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Enoki Surface
In mathematics, an Enoki surface is compact complex surface with positive second Betti number that has a global spherical shell and a non-trivial divisor ''D'' with ''H''0(O(''D'')) ≠ 0 and (''D'', ''D'') = 0. constructed some examples. They are surfaces of class VII In mathematics, surfaces of class VII are non-algebraic complex surfaces studied by that have Kodaira dimension −∞ and first Betti number 1. Minimal surfaces of class VII (those with no rational curves with self-intersection −1) a ..., so are non-Kähler and have Kodaira dimension −∞. References * Complex surfaces {{geometry-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compact Space
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", i.e. that the space not exclude any ''limiting values'' of points. For example, the open interval (0,1) would not be compact because it excludes the limiting values of 0 and 1, whereas the closed interval ,1would be compact. Similarly, the space of rational numbers \mathbb is not compact, because it has infinitely many "punctures" corresponding to the irrational numbers, and the space of real numbers \mathbb is not compact either, because it excludes the two limiting values +\infty and -\infty. However, the ''extended'' real number line ''would'' be compact, since it contains both infinities. There are many ways to make this heuristic notion precise. These ways usually agree in a metric space, but may not be equivalent in other topologic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a + bi, where and are real numbers. Because no real number satisfies the above equation, was called an imaginary number by René Descartes. For the complex number a+bi, is called the , and is called the . The set of complex numbers is denoted by either of the symbols \mathbb C or . Despite the historical nomenclature "imaginary", complex numbers are regarded in the mathematical sciences as just as "real" as the real numbers and are fundamental in many aspects of the scientific description of the natural world. Complex numbers allow solutions to all polynomial equations, even those that have no solutions in real numbers. More precisely, the fundamental theorem of algebra asserts that every non-constant polynomial equation with real or ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Surface (mathematics)
In mathematics, a surface is a mathematical model of the common concept of a surface. It is a generalization of a plane, but, unlike a plane, it may be curved; this is analogous to a curve generalizing a straight line. There are several more precise definitions, depending on the context and the mathematical tools that are used for the study. The simplest mathematical surfaces are planes and spheres in the Euclidean 3-space. The exact definition of a surface may depend on the context. Typically, in algebraic geometry, a surface may cross itself (and may have other singularities), while, in topology and differential geometry, it may not. A surface is a topological space of dimension two; this means that a moving point on a surface may move in two directions (it has two degrees of freedom). In other words, around almost every point, there is a ''coordinate patch'' on which a two-dimensional coordinate system is defined. For example, the surface of the Earth resembles (ideally) a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Betti Number
In algebraic topology, the Betti numbers are used to distinguish topological spaces based on the connectivity of ''n''-dimensional simplicial complexes. For the most reasonable finite-dimensional spaces (such as compact manifolds, finite simplicial complexes or CW complexes), the sequence of Betti numbers is 0 from some point onward (Betti numbers vanish above the dimension of a space), and they are all finite. The ''n''th Betti number represents the rank of the ''n''th homology group, denoted ''H''''n'', which tells us the maximum number of cuts that can be made before separating a surface into two pieces or 0-cycles, 1-cycles, etc. For example, if H_n(X) \cong 0 then b_n(X) = 0, if H_n(X) \cong \mathbb then b_n(X) = 1, if H_n(X) \cong \mathbb \oplus \mathbb then b_n(X) = 2, if H_n(X) \cong \mathbb \oplus \mathbb\oplus \mathbb then b_n(X) = 3, etc. Note that only the ranks of infinite groups are considered, so for example if H_n(X) \cong \mathbb^k \oplus \mathbb/(2) , where \mat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Global Spherical Shell
In mathematics, surfaces of class VII are non-algebraic complex surfaces studied by that have Kodaira dimension −∞ and first Betti number 1. Minimal surfaces of class VII (those with no rational curves with self-intersection −1) are called surfaces of class VII0. Every class VII surface is birational to a unique minimal class VII surface, and can be obtained from this minimal surface by blowing up points a finite number of times. The name "class VII" comes from , which divided minimal surfaces into 7 classes numbered I0 to VII0. However Kodaira's class VII0 did not have the condition that the Kodaira dimension is −∞, but instead had the condition that the geometric genus is 0. As a result, his class VII0 also included some other surfaces, such as secondary Kodaira surfaces, that are no longer considered to be class VII as they do not have Kodaira dimension −∞. The minimal surfaces of class VII are the class numbered "7" on the list of surfaces ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Surfaces Of Class VII
In mathematics, surfaces of class VII are non-algebraic complex surfaces studied by that have Kodaira dimension −∞ and first Betti number 1. Minimal surfaces of class VII (those with no rational curves with self-intersection −1) are called surfaces of class VII0. Every class VII surface is birational to a unique minimal class VII surface, and can be obtained from this minimal surface by blowing up points a finite number of times. The name "class VII" comes from , which divided minimal surfaces into 7 classes numbered I0 to VII0. However Kodaira's class VII0 did not have the condition that the Kodaira dimension is −∞, but instead had the condition that the geometric genus is 0. As a result, his class VII0 also included some other surfaces, such as secondary Kodaira surfaces, that are no longer considered to be class VII as they do not have Kodaira dimension −∞. The minimal surfaces of class VII are the class numbered "7" on the list of surfaces ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kodaira Dimension
In algebraic geometry, the Kodaira dimension ''κ''(''X'') measures the size of the canonical ring, canonical model of a projective variety ''X''. Igor Shafarevich, in a seminar introduced an important numerical invariant of surfaces with the notation ''κ''. Shigeru Iitaka extended it and defined the Kodaira dimension for higher dimensional varieties (under the name of canonical dimension), and later named it after Kunihiko Kodaira. The plurigenera The canonical bundle of a smooth scheme, smooth algebraic variety ''X'' of dimension ''n'' over a field is the line bundle of ''n''-forms, :\,\!K_X = \bigwedge^n\Omega^1_X, which is the ''n''th exterior power of the cotangent bundle of ''X''. For an integer ''d'', the ''d''th tensor power of ''K''''X'' is again a line bundle. For ''d'' ≥ 0, the vector space of global sections ''H''0(''X'',''K''''X''''d'') has the remarkable property that it is a birational invariant of smooth projective varieties ''X''. That is, this vector spa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
