Complex torus

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.

For example, we can write a normalized period matrix for a 2-dimensional complex torus as

$${\displaystyle {\begin{pmatrix}z_{1,1}&z_{1,2}&1&0\\z_{2,1}&z_{2,2}&0&1\end{pmatrix}}}$$

one such example is the normalized period matrix

$${\displaystyle {\begin{pmatrix}1+i&1-i&1&0\\1+2i&1+{\sqrt {2}}i&0&1\end{pmatrix}}}$$

since the determinant of ${\displaystyle {\text{Im}}(Z)}$ is nonzero, equal to ${\displaystyle 2+{\sqrt {2}}}$.

To get a period matrix which gives a projective complex manifold, hence an algebraic variety, the period matrix needs to further satisfy the Riemann bilinear relations.[3]

The class of homomorphic maps between complex tori have a very simple structure. Of course, every homomorphism induces a holomorphic map, but every holomorphic map is the composition of a special kind of holomorphic map with a homomorphism. For an element ${\displaystyle x_{0}\in X}$ we define the translation map

$${\displaystyle t_{x_{0}}:X\to X}$$

sending ${\displaystyle x\mapsto x+x_{0}}$ Then, if ${\displaystyle h}$ is a holomorphic map between complex tori ${\displaystyle X,X'}$, there is a unique homomorphism ${\displaystyle f:X\to X'}$ such that

$${\displaystyle h=t_{h(0)}\circ f}$$

showing the holomorphic maps are not much larger than the set of homomorphisms of complex tori.

One distinct class of homomorphisms of complex tori are called isogenies. These are endomorphisms of complex tori with a non-zero kernel. For example, if we let ${\displaystyle n\in \mathbb {Z} _{\neq 0}}$ be an integer, then there is an associated map

$${\displaystyle n_{X}:X\to X}$$

sending ${\displaystyle x\mapsto nx}$ which has kernel

$${\displaystyle X_{n}\cong (\mathbb {Z} /n\mathbb {Z} )^{2g}}$$

isomorphic to ${\displaystyle \Lambda /n\Lambda }$.

There is an isomorphism of complex structures on the real vector space ${\displaystyle \mathbb {R} ^{2g}}$ and the set

$${\displaystyle GL_{\mathbb {R} }(2g)/GL_{\mathbb {C} }(g)}$$

and isomorphic tori can be given by a change of basis of their lattices, hence a matrix in ${\displaystyle GL_{\mathbb {Z} }(2g)}$. This gives the set of isomorphism classes of complex tori of dimension ${\displaystyle g}$, ${\displaystyle {\mathcal {T}}_{g}}$, as the Double coset space

$${\displaystyle {\mathcal {T}}_{g}\cong GL_{\mathbb {Z} }(2g)\backslash GL_{\mathbb {R} }(2g)/GL_{\mathbb {C} }(g)}$$

Note that as a real manifold, this has dimension

$${\displaystyle 4g^{2}-2g^{2}=2g^{2}}$$

this is important when considering the dimensions of moduli of Abelian varieties, which shows there are far more complex tori than Abelian varieties.

Starting from the first group cohomology group

$${\displaystyle H^{1}(\pi _{1}(X),H^{0}({\mathcal {O}}_{\tilde {X}}^{*}))}$$

we recall how its elements can be represented. Since ${\displaystyle \pi _{1}(X)}$ acts on ${\displaystyle {\tilde {X}}}$ there is an induced action on all of its sheaves, hence on

$${\displaystyle H^{0}({\mathcal {O}}_{\tilde {X}}^{*})=\{f:{\tilde {X}}\to \mathbb {C} ^{*}\}}$$

The ${\displaystyle \pi _{1}(X)}$-action can then be represented as a holomorphic map ${\displaystyle f:\pi _{1}(X)\times {\tilde {X}}\to \mathbb {C} ^{*}}$. This map satisfies the cocycle condition if

$${\displaystyle f(a\cdot b,x)=f(a,b\cdot x)f(b,x)}$$

for every ${\displaystyle a,b\in \pi _{1}(X)}$ and ${\displaystyle x\in {\tilde {X}}}$. The abelian group of 1-cocycles ${\displaystyle Z^{1}(\pi _{1}(X),H^{0}({\mathcal {O}}_{\tilde {X}}^{*}))}$ is called the group of factors of automorphy. Note that such functions ${\displaystyle f}$ are also just called factors.

