Automorphism Groups of Affine Varieties and a Characterization of Affine n-Space

0.24 MB·English

by  Hanspeter Kraft

AUTOMORPHISM GROUPS OF AFFINE VARIETIES AND A
CHARACTERIZATION OF AFFINE n-SPACE
5
1
HANSPETERKRAFT
0
2
n Abstract. We show that the automorphism group of affine n-space An de-
a termines An up to isomorphism: If X is a connected affine variety such that
J Aut(X)≃Aut(An)asind-groups,thenX ≃An asvarieties.
6 We also show that every finite group and every torus appears as Aut(X)
2 for a suitable affine variety X, but that Aut(X) cannot be isomorphic to a
semisimplegroup.Infact,ifAut(X)isfinitedimensionalandifX 6≃A1,then
] theconnected component Aut(X)◦ isatorus.
G Concerning the structure of Aut(An) we prove that any homomorphism
Aut(An)→G ofind-groupseitherfactorsthroughjac: Aut(An)→k∗,oritis
A
aclosedimmersion.ForSAut(An):=ker(jac)⊆Aut(An)weshow thatevery
. nontrivialhomomorphismSAut(An)→G isaclosedimmersion.
h
Finally, we prove that every non-trivial homomorphism ϕ: SAut(An) →
t
a SAut(An) is an automorphism, and that ϕ is given by conjugation with an
m elementfromAut(An).
[

1
v
2 1. Introduction and main results
6
3 Ourbasefieldkisalgebraicallyclosedofcharacteristiczero.Foranaffinevariety
6 X the automorphism group Aut(X) has the structure of an ind-group. We will
0 shortly recall the basic definitions in the following section 2. The classicalexample
1. is Aut(An), the group of automorphism of affine n-space An =kn.
0 The first main result shows that An is determined by its automorphism group.
5
1 Theorem 1.1. Let X be a connected affine variety. If Aut(X) ≃ Aut(An) as
: ind-groups, then X ≃An as varieties.
v
i
X It is clear that X has to be connected, since the automorphism group does not
r change if we form the disjoint union of An with a variety Y with trivial automor-
a
phism group.
Another important question is which groups appear as automorphismgroups of
affine varieties. For finite groups we have the following result.
Theorem 1.2. For every finite group G there is a smooth affine curve C such that
Aut(C)≃G.
Moreover, there exist surfaces with infinite discrete automorphism groups (see
[FK14, Proposition 7.5.2]). As for algebraic groups, we will see that every torus
appears as Aut(X) (Example 6.4), but there are no examples where Aut(X) ≃
SL (k). In fact, this is not possible as the next result shows.
2
Date:January31,2015.
Theauthor waspartiallysupportedbySwissNationalScienceFoundation.
1
2 HANSPETERKRAFT
Theorem 1.3. Let X be a connected affine variety. If dimAut(X) < ∞, then
either X ≃A1 or Aut(X)0 is a torus.
ThelastresultsconcerntheautomorphismgroupAut(An)ofaffinen-space.This
group has a closed normal subgroup SAut(An) consisting of those automorphism
f =(f ,...,f ) whose Jacobian determinant jac(f):=det ∂fi is equal to 1:
1 n (cid:16)∂xj(cid:17)(i,j)
SAut(An):=ker(jac: Aut(An)→k∗).
One could expect that SAut(An) is simple as an ind-group,because its Lie algebra
issimple,andthatSAut(An)istheonlyclosedpropernormalsubgroupofAut(An).
This is claimed in [Sha66, Sha81], but the proofs turned out to be not correct (see
[FK14, Section 10]). What we can prove here are the following results.
Theorem 1.4.
(1) Let ϕ: Aut(An) → G be a homomorphism of ind-groups. Then either ϕ
factors through jac: Aut(An) → k∗, or ϕ is a closed immersion. This
means that ϕ(Aut(An)) ⊆ G is a closed ind-subgroup and ϕ: Aut(An) −∼→
ϕ(Aut(An)) is an isomorphism.
(2) Every nontrivial homomorphism SAut(An) → G of ind-groups is a closed
immersion.
Theorem 1.5.
