complex torus

One way to define complex tori{{Cite book|last=Mumford|first=David|url=https://www.worldcat.org/oclc/297809496|title=Abelian varieties|date=2008|publisher=Published for the Tata Institute of Fundamental Research|others=C. P. Ramanujam, I︠U︡. I. Manin|isbn=978-8185931869|oclc=297809496}} is as a compact connected complex Lie group $G$. 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 $\backslash mathfrak\{g\}\; =\; T\_0G$ whose covering map is the exponential map of a Lie algebra to its associated Lie group. The kernel of this map is a lattice $\backslash Lambda\; \backslash subset\; \backslash mathfrak\{g\}$ and $\backslash mathfrak\{g\}/\backslash Lambda\; \backslash cong\; G$.

Conversely, given a complex vector space $V$ and a lattice $\backslash Lambda\; \backslash subseteq\; V$ of maximal rank, the quotient complex manifold $V/\backslash Lambda$ 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{{Cite book|last=Birkenhake|first=Christina|url=https://www.worldcat.org/oclc/851380558|title=Complex Abelian Varieties|date=2004|publisher=Springer Berlin Heidelberg|others=Herbert Lange|isbn=978-3-662-06307-1|edition=Second, augmented|location=Berlin, Heidelberg|oclc=851380558}}{{rp|9}} is by using a $g\backslash times\; 2g$ matrix $\backslash Pi$ whose columns correspond to a basis $\backslash lambda\_1,\backslash ldots,\; \backslash lambda\_\{2g\}$ of the lattice $\backslash Lambda$ expanded out using a basis $e\_1,\backslash ldots,e\_g$ of $V$. That is, we write

$$\backslash Pi\; =\; \backslash begin\{pmatrix\}$$

\lambda_{1,1} & \cdots & \lambda_{1,2g} \\

\vdots & & \vdots \\

\lambda_{g,1} & \cdots & \lambda_{g,2g}

\end{pmatrix}

so

$$\backslash lambda\_i\; =\; \backslash sum\_\{j\}\backslash lambda\_\{ji\}e\_j$$

We can then write the torus $X\; =\; V/\backslash Lambda$ as

$$X\; =\; \backslash mathbb\{C\}^g/\backslash Pi\backslash mathbb\{Z\}^\{2g\}.$$

If we go in the reverse direction by selecting a matrix $\backslash Pi\; \backslash in\; Mat\_\backslash mathbb\{C\}(g,2g)$, it corresponds to a period matrix if and only if the corresponding matrix $P\; \backslash in\; Mat\_\backslash mathbb\{C\}(2g,2g)$ constructed by adjoining the complex conjugate matrix $\backslash overline\{\backslash Pi\}$ to $\backslash Pi$, so

$$P\; =\; \backslash begin\{pmatrix\}$$

\Pi \\

\overline{\Pi}

\end{pmatrix}

is nonsingular. This guarantees the column vectors of $\backslash Pi$ span a lattice in $\backslash mathbb\{C\}^g$ hence must be linearly independent vectors over $\backslash mathbb\{R\}$.

For a two-dimensional complex torus, it has a period matrix of the form

$$\backslash Pi\; =\; \backslash begin\{pmatrix\}$$

\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

$$\backslash Pi\; =\; \backslash begin\{pmatrix\}$$

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 $X\; =\; V/\backslash Lambda$ of dimension $g$ it has a period matrix $\backslash Pi$ of the form

$$(Z,\; 1\_g)$$where $1\_g$ is the identity matrix and $Z\; \backslash in\; Mat\_\backslash mathbb\{C\}(g)$ where $\backslash det\backslash text\{Im\}(Z)\; \backslash neq\; 0$. We can get this from taking a change of basis of the vector space $V$ giving a block matrix of the form above. The condition for $\backslash det\backslash text\{Im\}(Z)\; \backslash neq\; 0$ follows from looking at the corresponding $P$-matrix

$$\backslash begin\{pmatrix\}$$

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

$$\backslash begin\{align\}$$

\det P &= \det(1_g)\det(Z - 1_g1_g\overline{Z}) \\

&= \det(Z-\overline{Z}) \\

&\Rightarrow \det(\text{Im}(Z)) \neq 0

\end{align}

which gives the implication.

