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In mathematics, a complex torus is a particular kind of complex manifold ''M'' whose underlying smooth manifold is a torus in the usual sense (i.e. the cartesian product of some number ''N'' circles). Here ''N'' must be the even number 2''n'', where ''n'' is the complex dimension of ''M''. All such complex structures can be obtained as follows: take a lattice Λ in C''n'' considered as real vector space; then the quotient group :C''n''/Λ is a compact complex manifold. All complex tori, up to isomorphism, are obtained in this way. For ''n'' = 1 this is the classical period lattice construction of elliptic curves. For ''n'' > 1 Bernhard Riemann found necessary and sufficient conditions for a complex torus to be an algebraic variety; those that are varieties can be embedded into complex projective space, and are the abelian varieties. The actual projective embeddings are complicated (see equations defining abelian varieties) when ''n'' > 1, and are really coextensive with the theory of theta-functions of several complex variables (with fixed modulus). There is nothing as simple as the cubic curve description for ''n'' = 1. Computer algebra can handle cases for small ''n'' reasonably well. By Chow's theorem, no complex torus other than the abelian varieties can 'fit' into projective space. ==See also== *Complex Lie group 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「complex torus」の詳細全文を読む スポンサード リンク
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