Vanishing Theorem

	Vanishing Theorem In algebraic geometry, a vanishing theorem gives conditions for coherent cohomology groups to vanish. * Andreotti–Grauert vanishing theorem * Bogomolov–Sommese vanishing theorem * Grauert–Riemenschneider vanishing theorem * Kawamata–Viehweg vanishing theorem * Kodaira vanishing theorem * Le Potier's vanishing theorem * Mumford vanishing theorem * Nakano vanishing theorem * Ramanujam vanishing theorem * Serre's vanishing theorem In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaf cohomology is a technique for producing functions with specified properties. Many geometric questions can be formulated as questions about the exi ... {{Mathematical disambiguation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	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 topo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Coherent Sheaf In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a class of sheaves closely linked to the geometric properties of the underlying space. The definition of coherent sheaves is made with reference to a sheaf of rings that codifies this geometric information. Coherent sheaves can be seen as a generalization of vector bundles. Unlike vector bundles, they form an abelian category, and so they are closed under operations such as taking kernels, images, and cokernels. The quasi-coherent sheaves are a generalization of coherent sheaves and include the locally free sheaves of infinite rank. Coherent sheaf cohomology is a powerful technique, in particular for studying the sections of a given coherent sheaf. Definitions A quasi-coherent sheaf on a ringed space (X, \mathcal O_X) is a sheaf \mathcal F of \mathcal O_X- modules which has a local presentation, that is, every point in X has an open neighborhood U in which there is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Bogomolov–Sommese Vanishing Theorem In algebraic geometry, the Bogomolov–Sommese vanishing theorem is a result related to the Kodaira–Itaka dimension. It is named after Fedor Bogomolov and Andrew Sommese. Its statement has differing versions: This result is equivalent to the statement that: :H^\left(X,A^ \otimes \Omega ^_ (\log D) \right) = 0 for every complex projective snc pair (X, D) and every invertible sheaf#The Picard group, invertible sheaf A \in \mathrm(X) with \kappa(A) > p. Therefore, this theorem is called the vanishing theorem. See also Bogomolov–Miyaoka–Yau inequality Vanishing theorem (other) Notes References * * * * * * Further reading * * * * * * * * * * {{DEFAULTSORT:Bogomolov-Sommese vanishing theorem Theorems in algebraic geometry Theorems in complex geometry ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Grauert–Riemenschneider Vanishing Theorem In mathematics, the Grauert–Riemenschneider vanishing theorem is an extension of the Kodaira vanishing theorem on the vanishing of higher cohomology groups of coherent sheaves on a compact complex manifold In differential geometry and complex geometry, a complex manifold is a manifold with an atlas of charts to the open unit disc in \mathbb^n, such that the transition maps are holomorphic. The term complex manifold is variously used to mean a com ..., due to . Grauert–Riemenschneider conjecture The Grauert–Riemenschneider conjecture is a conjecture related to the Grauert–Riemenschneider vanishing theorem: This conjecture was proved by using the Riemann–Roch type theorem ( Hirzebruch–Riemann–Roch theorem) and by using Morse theory. Note References * * * * * Theorems in algebraic geometry {{algebraic-geometry-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	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 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 ...) on a complex projective manifold with canonical line bundle ''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]
	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 va ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Le Potier's Vanishing Theorem In algebraic geometry, Le Potier's vanishing theorem is an extension of the Kodaira vanishing theorem, on vector bundle In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to ev ...s. The theorem states the following In case of r = 1, and let E is an ample (or positive) line bundle on X, this theorem is equivalent to the Nakano vanishing theorem. Also, found another proof. generalizes Le Potier's vanishing theorem to k-ample and the statement as follows: gave a counterexample, which is as follows: See also * vanishing theorem * Barth–Lefschetz theorem Note References * * * * * * * * * * * * * * * * * * * Further reading * * * External links {{Citation , last=Demailly , first=Jean-Pierre , title=Complex Analytic and Differential Geometry , url=https://www-fourier. ... [...More Info...] [...Related Items...] OR:* [Wikipedia] [Google] [Baidu]
	Mumford Vanishing Theorem In algebraic geometry, the Mumford vanishing theorem proved by Mumford in 1967 states that if ''L'' is a semi-ample invertible sheaf with Iitaka dimension at least 2 on a complex projective manifold, then :H^i(X,L^)=0\texti = 0,1.\ The Mumford vanishing theorem is related to the Ramanujam vanishing theorem, and is generalized by the 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 * Theorems in algebraic geometry {{algebraic-geometry-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Nakano Vanishing Theorem In mathematics, specifically in the study of vector bundles over complex Kähler manifolds, the Nakano vanishing theorem, sometimes called the Akizuki–Nakano vanishing theorem, generalizes the Kodaira vanishing theorem. Given a compact complex manifold ''M'' with a holomorphic line bundle ''F'' over ''M'', the Nakano vanishing theorem provides a condition on when the cohomology groups H^q(M; \Omega^p(F)) equal zero. Here, \Omega^p(F) denotes the sheaf of holomorphic (''p'',0)-forms taking values on ''F''. The theorem states that, if the first Chern class of ''F'' is negative,H^q(M; \Omega^p(F)) = 0 \text q + p n. See also Le Potier's vanishing theorem In algebraic geometry, Le Potier's vanishing theorem is an extension of the Kodaira vanishing theorem, on vector bundle In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces p ... References Original publications * * Secondary sources Theor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Ramanujam Vanishing Theorem In algebraic geometry, the Ramanujam vanishing theorem is an extension of the Kodaira vanishing theorem due to , that in particular gives conditions for the vanishing of first cohomology groups of coherent sheaves on a surface. The 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 ... generalizes it. See also * Mumford vanishing theorem References * * * Theorems in algebraic geometry {{Algebraic-geometry-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]