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

$$\backslash begin\{pmatrix\}$$

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

$$\backslash begin\{pmatrix\}$$

1+i & 1 - i & 1 & 0\\

1+2i & 1+\sqrt{2}i & 0 & 1

\end{pmatrix}

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

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.{{Cite web|title=Riemann bilinear relations|url=https://www2.math.upenn.edu/~chai/papers_pdf/bilinear_relations.pdf|url-status=live|archive-url=https://web.archive.org/web/20210531181635/https://www2.math.upenn.edu/~chai/papers_pdf/bilinear_relations.pdf|archive-date=31 May 2021}}

If we have complex tori $X\; =\; V/\backslash Lambda$ and $X\text{'}\; =\; V\text{'}/\backslash Lambda\text{'}$ of dimensions $g,g\text{'}$, then a homomorphism{{rp|11}} of complex tori is a function

$$f:X\; \backslash to\; X\text{'}$$

such that the group structure is preserved. This has a number of consequences, such as every homomorphism induces a map of their covering spaces

$$F:V\; \backslash to\; V\text{'}$$

which is compatible with their covering maps. Furthermore, because $F$ induces a group homomorphism, it must restrict to a morphism of the lattices

$$F\_\backslash Lambda:\backslash Lambda\; \backslash to\; \backslash Lambda\; \text{'}$$In particular, there are injections

$$\backslash rho\_a:\backslash text\{Hom\}(X,X\text{'})\; \backslash to\; \backslash text\{Hom\}\_\backslash mathbb\{C\}(V,V\text{'})$$ and $\backslash rho\_r:\backslash text\{Hom\}(X,X\text{'})\; \backslash to\; \backslash text\{Hom\}\_\backslash mathbb\{Z\}(\backslash Lambda,\backslash Lambda\text{'})$

which are called the analytic and rational representations of the space of homomorphisms. These are useful to determining some information about the endomorphism ring $\backslash text\{End\}(X)\backslash otimes\backslash mathbb\{Q\}$ which has rational dimension $m\; \backslash leq\; 4gg\text{'}$.

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 $x\_0\; \backslash in\; X$ we define the translation map

$$t\_\{x\_0\}:X\backslash to\; X$$ sending $x\; \backslash mapsto\; x+x\_0$ Then, if $h$ is a holomorphic map between complex tori $X,X\text{'}$, there is a unique homomorphism $f:X\; \backslash to\; X\text{'}$ such that

$$h\; =\; t\_\{h(0)\}\backslash circ\; f$$

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 $n\; \backslash in\; \backslash mathbb\{Z\}\_\{\backslash neq\; 0\}$ be an integer, then there is an associated map

$$n\_X\; :X\backslash to\; X$$ sending $x\backslash mapsto\; nx$ which has kernel

$$X\_n\; \backslash cong\; (\backslash mathbb\{Z\}/n\backslash mathbb\{Z\})^\{2g\}$$

isomorphic to $\backslash Lambda\; /\; n\backslash Lambda$.

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

$$GL\_\backslash mathbb\{R\}(2g)/GL\_\backslash mathbb\{C\}(g)$$

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

$$\backslash mathcal\{T\}\_g\; \backslash cong\; GL\_\backslash mathbb\{Z\}(2g)\backslash backslash\; GL\_\backslash mathbb\{R\}(2g)$$

/GL_\mathbb{C}(g)

Note that as a real manifold, this has dimension

$$4g^2\; -\; 2g^2\; =\; 2g^2$$

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

For complex manifolds $X$, in particular complex tori, there is a construction{{rp|571}} relating the holomorphic line bundles $L\; \backslash to\; X$ whose pullback $\backslash pi^*L\; \backslash to\; \backslash tilde\{X\}$ are trivial using the group cohomology of $\backslash pi\_1(X)$. Fortunately for complex tori, every complex line bundle $\backslash pi^*L$ becomes trivial since $\backslash tilde\{X\}\; \backslash cong\; \backslash mathbb\{C\}^n$.