(1) Everyinjectivehomomorphism ϕ: Aut(An)→Aut(An)isanisomorphism,
and ϕ=Intg for a well-defined g∈Aut(An).
(2) Every nontrivial homomorphism ϕ: SAut(An) → SAut(An) is an isomor-
phism, and ϕ=Intg for a well-defined g∈Aut(An).
Let us point out the following example from [FK14] showing that bijective ho-
momorphisms of ind-groups are not necessarily isomorphisms. Denote by khx,yi
the freeassociativek-algebraintwogenerators.ThenAut(khx,yi) isanind-group,
and we have a canonical homomorphism π: Aut(khx,yi)→Aut(k[x,y]).
Proposition1.6(Furter-Kraft[FK14]). Themapπ: Aut(khx,yi)→Aut(k[x,y])
is a bijective homomorphism of ind-groups, but it is not an isomorphism, because
it is not an isomorphism on the Lie algebras.
2. Notation and preliminary results
The notion of an ind-group goes back to Shafarevich who called these objects
infinite dimensional groups, see [Sha66, Sha81]). We refer to [FK14] and [Kum02]
for basic notation in this context and to [KZ14] and [KR15] for some important
results.
Definition 2.1. An ind-variety V is a set together with an ascending filtration
V ⊆V ⊆V ⊆···⊆V such that the following holds:
0 1 2
(1) V = V ;
Sk∈N k
(2) Each V has the structure of an algebraic variety;
k
(3) For all k ∈N the subset V ⊆V is closed in the Zariski-topology.
k k+1
A CHARACTERIZATION OF AFFINE n-SPACE 3
Amorphismbetweenind-varietiesV = V andW = W isamapϕ: V →
Sk k Sm m
W suchthatforeveryk thereisanm suchthatϕ(V )⊆W andthatthe induced
k m
mapV →W isamorphismofvarieties.Isomorphismsofind-varietiesaredefined
k m
in the usual way.
Two filtrations V = V and V = V′ are called equivalent if for any
Sk∈N k Sk∈N k
k there is an m such that V ⊆ V′ is a closed subvariety as well as V′ ⊆ V .
k m k m
Equivalently,theidentitymapid: V = V →V = V′ isanisomorphism
Sk∈N k Sk∈N k
of ind-varieties.
An ind-variety V has a natural topology where S ⊆ V is open, resp. closed, if
S :=S∩V ⊆V is open, resp.closed,for all k. Obviously,a locally closedsubset
k k k
S ⊆ V has a natural structure of an ind-variety. It is called an ind-subvariety. An
ind-variety V is called affine if all V are affine. A subset X ⊆V is called algebraic
k
if it is locally closed and contained in some V . Such an X has a natural structure
k
of an algebraic variety.
Example 2.2. Any k-vector space V of countable dimension carries the structure
of an (affine) ind-variety by choosing an increasing sequence of finite dimensional
subspaces V such that V = V . Clearly, all these filtrations are equivalent.
If R is a kcommutative k-alSgekbrka of countable dimension, a⊆R a subspace, e.g.
an ideal, and S ⊆k[x ,...,x ] a set of polynomials, then the subset
1 n
{(a ,...,a )∈Rn |f(a ,...,a )∈a for all f ∈S}⊆Rn
1 n 1 n
is a closed ind-subvariety of Rn.
For any ind-variety V = V we can define the tangent space in x ∈ V in
Sk∈N k
the obvious way. We have x ∈ V for k ≥ k , and T V ⊆ T V for k ≥ k , and
k 0 x k x k+1 0
then define
T V := lim T V
x −→ x k
k≥k0
which is a vector space of countable dimension. A morphism ϕ: V → W induces
linear maps dϕ : T V → T W for every x ∈ X. Clearly, for a k-vector space V
x x ϕ(x)
of countable dimension and a for any v ∈V we have T V =V in a canonical way.
v
The product of two ind-varieties is defined in the obvious way. This allows to
define an ind-group as an ind-variety G with a group structure such that multipli-
cationG×G →G: (g,h)7→g·h,andinverseG →G: g 7→g−1,arebothmorphisms.
It is clear that a closed subgroup G of an ind-group G is an algebraic group if and
only if G is an algebraic subset of G.