For complex tori, these functions ${\displaystyle f}$ are given by functions

$${\displaystyle f:\mathbb {C} ^{n}\times \mathbb {Z} ^{2n}\to \mathbb {C} ^{*}}$$

which follow the cocycle condition. These are automorphic functions, more precisely, the automorphic functions used in the transformation laws for theta functions. Also, any such map can be written as

$${\displaystyle f=\exp(2\pi i\cdot g)}$$

for

$${\displaystyle g:V\times \Lambda \to \mathbb {C} }$$

which is useful for computing invariants related to the associated line bundle.

Given a factor of automorphy ${\displaystyle f}$ we can define a line bundle on ${\displaystyle X}$ as follows: the trivial line bundle ${\displaystyle {\tilde {X}}\times \mathbb {C} \to {\tilde {X}}}$ has a ${\displaystyle \pi _{1}(X)}$-action given by

$${\displaystyle a\cdot (x,t)=(a\cdot x,f(a,x)\cdot t)}$$

for the factor ${\displaystyle f}$. Since this action is free and properly discontinuous, the quotient bundle

$${\displaystyle L={\tilde {X}}\times \mathbb {C} /\pi _{1}(X)}$$

is a complex manifold. Furthermore, the projection ${\displaystyle p:L\to X}$ induced from the covering projection ${\displaystyle \pi$ :{\tilde {X}}\to X} . This gives a map

$${\displaystyle Z^{1}(\pi _{1}(X),H^{0}({\mathcal {O}}_{\tilde {X}}^{*}))\to H^{1}(X,{\mathcal {O}}_{X}^{*})}$$

which induces an isomorphism

$${\displaystyle H^{1}(\pi _{1}(X),H^{0}({\mathcal {O}}_{\tilde {X}}^{*}))\to \ker(H^{1}(X,{\mathcal {O}}_{X}^{*})\to H^{1}({\tilde {X}},{\mathcal {O}}_{\tilde {X}}^{*}))}$$

giving the desired result.

In the case of complex tori, we have ${\displaystyle H^{1}({\tilde {X}},{\mathcal {O}}_{\tilde {X}}^{*})\cong 0}$ hence there is an isomorphism

$${\displaystyle H^{1}(\pi _{1}(X),H^{0}({\mathcal {O}}_{\tilde {X}}^{*}))\cong H^{1}(X,{\mathcal {O}}_{X}^{*})}$$

representing line bundles on complex tori as 1-cocyles in the associated group cohomology. It is typical to write down the group ${\displaystyle \pi _{1}(X)}$ as the lattice ${\displaystyle \Lambda }$ defining ${\displaystyle X}$, hence

$${\displaystyle H^{1}(\Lambda ,H^{0}({\mathcal {O}}_{V}^{*}))}$$

contains the isomorphism classes of line bundles on ${\displaystyle X}$.

From the exponential exact sequence

$${\displaystyle 0\to \mathbb {Z} \to {\mathcal {O}}_{X}\to {\mathcal {O}}_{X}^{*}\to 0}$$

the connecting morphism

$${\displaystyle c_{1}:H^{1}({\mathcal {O}}_{X}^{*})\to H^{2}(X,\mathbb {Z} )}$$

is the first Chern class map, sending an isomorphism class of a line bundle to its associated first Chern class. It turns out there is an isomorphism between ${\displaystyle H^{2}(X,\mathbb {Z} )}$ and the module of alternating forms on the lattice ${\displaystyle \Lambda }$, ${\displaystyle Alt^{2}(\Lambda ,\mathbb {Z} )}$. Therefore, ${\displaystyle c_{1}(L)}$ can be considered as an alternating ${\displaystyle \mathbb {Z} }$-valued 2-form ${\displaystyle E_{L}}$ on ${\displaystyle \Lambda }$. If ${\displaystyle L}$ has factor of automorphy ${\displaystyle f=\exp(2\pi ig)}$ then the alternating form can be expressed as

$${\displaystyle E_{L}(\lambda ,\mu )=g(\mu ,v+\lambda )+g(\lambda ,v)-g(\lambda ,v+\mu )-g(\mu ,v)}$$

for ${\displaystyle \mu ,\lambda \in \Lambda }$ and ${\displaystyle v\in V}$.

