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Kawamata–Viehweg Vanishing Theorem
In algebraic geometry, the Kawamata–Viehweg vanishing theorem is an extension of the Kodaira vanishing theorem, on the vanishing of coherent cohomology groups, to logarithmic pairs, proved independently by Viehweg and Kawamata in 1982. The theorem states that if ''L'' is a big nef line bundle (for example, an ample line bundle) on a complex projective manifold with canonical line bundle In mathematics, the canonical bundle of a non-singular algebraic variety V of dimension n over a field is the line bundle \,\!\Omega^n = \omega, which is the ''n''th exterior power of the cotangent bundle Ω on ''V''. Over the complex numbers, it ... ''K'', then the coherent cohomology groups ''H''''i''(''L''⊗''K'') vanish for all positive ''i''. References * * Theorems in algebraic geometry {{abstract-algebra-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical problems about these sets of zeros. The fundamental objects of study in algebraic geometry are algebraic varieties, which are geometric manifestations of solutions of systems of polynomial equations. Examples of the most studied classes of algebraic varieties are: plane algebraic curves, which include lines, circles, parabolas, ellipses, hyperbolas, cubic curves like elliptic curves, and quartic curves like lemniscates and Cassini ovals. A point of the plane belongs to an algebraic curve if its coordinates satisfy a given polynomial equation. Basic questions involve the study of the points of special interest like the singular points, the inflection points and the points at infinity. More advanced questions involve the topology of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kodaira Vanishing Theorem
In mathematics, the Kodaira vanishing theorem is a basic result of complex manifold theory and complex algebraic geometry, describing general conditions under which sheaf cohomology groups with indices ''q'' > 0 are automatically zero. The implications for the group with index ''q'' = 0 is usually that its dimension — the number of independent global sections — coincides with a holomorphic Euler characteristic that can be computed using the Hirzebruch–Riemann–Roch theorem. The complex analytic case The statement of Kunihiko Kodaira's result is that if ''M'' is a compact Kähler manifold of complex dimension ''n'', ''L'' any holomorphic line bundle on ''M'' that is positive, and ''KM'' is the canonical line bundle, then ::: H^q(M, K_M\otimes L) = 0 for ''q'' > 0. Here K_M\otimes L stands for the tensor product of line bundles. By means of Serre duality, one also obtains the vanishing of H^q(M, L^) for ''q'' ''n''. The algebraic case The Kodaira vanishin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cohomology Group
In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewed as a method of assigning richer algebraic invariants to a space than homology. Some versions of cohomology arise by dualizing the construction of homology. In other words, cochains are functions on the group of chains in homology theory. From its beginning in topology, this idea became a dominant method in the mathematics of the second half of the twentieth century. From the initial idea of homology as a method of constructing algebraic invariants of topological spaces, the range of applications of homology and cohomology theories has spread throughout geometry and algebra. The terminology tends to hide the fact that cohomology, a contravariant theory, is more natural than homology in many applications. At a basic level, this has to do ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logarithmic Pair
In algebraic geometry, a logarithmic pair consists of a variety Variety may refer to: Arts and entertainment Entertainment formats * Variety (radio) * Variety show, in theater and television Films * ''Variety'' (1925 film), a German silent film directed by Ewald Andre Dupont * ''Variety'' (1935 film), ..., together with a divisor along which one allows mild logarithmic singularities. They were studied by . Definition A boundary Q-divisor on a variety is a Q-divisor ''D'' of the form Σ''d''''i''''D''''i'' where the ''D''''i'' are the distinct irreducible components of ''D'' and all coefficients are rational numbers with 0≤''d''''i''≤1. A logarithmic pair, or log pair for short, is a pair (''X'',''D'') consisting of a normal variety ''X'' and a boundary Q-divisor ''D''. The log canonical divisor of a log pair (''X'',''D'') is ''K''+''D'' where ''K'' is the canonical divisor of ''X''. A logarithmic 1-form on a log pair (''X'',''D'') is allowed to have logarithmic sin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eckart Viehweg
Eckart Viehweg (born 30 December 1948 in Zwickau, died 29 January 2010) was a German mathematician. He was a professor of algebraic geometry at the University of Duisburg-Essen. In 2003 he won the Gottfried Wilhelm Leibniz Prize with his wife, Hélène Esnault. See also *Kawamata–Viehweg vanishing theorem In algebraic geometry, the Kawamata–Viehweg vanishing theorem is an extension of the Kodaira vanishing theorem, on the vanishing of coherent cohomology groups, to logarithmic pairs, proved independently by Viehweg and Kawamata in 1982. The th ... References External links HomepageBook: ''Hélène Esnault, Eckart Viehweg'': "Lectures on Vanishing Theorems" (PDF, 1.3 MB)Book: ''Eckart Viehweg'': "Quasi-projective Moduli for Polarized Manifolds" (PDF, 1.5 MB) {{DEFAULTSORT:Viehweg, Eckart 1948 births 2010 deaths People from Zwickau Gottfried Wilhelm Leibniz Prize winners 20th-century German mathematicians 21st-century German mathematicians Academic staff of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal Für Die Reine Und Angewandte Mathematik
''Crelle's Journal'', or just ''Crelle'', is the common name for a mathematics journal, the ''Journal für die reine und angewandte Mathematik'' (in English: ''Journal for Pure and Applied Mathematics''). History The journal was founded by August Leopold Crelle (Berlin) in 1826 and edited by him until his death in 1855. It was one of the first major mathematical journals that was not a proceedings of an academy. It has published many notable papers, including works of Niels Henrik Abel, Georg Cantor, Gotthold Eisenstein, Carl Friedrich Gauss and Otto Hesse. It was edited by Carl Wilhelm Borchardt from 1856 to 1880, during which time it was known as ''Borchardt's Journal''. The current editor-in-chief is Rainer Weissauer (Ruprecht-Karls-Universität Heidelberg) Past editors * 1826–1856 August Leopold Crelle * 1856–1880 Carl Wilhelm Borchardt * 1881–1888 Leopold Kronecker, Karl Weierstrass * 1889–1892 Leopold Kronecker * 1892–1902 Lazarus Fuchs * 1903–1928 Kurt Hens ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Yujiro Kawamata
