abelian variety



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 kk to be a complete algebraic group over kk. 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)f(x,y) that is independent of xx for one value of yy is independent of xx for all values of yy.”

More precisely, in a category with products, say a morphism f:XYf: X \to Y is constant if it factors through the unique morphism X1X \to 1. Say a morphism f:X×YZf : X \times Y \to Z is independent of X X if it factors through the projection X×YYX \times Y \to Y.

Say a point of YY is a morphism p:1Yp: 1 \to Y. Say a morphism f:X×YZf: X \times Y \to Z is independent of X X at some point p p of Y Y if f(1 X×p):XYf \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×YZf: X \times Y \to Z that is independent of XX at some point of YY is in fact independent of XX.

Theorem 1. The category of complete algebraic varieties over an algebraically complete field kk 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,HG,H are group objects in a category CC with finite products obeying the rigidity lemma. Suppose f:GHf : G \to H is any morphism in CC preserving the identity. Then ff is a homomorphism.


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

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

Assume f(1)=1f(1) = 1. Then k(1,g)=1k(1,g') = 1 for all gGg' \in G, so by the rigidity lemma k(g,g)k(g,g') is independent of gg' and we can write k(g,g)=r(g)k(g,g') = r(g). Furthermore k(g,1)=1k(g,1) = 1 for all gGg \in G so k(g,g)k(g,g') is independent of gg. This means that r(g)r(g) is independent of gg, but r(1)=k(1,1)=1r(1) = k(1,1) = 1 so r(g)=1r(g) = 1 for all gg. This says that f(gg)=f(g)f(g)f(g \cdot g') = f(g) \cdot f(g'), so ff preserves multiplication. This in turn implies that ff preserves inverses, so ff 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(gg)(f(g)f(g)) 1f(g\cdot g')\cdot (f(g) \cdot f(g'))^{-1} compiles to a particular morphism G×GkHG\times G \xrightarrow{k} H. The fact that ff preserves the identity e G:1Ge_G:1\to G implies that both composites G(id,e G)G×GkHG \xrightarrow{(id,e_G)} G\times G \xrightarrow{k} H and G(e G,id)G×GkHG \xrightarrow{(e_G,id)} G\times G \xrightarrow{k} H are constant at the identity e H:1He_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, kk is independent of the first GG in its domain at the point e G:1Ge_G : 1\to G of the second GG in its domain. So by the rigidity lemma, there exists a morphism r:GHr : G\to H such that the composite G×Gπ 2GrHG\times G \xrightarrow{\pi_2} G \xrightarrow{r} H is equal to kk. Now precompose both of these with G(e G,id)G×GG \xrightarrow{(e_G,id)} G\times G: the first gives rπ 2(e G,id)=rr \circ \pi_2 \circ (e_G,id) = r and the second gives k(e G,id)=e H!k \circ (e_G,id) = e_H \circ !. Thus, rr is constant at the identity of HH, and hence so is kk. This implies ff preserves multiplication, and thus also inverses (again, by a calculation in internal logic or a diagram chase).

Corollary 1. If CC is a category with finite products obeying the rigidity lemma, any group object in CC is abelian.


If GG is a group object in CC, the inverse map inv:GGinv: G \to G preserves the identity, so by the above theorem it is a group homomorphism. This in turn implies that GG is abelian.

Corollary 2. Let Var *Var_* be the category of pointed complete varieties over an algebraically closed field kk, and let AbVarAbVar be the category of abelian varieties over kk. Then the forgetful functor U:AbVarVar *U : AbVar \to Var_* is full.


This follows immediately from the two theorems above.

A consequence of Corollary 2 is that if Alb:Var *AbVarAlb : Var_* \to AbVar is the left adjoint to UU, sending any connected pointed projective variety to its Albanese variety, the monad T=UAlbT = U \circ Alb is an idempotent monad. For more on this see Albanese variety.


  • 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

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

Revised on August 19, 2016 01:12:41 by John Baez (