For a normalized period matrix

$${\displaystyle \Pi ={\begin{pmatrix}z_{1,1}&z_{1,2}&1&0\\z_{2,1}&z_{2,2}&0&1\end{pmatrix}}}$$

expanded using the standard basis of ${\displaystyle \mathbb {C} ^{2}}$ we have the column vectors defining the lattice ${\displaystyle \Lambda \subset \mathbb {C} ^{2}}$. Then, any alternating form ${\displaystyle E_{L}}$ on ${\displaystyle \Lambda }$ is of the form

$${\displaystyle E_{L}={\begin{pmatrix}0&e_{2,1}&e_{3,1}&e_{4,1}\\-e_{2,1}&0&e_{3,2}&e_{4,2}\\-e_{3,1}&-e_{2,3}&0&e_{4,3}\\-e_{4,1}&-e_{4,2}&-e_{4,3}&0\end{pmatrix}}}$$

where a number of compatibility conditions must be satisfied.

For a line bundle ${\displaystyle L}$ given by a factor of automorphy ${\displaystyle f:\Lambda \times V\to \mathbb {C} ^{*}}$, so ${\displaystyle [f]\in H^{1}(\Lambda ,H^{0}(V,{\mathcal {O}}_{V}^{*}))}$ and ${\displaystyle \phi _{1}[f]=[L]\in {\text{Pic}}(X)}$, there is an associated sheaf of sections ${\displaystyle {\mathcal {L}}}$ where

$${\displaystyle {\mathcal {L}}(U)=\left\{\theta$$ :\pi ^{-1}(U)\to \mathbb {C} :{\begin{matrix}\theta {\text{ holomorphic with }}\theta (v+\lambda )=f(\lambda ,v)\theta (v)\\{\text{for all }}(\lambda ,v)\in \Lambda \times \pi ^{-1}(U)\end{matrix}}\right\}}

with ${\displaystyle U\subset X}$ open. Then, evaluated on global sections, this is the set of holomorphic functions ${\displaystyle \theta :V\to \mathbb {C} }$ such that

$${\displaystyle \theta (v+\lambda )=f(\lambda ,v)\theta (v)}$$

which are exactly the theta functions on the plane. Conversely, this process can be done backwards where the automorphic factor in the theta function is in fact the factor of automorphy defining a line bundle on a complex torus.

For a complex torus ${\displaystyle X=V/\Lambda }$ we can define the Neron-Serveri group ${\displaystyle NS(X)}$ as the group of Hermitian forms ${\displaystyle H}$ on ${\displaystyle V}$ with

$${\displaystyle {\text{Im}}H(\Lambda ,\Lambda )\subseteq \mathbb {Z} }$$

Equivalently, it is the image of the homomorphism

$${\displaystyle c_{1}:H^{1}({\mathcal {O}}_{X}^{*})\to H^{2}(X,\mathbb {Z} )}$$

from the first Chern class. We can also identify it with the group of alternating real-valued alternating forms ${\displaystyle E}$ on ${\displaystyle V}$ such that ${\displaystyle E(\Lambda ,\Lambda )\subseteq \mathbb {Z} }$.

For[4] an elliptic curve ${\displaystyle {\mathcal {E}}}$ given by the lattice ${\displaystyle {\begin{pmatrix}1&\tau \end{pmatrix}}}$ where ${\displaystyle \tau \in \mathbb {H} }$ we can find the integral form ${\displaystyle E\in {\text{Alt}}^{2}(\Lambda ,\mathbb {Z} )}$ by looking at a generic alternating matrix and finding the correct compatibility conditions for it to behave as expected. If we use the standard basis ${\displaystyle x_{1},y_{1}}$ of ${\displaystyle \mathbb {C} }$ as a real vector space (so ${\displaystyle z=z_{1}+iz_{2}=z_{1}x_{1}+z_{2}y_{1}}$), then we can write out an alternating matrix

$${\displaystyle E={\begin{pmatrix}0&e\\-e&0\end{pmatrix}}}$$

and calculate the associated products on the vectors associated to ${\displaystyle 1,\tau }$. These are

$${\displaystyle {\begin{aligned}E\cdot {\begin{pmatrix}1\\0\end{pmatrix}}={\begin{pmatrix}0\\-e\end{pmatrix}}&&E\cdot {\begin{pmatrix}\tau _{1}\\\tau _{2}\end{pmatrix}}={\begin{pmatrix}e\tau _{2}\\-e\tau _{1}\end{pmatrix}}\end{aligned}}}$$

