Translator’s note

This page is a translation into English of the following:

Douady, A. “Variétés abéliennes.” Séminaire Claude Chevalley 4 (1958–59), Talk no. 9. http://www.numdam.org/item/SCC_1958-1959__4__A9_0

The translator (Tim Hosgood) takes full responsibility for any errors introduced, and claims no rights to any of the mathematical content herein.

Version: f36a214

1 Algebraic groups

An algebraic group is a pair (G,\varphi), where G is an algebraic variety and \varphi is a morphism from G\times G to G that endows the set of points of G with the structure of a group.

For every point a of G, the translations l_a and r_a defined by l_a(x)=\varphi(a,x) and r_a(x)=\varphi(x,a) are automorphisms of the algebraic variety structure of G.

Every point x of G is simple (indeed, if y is a simple point of G, then there exists an automorphism of G that sends y to x).

If G is connected, then G is irreducible. The map \pi\colon G\to G that, to any point, associates its inverse under the group law is a morphism (and thus an automorphism) of the algebraic variety structure. The proof of this property uses the fact that a bijective (and thus radicial) morphism from one variety to another is birational if it is unramified ([1, Corollary 2 to Proposition 3, Section II, Chapter VI]).

(There would be no problem with asking for G to be connected as part of the definition of algebraic groups).

An abelian variety is an algebraic group whose variety is connected (and thus irreducible) and complete.

We will show that this implies that the group is commutative.

2 A property of complete varieties

Recall that a variety V is said to be complete if, for every variety T and every closed subset F\subset T\times V, the projection from F to T is closed. In the classical case, this property is equivalent to compactness.

2.1 Proposition 0

Let V be a complete connected variety, T a connected variety, and f a morphism from T\times V to another variety U. Then, if, for some t_0\in T, f(t_0,v) does not depend on v, then f(t,v)=\varphi(t) \quad\text{for all }t,v where \varphi is a morphism from T to U.

Proof. If v_1,v_2\in V, then the set P_{v_1,v_2} of t\in T such that f(t,v_1)=f(t,v_2) is closed in T. The set P=\bigcap P_{v_1,v_2} of t\in T for which f(t,v) does not depend on v is thus also closed. We will now show that it is also open. If t_1\in P, then f(t_1,v)=u_1 for all v. Let U\setminus F be an affine neighbourhood of u_1, with F closed, so f^{-1}(F) is closed, G=\operatorname{pr}_T(f^{-1}(F)) is closed, and T\setminus G is a neighbourhood of t_1. For t'\in T\setminus G, f defines a morphism v\mapsto f(t',v) from V, which is complete and connected, to U\setminus F, which is affine. This map is necessarily constant. Thus P\supset T\setminus G is a neighbourhood of each of its points, i.e. an open subset.

Since T is connected, if t_0\in P, then P=T, which finishes the proof. (To see that \varphi is a morphism, it suffices to take a point v_0\in V, without worrying about the case where V=\varnothing).

There is an analogous statement in analytic geometry: let V be a compact connected complex-analytic space, T a connected topological space, and f a continuous map from T\times V to another analytic space U, such that f induces, for all t\in T, a holomorphic map f_t from \{t\}\times V to U. Then, if f_t is constant for t=t_0, then it is constant for all t\in T. In other words, a holomorphic map from V to U that is homotopic to a constant map amongst holomorphic maps is constant. The hypothesis that f_t be holomorphic for all t is essential: there are counter-examples with non-Kähler varieties V.

2.2 Consequences of Proposition 0

If V is a complete connected variety, T a connected variety, and G an algebraic group, then every morphism f\colon T\times V\to G is of the form f(t,v) = \varphi_1(t)\times\varphi_2(v) where \varphi_1 and \varphi_2 are morphisms from T and V (respectively) to G.

Proof. Let t_0\in T. Consider f(t,v)\cdot f(t_0,v)^{-1}; this is a morphism from T\times V to G that, for t=t_0 and arbitrary v, takes the value e (the identity element in G). We thus have \begin{aligned} f(t,v)\cdot f(t_0,v)^{-1} &= \varphi_1(t) \\\implies f(t,v) &= \varphi_1(t)\cdot f(t_0,v). \end{aligned} which finishes the proof.

