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Kodaira–Nakano Vanishing Theorem
In mathematics, specifically in the study of vector bundles over Kähler manifold, 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 (mathematics), 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 References Original publications * * * Secondary sources Theorems in complex geometry Topological methods of algebraic geometry Theorems in algebraic geometry {{analysis-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 every point x of the space X we associate (or "attach") a vector space V(x) in such a way that these vector spaces fit together to form another space of the same kind as X (e.g. a topological space, manifold, or algebraic variety), which is then called a vector bundle over X. The simplest example is the case that the family of vector spaces is constant, i.e., there is a fixed vector space V such that V(x)=V for all x in X: in this case there is a copy of V for each x in X and these copies fit together to form the vector bundle X\times V over X. Such vector bundles are said to be ''trivial''. A more complicated (and prototypical) class of examples are the tangent bundles of smooth (or differentiable) manifolds: to every point of such a man ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