Then, taking the inner products (with the standard inner product) of these vectors with the vectors ${\displaystyle 1,\tau }$ we get

$${\displaystyle {\begin{aligned}{\begin{pmatrix}1\\0\end{pmatrix}}\cdot {\begin{pmatrix}0\\-e\end{pmatrix}}=0&&{\begin{pmatrix}\tau _{1}\\\tau _{2}\end{pmatrix}}\cdot {\begin{pmatrix}0\\-e\end{pmatrix}}=-e\tau _{2}\\{\begin{pmatrix}1\\0\end{pmatrix}}\cdot {\begin{pmatrix}e\tau _{2}\\-e\tau _{1}\end{pmatrix}}=e\tau _{2}&&{\begin{pmatrix}\tau _{1}\\\tau _{2}\end{pmatrix}}\cdot {\begin{pmatrix}e\tau _{2}\\-e\tau _{1}\end{pmatrix}}=0\end{aligned}}}$$

so if ${\displaystyle E(\Lambda ,\Lambda )\subset \mathbb {Z} }$, then

$${\displaystyle e=a{\frac {1}{{\text{Im}}(\tau )}}}$$

We can then directly verify ${\displaystyle E(v,w)=E(iv,iw)}$, which holds for the matrix above. For a fixed ${\displaystyle a}$, we will write the integral form as ${\displaystyle E_{a}}$. Then, there is an associated Hermitian form

$${\displaystyle H_{a}:\mathbb {C} \times \mathbb {C} \to \mathbb {C} }$$

given by

$${\displaystyle H_{a}(z,w)=a\cdot {\frac {z{\overline {w}}}{{\text{Im}}(\tau )}}}$$

where ${\displaystyle a\in \mathbb {Z} }$

For a Hermitian form ${\displaystyle H}$ a semi-character is a map ${\displaystyle \chi$ :\Lambda \to U(1)} such that

$${\displaystyle \chi (\lambda +\mu )=\chi (\lambda )\chi (\mu )\exp(i\pi {\text{Im}}H(\lambda ,\mu ))}$$

hence the map ${\displaystyle \chi }$ behaves like a character twisted by the Hermitian form. Note that if ${\displaystyle H}$ is the zero element in ${\displaystyle NS(X)}$, so it corresponds to the trivial line bundle ${\displaystyle \mathbb {C} \times X\to X}$, then the associated semi-characters are the group of characters on ${\displaystyle \Lambda }$. It will turn out this corresponds to the group ${\displaystyle {\text{Pic}}^{0}(X)}$ of degree ${\displaystyle 0}$ line bundles on ${\displaystyle X}$, or equivalently, its dual torus, which can be seen by computing the group of characters ${\displaystyle {\text{Hom}}(\Lambda ,U(1))}$ whose elements can be factored as maps ${\displaystyle \Lambda \to \mathbb {R} \to \mathbb {R} /\mathbb {Z} \cong U(1)}$ showing a character is of the form ${\displaystyle \chi (\cdot )=\exp \left(2\pi iv^{*}(\cdot )\right)}$ for some fixed dual lattice vector ${\displaystyle v^{*}\in \Lambda ^{*}}$. This gives the isomorphism ${\displaystyle {\text{Hom}}(\Lambda ,U(1))\cong \mathbb {R} ^{2g}/\mathbb {Z} ^{2g}}$ of the set of characters with a real torus. The set of all pairs of semi-characters and their associated Hermitian form ${\displaystyle (\chi ,H)}$, or semi-character pairs, forms a group ${\displaystyle {\mathcal {P}}(\Lambda )}$ where

$${\displaystyle (H_{1},\chi _{1})*(H_{2},\chi _{2})=(H_{1}+H_{2},\chi _{1}\chi _{2})}$$

This group structure comes from applying the previous commutation law for semi-characters to the new semicharacter ${\displaystyle \chi _{1}\chi _{2}}$:

$${\displaystyle {\begin{aligned}\chi _{1}\chi _{2}(\lambda +\mu )&=\chi _{1}(\lambda +\mu )\chi _{2}(\lambda +\mu )\\&=\chi _{1}(\lambda )\chi _{1}(\mu )\chi _{2}(\lambda )\chi _{2}(\mu )\exp(i\pi {\text{Im}}H_{1}(\lambda ,\mu ))\exp(i\pi {\text{Im}}H_{2}(\lambda ,\mu ))\\&=\chi _{1}\chi _{2}(\lambda )\chi _{1}\chi _{2}(\mu )\exp(i\pi {\text{Im}}H_{1}(\lambda ,\mu )+i\pi {\text{Im}}H_{2}(\lambda ,\mu ))\end{aligned}}}$$

It turns out this group surjects onto ${\displaystyle NS(X)}$ and has kernel ${\displaystyle {\text{Hom}}(\Lambda ,U(1))}$, giving a short exact sequence

$${\displaystyle 1\to {\text{Hom}}(\Lambda ,U(1))\to {\mathcal {P}}(\Lambda )\to NS(X)\to 1}$$

This surjection can be constructed through associating to every semi-character pair a line bundle ${\displaystyle L(H,\chi )}$.