Yujiro Kawamata (born 1952) is a Japanese mathematician working in algebraic geometry. Career Kawamata completed the master's course at the University of Tokyo in 1977. He was an Assistant at the University of Mannheim from 1977 to 1979 and a Miller Fellow at the University of California, Berkeley from 1981 to 1983. Kawamata is now a professor at the University of Tokyo. He won the Mathematical Society of Japan Autumn award (1988) and the Japan Academy of Sciences award (1990) for his work in algebraic geometry. Research Kawamata was involved in the development of the minimal model program in the 1980s. The program aims to show that every algebraic variety is birational to one of an especially simple type: either a minimal model or a Fano fiber space. The Kawamata-Viehweg vanishing theorem, strengthening the Kodaira vanishing theorem, is a method. Building on that, Kawamata proved the basepoint-free theorem. The cone theorem and contraction theorem, central results in the theor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematische Annalen
''Mathematische Annalen'' (abbreviated as ''Math. Ann.'' or, formerly, ''Math. Annal.'') is a German mathematical research journal founded in 1868 by Alfred Clebsch and Carl Neumann. Subsequent managing editors were Felix Klein, David Hilbert, Otto Blumenthal, Erich Hecke, Heinrich Behnke, Hans Grauert, Heinz Bauer, Herbert Amann, Jean-Pierre Bourguignon, Wolfgang Lück, and Nigel Hitchin. Currently, the managing editor of Mathematische Annalen is Thomas Schick. Volumes 1–80 (1869–1919) were published by Teubner. Since 1920 (vol. 81), the journal has been published by Springer. In the late 1920s, under the editorship of Hilbert, the journal became embroiled in controversy over the participation of L. E. J. Brouwer on its editorial board, a spillover from the foundational Brouwer–Hilbert controversy. Between 1945 and 1947 the journal briefly ceased publication. References External links''Mathematische Annalen''homepage at Springer''Mathematische Annalen''archive (1869†... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Iitaka Dimension
In algebraic geometry, the Iitaka dimension of a line bundle ''L'' on an algebraic variety ''X'' is the dimension of the image of the rational map to projective space determined by ''L''. This is 1 less than the dimension of the section ring of ''L'' :R(X, L) = \bigoplus_^\infty H^0(X, L^). The Iitaka dimension of ''L'' is always less than or equal to the dimension of ''X''. If ''L'' is not effective, then its Iitaka dimension is usually defined to be -\infty or simply said to be negative (some early references define it to be −1). The Iitaka dimension of ''L'' is sometimes called L-dimension, while the dimension of a divisor D is called D-dimension. The Iitaka dimension was introduced by . Big line bundles A line bundle is big if it is of maximal Iitaka dimension, that is, if its Iitaka dimension is equal to the dimension of the underlying variety. Bigness is a birational invariant: If is a birational morphism of varieties, and if ''L'' is a big line bundle on ''X'', th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nef Line Bundle
In algebraic geometry, a line bundle on a projective variety is nef if it has nonnegative degree on every curve in the variety. The classes of nef line bundles are described by a convex cone, and the possible contractions of the variety correspond to certain faces of the nef cone. In view of the correspondence between line bundles and divisors (built from codimension-1 subvarieties), there is an equivalent notion of a nef divisor. Definition More generally, a line bundle ''L'' on a proper scheme ''X'' over a field ''k'' is said to be nef if it has nonnegative degree on every (closed irreducible) curve in ''X''. (The degree of a line bundle ''L'' on a proper curve ''C'' over ''k'' is the degree of the divisor (''s'') of any nonzero rational section ''s'' of ''L''.) A line bundle may also be called an invertible sheaf. The term "nef" was introduced by Miles Reid as a replacement for the older terms "arithmetically effective" and "numerically effective", as well as for the phrase "num ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Line Bundle
In mathematics, a line bundle expresses the concept of a line that varies from point to point of a space. For example, a curve in the plane having a tangent line at each point determines a varying line: the ''tangent bundle'' is a way of organising these. More formally, in algebraic topology and differential topology, a line bundle is defined as a ''vector bundle'' of rank 1. Line bundles are specified by choosing a one-dimensional vector space for each point of the space in a continuous manner. In topological applications, this vector space is usually real or complex. The two cases display fundamentally different behavior because of the different topological properties of real and complex vector spaces: If the origin is removed from the real line, then the result is the set of 1×1 invertible real matrices, which is homotopy-equivalent to a discrete two-point space by contracting the positive and negative reals each to a point; whereas removing the origin from the complex plane ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ample Line Bundle
In mathematics, a distinctive feature of algebraic geometry is that some line bundles on a projective variety can be considered "positive", while others are "negative" (or a mixture of the two). The most important notion of positivity is that of an ample line bundle, although there are several related classes of line bundles. Roughly speaking, positivity properties of a line bundle are related to having many global sections. Understanding the ample line bundles on a given variety ''X'' amounts to understanding the different ways of mapping ''X'' into projective space. In view of the correspondence between line bundles and divisors (built from codimension-1 subvarieties), there is an equivalent notion of an ample divisor. In more detail, a line bundle is called basepoint-free if it has enough sections to give a morphism to projective space. A line bundle is semi-ample if some positive power of it is basepoint-free; semi-ampleness is a kind of "nonnegativity". More strongly, a line bun ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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