Starting from the first group cohomology group

$$H^1(\backslash pi\_1(X),H^0(\backslash mathcal\{O\}\_\{\backslash tilde\{X\}\}^*))$$we recall how its elements can be represented. Since $\backslash pi\_1(X)$ acts on $\backslash tilde\{X\}$ there is an induced action on all of its sheaves, hence on

$$H^0(\backslash mathcal\{O\}^*\_\{\backslash tilde\{X\}\})\; =\; \backslash \{\; f:\; \backslash tilde\{X\}\; \backslash to\; \backslash mathbb\{C\}^*\; \backslash \}$$The $\backslash pi\_1(X)$-action can then be represented as a holomorphic map $f:\backslash pi\_1(X)\backslash times\backslash tilde\{X\}\; \backslash to\; \backslash mathbb\{C\}^*$. This map satisfies the cocycle condition if

$$f(a\backslash cdot\; b,\; x)\; =\; f(a,b\backslash cdot\; x)f(b,\; x)$$

for every $a,b\; \backslash in\; \backslash pi\_1(X)$ and $x\; \backslash in\; \backslash tilde\{X\}$. The abelian group of 1-cocycles $Z^1(\backslash pi\_1(X),H^0(\backslash mathcal\{O\}\_\{\backslash tilde\{X\}\}^*))$ is called the group of factors of automorphy. Note that such functions $f$ are also just called factors.

For complex tori, these functions $f$ are given by functions

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

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

$$f\; =\; \backslash exp(2\; \backslash pi\; i\; \backslash cdot\; g)$$

for

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

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

Given a factor of automorphy $f$ we can define a line bundle on $X$ as follows: the trivial line bundle $\backslash tilde\{X\}\backslash times\backslash mathbb\{C\}\; \backslash to\; \backslash tilde\{X\}$ has a $\backslash pi\_1(X)$-action given by

$$a\backslash cdot\; (x,t)\; =\; (a\backslash cdot\; x,\; f(a,x)\backslash cdot\; t)$$

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

$$L\; =\; \backslash tilde\{X\}\backslash times\; \backslash mathbb\{C\}/\backslash pi\_1(X)$$

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

$$Z^1(\backslash pi\_1(X),H^0(\backslash mathcal\{O\}\_\backslash tilde\{X\}^*))\; \backslash to\; H^1(X,\backslash mathcal\{O\}\_X^*)$$

which induces an isomorphism

$$H^1(\backslash pi\_1(X),H^0(\backslash mathcal\{O\}\_\backslash tilde\{X\}^*))\; \backslash to\; \backslash ker(H^1(X,\backslash mathcal\{O\}\_X^*)\; \backslash to\; H^1(\backslash tilde\{X\},\backslash mathcal\{O\}\_\backslash tilde\{X\}^*))$$

giving the desired result.

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

$$H^1(\backslash pi\_1(X),H^0(\backslash mathcal\{O\}\_\backslash tilde\{X\}^*))\; \backslash cong\; H^1(X,\backslash mathcal\{O\}\_X^*)$$

representing line bundles on complex tori as 1-cocyles in the associated group cohomology. It is typical to write down the group $\backslash pi\_1(X)$ as the lattice $\backslash Lambda$ defining $X$, hence

$$H^1(\backslash Lambda,H^0(\backslash mathcal\{O\}\_V^*))$$

contains the isomorphism classes of line bundles on $X$.

From the exponential exact sequence

$$0\; \backslash to\; \backslash mathbb\{Z\}\; \backslash to\; \backslash mathcal\{O\}\_X\; \backslash to\; \backslash mathcal\{O\}\_X^*\; \backslash to\; 0$$the connecting morphism

$$c\_1:H^1(\backslash mathcal\{O\}\_X^*)\; \backslash to\; H^2(X,\backslash mathbb\{Z\})$$

is the first Chern class map, sending an isomorphism class of a line bundle to its associated first Chern class. It turns out that there is an isomorphism between $H^2(X,\backslash mathbb\{Z\})$ and the module of alternating forms on the lattice $\backslash Lambda$, $Alt^2(\backslash Lambda,\; \backslash mathbb\{Z\})$. Therefore, $c\_1(L)$ can be considered as an alternating $\backslash mathbb\{Z\}$-valued 2-form $E\_L$ on $\backslash Lambda$. If $L$ has factor of automorphy $f\; =\; \backslash exp(2\backslash pi\; i\; g)$ then the alternating form can be expressed as

$$E\_L(\backslash lambda,\; \backslash mu)\; =\; g(\backslash mu,\; v+\backslash lambda)\; +\; g(\backslash lambda,\; v)\; -\; g(\backslash lambda,\; v\; +\; \backslash mu)\; -\; g(\backslash mu,\; v)$$for $\backslash mu,\backslash lambda\; \backslash in\; \backslash Lambda$ and $v\; \backslash in\; V$.

