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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 Segre class ： ウィキペディア英語版 Segre class In mathematics, the Segre class is a characteristic class used in the study of singular vector bundles. The total Segre class is inverse to the total Chern class, and thus provides equivalent information; the advantage of the Segre class is that it generalizes to singular vector bundles, while the Chern class does not. The Segre class is named after Beniamino Segre. == Definition == For a holomorphic vector bundle $E$ over a complex manifold $M$ a total Segre class $s(E)$ is the inverse to the total Chern class $c(E)$, see e.g.〔Fulton W. (1998). ''Intersection theory'', p.50. Springer, 1998.〕 Explicitly, for a total Chern class : $$ c(E) = 1+c_1(E) + c_2(E) + \cdots \, one gets the total Segre class : $$ s(E) = 1 + s_1 (E) + s_2 (E) + \cdots \, where : $$ c_1(E) = -s_1(E), \quad c_2(E) = s_1(E)^2 - s_2(E), \quad \dots, \quad c_n(E) = -s_1(E)c_(E) - s_2(E) c_(E) - \cdots - s_n(E) Let $x\_1,\; \backslash dots,\; x\_k$ be Chern roots, i.e. formal eigenvalues of $\backslash frac$ where $\backslash Omega$ is a curvature of a connection on $E$. While the Chern class s(E) is written as :$c(E)\; =\; \backslash prod\_^\; (1+x\_i)\; =\; c\_0\; +\; c\_1\; +\; \backslash cdots\; +\; c\_k\; \backslash ,$ where $c\_i$ is an elementary symmetric polynomial of degree $i$ in variables $x\_1,\; \backslash dots,\; x\_k$ the Segre for the dual bundle $E^\backslash vee$ which has Chern roots $-x\_1,\; \backslash dots,\; -x\_k$ is written as :$s(E)\; =\; \backslash prod\_^\; \backslash frac\; =\; s\_0\; +\; s\_1\; +\; \backslash cdots$ Expanding the above expression in powers of $x\_1,\; \backslash dots\; x\_k$ one can see that $s\_i\; (E^\backslash vee)$ is represented by a complete homogeneous symmetric polynomial of $x\_1,\; \backslash dots\; x\_k$ 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「Segre class」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.