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\section*{abelian variety}

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{automatic_abelianness}{Automatic abelianness}\dotfill \pageref*{automatic_abelianness} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{literature}{Literature}\dotfill \pageref*{literature} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

Abelian varieties are higher dimensional analogues of [[elliptic curve]]s (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.

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

In his book \emph{Abelian Varieties}, David Mumford defines an \textbf{abelian variety} over an algebraically closed field $k$ to be a [[complete algebraic variety|complete]] [[algebraic group]] over $k$. Remarkably, any such thing is an \emph{abelian} algebraic group. The assumption of connectedness is necessary for that conclusion.

\hypertarget{automatic_abelianness}{}\subsection*{{Automatic abelianness}}\label{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  if it factors through the unique morphism $X \to 1$. Say a morphism $f : X \times Y \to Z$ is  if it factors through the projection $X \times Y \to Y$.

Say a  of $Y$ is a morphism $p: 1 \to Y$. Say a morphism $f: X \times Y \to Z$ is  if $f \circ (1_X \times p) : X \to Y$ is constant.

\textbf{Definition.} A category with finite products \textbf{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 $X$.

\textbf{Theorem 1.} The category of complete algebraic varieties over an algebraically complete field $k$ has finite products and obeys the rigidity lemma.

\begin{proof}
The hard part, the rigidity lemma, is proved for complete algebraic varieties on page 43 of Mumford's \emph{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.

\end{proof}
\textbf{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.

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

\begin{displaymath}
k(g,g') = f(g\cdot g')\cdot (f(g) \cdot f(g'))^{-1}
\end{displaymath}
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).

\end{proof}
\textbf{Corollary 1.} If $C$ is a category with finite products obeying the rigidity lemma, any group object in $C$ is abelian.

\begin{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.

\end{proof}
\textbf{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.

\begin{proof}
This follows immediately from the two theorems above.

\end{proof}
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]].

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[theta function]]


\item [[Cartier duality]]


\item [[dual abelian group scheme]]


\item [[Albanese variety]]



\end{itemize}
\hypertarget{literature}{}\subsection*{{Literature}}\label{literature}

\begin{itemize}%
\item C. Bartocci, Ugo Bruzzo, D. Hernandez Ruiperez, Fourier-Mukai and Nahm transforms in geometry and mathematical physics, Progress in Mathematics 276, Birkhauser 2009.


\item [[M. Demazure]], [[P. Gabriel]], \emph{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, \href{http://dx.doi.org/10.1112/blms/12.6.476b}{doi}


\item [[Daniel Huybrechts]], \emph{Fourier-Mukai transforms in algebraic geometry}, Oxford Mathematical Monographs. 2006. 307 pages.


\item J. S. Milne, \emph{Abelian varieties}, course notes, \href{http://www.jmilne.org/math/CourseNotes/AV.pdf}{pdf}


\item David Mumford, \emph{Abelian varieties}, Oxford Univ. Press 1970.


\item [[Alexander Polishchuk]], \emph{Abelian varieties, theta functions and the Fourier transform}, Cambridge Univ. Press 2003.


\item Goro Shimura, \emph{Abelian varieties with complex multiplication and modular functions}, Princeton Univ. Press 1997.


\item [[André Weil]], \emph{Courbes alg\'e{}briques et vari\'e{}t\'e{}s ab\'e{}liennes}, Paris: Hermann 1971



\end{itemize}
In [[E-infinity geometry]]:

\begin{itemize}%
\item [[Jacob Lurie]], \emph{Elliptic Cohomology I: Spectral Abelian Varieties} (\href{http://www.math.harvard.edu/~lurie/papers/Elliptic-I.pdf}{pdf})

\end{itemize}
For a discussion of how the rigidity lemma gives `automatic abelianness' see:

\begin{itemize}%
\item \href{https://golem.ph.utexas.edu/category/2016/08/the_magic_of_algebraic_geometr.html}{Two miracles in algebraic geometry}, \emph{The n-Category Caf\&eacute}.

\end{itemize}
[[!redirects abelian varieties]] [[!redirects abelian group scheme]] [[!redirects abelian group schemes]]



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