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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 Calabi conjecture ： ウィキペディア英語版 Calabi conjecture In mathematics, the Calabi conjecture was a conjecture about the existence of certain "nice" Riemannian metrics on certain complex manifolds, made by and proved by . Yau received the Fields Medal in 1982 in part for this proof. The Calabi conjecture states that a compact Kähler manifold has a unique Kähler metric in the same class whose Ricci form is any given 2-form representing the first Chern class. In particular if the first Chern class vanishes there is a unique Kähler metric in the same class with vanishing Ricci curvature; these are called Calabi–Yau manifolds. More formally, the Calabi conjecture states: :If ''M'' is a compact Kähler manifold with Kähler metric $g\backslash ;$ and Kähler form $\backslash omega\backslash ;$, and ''R'' is any (1,1)-form representing the manifold's first Chern class, then there exists a unique Kähler metric $\backslash tilde$ on ''M'' with Kähler form $\backslash tilde$ such that $\backslash omega\backslash ;$ and $\backslash tilde$ represent the same class in cohomology H2(''M'',R) and the Ricci form of $\backslash tilde$ is ''R''. The Calabi conjecture is closely related to the question of which Kähler manifolds have Kähler–Einstein metrics. ==Kähler–Einstein metrics== A conjecture closely related to the Calabi conjecture states that if a compact Kähler variety has a negative, zero, or positive first Chern class then it has a Kähler–Einstein metric in the same class as its Kähler metric, unique up to rescaling. This was proved for negative first Chern classes independently by Thierry Aubin and Shing-Tung Yau in 1976. When the Chern class is zero it was proved by Yau as an easy consequence of the Calabi conjecture. It was disproved for positive first Chern classes by Yau, who observed that the complex projective plane blown up at 2 points has no Kähler–Einstein metric and so is a counterexample. Also even when Kähler–Einstein metric exists it need not be unique. There has been a lot of further work on the positive first Chern class case. A necessary condition for the existence of a Kähler–Einstein metric is that the Lie algebra of holomorphic vector fields is reductive. Yau conjectured that when the first Chern class is positive, a Kähler variety has a Kähler–Einstein metric if and only if it is stable in the sense of geometric invariant theory. The case of complex surfaces has been settled by Gang Tian. The complex surfaces with positive Chern class are either a product of two copies of a projective line (which obviously has a Kähler–Einstein metric) or a blowup of the projective plane in at most 8 points in "general position", in the sense that no 3 lie on a line and no 6 lie on a quadric. The projective plane has a Kähler–Einstein metric, and the projective plane blown up in 1 or 2 points does not, as the Lie algebra of holomorphic vector fields is not reductive. Tian showed that the projective plane blown up in 3, 4, 5, 6, 7, or 8 points in general position has a Kähler–Einstein metric. 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「Calabi conjecture」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.