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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. Here N must be the even number 2n, where n is the complex dimension of M.
All such complex structures can be obtained as follows: take a lattice Λ in a vector space V isomorphic to C considered as real vector space; then the quotient group
The actual projective embeddings are complicated when n > 1, and are really coextensive with the theory of theta-functions of several complex variables. 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.