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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 Kuranishi structure ： ウィキペディア英語版 Kuranishi structure In mathematics, especially in topology, a Kuranishi structure is a smooth analogue of scheme structure. If a topological space is endowed with a Kuranishi structure, then locally it can be identified with the zero set of a smooth map $(f\_1,\; \backslash ldots,\; f\_k):\; ^\; \backslash to\; ^k.$ Kuranishi structure was introduced by Japanese mathematicians Kenji Fukaya and Kaoru Ono in the study of Gromov–Witten invariants in symplectic geometry.〔Fukaya, K. and Ono, K., "Arnold Conjecture and Gromov–Witten Invariant", ''Topology'' 38 (1999), no. 5, 933–1048〕 ==Definition== Let $X$ be a compact metrizable topological space. Let $p\; \backslash in\; X$ be a point. A Kuranishi neighborhood of $p$ (of dimension $k$) is a 5-tuple :::$K\_p\; =\; (U\_p,\; E\_p,\; S\_p,\; \backslash psi\_p,\; F\_p)$ where * $U\_p$ is a smooth orbifold; * $E\_p\; \backslash to$ is a smooth orbifold vector bundle; * $S\_p:\; U\_p\; \backslash to\; E\_p$ is a smooth section; * $\backslash psi\_p:\; S\_p^(0)\; \backslash to\; X$ is a continuous map and is homeomorphic onto its image $F\_p\; \backslash subset\; X$. They should satisfy that $\backslash dim\; U\_p\; -\; \backslash operatorname\; E\_p\; =\; k$. If $p,\; q\; \backslash in\; X$ and $K\_p\; =\; (U\_p,\; E\_p,\; S\_p,\; \backslash psi\_p,\; F\_p)$, $K\_q\; =\; (U\_q,\; E\_q,\; S\_q,\; \backslash psi\_q,\; F\_q)$ are their Kuranishi neighborhoods respectively, then a coordinate change from $K\_q$ to $K\_p$ is a triple :::$T\_\; =\; (\; U\_,\; \backslash phi\_,\; \backslash hat\backslash phi\_)$ where * $U\_\; \backslash subset\; U\_q$ is an open sub-orbifold; * $\backslash phi\_:\; U\_\; \backslash to\; U\_p$ is an orbifold embedding; * $\backslash hat\backslash phi\_:\; E\_q|\_$. In addition, they must satisfy the compatibility condition: * $S\_p\; \backslash circ\; \backslash phi\_\; =\; \backslash hat\backslash phi\_\; \backslash circ\; S\_q|\_|\_\}\; =\; \backslash psi\_q|\_\}$. A Kuranishi structure on $X$ of dimension $k$ is a collection ::: $\backslash Big(\; \backslash ,\backslash \; \backslash ,\; \backslash phi\_,\; \backslash hat\backslash phi\_\; )\; \backslash \; |\backslash \; p\; \backslash in\; X,\backslash \; q\; \backslash in\; F\_p\backslash \}\; \backslash Big)$ where * $K\_p$ is a Kuranishi neighborhood of $p$ of dimension $k$; * $T\_$ is a coordinate change from $K\_q$ to $K\_p$. In addition, the coordinate changes must satisfy the cocycle condition, namely, whenever $q\backslash in\; F\_p,\backslash \; r\; \backslash in\; F\_q$, we require that ::: $\backslash phi\_\; \backslash circ\; \backslash phi\_\; =\; \backslash phi\_,\backslash \; \backslash hat\backslash phi\_\; \backslash circ\; \backslash hat\backslash phi\_\; =\; \backslash hat\backslash phi\_$ over the regions where both sides are defined. 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「Kuranishi structure」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.