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Noether Inequality
In mathematics, the Noether inequality, named after Max Noether, is a property of compact minimal complex surfaces that restricts the topological type of the underlying topological 4-manifold. It holds more generally for minimal projective surfaces of general type over an algebraically closed field. Formulation of the inequality Let ''X'' be a smooth minimal projective surface of general type defined over an algebraically closed field (or a smooth minimal compact complex surface of general type) with canonical divisor ''K'' = −''c''1(''X''), and let ''p''g = ''h''0(''K'') be the dimension of the space of holomorphic two forms, then : p_g \le \frac c_1(X)^2 + 2. For complex surfaces, an alternative formulation expresses this inequality in terms of topological invariants of the underlying real oriented four manifold. Since a surface of general type is a Kähler surface, the dimension of the maximal positive subspace in intersection form on the second cohomology is given by ''b ... [...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] |
Euler Characteristic
In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent. It is commonly denoted by \chi ( Greek lower-case letter chi). The Euler characteristic was originally defined for polyhedra and used to prove various theorems about them, including the classification of the Platonic solids. It was stated for Platonic solids in 1537 in an unpublished manuscript by Francesco Maurolico. Leonhard Euler, for whom the concept is named, introduced it for convex polyhedra more generally but failed to rigorously prove that it is an invariant. In modern mathematics, the Euler characteristic arises from homology and, more abstractly, homological algebra. Polyhedra The Euler characteristic \chi was classically defined for the surfaces of polyhedra, acc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Clifford's Theorem On Special Divisors
In mathematics, Clifford's theorem on special divisors is a result of on algebraic curves, showing the constraints on special linear systems on a curve ''C''. Statement A divisor on a Riemann surface ''C'' is a formal sum \textstyle D = \sum_P m_P P of points ''P'' on ''C'' with integer coefficients. One considers a divisor as a set of constraints on meromorphic functions in the function field of ''C,'' defining L(D) as the vector space of functions having poles only at points of ''D'' with positive coefficient, ''at most as bad'' as the coefficient indicates, and having zeros at points of ''D'' with negative coefficient, with ''at least'' that multiplicity. The dimension of L(D) is finite, and denoted \ell(D). The linear system of divisors attached to ''D'' is the corresponding projective space of dimension \ell(D)-1. The other significant invariant of ''D'' is its degree ''d'', which is the sum of all its coefficients. A divisor is called ''special'' if ''ℓ''(''K'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Special Divisor
Special or specials may refer to: Policing * Specials, Ulster Special Constabulary, the Northern Ireland police force * Specials, Special Constable, an auxiliary, volunteer, or temporary; police worker or police officer Literature * ''Specials'' (novel), a novel by Scott Westerfeld * ''Specials'', the comic book heroes, see ''Rising Stars'' (comic) Film and television * Special (lighting), a stage light that is used for a single, specific purpose * ''Special'' (film), a 2006 scifi dramedy * ''The Specials'' (2000 film), a comedy film about a group of superheroes * ''The Specials'' (2019 film), a film by Olivier Nakache and Éric Toledano * Television special, television programming that temporarily replaces scheduled programming * ''Special'' (TV series), a 2019 Netflix Original TV series * ''Specials'' (TV series), a 1991 TV series about British Special Constables * ''The Specials'' (TV series), an internet documentary series about 5 friends with learning disabilities ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Adjunction Formula
In mathematics, especially in algebraic geometry and the theory of complex manifolds, the adjunction formula relates the canonical bundle of a variety and a hypersurface inside that variety. It is often used to deduce facts about varieties embedded in well-behaved spaces such as projective space or to prove theorems by induction. Adjunction for smooth varieties Formula for a smooth subvariety Let ''X'' be a smooth algebraic variety or smooth complex manifold and ''Y'' be a smooth subvariety of ''X''. Denote the inclusion map by ''i'' and the ideal sheaf of ''Y'' in ''X'' by \mathcal. The conormal exact sequence for ''i'' is :0 \to \mathcal/\mathcal^2 \to i^*\Omega_X \to \Omega_Y \to 0, where Ω denotes a cotangent bundle. The determinant of this exact sequence is a natural isomorphism :\omega_Y = i^*\omega_X \otimes \operatorname(\mathcal/\mathcal^2)^\vee, where \vee denotes the dual of a line bundle. The particular case of a smooth divisor Suppose that ''D'' is a smooth diviso ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Surfaces Of General Type
In algebraic geometry, a surface of general type is an algebraic surface with Kodaira dimension 2. Because of Algebraic geometry and analytic geometry#Chow.27s theorem, Chow's theorem any compact complex manifold of dimension 2 and with Kodaira dimension 2 will actually be an algebraic surface, and in some sense most surfaces are in this class. Classification Gieseker showed that there is a coarse moduli scheme for surfaces of general type; this means that for any fixed values of the Chern numbers c_1^2, c_2, there is a quasi-projective scheme classifying the surfaces of general type with those Chern numbers. It remains a very difficult problem to describe these schemes explicitly, and there are few pairs of Chern numbers for which this has been done (except when the scheme is empty). There are some indications that these schemes are in general too complicated to write down explicitly: the known upper bounds for the number of components are very large, some components can be red ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Irregularity Of A Surface