IfGisanaffineind-group,thenT G hasanaturalstructureofaLiealgebrawhich
e
willbe denotedby LieG. The structure is obtainedby showing that everyA∈T G
e
defines a unique left-invariant vector field δ on G, see [Kum02, Proposition 4.2.2,
A
p. 114].
Remark 2.3. It is knownthat forn≥2 the Lie algebraLieSAut(An)is simple and
that LieSAut(An) ⊆ LieAut(An) is the only proper ideal, see [Sha81, Lemma 3].
Moreover,both Lie algebras are generated by the subalgebras LieG where G is an
algebraic subgroup.
Definition 2.4. An ind-group G = G is called discrete if G is finite for all k.
Sk k k
Clearly, G is discrete if and only if LieG is trivial.
Anind-groupG iscalledconnectedifforeveryg ∈G thereisanirreduciblecurve
D and a morphism D →G whose image contains e and g.
4 HANSPETERKRAFT
The next resultfollows from [FK14, Theorem3.1.1]and [KZ14, Theorem 4.3.2].
HereVec(X)denotestheLiealgebraof(algebraic)vectorfieldsonX,i.e.Vec(X)=
Der(O(X)), the Lie algebra of derivations of O(X).
Proposition2.5. LetX beanaffinevariety. Then Aut(X)has anaturalstructure
ofanaffineind-group,andthereisacanonicalembeddingξ: LieAut(X)֒→Vec(X)
of Lie algebras.
Remark 2.6. IncaseX =AntheembeddingξidentifiesLieAut(An))withVecc(An),
the vector fields δ = f ∂ with constant divergence divδ = ∂fi ∈ k, see
Pi i∂xi Pi ∂xi
[FK14, Proposition 3.5.1].
Another result which we will need is proved in [KZ14, Proposition 6.5.2].
Proposition 2.7. Let ϕ,ψ: G →H be two homomorphisms of ind-groups. Assume
that G is connected and that dϕ =dψ : LieG →LieH. Then ϕ=ψ.
e e
Afinalresultwhichwewilluseisthefollowing(see[KRZ14,Lemma6.1]).Denote
by Aff ⊆ Aut(An) the subgroup of affine transformations, i.e. Aff = GL (k)⋉
n n n
(kn)+. Similarly, the subgroup SAff ⊆ Aff consists of the transformations with
n n
determinant 1, i.e. SAff =SL (k)⋉(kn)+.
n n
Proposition 2.8. Let X be an affine variety with a faithful action of SAff . If
n
dimX ≤n, then X is SAff -isomorphic to An.
n
Remark 2.9. It is shown in [KRZ14, Lemma 6.1] that the same holds if we replace
SAff by Aff . Using Theorem 1.5 we see that we can replace SAff by Aut(An)
n n n
or SAut(An) as well.
3. The adjoint representation
IfLisafinitely generatedLie algebra,thenAut (L)hasanaturalstructureof
Lie
anind-groupdefinedinthefollowingway(see[FK14]).Chooseafinite-dimensional
subspace L ⊆ L which generates L as a Lie algebra. Then the restriction map
0
End (L)→Hom(L ,L)isinjectiveandtheimageisaclosedaffineind-subvariety.
Lie 0
(To see this write L as the quotient of the free Lie algebra F(L ) over L modulo
0 0
an ideal I.) Choosing a filtration L= L by finite-dimensional subspaces, we
Sk≥0 k
set End (L) := {α ∈ End (L) | α(L ) ⊆ L } which is a closed subvariety of
Lie k Lie 0 k
Hom(L ,L ) (see Example 2.2). Then we define the ind-structure on Aut (L) by
0 k Lie
identifying Aut (L) with the closed subset
Lie
{(α,β)∈End (L)×End (L)|α◦β =β◦α=id }⊆End (L)×End (L),
Lie Lie L Lie Lie
i.e.
Aut (L) :={α∈Aut (L)|α,α−1 ∈End (L) }.
Lie k Lie Lie k
ItfollowsthatAut (L)isanaffineind-groupwiththeusualfunctorialproperties.
Lie
Lemma 3.1. Let ρ: G → Aut (L) be an abstract homomorphism where L is
Lie
a finitely generated Lie algebra. Then ρ is a homomorphism of ind-groups if and
only if ρ is an ind-representation, i.e. the map ρ: G ×L → L is a morphism of
ind-varieties.
