\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 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 $x_0 \in X(K)$, $y \in Y(K)$ and $z_0 \in Z(K)$, such that  
$$
f(X\times \{y_0\}) = f(\{x_0\}\times Y) = \{z_0\}. 
$$
Then $f(X \times Y) = \{z_0\}$. 
\end{thm}

\begin{proof}

\end{proof}