For a normalized period matrix

$$\backslash Pi\; =\; \backslash begin\{pmatrix\}$$

z_{1,1} & z_{1,2} & 1 & 0 \\

z_{2,1} & z_{2,2} & 0 & 1

\end{pmatrix}

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

$$E\_L\; =\; \backslash begin\{pmatrix\}$$

0 & e_{2,1} & e_{3,1} & e_{4,1} \\

-e_{2,1} & 0 & e_{3,2} & e_{4,2} \\

-e_{3,1} & -e_{3,2} & 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 $L$ given by a factor of automorphy $f:\backslash Lambda\backslash times\; V\; \backslash to\; \backslash mathbb\{C\}^\{*\}$, so $[f]\; \backslash in\; H^1(\backslash Lambda,\; H^0(V,\backslash mathcal\{O\}\_V^*))$ and $\backslash phi\_1[f]\; =\; [L]\; \backslash in\; \backslash text\{Pic\}(X)$, there is an associated sheaf of sections $\backslash mathcal\{L\}$ where

$$\backslash mathcal\{L\}(U)\; =\; \backslash left\backslash \{$$

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

$$\backslash theta(v\; +\; \backslash lambda)\; =\; f(\backslash lambda,\; v)\backslash theta(v)$$

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.

{{See also|Appell–Humbert theorem}}

For the alternating $\backslash mathbb\{Z\}$-valued 2-form $E\_L$ associated to the line bundle $L\; \backslash to\; X$, it can be extended to be $\backslash mathbb\{R\}$-valued. Then, it turns out that any $\backslash mathbb\{R\}$-valued alternating form $E:V\backslash times\; V\; \backslash to\; \backslash mathbb\{R\}$ satisfying the following conditions

1. $E(\backslash Lambda,\backslash Lambda)\backslash subseteq\; \backslash mathbb\{Z\}$
2. $E(iv,iw)\; =\; E(v,w)$ for any $v,w\backslash in\; V$

is the extension of some first Chern class $c\_1(L)$ of a line bundle $L\; \backslash to\; X$. Moreover, there is an associated Hermitian form $H:V\backslash times\; V\; \backslash to\; \backslash mathbb\{C\}$ satisfying

1. $\backslash text\{Im\}H(v,w)\; =\; E(v,w)$
2. $H(v,w)\; =\; E(iv,w)\; +\; iE(v,w)$

for any $v,w\; \backslash in\; V$.

{{See also|Néron–Severi group}}

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

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

Equivalently, it is the image of the homomorphism

$$c\_1:H^1(\backslash mathcal\{O\}\_X^*)\; \backslash to\; H^2(X,\backslash mathbb\{Z\})$$

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

For{{Cite web|title=How Appell-Humbert theorem works in the simplest case of an elliptic curve|url=https://math.stackexchange.com/a/1113513}} an elliptic curve $\backslash mathcal\{E\}$ given by the lattice where $\backslash tau\; \backslash in\; \backslash mathbb\{H\}$ we can find the integral form $E\; \backslash in\; \backslash text\{Alt\}^2(\backslash Lambda,\backslash mathbb\{Z\})$ 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 $x\_1,y\_1$ of $\backslash mathbb\{C\}$

as a real vector space (so $z\; =\; z\_1\; +\; iz\_2\; =\; z\_1x\_1\; +\; z\_2y\_1$), then we can write out an alternating matrix

$$E\; =\; \backslash begin\{pmatrix\}$$

0 & e \\

-e & 0

\end{pmatrix}

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

$$\backslash begin\{align\}$$

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{align}

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

$$\backslash begin\{align\}$$

\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{align}

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

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

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

$$H\_a:\backslash mathbb\{C\}\backslash times\backslash mathbb\{C\}\backslash to\backslash mathbb\{C\}$$

given by

$$H\_a(z,w)\; =\; a\backslash cdot\; \backslash frac\{z\backslash overline\{w\}\}\{\backslash text\{Im\}(\backslash tau)\}$$

where $a\; \backslash in\; \backslash mathbb\{Z\}$

For a Hermitian form $H$ a semi-character is a map

$\backslash chi:\backslash Lambda\; \backslash to\; U(1)$

such that

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

hence the map $\backslash chi$ behaves like a character twisted by the Hermitian form. Note that if $H$ is the zero element in $NS(X)$, so it corresponds to the trivial line bundle $\backslash mathbb\{C\}\backslash times\; X\; \backslash to\; X$, then the associated semi-characters are the group of characters on $\backslash Lambda$. It will turn out that this corresponds to the group $\backslash text\{Pic\}^0(X)$ of degree $0$ line bundles on $X$, or equivalently, its dual torus, which can be seen by computing the group of characters