A CHARACTERIZATION OF AFFINE n-SPACE 5
Proof. Assume that L is generated by the finite dimensional subspace L ⊆ L. If
0
G = G and if ρ: G×L→L is a morphism, then, for any j, there is a k =k(j)
Sj j
suchthat ρ(G ×L )⊆L and ρ(G−1×L )⊆L . Hence, ρ(G )⊆Aut (L) , and
j 0 k j 0 k j Lie k
the map G →Hom(L ,L ) is clearly a morphism.
j 0 k
Now assume that G → Aut (L) is a homomorphism of ind-groups. Then, for
Lie
any j, there is a k = k(j) such that ρ(G ) ⊆ Aut (L) ֒→ Hom(L ,L ). Hence,
j Lie k 0 k
ρ(G ×L )⊆L , and G ×L →L is a morphism. (cid:3)
j 0 k j 0 k
The adjoint representation Ad: G → Aut (LieG) of an ind-group G is defined
Lie
∼
in the usual way: Adg := (dIntg) : LieG −→ LieG where Intg is the inner auto-
e
morphism h7→ghg−1.
Proposition 3.2. For any ind-group G the canonical map Ad: G →Aut (LieG)
Lie
is a homomorphism of ind-groups.
Proof. Let γ: G×G →G denote the morphism (g,h)7→ghg−1. For any g ∈G, the
map γ : G → G, h 7→ ghg−1, is an isomorphism of ind-groups, and its differential
g
Ad(g) = (dγ ) : LieG → LieG is an isomorphism of Lie algebras. If G = G ,
thenforanypg,qe ∈Nthereisanm∈Nsuchthatγ: G ×G →G .Clearly,ASdkgfokr
p p m
g ∈G is then given by (dγ ) : T G →T G , and the map G →Hom(T G ,G )
p g e e q e m k e q m
isamorphism,bythefollowinglemma.NowtheclaimfollowsfromLemma3.1. (cid:3)
Lemma3.3. LetΦ: X×Y →Z beamorphism ofaffinevarieties andsetΦ (y):=
x
Φ(x,y). Assume that there exist y ∈ Y and z ∈ Z such that Φ (y ) = z for all
0 0 x 0 0
x∈X. Then the induced map X →Hom(T Y,T Z), x7→d Φ , is a morphism.
y0 z0 y0 x
Proof. We can assume that Y,Z are vector spaces, Y = W and Z = V. Choose
bases (w ,...,w ) and (v ,...,v ). Then Φ is given by an element of the form
1 m 1 n
n
f ⊗h ⊗v , where f ∈O(X) and h ∈O(Y)=k[y ,...,y ],
XX ij ij i ij ij 1 m
i=1 j
and so the differential (dΦ ) : W →V is given by the matrix
x y0
 ∂hij 
f (x) (y )
X ij ∂y 0
 j k 
(i,k)
whose entries are regular functions on x. The claim follows. (cid:3)
Remark 3.4. In [KR15] we show that the canonical homomorphisms
Ad: Aut(An)→Aut (LieAut(An)) and Ad: Aut(An)→Aut (LieSAut(An))
Lie Lie
are both bijective. This can be improved.
Proposition 3.5.
(1) The adjoint representation Ad: Aut(An)→Aut (LieAut(An)) is an iso-
Lie
morphism of ind-groups.
(2) The induced map ρ: Aut (LieAut(An)) → Aut (LieSAut(An)) is an
Lie Lie
isomorphism of ind-groups.
Proof. We will use here the identification of LieAut(An) with Vecc(An), see Re-
mark 2.6. Put ∂ := ∂ .
xi ∂xi
6 HANSPETERKRAFT
(1) Let f = (f ,...,f ) ∈ Aut(An) and set θ := Ad(f−1) ∈ Aut (Vecc(An)).
