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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 Bogomolov–Miyaoka–Yau inequality ： ウィキペディア英語版 Bogomolov–Miyaoka–Yau inequality In mathematics, the Bogomolov–Miyaoka–Yau inequality is the inequality :$c\_1^2\; \backslash le\; 3\; c\_2\backslash$ between Chern numbers of compact complex surfaces of general type. Its major interest is the way it restricts the possible topological types of the underlying real 4-manifold. It was proved independently by and , after and proved weaker versions with the constant 3 replaced by 8 and 4. Borel and Hirzebruch showed that the inequality is best possible by finding infinitely many cases where equality holds. The inequality is false in positive characteristic: and gave examples of surfaces in characteristic ''p'', such as generalized Raynaud surfaces, for which it fails. ==Formulation of the inequality== The conventional formulation of the Bogomolov–Miyaoka–Yau inequality is Let ''X'' be a compact complex surface of general type, and let ''c''1 = ''c''1(''X'') and ''c''2 = ''c''2(''X'') be the first and second Chern class of the complex tangent bundle of the surface. Then : $c\_1^2\; \backslash le\; 3\; c\_2.\; \backslash ,$ moreover if equality holds then ''X'' is a quotient of a ball. The latter statement is a consequence of Yau's differential geometric approach which is based on his resolution of the Calabi conjecture. Since $c\_2(X)\; =\; e(X)$ is the topological Euler characteristic and by the Thom–Hirzebruch signature theorem $c\_1^2(X)\; =\; 2\; e(X)\; +\; 3\backslash sigma(X)$ where $\backslash sigma(X)$ is the signature of the intersection form on the second cohomology, the Bogomolov–Miyaoka–Yau inequality can also be written as a restriction on the topological type of the surface of general type: :$\backslash sigma(X)\; \backslash le\; \backslash frac\; e(X),$ moreover if $\backslash sigma(X)\; =\; (1/3)e(X)$ then the universal covering is a ball. Together with the Noether inequality the Bogomolov–Miyaoka–Yau inequality sets boundaries in the search for complex surfaces. Mapping out the topological types that are realized as complex surfaces is called geography of surfaces. see surfaces of general type. 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「Bogomolov–Miyaoka–Yau inequality」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.