Complex torus

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a complex torus is a particular kind of <a href="/facts/Complex_manifold/DlvPnpzm">complex manifold</a> M whose underlying <a href="/facts/Smooth_manifold/aSnbXKcV">smooth manifold</a> is a <a href="/facts/Torus/J9SkaV1W">torus</a> in the usual sense (i.e. the <a href="/facts/Cartesian_product/BfqBWzdL">cartesian product</a> of some number N <a href="/facts/Circle/9fy5ggKn">circles</a>). Here N must be the even number 2n, where n is the <a href="/facts/Complex_dimension/e7M7pctQ">complex dimension</a> of M.
All such complex structures can be obtained as follows: take a <a href="/facts/Lattice_(group)/2ls9Iv9N">lattice</a> Λ in a vector space V isomorphic to Cn considered as real vector space; then the <a href="/facts/Quotient_group/HlwpHtjb">quotient group</a>

V
        
          /
        
        Λ
      
    
    {\displaystyle V/\Lambda }

is a <a href="/facts/Compact_space/d0cgXJH7">compact</a> complex manifold. All complex tori, up to isomorphism, are obtained in this way. For n = 1 this is the classical <a href="/facts/Period_lattice/DsPofQqG">period lattice</a> construction of <a href="/facts/Elliptic_curve/60jvaHSA">elliptic curves</a>. For n > 1 <a href="/facts/Bernhard_Riemann/oV2fL0k8">Bernhard Riemann</a> found necessary and sufficient conditions for a complex torus to be an <a href="/facts/Algebraic_variety/Vk7f97vn">algebraic variety</a>; those that are varieties can be embedded into <a href="/facts/Complex_projective_space/QmgneIzS">complex projective space</a>, and are the <a href="/facts/Abelian_varieties/dGSDVruG">abelian varieties</a>.
The actual projective embeddings are complicated (see <a href="/facts/Equations_defining_abelian_varieties/6abzQqjG">equations defining abelian varieties</a>) when n > 1, and are really coextensive with the theory of <a href="/facts/Theta-function/g09xvoQQ">theta-functions</a> of <a href="/facts/Several_complex_variable/axwNpV4a">several complex variables</a> (with fixed modulus). There is nothing as simple as the <a href="/facts/Cubic_curve/FSU8tXAb">cubic curve</a> description for n = 1. <a href="/facts/Computer_algebra/hjun1KAX">Computer algebra</a> can handle cases for small n reasonably well. By <a href="/facts/Algebraic_geometry_and_analytic_geometry/k2ltJ1Ue">Chow's theorem</a>, no complex torus other than the abelian varieties can 'fit' into <a href="/facts/Projective_space/7MSpA9cM">projective space</a>.

Complex torus open-in-new

Complex torus