1 n Lie
Then the matrix θ(∂ )x is invertible, and
(cid:16) xk j(cid:17)(j,k)
−1 ∂f
(∗) θ(∂ )x =Jac(f)= j ,
(cid:16) xk j(cid:17)(j,k) (cid:16)∂xi(cid:17)(i,j)
see [KR15, Remark 4.2]. We now claim that the map
−1
θ 7→ θ(∂ )x :Aut (Vecc(An))→M (k[x ,...,x ])
(cid:16) xk j(cid:17)(j,k) Lie n 1 n
isawell-definedmorphismofind-varieties.Infact,θ 7→θ(∂ )x isthecomposition
xk j
of the orbit map θ 7→ θ(∂ ): Aut (Vecc(An)) → Vecc(An) and the evaluation
xk Lie
map δ 7→ δ(x ): Vecc(An) → k[x ,...,x ], hence θ 7→ Θ := θ(∂ )x is a
j 1 n (cid:16) xk j(cid:17)(j,k)
morphism. Since jac(Θ)∈k∗ the claim follows.
Nowrecallthatthegradientk[x ,...,x ]→k[x ,...,x ]n, f 7→( ∂f ,..., ∂f ),
1 n 1 n ∂x1 ∂xn
defines an isomorphism
∂h ∂h
γ: k[x ,...,x ] −∼→Γ:={(h ,...,h )| i = j for all i<j}.
1 n ≥1 1 n
∂x ∂x
j i
−1
It follows from (∗) that the rows of the matrix (h ) := θ(∂ )x belong
ij (i,j) (cid:16) xk j(cid:17)(j,k)
to Γ, so that we get a morphism
ψ: Aut (Vecc(An))→k[x ,...,x ]n, θ 7→(f ,...,f ),
Lie 1 n 1 n
where f :=γ−1(h ,...,h )∈k[x ,...,x ] . By construction, we have
i i1 in 1 n ≥1
(∗∗) ψ(θ)=ψ(Ad(f−1))=f0 :=(f1−f1(0),...,fn−fn(0))=t−f(0)◦f
wheret isthetranslationv 7→v+a.LetS ⊆Aff bethesubgroupoftranslations,
a n
andsetS˜:=Ad(S).ThenS˜⊆Aut (Vecc(An))isaclosedalgebraicsubgroupand
Lie
Ad: S →S˜isanisomorphism.Itfollowsfrom(∗∗)thatAd(ψ(θ))·θ =Ad(t−f(0))∈
S˜, and so
ψ˜(θ):=ψ(θ)−1·(Ad| )−1(Ad(ψ(θ))·θ)
S
isawell-definedmorphismψ˜: Aut (Vecc(An))→Aut(An)withthepropertythat
Lie
Ad(ψ˜(θ))=Ad(ψ(θ)−1)·Ad(ψ(θ))·θ =θ.
Thus Ad: Aut(An)→Aut (LieAut(An)) is an isomorphism, with inverse ψ˜.
Lie
(2) Clearly, the restriction ρ: Aut (LieAut(An)) → Aut (LieSAut(An)) is
Lie Lie
a homomorphism of ind-groups, and it is bijective (Remark 3.4). It follows from
(1) that the composition ρ◦Ad: Aut(An) → Aut (LieSAut(An)) is a bijective
Lie
homomorphism of ind-groups. Now we use Theorem 1.4(1) to conclude that ρ◦
Ad is an isomorphism, hence ρ is an isomorphism, too. Note that in the proof of
Theorem 1.4(1) below we will only use Proposition 3.5(1). (cid:3)
Proof of Theorem 1.4. (1) Let ϕ: Aut(An) → G be an homomorphism of ind-
groups such that dϕ is injective. We can assume that G = ϕ(Aut(An)), and we
will show that ϕ is an isomorphism. The basic idea is to construct a homomor-
phismψ: G →Aut(An)suchthatψ◦ϕ=id.ByProposition3.6below thisimplies
that ϕ is a closed immersion, hence an isomorphism.
