|
Inoue Surface
In complex geometry, an Inoue surface is any of several complex surfaces of Kodaira class VII. They are named after Masahisa Inoue, who gave the first non-trivial examples of Kodaira class VII surfaces in 1974. The Inoue surfaces are not Kähler manifolds. Inoue surfaces with ''b''2 = 0 Inoue introduced three families of surfaces, ''S''0, ''S''+ and ''S''−, which are compact quotients of \Complex \times \mathbb (a product of a complex plane by a half-plane). These Inoue surfaces are solvmanifolds. They are obtained as quotients of \Complex \times \mathbb by a solvable discrete group which acts holomorphically on \Complex \times \mathbb. The solvmanifold surfaces constructed by Inoue all have second Betti number b_2=0. These surfaces are of Kodaira class VII, which means that they have b_1=1 and Kodaira dimension -\infty. It was proven by Bogomolov, Li– Yau and Teleman that any surface of class VII with b_2=0 is a Hopf surface or an Inoue-type solvmanifold. These surfa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Geometry
In mathematics, complex geometry is the study of geometric structures and constructions arising out of, or described by, the complex numbers. In particular, complex geometry is concerned with the study of spaces such as complex manifolds and complex algebraic varieties, functions of several complex variables, and holomorphic constructions such as holomorphic vector bundles and coherent sheaves. Application of transcendental methods to algebraic geometry falls in this category, together with more geometric aspects of complex analysis. Complex geometry sits at the intersection of algebraic geometry, differential geometry, and complex analysis, and uses tools from all three areas. Because of the blend of techniques and ideas from various areas, problems in complex geometry are often more tractable or concrete than in general. For example, the classification of complex manifolds and complex algebraic varieties through the minimal model program and the construction of moduli spaces ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hopf Surface
In complex geometry, a Hopf surface is a compact complex surface obtained as a quotient of the complex vector space (with zero deleted) \Complex^2\setminus \ by a free action of a discrete group. If this group is the integers the Hopf surface is called primary, otherwise it is called secondary. (Some authors use the term "Hopf surface" to mean "primary Hopf surface".) The first example was found by , with the discrete group isomorphic to the integers, with a generator acting on \Complex^2 by multiplication by 2; this was the first example of a compact complex surface with no Kähler metric. Higher-dimensional analogues of Hopf surfaces are called Hopf manifolds. Invariants Hopf surfaces are surfaces of class VII and in particular all have Kodaira dimension -\infty, and all their plurigenera vanish. The geometric genus is 0. The fundamental group has a normal central infinite cyclic subgroup of finite index. The Hodge diamond is In particular the first Betti number is 1 and the s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Iku Nakamura , Kyrgyzstan (IATA: IKU)
{{disambiguation, surname ...
Iku or IKU may refer to: Languages * Iku language (ISO 639: ikv), a Plateau language of Nigeria * Arhuaco language (ISO 639: arh), also known as Ikʉ, a Chibchan language of Colombia * Inuktitut (ISO 639: iku), an Inuit language of Canada Other uses * '' I.K.U.'', a 2001 Japanese erotic cyberpunk film * Iku (singer), a Japanese singer * Istanbul Kültür University (İKÜ), in Istanbul, Turkey * Irène K:son Ullberg (born 1930), Swedish painter * Issyk-Kul International Airport Issyk-Kul International Airport (Kyrgyz: Ысык-Көл эл аралык аэропорту, ''Isıq-Köl el aralıq aeroportu'', ىسىق-كۅل ەل ارالىق اەروپورتۇ; Russian: Международный аэропорт «Исс ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lie Group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additional properties it must have to be thought of as a "transformation" in the abstract sense, for instance multiplication and the taking of inverses (division), or equivalently, the concept of addition and the taking of inverses (subtraction). Combining these two ideas, one obtains a continuous group where multiplying points and their inverses are continuous. If the multiplication and taking of inverses are smooth (differentiable) as well, one obtains a Lie group. Lie groups provide a natural model for the concept of continuous symmetry, a celebrated example of which is the rotational symmetry in three dimensions (given by the special orthogonal group \text(3)). Lie groups are widely used in many parts of modern mathematics and physics. Lie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Solvable Group
In mathematics, more specifically in the field of group theory, a solvable group or soluble group is a group that can be constructed from abelian groups using extensions. Equivalently, a solvable group is a group whose derived series terminates in the trivial subgroup. Motivation Historically, the word "solvable" arose from Galois theory and the proof of the general unsolvability of quintic equation. Specifically, a polynomial equation is solvable in radicals if and only if the corresponding Galois group is solvable (note this theorem holds only in characteristic 0). This means associated to a polynomial f \in F /math> there is a tower of field extensionsF = F_0 \subseteq F_1 \subseteq F_2 \subseteq \cdots \subseteq F_m=Ksuch that # F_i = F_ alpha_i/math> where \alpha_i^ \in F_, so \alpha_i is a solution to the equation x^ - a where a \in F_ # F_m contains a splitting field for f(x) Example For example, the smallest Galois field extension of \mathbb containing the elemen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kodaira Surface
