Cartan ( 1953, p. 51) attributes to J.-P. Serre): Analogous properties were established by Serre (1957) for coherent sheaves in algebraic geometry, when is an affine scheme. The analogue of Theorem B in this context is as follows : These theorems have many important applications. For instance, they imply that a holomorphic function on a closed complex submanifold, , of a Stein manifold can be extended to a holomorphic function on all of . At a deeper level, these theorems were used by Jean-Pierre Serre to prove the GAGA theorem. Theorem B is sharp in the sense that if for all coherent sheaves on a complex manifold (resp. quasi-coherent she ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sheaf Cohomology
In mathematics, sheaf cohomology is the application of homological algebra to analyze the global sections of a sheaf on a topological space. Broadly speaking, sheaf cohomology describes the obstructions to solving a geometric problem globally when it can be solved locally. The central work for the study of sheaf cohomology is Grothendieck's 1957 Tôhoku paper. Sheaves, sheaf cohomology, and spectral sequences were introduced by Jean Leray at the prisoner-of-war camp Oflag XVII-A in Austria. From 1940 to 1945, Leray and other prisoners organized a "université en captivité" in the camp. Leray's definitions were simplified and clarified in the 1950s. It became clear that sheaf cohomology was not only a new approach to cohomology in algebraic topology, but also a powerful method in complex analytic geometry and algebraic geometry. These subjects often involve constructing global functions with specified local properties, and sheaf cohomology is ideally suited to such problems. Man ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symplectic Filling
In mathematics, a filling of a manifold ''X'' is a cobordism ''W'' between ''X'' and the empty set. More to the point, the ''n''-dimensional topological manifold ''X'' is the boundary of an (''n'' + 1)-dimensional manifold ''W''. Perhaps the most active area of current research is when ''n'' = 3, where one may consider certain types of fillings. There are many types of fillings, and a few examples of these types (within a probably limited perspective) follow. *An oriented filling of any orientable manifold ''X'' is another manifold ''W'' such that the orientation of ''X'' is given by the boundary orientation of ''W'', which is the one where the first basis vector of the tangent space at each point of the boundary is the one pointing directly out of ''W'', with respect to a chosen Riemannian metric. Mathematicians call this orientation the ''outward normal first'' convention. All the following cobordisms are oriented, with the orientation on ''W'' given by a sy ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Contact Geometry
In mathematics, contact geometry is the study of a geometric structure on smooth manifolds given by a hyperplane distribution (differential geometry), distribution in the tangent bundle satisfying a condition called 'complete non-integrability'. Equivalently, such a distribution may be given (at least locally) as the kernel of a differential one-form, and the non-integrability condition translates into a maximal non-degeneracy condition on the form. These conditions are opposite to two equivalent conditions for 'integrable system, complete integrability' of a hyperplane distribution, i.e. that it be tangent to a codimension one foliation on the manifold, whose equivalence is the content of the Frobenius theorem (differential topology), Frobenius theorem. Contact geometry is in many ways an odd-dimensional counterpart of symplectic geometry, a structure on certain even-dimensional manifolds. Both contact and symplectic geometry are motivated by the mathematical formalism of class ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eugenio Elia Levi
Eugenio Elia Levi (18 October 1883 – 28 October 1917) was an Italian mathematician, known for his fundamental contributions in group theory, in the theory of partial differential operators and in the theory of functions of several complex variables. He was a younger brother of Beppo Levi and was killed in action during First World War. Work Research activity He wrote 33 papers, classified by his colleague and friend Mauro Picone according to the scheme reproduced in this section. Differential geometry Group theory He wrote only three papers in group theory: in the first one, discovered what is now called Levi decomposition, which was conjectured by Wilhelm Killing and proved by Élie Cartan in a special case. Function theory In the theory of functions of several complex variables he introduced the concept of pseudoconvexity during his investigations on the domain of existence of such functions: it turned out to be one of the key concepts of the theory. Cauchy and Gour ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Morse Theory
In mathematics, specifically in differential topology, Morse theory enables one to analyze the topology of a manifold by studying differentiable functions on that manifold. According to the basic insights of Marston Morse, a typical differentiable function on a manifold will reflect the topology quite directly. Morse theory allows one to find CW structures and handle decompositions on manifolds and to obtain substantial information about their homology. Before Morse, Arthur Cayley and James Clerk Maxwell had developed some of the ideas of Morse theory in the context of topography. Morse originally applied his theory to geodesics ( critical points of the energy functional on the space of paths). These techniques were used in Raoul Bott's proof of his periodicity theorem. The analogue of Morse theory for complex manifolds is Picard–Lefschetz theory. Basic concepts To illustrate, consider a mountainous landscape surface M (more generally, a manifold). If f is the function M ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Plurisubharmonic Function