$\backslash text\{Hom\}(\backslash Lambda,\; U(1))$

whose elements can be factored as maps

$\backslash Lambda\; \backslash to\; \backslash mathbb\{R\}\; \backslash to\; \backslash mathbb\{R\}/\backslash mathbb\{Z\}\; \backslash cong\; U(1)$

showing a character is of the form

$\backslash chi(\backslash cdot)\; =\; \backslash exp\backslash left(2\backslash pi\; i\; v^*(\backslash cdot)\; \backslash right)$

for some fixed dual lattice vector $v^*\; \backslash in\; \backslash Lambda^*$. This gives the isomorphism

$\backslash text\{Hom\}(\backslash Lambda,\; U(1))\; \backslash cong\; \backslash mathbb\{R\}^\{2g\}/\backslash mathbb\{Z\}^\{2g\}$

of the set of characters with a real torus. The set of all pairs of semi-characters and their associated Hermitian form $(\backslash chi,\; H)$, or semi-character pairs, forms a group $\backslash mathcal\{P\}(\backslash Lambda)$ where

$$(H\_1,\backslash chi\_1)*(H\_2,\backslash chi\_2)\; =\; (H\_1\; +\; H\_2,\backslash chi\_1\backslash chi\_2)$$

This group structure comes from applying the previous commutation law for semi-characters to the new semicharacter $\backslash chi\_1\backslash chi\_2$:

$$\backslash begin\{align\}$$

\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{align}

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

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

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

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

$$a\_\{(H,\backslash chi)\}:\backslash Lambda\backslash times\; V\; \backslash to\; \backslash mathbb\{C\}^*$$defined as

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

The cocycle relation

$$a(\backslash lambda+\backslash mu,\; v)\; =\; a(\backslash lambda,\; v+\backslash mu)a(\backslash mu,v)$$

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

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

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

$$\backslash lambda\backslash circ(v,t)=\; (v+t,\; a\_\{(H,\backslash chi)\}(\backslash lambda,\; v)t)$$

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

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

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

$$\backslash begin\{matrix\}$$

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.

{{See also|Poincare bundle|Dual abelian variety}}

As mentioned before, a character on the lattice can be expressed as a function

$$\backslash chi(\backslash cdot)\; =\; \backslash exp\backslash left(2\backslash pi\; i\; v^*(\backslash cdot)\; \backslash right)$$

for some fixed dual vector $v^*\backslash in\backslash Lambda^*$. If we want to put a complex structure on the real torus of all characters, we need to start with a complex vector space which $\backslash Lambda^*$ embeds into. It turns out that the complex vector space

$$\backslash overline\{\backslash Omega\}\; =\; \backslash text\{Hom\}\_\{\backslash overline\{\backslash mathbb\{C\}\}\}(V,\backslash mathbb\{C\})$$

of complex antilinear maps, is isomorphic to the real dual vector space $\backslash text\{Hom\}\_\backslash mathbb\{R\}(V,\backslash mathbb\{R\})$, which is part of the factorization for writing down characters. Furthermore, there is an associated lattice

$$\backslash hat\{\backslash Lambda\}\; =\; \backslash \{\; l\; \backslash in\; \backslash overline\{\backslash Omega\}\; :\; \backslash langle\; l,\; \backslash Lambda\; \backslash rangle\; \backslash subseteq\; \backslash mathbb\{Z\}\; \backslash \}$$

called the dual lattice of $\backslash Lambda$. Then, we can form the dual complex torus

$$\backslash hat\{X\}\; \backslash cong\; \backslash overline\{\backslash Omega\}/\backslash hat\{\backslash Lambda\}$$

which has the special property that that dual of the dual complex torus is the original complex torus. Moreover, from the discussion above, we can identify the dual complex torus with the Picard group of $X$

$$\backslash hat\{X\}\; \backslash cong\; \backslash text\{Pic\}^0(X)$$

by sending an anti-linear dual vector $l$ to

$$l\; \backslash mapsto\; \backslash exp(2\backslash pi\; i\; \backslash langle\; l,\; \backslash cdot\; \backslash rangle)$$

giving the map

$$\backslash overline\{\backslash Omega\}\; \backslash to\; \backslash text\{Hom\}(\backslash Lambda,U(1))$$

which factors through the dual complex torus. There are other constructions of the dual complex torus using techniques from the theory of Abelian varieties.{{rp|123–125}} Essentially, taking a line bundle $L$ over a complex torus (or Abelian variety) $X$, there is a closed subset $K(L)$ of $X$ defined as the points of $x\backslash in\; X$ where their translations are invariant, i.e.