“By an analogous argument we can show,” or “by considering the dual group of G, we deduce” that f(t,v) can also be written in the form \psi_1(v)\psi_2(t).

If \varphi_1(t)\varphi_2(v)=\varphi'_1(t)\varphi'_2(v), then \varphi'_1(t)=\varphi_1(t)\cdot a and \varphi'_2(v)=\varphi_2(v), where a is some fixed element of G.

Let G be a connected group, and V a complete connected variety; suppose that e\in V\subset G. Then V is contained in the centre of G.

Proof. Consider f\colon G\times V\to G defined by f(g,v)=v\cdot g\cdot v^{-1}. If g=e, then f does not depend on v. So f(g,v)=\varphi(g). Setting v=e, we find that \varphi(g)=g, and so vgv^{-1}=g, which proves the proposition.

In particular:

The underlying group of an abelian variety is abelian.

(For another proof of this result, see the Appendix).

3 Functions with values in an abelian variety

Every function f on a non-singular variety U with values in an abelian variety A is a morphism.

This theorem results from the combination of two lemmas.

If f is a function defined on a non-singular variety U with values in an algebraic group G, then the set S of points of U where f is not defined is of pure codimension 1.

Proof. Let \varphi be the function from U\times U to G defined by \varphi(u,u')=f(u)f(u')^{-1}. Let X be an affine neighbourhood of e in G, and \varphi_0 the function from U\times U to X that only differs from \varphi in the definition of its domain; there is no worry that \varphi(U) might not be contained in G\setminus X, since, if f is defined at u, then \varphi is defined at (u,u), and there it takes the value e; more precisely, we will show that the following three properties are equivalent:

  1. f is defined at u;
  2. \varphi is defined at (u,u);
  3. \varphi_0 is defined at (u,u).

Firstly, (a)\implies(b)\iff(c) is evident.

We now show that (b)\implies(a). If \varphi is defined at (u,u), let v\in U be such that f is defined at v, and such that \varphi is defined at (u,v); then, for all u' where f is defined, f(u') = f(u')\cdot f(v)^{-1}\cdot f(v) = \varphi(u',v)\cdot f(v). The function f_0 defined by f_0(u')=\varphi(u',v)\cdot f(v) agrees with f, and is defined at the point u, and so (b)\implies(a).

This shows that the intersection of with the diagonal of the set H of points of U\times U where \varphi_0 is not defined is S, or rather the image of S under the diagonal map. But the sets of points at which a numerical function on a normal variety is not defined is of pure codimension 1; this is thus also true if we replace “numerical function” with “function with values in an affine variety.” So H is of pure codimension 1. Since U\times U is not singular, the codimension in U\times U of H\cap\Delta is \leqslant\operatorname{codim}H+\operatorname{codim}\Delta = \dim U+1, which shows that every component of H\cap\Delta is of codimension \leqslant 1 in \Delta.

The hypothesis that U be non-singular is essential, both for Lemma 1 and for Theorem 1.

Let U be a cone in K^3 that has a cubic G of genus 1 in the 2-dimensional projective space as its base. Then G can be endowed with a group structure. The projection f from U to G is defined at every point except for the origin; S is thus of codimension 2.

If f is a function defined on a normal variety U with values in a complete variety V, then the set S of points of U where f is not defined is of codimension >1.

Proof. Since V is complete, there exists a variety W contained in D^r (where D denotes the projective line), a morphism p from W to V, and a function s from V to W such that p\circ s=I_V. Then f=p\circ(s\circ f) will be defined whenever s\circ f is defined. But s\circ f can be considered as taking values in D^r, since W is closed in D^r, and will thus be defined whenever the r coordinate functions of f are defined. These functions take values in D, and so, since U is normal, the set of points where they are not defined is of codimension >1 [1, Corollary to Proposition 2, Section 1, Chapter V].

4 Functions defined on a product with values in an abelian variety

Let X and Y be irreducible varieties, and f a function defined on X\times Y with values in an abelian variety A (whose group law is written additively). Then f is of the form f(x,y)=f_1(x)+f_2(y), where f_1 and f_2 are functions from X and Y (respectively) to A.

This implies that f is defined at the points where f_1 and f_2 are both defined, and at these points only.