In mathematics, plurisubharmonic functions (sometimes abbreviated as psh, plsh, or plush functions) form an important class of functions used in complex analysis. On a Kähler manifold, plurisubharmonic functions form a subset of the subharmonic functions. However, unlike subharmonic functions (which are defined on a Riemannian manifold) plurisubharmonic functions can be defined in full generality on complex analytic spaces. Formal definition A function :f \colon G \to \cup\, with ''domain'' G \subset ^n is called plurisubharmonic if it is upper semi-continuous, and for every complex line :\\subset ^n with a, b \in ^n the function z \mapsto f(a + bz) is a subharmonic function on the set :\. In ''full generality'', the notion can be defined on an arbitrary complex manifold or even a complex analytic space X as follows. An upper semi-continuous function :f \colon X \to \cup \ is said to be plurisubharmonic if and only if for any holomorphic map \varphi\colon\Delta\to X the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Runge Theorem
In complex analysis, Runge's theorem (also known as Runge's approximation theorem) is named after the German mathematician Carl Runge who first proved it in the year 1885. It states the following: Denoting by C the set of complex numbers, let ''K'' be a compact set, compact subset of C and let ''f'' be a function (mathematics), function which is holomorphic function, holomorphic on an open set containing ''K''. If ''A'' is a set containing Existential quantification, at least one complex number from every bounded set, bounded connected set, connected component of C\''K'' then there exists a sequence (r_n)_ of rational functions which uniform convergence, converges uniformly to ''f'' on ''K'' and such that all the pole (complex analysis), poles of the functions (r_n)_ are in ''A.'' Note that not every complex number in ''A'' needs to be a pole of every rational function of the sequence (r_n)_. We merely know that for all members of (r_n)_ that do have poles, those poles lie in ''A' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
If And Only If
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is biconditional (a statement of material equivalence), and can be likened to the standard material conditional ("only if", equal to "if ... then") combined with its reverse ("if"); hence the name. The result is that the truth of either one of the connected statements requires the truth of the other (i.e. either both statements are true, or both are false), though it is controversial whether the connective thus defined is properly rendered by the English "if and only if"—with its pre-existing meaning. For example, ''P if and only if Q'' means that ''P'' is true whenever ''Q'' is true, and the only case in which ''P'' is true is if ''Q'' is also true, whereas in the case of ''P if Q'', there could be other scenarios where ''P'' is true and ''Q'' is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ambient Space
An ambient space or ambient configuration space is the space surrounding an object. While the ambient space and hodological space are both considered ways of perceiving penetrable space, the former perceives space as ''navigable'', while the latter perceives it as ''navigated''. Mathematics In mathematics, especially in geometry and topology, an ''ambient space'' is the space surrounding a mathematical object along with the object itself. For example, a 1-dimensional line (l) may be studied in isolation —in which case the ambient space of l is l, or it may be studied as an object embedded in 2-dimensional Euclidean space (\mathbb^2)—in which case the ambient space of l is \mathbb^2, or as an object embedded in 2-dimensional hyperbolic space (\mathbb^2)—in which case the ambient space of l is \mathbb^2. To see why this makes a difference, consider the statement "Parallel lines never intersect." This is true if the ambient space is \mathbb^2, but false if the ambient sp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proper Map
In mathematics, a function between topological spaces is called proper if inverse images of compact subsets are compact. In algebraic geometry, the analogous concept is called a proper morphism. Definition There are several competing definitions of a "proper function". Some authors call a function f : X \to Y between two topological spaces if the preimage of every compact set in Y is compact in X. Other authors call a map f if it is continuous and ; that is if it is a continuous closed map and the preimage of every point in Y is compact. The two definitions are equivalent if Y is locally compact and Hausdorff. Let f : X \to Y be a closed map, such that f^(y) is compact (in X) for all y \in Y. Let K be a compact subset of Y. It remains to show that f^(K) is compact. Let \left\ be an open cover of f^(K). Then for all k \in K this is also an open cover of f^(k). Since the latter is assumed to be compact, it has a finite subcover. In other words, for every k \in K, there exis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
