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From here: https://www.encyclopediaofmath.org/index.php/Complex_torus

A complex torus is a complex Abelian Lie group obtained from the $n$-dimensional complex space $\mathbb{C}^n$ by factorizing with respect to a lattice $\def\G{\Gamma}\G\subset \mathbb{C}^n $ of rank $2n$.

A basis for a lattice $\Gamma\subset \mathbb{C}^n$ can be given by a matrix $\def\O{\Omega}\O$ of dimension $n\times 2n$, called the period matrix of the torus $T=\mathbb{C}^n/\G$. Tori $T_i = \mathbb{C}^n/\Gamma_i$ with period matrices $\O_i$ ($i=1,2$) are isomorphic (as complex Lie groups or as complex manifolds) if and only if there exist matrices $C\in \textrm{GL}(n,\mathbb{C}) $ and $Z\in \textrm{GL}(2n,\mathbb{Z})$ such that $\O_2 = C\O_1 Z$.

My questions are:

  1. What is the role of these two actions ($\textrm{GL}(n,\mathbb{C}) $ and $\textrm{GL}(2n,\mathbb{Z})$)? Is one of them related to the lattice (isomorphism of the lattices?) and another to the complex structure of the torus itself? Where can I find a detailed proof for the n-dimensional case?

  2. Let us now for a moment forget a complex structure and regard $\mathbb{C}^n$ as $\mathbb{R}^{2n}$ with coordinates $\{x^i, y^i\}, i=\overline{1,n}$. Suppose we have a $n$-dimensional foliation on $\mathbb{R}^{2n}$ given as $y^i=const$. This foliation induces foliations on tori $T_i$, obtained from different lattices. As far as I understand biholomorphic isomorphisms mentioned above do not take into consideration the structure of these foliations on tori. Do someone have the idea how to find the relation (isomorphism) between two different foliated n-tori (obtained from two different lattices), in the terms of period matrices of the lattices?

I see there perhaps will be much of number theory (in 2-dim case rational and irrational slopes will obviously give closed and dense leaves corresp.), but I am not sure how to express it correctly in n-dim case.

Thanks to all of you in advance! ♥

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1.

Of course, the set of Lie group isomorphisms of $\mathbb{C}^n$ which preserves the complex structure is precisely $GL(n,\mathbb{C})$. The $GL(n,\mathbb{C})$ matrices are, as you say, related to the isomorphisms of the underlying vector space. The $GL(2n,\mathbb{Z})$ are not so much related to the isomorphisms of the lattice itself, but rather to the freedom to choose from many different bases for the sane lattice. We can make this more precise with two propositions:

I. Two complex tori $\mathbb{C}^n/\Gamma_1$ and $\mathbb{C}^n/\Gamma_2$ are isomorphic iff there exists a transformation $C\in GL(n,\mathbb{C})$ such that $\Gamma_2=C(\Gamma_1)$.

One direction, showing the given transformation is indeed an isomorphism, is straightforward. The other can be shown by lifting any isomorphism of complex tori onto an isomorphism of universal covering groups $C:\mathbb{C}^n\to\mathbb{C}^n$, which can be shown to have all the stated properties.

II. Two bases $A=[a_1,...,a_{2n}],\ B=[b_1,...,b_{2n}]\ \subset\mathbb{C}^n$ generate the same lattice iff there is a transformation $Z\in GL(2n,\mathbb{Z})$ such that $A=BH$, regarding $A,B$ as period matrices.

Forgetting for a moment the complex structure and regarding $\mathbb{C}^n$ as a real vector space, a basis is just a set of $2n$ $\mathbb{R}$-linearly independent vectors, the integer combinations of which generate the lattice.

Suppose $A$ and $B$ as above generate the same lattice. It follows immediately that each element of $A$ can be written as a unique integer combination of elements of $B$, and likewise for the reverse. Denote these combinations $$ b_i=\sum_ja_jZ_{ji},\ \ \ a_i=\sum_jb_jZ'_{ji},\ \ \ Z_{ij},Z'_{ij}\in\mathbb{Z} $$ Reading the above as matrix multiplication $A=BZ$, $B=AZ'$, we see that $Z'=Z^{-1}$ and thus $Z\in GL(2n,\mathbb{Z})$. For the reverse, suppose $B=AZ$. The set of integer combinations can be written $\text{span}_{\mathbb{Z}}(A)=\{Az:z\in\mathbb{Z}^{2n}\}$, $\text{span}_{\mathbb{Z}}(B)=\{AZz:z\in\mathbb{Z}^{2n}\}$. Since $Z$ bijectively maps $\mathbb{Z}^{2n}$ to itself, it is straightforward to show these two integer spans are identical.

If we describe complex tori in terms of period matrices, we can combine I and II, giving the matrix equation you describe, which states that $\mathbb{C}^n/\text{span}_\mathbb{Z}(\Omega_1)$ and $\mathbb{C}^n/\text{span}_\mathbb{Z}(\Omega_2)$ are equivalent iff there exists an element of $GL(n\mathbb{C})$ which transforms the $\Omega_1$ into a period matrix equivalent to $\Omega_2$.

2.

If we look at real tori, then things are considerably simpler: the quotients $\mathbb{R}^{2n}/\Gamma$ are isomorphic Lie groups for all lattices $\Gamma$ of rank $2n$: in particular, they all have a canonical form $\mathbb{R}^{2n}/\mathbb{Z}^{2n}$, unique up to $GL(2n,\mathbb{Z})$, obtained by the transformation which is the inverse of the period matrix. If we equip such tori with bi-invariant foliations/distributions of dimension $n$, two will be isomorphic if and only if this extra structure is identical when we write both tori in some canonical form.

As an example, we can describe the foliated tori with a subspace $X\subset\mathbb{R}^n$ with dimension $n$. which induces the foliation. Two such tori $(X\hookrightarrow\mathbb{R}^{2n})/\text{span}_{\mathbb{Z}}(\Omega_1)$, $(X\hookrightarrow\mathbb{R}^{2n})/\text{span}_{\mathbb{Z}}(\Omega_2)$ are isomorphic iff $\Omega_1^{-1}(X)=Z(\Omega_2^{-1}(X))$ for some $Z\in GL(2n,\mathbb{Z})$, where $\Omega_i, Z$, are understood to act by elementwise left (or right) multiplication.

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