Proof. Let (x_0,y_0) be a simple point of X\times Y. By considering the function g on X\times Y, defined by g(x,y) = f(x,y) - f(x_0,y) - f(x,y_0) + f(x_0,y_0), we can reduce to showing that, if a function g with values in A is zero on X\vee Y = \{x_0\}\times Y \cup X\times\{y_0\}, then it is zero on X\times Y.

We will successively reduce to the following particular cases:

  1. X is a curve;
  2. X is a complete non-singular curve, and Y is non-singular.

Reduction to (a). The set of points x_1 of X such that there exists an irreducible curve containing x_0 and x_1, where x_0 is a simple point, is dense in X. But (a) implies that g is zero at every point (x_1,y) of the open set on which it is defined, and thus at every point of a dense set, and thus everywhere.

Reduction to (b). If X is an irreducible curve, then there exists a curve X_1 that is both complete and normal (and thus non-singular), as well as a rational equivalence from X to X_1, defined at x_0. Also, if Y_1 is the open subset of simple points of Y, then Y_1 is birationally equivalent to Y. The function g, defined on X\times Y, has a corresponding function g_1, defined on X_1\times Y_1, and it clearly suffices to prove the theorem for the function g_1.

Proof in case (b). The variety X\times Y is a product of two non-singular varieties, and thus itself non-singular, and so g is a morphism, by Theorem 1. Its value does not depend on x for y=y_0, and thus also for all y, by Proposition 0, since X is complete. It is zero for x=x_0, and thus for all x.

This concludes the proof of the theorem.

Every function f defined on an algebraic group G with values in an abelian variety A is of the form h+a, where h is a homomorphism from G to A, and a is a constant.

Proof. Set h(x)=f(x)-f(e). Then the function g\colon G\times G\to A defined by g(x,y)=h(x\cdot y) is of the form g_1(x)+g_2(y). We can impose that g_1(e)=0, and then g_2(e)=0. By successively taking x=e and then y=e, we find that g_1=g_2=h. Whence h(x,y)=h(x)+h(y).

Every function f defined on a line D with values in an abelian variety A is constant.

Proof. By Theorem 1, f is everywhere defined, and by Corollary 1 applied to the multiplicative group, we have that f(xy)=f(x)+f(y)+a. Setting x=0, we have f(0)=f(0)+f(y)+a, whence f(y)=-a.

Appendix: adjoint representations

We can also obtain Theorem 0 from the following proposition:

Let G be a connected algebraic group, and let C be the centre of G. Then there exists a linear group L=\mathrm{GL}(m) along with an algebraic homomorphism f\colon G\to L such that f^{-1}(e)=C.

Proof. Let {\mathfrak{a}} be the local ring of functions on G defined at the identity element e, and let {\mathscr{J}} be its maximal ideal; set T_n={\mathscr{J}}/{\mathscr{J}}^n. For all n, T_n is a finite-dimensional vector space. Every element x\in G defines an inner automorphism \alpha(x)\colon G\to G that induces an automorphism \operatorname{Ad}_n(x)\colon T_n\to T_n. Let C_n be the kernel of \operatorname{Ad}_n\colon G\to\mathrm{GL}(T_n). The C_n form a decreasing sequence of subvarieties of G. Such a sequence is stationary, and so there exists some n_0 such that C_n=C_{n_0} for all n\geqslant n_0. We now show that C_{n_0}=C: if x\in C_{n_0}, then \operatorname{Ad}_n(x) is the identity for all n, and so the automorphism of {\mathfrak{a}} defined by the inner automorphism \alpha(x) of G is the identity, since the local ring {\mathfrak{a}} is separated. Consequently, since G is connected (and thus irreducible), \alpha(x) is the identity. Whence the proposition, taking L=\mathrm{GL}(T_{n_0}).

In characteristic p\neq0, the monomorphism G/C\to L is not necessarily an isomorphism from G/C to its image. For example, consider G=k^*\times k endowed with the group law \varphi((a,b),(a',b))=(aa',b+a^pb').

We deduce Proposition 2 and Theorem 0 from Proposition 3 by noting that L is an affine variety, and that, if V is complete and connected, then every morphism from V to L is constant.

Bibliography

[1]
C. Chevalley. Fondements de la Géométrie algébrique. Paris, Secrétariat mathématique, 1958.