$$T^*\_x(L)\; \backslash cong\; L$$

Then, the dual complex torus can be constructed as

$$\backslash hat\{X\}\; :=\; X/K(L)$$

presenting it as an isogeny. It can be shown that defining $\backslash hat\{X\}$ this way satisfied the universal properties of $\backslash text\{Pic\}^0(X)$, hence is in fact the dual complex torus (or Abelian variety).

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

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

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

$$\backslash begin\{matrix\}$$

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

$$\backslash begin\{matrix\}$$

\chi:\Lambda\times\hat{\Lambda} \to U(1) \\

\chi(\lambda,l_0) = \exp(i\pi \text{Im}l_0(\lambda))

\end{matrix}

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

• Poincare bundle
• Complex Lie group
• Automorphic function
• Intermediate Jacobian
• Elliptic gamma function

• {{Citation | last1=Birkenhake | first1=Christina | last2=Lange | first2=Herbert | title=Complex tori | publisher=Birkhäuser Boston | location=Boston, MA | series=Progress in Mathematics | isbn=978-0-8176-4103-0 | mr=1713785 | year=1999 | volume=177}}

• {{cite journal |doi=10.1007/BF02570737 |title=When is an abelian surface isomorphic or isogeneous to a product of elliptic curves? |year=1990 |last1=Ruppert |first1=Wolfgang M. |journal=Mathematische Zeitschrift |volume=203 |pages=293–299 |s2cid=120799085|url=http://gdz.sub.uni-goettingen.de/dms/resolveppn/?PPN=GDZPPN002437074|url-access=subscription }} - Gives tools to find complex tori which are not Abelian varieties
• {{cite journal|url=https://eudml.org/doc/195875|title=Abelian surfaces and products of elliptic curves |journal=Bollettino dell'unione Matematica Italiana |year=1998 |volume=1-B |issue=2 |pages=407–427 |last1=Marchisio |first1=Marina Rosanna }}

• {{cite journal|arxiv=0811.2746|doi=10.4171/JNCG/96 |title=Gerbes and the holomorphic Brauer group of complex tori |year=2012 |last1=Ben-Bassat |first1=Oren |journal=Journal of Noncommutative Geometry |volume=6 |issue=3 |pages=407–455 |s2cid=15049025 }} - Extends idea of using alternating forms on the lattice to $\backslash text\{Alt\}^3(\backslash Lambda,\backslash mathbb\{Z\})$, to construct gerbes on a complex torus
• {{cite journal|arxiv=0803.1529v2|last1=Block |first1=Jonathan |last2=Daenzer |first2=Calder |title=Mukai duality for gerbes with connection |year=2008 |journal=Crelle's Journal}} - includes examples of gerbes on complex tori
• {{cite journal|arxiv=1102.2312|doi=10.1016/j.geomphys.2012.10.012 |title=Equivariant gerbes on complex tori |year=2013 |last1=Ben-Bassat |first1=Oren |journal=Journal of Geometry and Physics |volume=64 |pages=209–221 |bibcode=2013JGP....64..209B |s2cid=119599648 }}
• {{cite journal|arxiv=math/0601337|doi=10.1215/S0012-7094-08-14111-0 |title=A gerbe for the elliptic gamma function |year=2008 |last1=Felder |first1=Giovanni |last2=Henriques |first2=André |last3=Rossi |first3=Carlo A. |last4=Zhu |first4=Chenchang |journal=Duke Mathematical Journal |volume=141 |s2cid=817920 }} - could be extended to complex tori

• [https://spectrum.library.concordia.ca/984074/1/Colo%CC%80_MSc_F2018.pdf p-adic Abelian Integrals: from Theory to Practice]

Category:Complex manifolds

Category:Complex surfaces

Category:Abelian varieties