DenotebyL⊆LieG theimageofdϕ.Foranyg ∈Aut(An)wehavedϕ◦Ad(g)=
Ad(ϕ(g))◦dϕ.Inparticular,LisstableunderAd(ϕ(g)),hencestableunderAd(G),
A CHARACTERIZATION OF AFFINE n-SPACE 7
becauseϕ(Aut(An))isdenseinG.Thuswegetthe followingcommutativediagram
of homomorphisms of ind-groups
Aut(An) −−−ϕ−→ G
AdAut(An)≃ AdG
 
Aut (LieyAut(An))) −−−≃−→ Auty (L)
Lie Lie
where the first vertical map is an isomorphism by Proposition 3.5(1). Thus, the
composition Ad ◦ϕ: Aut(An)→Aut (L)≃Aut(An) is an isomorphism, and so
G Lie
ϕ is also an isomorphism, by Proposition 3.6 below.
If dϕ is not injective, then kerdϕ ⊇ LieSAut(An) (Remark 2.3) and so dϕ =
f◦djacwheref: k→LieG isaLiealgebrahomomorphism.Ifk∗ ⊆GL (k)denotes
n
thecenter,thenϕ|k∗: k∗ →G factorthrough?n: k∗ →k∗,becauseSLn(k)⊆kerϕ,
i.e. ϕ(z) = ρ(zn) for any z ∈ k∗ and a suitable homomorphism ρ: k∗ → G of ind-
groups. By construction, dρ = f: k → LieG, and so the two homomorphisms ϕ
e
andρ◦jac have the same differential.Thus, by Proposition2.7, we get ϕ=ρ◦jac,
and we are done.
(2) Let ϕ: SAut(An) → G be a homomorphism of ind-groups. If dϕ is not
e
injective,thendϕ isthetrivialmap(Remark2.3),hencedϕ =dϕ¯ whereϕ¯: g7→
e e e
e is the constant homomorphism. Again by Proposition 2.7 we get ϕ=ϕ¯.
If dϕ is injective, set L := dϕ (LieSAut) ⊆ LieG. Again we can assume that
e e
G = ϕ(SAut(An)). Since L is stable under Adϕ(g) for all g ∈ Aut(An) it is also
stable under G, and we get, as above, the following commutative diagram
Aut(An) ←−⊇−−− SAut(An) −−−ϕ−→ G
AdAut(An)≃ AdSAut(An)⊆ AdG
  
Aut (LieyAut(An))) −−−Ψ−→ Aut (LieySAut(An))) −−−Φ−→ Auty (L)
Lie Lie Lie
bij ≃
whereAdAut(An) isanisomorphismandΨisabijectivehomomorphism(seePropo-
sition 3.5 and Remark 3.4). The image A ⊆ Aut (LieSAut(An)) of SAut(An) is
Lie
a closed subgroup isomorphic to SAut(An), and A −∼→ Φ(A) = Ad (ϕ(SAut(An)).
G
But ϕ(SAut(An)) ⊆ G is dense, and so Ad (G) = Φ(A). Thus, the composition
G
Ad ◦ϕ: SAut(An) → Φ(A) is an isomorphism, and so ϕ is an isomorphism, by
G
Proposition 3.6 below. (cid:3)
Proposition 3.6. Let H,G be two ind-groups, and let ϕ: H → G, ψ: G → H be
two homomorphisms. If ψ◦ϕ=id , then ϕ is a closed immersion, i.e. ϕ(H)⊆G
H
is a closed subgroup and ϕ induces an isomorphism H−∼→ϕ(H).
Proof. By base change we can assume that the base field k is uncountable. Let
H = H and G = G where we can assume that H ⊆ G for all i. Moreover,
Si i Sj j i i
foreveryithereisak=k(i)suchthatψ(G )⊆H .Byassumption,thecomposition
i k
ψ◦ϕ: H → G → H is the closed embedding H ֒→ H , hence the first map is
i i k i k
a closed embedding. Thus H := ϕ(H ) is a closed subset of G and H := ϕ(H) =
i i i
H . Now the claim follows from Lemma 3.7 below by setting S :=kerψ. (cid:3)
Si i
Recall that a subset S ⊆ V of an ind-variety V is called ind-constructible if
S = S where S ⊆S are constructible subsets of V.
Si i i i+1
8 HANSPETERKRAFT
Lemma 3.7. Let G be an ind-group, H ⊆ G a subgroup and S ⊆ G an ind-
constructible subset. Assume that k is uncountable and that
(1) H = H where H ⊆H ⊆G are closed algebraic subsets,
Si i i i+1
(2) the multiplication map S×H →G is bijective.
