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 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 Cⁿ considered as real vector space; then the quotient group $${\displaystyle V/\Lambda }$$ 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.

One way to define complex tori is as a compact connected complex Lie group ${\displaystyle G}$. These are Lie groups where the structure maps are holomorphic maps of complex manifolds. It turns out that all such compact connected Lie groups are commutative, and are isomorphic to a quotient of their Lie algebra ${\displaystyle {\mathfrak {g}}=T_{0}G}$ whose covering map is the exponential map of a Lie algebra to its associated Lie group. The kernel of this map is a lattice ${\displaystyle \Lambda \subset {\mathfrak {g}}}$ and ${\displaystyle {\mathfrak {g}}/\Lambda \cong G}$.

Conversely, given a complex vector space ${\displaystyle V}$ and a lattice ${\displaystyle \Lambda \subseteq V}$ of maximal rank, the quotient complex manifold ${\displaystyle V/\Lambda }$ has a complex Lie group structure, and is also compact and connected. This implies that the two definitions for complex tori are equivalent.

One way to describe a g-dimensional complex torus is by using a ${\displaystyle g\times 2g}$ matrix ${\displaystyle \Pi }$ whose columns correspond to a basis ${\displaystyle \lambda _{1},\ldots ,\lambda _{2g}}$ of the lattice ${\displaystyle \Lambda }$ expanded out using a basis ${\displaystyle e_{1},\ldots ,e_{g}}$ of ${\displaystyle V}$. That is, we write $${\displaystyle \Pi ={\begin{pmatrix}\lambda _{1,1}&\cdots &\lambda _{1,2g}\\\vdots &&\vdots \\\lambda _{g,1}&\cdots &\lambda _{g,2g}\end{pmatrix}}}$$ so $${\displaystyle \lambda _{i}=\sum _{j}\lambda _{ji}e_{j}}$$ We can then write the torus ${\displaystyle X=V/\Lambda }$ as $${\displaystyle X=\mathbb {C} ^{g}/\Pi \mathbb {Z} ^{2g}.}$$ If we go in the reverse direction by selecting a matrix ${\displaystyle \Pi \in Mat_{\mathbb {C} }(g,2g)}$, it corresponds to a period matrix if and only if the corresponding matrix ${\displaystyle P\in Mat_{\mathbb {C} }(2g,2g)}$ constructed by adjoining the complex conjugate matrix ${\displaystyle {\overline {\Pi }}}$ to ${\displaystyle \Pi }$, so $${\displaystyle P={\begin{pmatrix}\Pi \\{\overline {\Pi }}\end{pmatrix}}}$$ is nonsingular. This guarantees the column vectors of ${\displaystyle \Pi }$ span a lattice in ${\displaystyle \mathbb {C} ^{g}}$ hence must be linearly independent vectors over ${\displaystyle \mathbb {R} }$.

For a two-dimensional complex torus, it has a period matrix of the form $${\displaystyle \Pi ={\begin{pmatrix}\lambda _{1,1}&\lambda _{1,2}&\lambda _{1,3}&\lambda _{1,4}\\\lambda _{2,1}&\lambda _{2,2}&\lambda _{2,3}&\lambda _{2,4}\end{pmatrix}}}$$ for example, the matrix $${\displaystyle \Pi ={\begin{pmatrix}1&0&i&2i\\1&-i&1&1\end{pmatrix}}}$$ forms a period matrix since the associated period matrix has determinant 4.

For any complex torus ${\displaystyle X=V/\Lambda }$ of dimension ${\displaystyle g}$ it has a period matrix ${\displaystyle \Pi }$ of the form $${\displaystyle (Z,1_{g})}$$where ${\displaystyle 1_{g}}$ is the identity matrix and ${\displaystyle Z\in Mat_{\mathbb {C} }(g)}$ where ${\displaystyle \det {\text{Im}}(Z)\neq 0}$. We can get this from taking a change of basis of the vector space ${\displaystyle V}$ giving a block matrix of the form above. The condition for ${\displaystyle \det {\text{Im}}(Z)\neq 0}$ follows from looking at the corresponding ${\displaystyle P}$-matrix $${\displaystyle {\begin{pmatrix}Z&1_{g}\\{\overline {Z}}&1_{g}\end{pmatrix}}}$$ since this must be a non-singular matrix. This is because if we calculate the determinant of the block matrix, this is simply $${\displaystyle {\begin{aligned}\det P&=\det(1_{g})\det(Z-1_{g}1_{g}{\overline {Z}})\\&=\det(Z-{\overline {Z}})\\&\Rightarrow \det({\text{Im}}(Z))\neq 0\end{aligned}}}$$ which gives the implication.