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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Hirzebruch–Riemann–Roch theorem ： ウィキペディア英語版 Hirzebruch–Riemann–Roch theorem In mathematics, the Hirzebruch–Riemann–Roch theorem, named after Friedrich Hirzebruch, Bernhard Riemann, and Gustav Roch, is Hirzebruch's 1954 result contributing to the Riemann–Roch problem for complex algebraic varieties of all dimensions. It was the first successful generalisation of the classical Riemann–Roch theorem on Riemann surfaces to all higher dimensions, and paved the way to the Grothendieck–Hirzebruch–Riemann–Roch theorem proved about three years later. ==Statement of Hirzebruch–Riemann–Roch theorem== The Hirzebruch–Riemann–Roch theorem applies to any holomorphic vector bundle ''E'' on a compact complex manifold ''X'', to calculate the holomorphic Euler characteristic of ''E'' in sheaf cohomology, namely the alternating sum :$\backslash chi(X,E)\; =\; \backslash sum\_^\; (-1)^\; \backslash dim\_(X,E)$ of the dimensions as complex vector spaces. Hirzebruch's theorem states that χ(''X'', ''E'') is computable in terms of the Chern classes ''C''''j''(''E'') of ''E'', and the Todd polynomials ''T''''j'' in the Chern classes of the holomorphic tangent bundle of ''X''. These all lie in the cohomology ring of ''X''; by use of the fundamental class (or, in other words, integration over ''X'') we can obtain numbers from classes in ''H''2''n''(''X''). The Hirzebruch formula asserts that :$\backslash chi(X,E)\; =\; \backslash sum\; \backslash operatorname\_(E)\; \backslash frac$ taken over all relevant ''j'' (so 0 ≤ ''j'' ≤ ''n''), using the Chern character ch(''E'') in cohomology. In other words, the cross products are formed in cohomology ring of all the 'matching' degrees that add up to 2''n'', where to 'massage' the ''C''''j''(''E'') a formal manipulation is done, setting :$\backslash operatorname(E)\; =\; \backslash sum\; \backslash exp(x\_)$ and the total Chern class :$C(E)\; =\; \backslash sum\; C\_(E)\; =\; \backslash prod\; (1\; +\; x\_).$ Formulated differently the theorem gives the equality :$\backslash chi(X,E)\; =\; \backslash int\_X\; \backslash operatorname(E)\; \backslash operatorname(X)$ where ''td(X)'' is the Todd class of the tangent bundle of ''X''. Significant special cases are when ''E'' is a complex line bundle, and when ''X'' is an algebraic surface (Noether's formula). Weil's Riemann–Roch theorem for vector bundles on curves, and the Riemann–Roch theorem for algebraic surfaces (see below), are included in its scope. The formula also expresses in a precise way the vague notion that the Todd classes are in some sense reciprocals of characteristic classes. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Hirzebruch–Riemann–Roch theorem」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.