Abelian varieties are higher dimensional analogues of elliptic curves (which are included) – they are varieties equipped with a structure of an abelian group, hence abelian group schemes, whose multiplication and inverse are regular maps.
In his book Abelian Varieties, David Mumford defines an abelian variety over an algebraically closed field $k$ to be a complete algebraic group over $k$. Remarkably, any such thing is an abelian algebraic group. The assumption of connectedness is necessary for that conclusion.
David Mumford gives at two proofs that every complete algebraic group over an algebraically closed field is automatically abelian. One of them uses a ‘rigidity lemma’ which has an interesting category-theoretic interpretation. We outline this here:
In simple terms, the rigidity lemma says that under certain circumstances “a 2-variable function $f(x,y)$ that is independent of $x$ for one value of $y$ is independent of $x$ for all values of $y$.”
More precisely, in a category with products, say a morphism $f: X \to Y$ is constant if it factors through the unique morphism $X \to 1$. Say a morphism $f : X \times Y \to Z$ is independent of $X$ if it factors through the projection $X \times Y \to Y$.
Say a point of $Y$ is a morphism $p: 1 \to Y$. Say a morphism $f: X \times Y \to Z$ is independent of $X$ at some point $p$ of $Y$ if $f \circ (1_X \times p) : X \to Y$ is constant.
Definition. A category with finite products obeys the rigidity lemma if any morphism $f: X \times Y \to Z$ that is independent of $X$ at some point of $Y$ is in fact independent of $X$.
Theorem 1. The category of complete algebraic varieties over an algebraically complete field $k$ has finite products and obeys the rigidity lemma.
The hard part, the rigidity lemma, is proved for complete algebraic varieties on page 43 of Mumford’s Abelian Varieties. Mumford mentions in a footnote that complete algebraic varieties are automatically irreducible, and he later seems to assume without much explanation that they are connected: these points could use some clarification, at least for amateurs.
Theorem 2. Suppose $G,H$ are group objects in a category $C$ with finite products obeying the rigidity lemma. Suppose $f : G \to H$ is any morphism in $C$ preserving the identity. Then $f$ is a homomorphism.
The idea is this: suppose $C$ is a concrete category and look at the function $k : G \times G \to H$ given by
Assume $f(1) = 1$. Then $k(1,g') = 1$ for all $g' \in G$, so by the rigidity lemma $k(g,g')$ is independent of $g'$ and we can write $k(g,g') = r(g)$. Furthermore $k(g,1) = 1$ for all $g \in G$ so $k(g,g')$ is independent of $g$. This means that $r(g)$ is independent of $g$, but $r(1) = k(1,1) = 1$ so $r(g) = 1$ for all $g$. This says that $f(g \cdot g') = f(g) \cdot f(g')$, so $f$ preserves multiplication. This in turn implies that $f$ preserves inverses, so $f$ is a group homomorphism.
In fact a version of this argument works in any category with finite products obeying the rigidity lemma. The expression $f(g\cdot g')\cdot (f(g) \cdot f(g'))^{-1}$ compiles to a particular morphism $G\times G \xrightarrow{k} H$. The fact that $f$ preserves the identity $e_G:1\to G$ implies that both composites $G \xrightarrow{(id,e_G)} G\times G \xrightarrow{k} H$ and $G \xrightarrow{(e_G,id)} G\times G \xrightarrow{k} H$ are constant at the identity $e_H:1\to H$. (This is a straightforward calculation in the internal logic of a category with products, or alternatively a slightly tedious diagram chase.)
In particular, $k$ is independent of the first $G$ in its domain at the point $e_G : 1\to G$ of the second $G$ in its domain. So by the rigidity lemma, there exists a morphism $r : G\to H$ such that the composite $G\times G \xrightarrow{\pi_2} G \xrightarrow{r} H$ is equal to $k$. Now precompose both of these with $G \xrightarrow{(e_G,id)} G\times G$: the first gives $r \circ \pi_2 \circ (e_G,id) = r$ and the second gives $k \circ (e_G,id) = e_H \circ !$. Thus, $r$ is constant at the identity of $H$, and hence so is $k$. This implies $f$ preserves multiplication, and thus also inverses (again, by a calculation in internal logic or a diagram chase).
Corollary 1. If $C$ is a category with finite products obeying the rigidity lemma, any group object in $C$ is abelian.
If $G$ is a group object in $C$, the inverse map $inv: G \to G$ preserves the identity, so by the above theorem it is a group homomorphism. This in turn implies that $G$ is abelian.
Corollary 2. Let $Var_*$ be the category of pointed complete varieties over an algebraically closed field $k$, and let $AbVar$ be the category of abelian varieties over $k$. Then the forgetful functor $U : AbVar \to Var_*$ is full.
This follows immediately from the two theorems above.
A consequence of Corollary 2 is that if $Alb : Var_* \to AbVar$ is the left adjoint to $U$, sending any connected pointed projective variety to its Albanese variety, the monad $T = U \circ Alb$ is an idempotent monad. For more on this see Albanese variety.
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Daniel Huybrechts, Fourier-Mukai transforms in algebraic geometry, Oxford Mathematical Monographs. 2006. 307 pages.
J. S. Milne, Abelian varieties, course notes, pdf
David Mumford, Abelian varieties, Oxford Univ. Press 1970.
Alexander Polishchuk, Abelian varieties, theta functions and the Fourier transform, Cambridge Univ. Press 2003.
Goro Shimura, Abelian varieties with complex multiplication and modular functions, Princeton Univ. Press 1997.
André Weil, Courbes algébriques et variétés abéliennes, Paris: Hermann 1971
For a discussion of how the rigidity lemma gives ‘automatic abelianness’ see: