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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 : 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 generating it. On the other hand, another, more algebro-geometric way of saying that the genus of ''C'' is ''g'', is that : 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 . Given forms and closed loops we can integrate, and we define 2''g'' vectors : It follows from the Riemann bilinear relations that the generate a nondegenerate lattice (that is, they are a real basis for ), and the Jacobian is defined by : The Abel–Jacobi map is then defined as follows. We pick some base point and, nearly mimicking the definition of , define the map : Although this is seemingly dependent on a path from to any two such paths define a closed loop in and, therefore, an element of so integration over it gives an element of Thus the difference is erased in the passage to the quotient by . Changing base-point 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
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