\chapter{Abelian Varieties}
The purpose of this chapter is to introduce the basic theory of abelian varieties.
In $\S1$ we discuss abelian varieties over $\C$, considered as complex tori,
equipped with Hermitian forms. We introduce the notion of an abelian
manifold as a complex torus $T$ with enough meromorphic functions.
\section{Abelian Varieties over Arbitrary Fields}
Abelian varieties are the main objects of study of this paper.
\begin{defn}
An \emph{abelian variety} over a field $K$ is a smooth, proper, algebraic variety $X$ over
$K$, together with multiplication and inverse morphisms
$$
m : X \times X \ra X\ \ \textrm{(multiplication)}
$$
$$
i : X \ra X\ \ \textrm{(inverse)},
$$
and an identity element $e \in X(K)$, such that the maps $m, i$ and the element $e$ define
a group structure on $X(\overline{K})$.
\end{defn}
\begin{example}
The obvious examples are elliptic curves, since they are smooth as algebraic varieties
and have a group structure (the group law is defined by the usual addition of points law on
elliptic curves).
\end{example}
It is not clear \emph{\`a priori} whether multiplication on the group variety is commutative.
For elliptic curves, commutativity is straightforward from the definition of the group law.
To prove commutativity in general, we use the following
\begin{thm}[Rigidity Theorem]
Let $f : X \times Y \ra Z$ be a morphism of varieties over $K$. Suppose that $X$ is smooth
and there exist $y_0 \in Y(K)$ and $z_0 \in Z(K)$, such that
$$
f(X\times \{y_0\}) = \{z_0\}.
$$
Then there exists a morphism $g : Y \ra Z$, such that $f = g \circ \pi$, where
$\pi : X \times Y \ra Y$ is the projection morphism.
\end{thm}
\begin{proof}
Choose a point $x_0 \in X$ and define $g(y) = f(x_0, y)$. Choose an open affine neighborhood
$U_0$ of $z_0$ in $Z$. Since $X$ is proper over
$K$, then $\pi$ is closed. Then $W = \pi(f^{-1}(Z - U_0))$ is closed in $Y$. Then $Y - W$ is
an open set of $Y$, which is nonempty, because $y_0 \in Z - W$. Indeed,
$y \in Y - W$ if and only if $f(X \times \{y\}) \subset U_0$. Therefore, whenever $y$ is a
closed point of $X$, $f$ maps the complete variety $X \times \{y\}$ to the affine
variety $U_0$, so it must be a constant map. Therefore, for any $x \in X$ and $y \in Y$,
$$
f(x, y) = f(x_0, y) = g(y) = (g \circ \pi)(x, y).
$$
This means that $f$ and $g \circ \pi$ agree on an open dense subset of $X \times Y$ and so
they coincide everywhere.
\end{proof}
Rigidity theorem allows us to express morphisms of abelian varieties as a composition of
homomorphisms and translations.
\begin{cor}
Let $X$ and $Y$ be abelian varieties and $f : X \ra Y$ be any morphism. There is a
homomorphism $g : X \ra Y$ and $a \in Y$, such that $f(x) = g(x) + a$.
\end{cor}
\begin{proof}
Let $a = f(0)$. By replacing $f$ with $f - a$, we can assume that $f : X \ra Y$ satisfies
$f(0) = 0$. We will show that $f$ is a homomorphism. Consider $\phi : X \times X \ra Y$,
defined by $\phi(x', x'') = f(x' + x'') - f(x') - f(x'')$. For fixed $x'' \in X$,
$\phi(x',x'')$ is independent of the choice of $x'$, so $\phi(x',x'') = \phi(0, x') = 0$.
Thus, $\phi \equiv 0$ and so $f$ is a homomorphism.\footnote{Although we used additive
notation for the group law, we do not make any use of the commutativity so far.}
\end{proof}
Finally, we conclude that any abelian varieties are commutative.
\begin{cor}
If $X$ is an abelian variety, then $X$ is commutative.
\end{cor}
\begin{proof}
Consider the morphism $x \mapsto x^{-1}$. It maps the identity element to itself, so by the
previous corollary, it must be a homomorphism. Thus, $x^{-1}y^{-1} = y^{-1}x^{-1}$, so $X$
is commutative.
\end{proof}
\section{The Dual Abelian Variety in Characteristic Zero}
One of the main problems from the theory of abelian varieties
deals with studying the isomorphism classes of invertible sheaves
on the varieties (the structure of the Picard group). The goal of
this section is to endow the group of isomorphism classes of
invertible sheaves of degree 0 on $A$, considered over the closure
of $K$ (or $\textrm{Pic}^0(A_{\overline{K}})$) with the structure
of an abelian variety over $K$. We will call this variety
$A^{\vee}$ the dual of $A$ (or the Picard variety $A$).