For a semi-character pair ${\displaystyle (H,\chi )}$ we can construct a 1-cocycle ${\displaystyle a_{(H,\chi )}}$ on ${\displaystyle \Lambda }$ as a map

$${\displaystyle a_{(H,\chi )}:\Lambda \times V\to \mathbb {C} ^{*}}$$

defined as

$${\displaystyle a(\lambda ,v)=\chi (\lambda )\exp(\pi H(v,\lambda )+{\frac {\pi }{2}}H(\lambda ,\lambda ))}$$

The cocycle relation

$${\displaystyle a(\lambda +\mu ,v)=a(\lambda ,v+\mu )a(\mu ,v)}$$

can be easily verified by direct computation. Hence the cocycle determines a line bundle

$${\displaystyle L(H,\chi )\cong V\times \mathbb {C} /\Lambda }$$

where the ${\displaystyle \Lambda }$-action on ${\displaystyle V\times \mathbb {C} }$ is given by

$${\displaystyle \lambda \circ (v,t)=(v+t,a_{(H,\chi )}(\lambda ,v)t)}$$

Note this action can be used to show the sections of the line bundle ${\displaystyle L(H,\chi )}$ are given by the theta functions with factor of automorphy ${\displaystyle a_{(H,\chi )}}$. Sometimes, this is called the canonical factor of automorphy for ${\displaystyle L}$. Note that because every line bundle ${\displaystyle L\to X}$ has an associated Hermitian form ${\displaystyle H}$, and a semi-character can be constructed using the factor of automorphy for ${\displaystyle L}$, we get a surjection

$${\displaystyle {\mathcal {P}}(\Lambda )\to {\text{Pic}}(X)}$$

Moreover, this is a group homomorphism with a trivial kernel. These facts can all be summarized in the following commutative diagram

$${\displaystyle {\begin{matrix}1&\to &{\text{Hom}}(\Lambda ,U(1))&\to &{\mathcal {P}}(\Lambda )&\to &NS(X)&\to 0\\&&\downarrow &&\downarrow &&\downarrow \\1&\to &{\text{Pic}}^{0}(X)&\to &{\text{Pic}}(X)&\to &{\text{NS}}(X)&\to 0\end{matrix}}}$$

where the vertical arrows are isomorphisms, or equality. This diagram is typically called the Appell-Humbert theorem.

From the construction of the dual complex torus, it is suggested there should exist a line bundle ${\displaystyle {\mathcal {P}}}$ over the product of the torus ${\displaystyle X}$ and its dual which can be used to present all isomorphism classes of degree 0 line bundles on ${\displaystyle X}$. We can encode this behavior with the following two properties

1. ${\displaystyle {\mathcal {P}}|_{X\times \{[L]\}}\cong L}$ for any point ${\displaystyle [L]\in {\hat {X}}}$ giving the line bundle ${\displaystyle L}$
2. ${\displaystyle {\mathcal {P}}|_{\{0\}\times {\hat {X}}}}$ is a trivial line bundle

where the first is the property discussed above, and the second acts as a normalization property. We can construct ${\displaystyle {\mathcal {P}}}$ using the following hermitian form

$${\displaystyle {\begin{matrix}H:(V\times {\overline {\Omega }})\times (V\times {\overline {\Omega }})\to \mathbb {C} \\H((v_{1},l_{1}),(v_{2},l_{2}))={\overline {l_{2}(v_{1})}}+l_{1}(v_{2})\end{matrix}}}$$

and the semi-character

$${\displaystyle {\begin{matrix}\chi$$ :\Lambda \times {\hat {\Lambda }}\to U(1)\\\chi (\lambda ,l_{0})=\exp(i\pi {\text{Im}}l_{0}(\lambda ))\end{matrix}}}

for ${\displaystyle H}$. Showing this data constructs a line bundle with the desired properties follows from looking at the associated canonical factor of ${\displaystyle (H,\chi )}$, and observing its behavior at various restrictions.

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