In mathematics, a Kodaira surface is a compact space, compact algebraic surface, complex surface of Kodaira dimension 0 and odd first Betti number. The concept is named after Kunihiko Kodaira. These are never algebraic, though they have non-constant meromorphic functions. They are usually divided into two subtypes: primary Kodaira surfaces with trivial canonical bundle, and secondary Kodaira surfaces which are quotients of these by finite groups of orders 2, 3, 4, or 6, and which have non-trivial canonical bundles. The secondary Kodaira surfaces have the same relation to primary ones that Enriques surfaces have to K3 surfaces, or bielliptic surfaces have to abelian surfaces. Invariants: If the surface is the quotient of a primary Kodaira surface by a group of order ''k'' = 1,2,3,4,6, then the plurigenera ''P''''n'' are 1 if ''n'' is divisible by ''k'' and 0 otherwise. Hodge diamond: Examples: Take a non-trivial line bundle over an elliptic curve, remove the zero sect ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperelliptic Surface
In mathematics, a hyperelliptic surface, or bi-elliptic surface, is a surface whose Albanese morphism is an elliptic fibration. Any such surface can be written as the quotient of a product of two elliptic curves by a finite abelian group. Hyperelliptic surfaces form one of the classes of surfaces of Kodaira dimension 0 in the Enriques–Kodaira classification. Invariants The Kodaira dimension is 0. Hodge diamond: Classification Any hyperelliptic surface is a quotient (''E''×''F'')/''G'', where ''E'' = C/Λ and ''F'' are elliptic curves, and ''G'' is a subgroup of ''F'' ( acting on ''F'' by translations). There are seven families of hyperelliptic surfaces as in the following table. Here ω is a primitive cube root of 1 and i is a primitive 4th root of 1. Quasi hyperelliptic surfaces A quasi-hyperelliptic surface is a surface whose canonical divisor is numerically equivalent to zero, the Albanese mapping maps to an elliptic curve, and all its fibers are rational with a cus ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Torus
In mathematics, a complex torus is a particular kind of complex manifold ''M'' whose underlying smooth manifold is a torus in the usual sense (i.e. the cartesian product of some number ''N'' circles). Here ''N'' must be the even number 2''n'', where ''n'' is the complex dimension of ''M''. All such complex structures can be obtained as follows: take a lattice Λ in a vector space V isomorphic to C''n'' considered as real vector space; then the quotient group V/\Lambda is a compact complex manifold. All complex tori, up to isomorphism, are obtained in this way. For ''n'' = 1 this is the classical period lattice construction of elliptic curves. For ''n'' > 1 Bernhard Riemann found necessary and sufficient conditions for a complex torus to be an algebraic variety; those that are varieties can be embedded into complex projective space, and are the abelian varieties. The actual projective embeddings are complicated (see equations defining abelian varieties) when ''n'' > 1, and are real ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Shing-Tung Yau
Shing-Tung Yau (; ; born April 4, 1949) is a Chinese-American mathematician and the William Caspar Graustein Professor of Mathematics at Harvard University. In April 2022, Yau announced retirement from Harvard to become Chair Professor of mathematics at Tsinghua University. Yau was born in Shantou, China, moved to Hong Kong at a young age, and to the United States in 1969. He was awarded the Fields Medal in 1982, in recognition of his contributions to partial differential equations, the Calabi conjecture, the positive energy theorem, and the Monge–Ampère equation. Yau is considered one of the major contributors to the development of modern differential geometry and geometric analysis. The impact of Yau's work can be seen in the mathematical and physical fields of differential geometry, partial differential equations, convex geometry, algebraic geometry, enumerative geometry, mirror symmetry, general relativity, and string theory, while his work has also touched upon applied ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Surface
Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each other * Complex (psychology), a core pattern of emotions etc. in the personal unconscious organized around a common theme such as power or status Complex may also refer to: Arts, entertainment and media * Complex (English band), formed in 1968, and their 1971 album ''Complex'' * Complex (band), a Japanese rock band * Complex (album), ''Complex'' (album), by Montaigne, 2019, and its title track * Complex (EP), ''Complex'' (EP), by Rifle Sport, 1985 * Complex (song), "Complex" (song), by Gary Numan, 1979 * Complex Networks, publisher of magazine ''Complex'', now online Biology * Protein–ligand complex, a complex of a protein bound with a ligand * Exosome complex, a multi-protein intracellular complex * Protein complex, a group of two or ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fedor Bogomolov
Fedor Alekseyevich Bogomolov (born 26 September 1946) (Фёдор Алексеевич Богомолов) is a Russian and American mathematician, known for his research in algebraic geometry and number theory. Bogomolov worked at the Steklov Institute in Moscow before he became a professor at the Courant Institute in New York. He is most famous for his pioneering work on hyperkähler manifolds. Born in Moscow, Bogomolov graduated from Moscow State University, Faculty of Mechanics and Mathematics, and earned his doctorate (''"candidate degree"'') in 1973, at the Steklov Institute. His doctoral advisor was Sergei Novikov. Geometry of Kähler manifolds Bogomolov's Ph.D. thesis was entitled ''Compact Kähler varieties''. In his early papers Bogomolov studied the manifolds which were later called Calabi–Yau and hyperkähler. He proved a decomposition theorem, used for the classification of manifolds with trivial canonical class. It has been re-proven using the Calabi–Y ... [...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] |