In mathematics, the irregularity of a complex surface ''X'' is the Hodge number h^= \dim H^1(\mathcal_X), usually denoted by ''q.'' The irregularity of an algebraic surface is sometimes defined to be this Hodge number, and sometimes defined to be the dimension of the Picard variety, which is the same in characteristic 0 but can be smaller in positive characteristic. The name "irregularity" comes from the fact that for the first surfaces investigated in detail, the smooth complex surfaces in P3, the irregularity happens to vanish. The irregularity then appeared as a new "correction" term measuring the difference p_g - p_a of the geometric genus and the arithmetic genus of more complicated surfaces. Surfaces are sometimes called regular or irregular depending on whether or not the irregularity vanishes. For a complex analytic manifold ''X'' of general dimension, the Hodge number h^= \dim H^1(\mathcal_X) is called the irregularity of X, and is denoted by ''q''. Complex surfaces For no ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Noether Formula
Noether is the family name of several mathematicians (particularly, the Noether family), and the name given to some of their mathematical contributions: * Max Noether (1844–1921), father of Emmy and Fritz Noether, and discoverer of: ** Noether inequality ** Max Noether's theorem, several theorems * Emmy Noether (1882–1935), professor at the University of Göttingen and at Bryn Mawr College ** Noether's theorem (or Noether's first theorem) ** Noether's second theorem ** Noether normalization lemma ** Noetherian rings ** Nöther crater, on the far side of the moon, named after Emmy Noether * Fritz Noether Fritz Alexander Ernst Noether (7 October 1884 – 10 September 1941) was a Jewish German mathematician who emigrated from Nazi Germany to the Soviet Union. He was later executed by the NKVD. Biography Fritz Noether's father Max Noethe ... (1884–1941), professor at the University of Tomsk * Gottfried E. Noether (1915–1991), son of Fritz Noether, statistician ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Intersection Form (4-manifold)
In mathematics, the intersection form of an oriented compact 4-manifold is a special symmetric bilinear form on the 2nd (co)homology group of the 4-manifold. It reflects much of the topology of the 4-manifolds, including information on the existence of a smooth structure. Definition using intersection Let ''M'' be a closed 4-manifold (PL or smooth). Take a triangulation ''T'' of ''M''. Denote by T^* the dual cell subdivision. Represent classes a,b\in H_2(M;\Z/2\Z) by 2-cycles ''A'' and ''B'' modulo 2 viewed as unions of 2-simplices of ''T'' and of T^*, respectively. Define the intersection form modulo 2 :\cap_: H_2(M;\Z/2\Z) \times H_2(M;\Z/2\Z) \to \Z/2\Z by the formula :a\cap_ b = , A\cap B, \bmod2. This is well-defined because the intersection of a cycle and a boundary consists of an even number of points (by definition of a cycle and a boundary). If ''M'' is oriented, analogously (i.e. counting intersections with signs) one defines the intersection form on the 2n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hirzebruch Signature Theorem
In differential topology, an area of mathematics, the Hirzebruch signature theorem (sometimes called the Hirzebruch index theorem) is Friedrich Hirzebruch's 1954 result expressing the signature of a smooth closed oriented manifold by a linear combination of Pontryagin numbers called the L-genus. It was used in the proof of the Hirzebruch–Riemann–Roch theorem. Statement of the theorem The L-genus is the genus for the multiplicative sequence of polynomials associated to the characteristic power series : = \sum_ = 1 + - +\cdots . The first two of the resulting L-polynomials are: * L_1 = \tfrac13 p_1 * L_2 = \tfrac1(7p_2 - p_1^2) By taking for the p_i the Pontryagin classes p_i(M) of the tangent bundle of a 4''n'' dimensional smooth closed oriented manifold M one obtains the L-classes of M. Hirzebruch showed that the n-th L-class of M evaluated on the fundamental class of M, /math>, is equal to \sigma(M), the signature of M (i.e. the signature of the intersection form on th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Max Noether
Max Noether (24 September 1844 – 13 December 1921) was a German mathematician who worked on algebraic geometry and the theory of algebraic functions. He has been called "one of the finest mathematicians of the nineteenth century". He was the father of Emmy Noether. Biography Max Noether was born in Mannheim in 1844, to a Jewish family of wealthy wholesale hardware dealers. His grandfather, Elias Samuel, had started the business in Bruchsal in 1797. In 1809 the Grand Duchy of Baden established a "Tolerance Edict", which assigned a hereditary surname to the male head of every Jewish family which did not already possess one. Thus the Samuels became the Noether family, and as part of this Christianization of names, their son Hertz (Max's father) became Hermann. Max was the third of five children Hermann had with his wife Amalia Würzburger. At 14, Max contracted polio and was afflicted by its effects for the rest of his life. Through self-study, he learned advanced mathematics ... [...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] |
