Contents

# Contents

## Idea

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.

## Definition

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.

## Automatic abelianness

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 $f \circ (1_X \times p) : X \to Z$ 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.

###### Proof

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.

###### Proof

The idea is this: suppose $C$ is a concrete category and look at the function $k : G \times G \to H$ given by

$k(g,g') = f(g\cdot g')\cdot (f(g) \cdot f(g'))^{-1}$

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.

###### Proof

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.

###### Proof

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.

## Literature

• C. Bartocci, Ugo Bruzzo, D. Hernandez Ruiperez, Fourier-Mukai and Nahm transforms in geometry and mathematical physics, Progress in Mathematics 276, Birkhauser 2009.

• M. Demazure, P. Gabriel, Groupes algebriques, tome 1 (later volumes never appeared), Mason and Cie, Paris 1970 – has functor of points point of view; for review see Bull. London Math. Soc. (1980) 12 (6): 476-478, doi

• 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

• Jacob Lurie, Elliptic Cohomology I: Spectral Abelian Varieties (pdf)

For a discussion of how the rigidity lemma gives ‘automatic abelianness’ see:

Last revised on October 10, 2020 at 17:04:11. See the history of this page for a list of all contributions to it.