We start with an important result about invertible sheaves on
abelian varieties.
\begin{thm}[Theorem of the Square]
Let $A$ be an abelian variety over $K$ and $\Lcal$ be an
invertible sheaf on the variety $A$. For any point $c \in A$, we
denote by $t_c : A \ra A$ the translation map $x \mapsto c + x$.
Then
$$
t_{a+b}^* \Lcal \otimes \Lcal\cong t_a^* \Lcal \otimes t_b^*
\Lcal,
$$
for arbitrary points $a, b \in A(K)$.\footnote{By $t_c^* \Lcal$ we
mean the pullback of the sheaf $\Lcal$ under the map $t_c : A \ra
A$.}
\end{thm}
\begin{proof}
This is proved in \cite{}.
\end{proof}
The above theorem is very important, because it can be used to
construct a homomorphism $A \ra \textrm{Pic}(A)$ in the following
way: fix an invertible sheaf $\Lcal$ on $A$ and define a map
$$
\varphi_\Lcal : A \ra \textrm{Pic}(A),\ a \mapsto t_a^*\Lcal
\otimes \Lcal^{-1}.
$$
The theorem of the square implies that
$$
t_{a+b}^*\Lcal \otimes \Lcal^{-1} \cong (t_a^*\Lcal \otimes
\Lcal^{-1}) \otimes (t_b^*\Lcal \otimes \Lcal^{-1}),
$$
so $\varphi_\Lcal$ is a homomorphism.
Next, we will show that the image of $\varphi_\Lcal$ is contained
in the subgroup Pic$^0(A)$ - the subgroup of isomorphism classes
of invertible sheaves of degree 0. To check this, it suffices to
check for any $a \in A(\overline{K})$ and $b \in A(K)$,
$$
t_a^*(\varphi_\Lcal(b)) \cong \varphi_\Lcal(b).
$$
This follows, since
$$
t_a^*(\varphi_\Lcal(b)) \cong t_a^*(t_b^*\Lcal \otimes \Lcal^{-1})
\cong t_{a+b}^*\Lcal \otimes (t_a^*\Lcal)^{-1}.
$$
The theorem of the square implies that the last sheaf is
isomorphic to $t_b^*\Lcal \otimes \Lcal^{-1} = \varphi_\Lcal(b)$.
Therefore, $\varphi_\Lcal(b) \in \textrm{Pic}^0(A)$ and we are
done.
Our next goal is to view Pic$^0(A)$ as an abelian variety, which
is a quotient of $A$. So far, we did not make any extra
assumptions about the sheaf $\Lcal$. It turns out that if $\Lcal$
is chosen to be an ample invertible sheaf and if $K$ is
algebraically closed, then $\vaprhi_\Lcal : A \ra
\textrm{Pic}^0(A)$ is surjective. This would allow us to endow
Pic$^0(A)$ with a structure of an abelian variety. The result is
contained in the following theorem, which is proved in \cite{}.
\begin{thm}
If $\Lcal$ is an ample invertible sheaf and $K$ is algebraically
closed, then the map $\varphi_\Lcal : A \ra \textrm{Pic}^0(A)$ is
surjective.
\end{thm}
\section{The Dual Isogeny and the Dual Exact Sequence}
In the previous section, we explained how to dualize abelian
varieties. The next important construction is the dualization of
homomorphisms of abelian varieties.
Suppose that $f : A \ra B$ is a homomorphism of abelian varieties
and consider the induced map $f^{\vee} : \textrm{Pic } B \ra
\textrm{Pic }A$ on isomorphism classes of invertible sheaves on
$A$. Since sheaves of zero degree are mapped to sheaves of zero
degree, then we get a natural map on points $f^{\vee} : A^{\vee}
\ra B^{\vee}$, which is in fact a morphism. To give an argument
for the last statement, let $\mathcal P_B$ be the Poincar\'e sheaf
on $B \times B^{\vee}$ and consider the pullback sheaf $(f \times
1)^* \mathcal P_B$, which is a sheaf on $A \times B^{\vee}$. The
fact that $f^{\vee}$ is a morphism follows from the universal
mapping property, because $(f \times 1)^* \mathcal P_B|_{X \times
\{\tilde{y}\}}$ represents $f^{\vee}(\tilde{y})$ for any
$\tilde{y} \in Y^{\vee}$. Thus, every homomorphism of abelian
varieties induces a homomorphism on the dual varieties.
Our next goal is to study dual homomorphisms to isogenies.
\begin{prop}
Let $f : A \ra B$ be an isogeny with finite kernel $N$.
\end{prop}