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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 Abel–Jacobi map ： ウィキペディア英語版 Abel–Jacobi map In mathematics, the Abel–Jacobi map is a construction of algebraic geometry which relates an algebraic curve to its Jacobian variety. In Riemannian geometry, it is a more general construction mapping a manifold to its Jacobi torus. The name derives from the theorem of Abel and Jacobi that two effective divisors are linearly equivalent if and only if they are indistinguishable under the Abel–Jacobi map. ==Construction of the map== In complex algebraic geometry, the Jacobian of a curve ''C'' is constructed using path integration. Namely, suppose ''C'' has genus ''g'', which means topologically that : $H\_1(C,\; \backslash mathbb)\; \backslash cong\; \backslash mathbb^.$ Geometrically, this homology group consists of (homology classes of) ''cycles'' in ''C'', or in other words, closed loops. Therefore, we can choose 2''g'' loops $\backslash gamma\_1,\; \backslash dots,\; \backslash gamma\_$ generating it. On the other hand, another, more algebro-geometric way of saying that the genus of ''C'' is ''g'', is that : $H^0(C,\; K)\; \backslash cong\; \backslash mathbb^g,$ where ''K'' is the canonical bundle on ''C''. By definition, this is the space of globally defined holomorphic differential forms on ''C'', so we can choose ''g'' linearly independent forms $\backslash omega\_1,\; \backslash dots,\; \backslash omega\_g$. Given forms and closed loops we can integrate, and we define 2''g'' vectors : $\backslash Omega\_j\; =\; \backslash left(\backslash int\_\; \backslash omega\_1,\; \backslash dots,\; \backslash int\_\; \backslash omega\_g\backslash right)\; \backslash in\; \backslash mathbb^g.$ It follows from the Riemann bilinear relations that the $\backslash Omega\_j$ generate a nondegenerate lattice $\backslash Lambda$ (that is, they are a real basis for $\backslash mathbb^g\; \backslash cong\; \backslash mathbb^$), and the Jacobian is defined by : $J(C)\; =\; \backslash mathbb^g/\backslash Lambda.$ The Abel–Jacobi map is then defined as follows. We pick some base point $p\_0\; \backslash in\; C$ and, nearly mimicking the definition of $\backslash Lambda$, define the map : $u\; \backslash colon\; C\; \backslash to\; J(C),\; u(p)\; =\; \backslash left(\; \backslash int\_^p\; \backslash omega\_1,\; \backslash dots,\; \backslash int\_^p\; \backslash omega\_g\backslash right)\; \backslash bmod\; \backslash Lambda.$ Although this is seemingly dependent on a path from $p\_0$ to $p,$ any two such paths define a closed loop in $C$ and, therefore, an element of $H\_1(C,\; \backslash mathbb),$ so integration over it gives an element of $\backslash Lambda.$ Thus the difference is erased in the passage to the quotient by $\backslash Lambda$. Changing base-point $p\_0$ does change the map, but only by a translation of the torus. 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「Abel–Jacobi map」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.