Then H is a closed subgroup of G.
Proof. LetG = G .Wehavetoshowthatforeverykthereexistsani=i(k)such
Sk k
that H ∩G =H ∩G . We can assume that e∈S = S . Then, by assumption,
k i k Si i
G = S H . Since S H ∩G is a constructible subset of G it follows that there
Sj j j j j k k
existsa j =j(k)suchthat G ⊆S H ([FK14,Lemma 3.6.4]).Setting S˙ :=S\{e}
k j j
we get S˙H∩H =∅. Thus, G =(S˙ H ∩G )∪(H ∩G ) and H ∩S˙ H =∅, hence
k i i k i k i i
H ∩G =H ∩G . (cid:3)
k i k
Finally, we can prove Theorem 1.5.
Proof of Theorem 1.5. (1) We already know from Theorem 1.4 that an injective
homomorphism ϕ: Aut(An) → Aut(An) is a closed immersion. We claim that
dϕ : LieAut(An) → LieAut(An) is an isomorphism. To show this, consider the
e
linear action of GL (k) on LieAut(An). We then have
n
LieAut(An)⊆Vec(An)≃kn⊗k[x ,...,x ]= kn⊗k[x ,...,x ]
1 n M 1 n d
d
andthelatterismultiplicity-freeasaGL (k)-moduleaswellasanSL (k)-module.
n n
Now ϕ(GL (k)) ⊆ Aut(An) is a closed subgroup isomorphic to GL (k). More-
n n
over, dϕ : LieAut(An) → LieAut(An) is an injective linear map which is equi-
e
variant with respect to ϕ: GL (k) −∼→ ϕ(GL (k)). Since ϕ(GL (k)) is conju-
n n n
gate to the standard GL (k) ⊆ Aut(An) and since the representation of GL (k)
n n
on LieAut(An) is multiplicity-free, it follows that dϕ is an isomorphism. Thus
e
G := ϕ(Aut(An)) ⊆ Aut(An) is a closed subgroup with the same Lie algebra as
Aut(An),andwegetthefollowingcommutativediagram(seeproofofTheorem1.4):
Aut(An) −−−ϕ−→ G −−−⊆−→ Aut(An)
AdAut(An)≃ AdG AdAut(An)
  
Aut (LieyAut(An))) −−−≃−→ Aut y(LieG) Aut (LiyeAut(An))
Lie Lie Lie
As a consequence, all maps are isomorphisms, and so G = Aut(An) and ϕ is an
isomorphism.
It remains to see that every automorphism ϕ ∈ Aut(An) is inner. Since dϕ ∈
e
Aut (LieAut(An)) we get dϕ = Ad(g) for some g ∈ Aut(An), by Remark 3.4.
Lie e
This means that dϕ =(dIntg) and so ϕ=Intg, by Proposition 2.7.
e e
(2) The same argument as above shows that every nontrivial homomorphism
SAut(An)→SAut(An) is an isomorphismwhere we use the fact that the action of
SL (k) on LieSAut(An) is multiplicity-free.
n
Moreover,the homomorphismAd: Aut(An)→Aut (LieSAut(An)) is a bijec-
Lie
tive homomorphism of ind-groups (Remark 3.4). Hence, for every ϕ ∈ SAut(An)
there is a g∈Aut(An) such that dϕ =Adg which implies that ϕ=Intg. (cid:3)
e
A CHARACTERIZATION OF AFFINE n-SPACE 9
4. A special subgroup of Aut(X)
Our Theorem 1.1 will follow from a more general result which we will describe
now.ForanyaffinevarietyX considerthenormalsubgroupU(X)ofAut(X)gener-
atedbytheunipotentelements,or,equivalently,bytheclosedsubgroupsisomorphic
to k+. This is an instance of a so-calledalgebraically generated subgroup of an ind-
group, see [KZ14]. The group U(X) was introduced and studied in [AFK13] where
the authors called it the group of special automorphisms of X. In particular, they
proved a very interesting connection between transitivity properties of U(X) and
the flexibility of X.
Let us define the following notion of an “algebraic” homomorphism between
these groups.
Definition 4.1. Ahomomorphismϕ: U(X)→U(Y)is algebraic, if foranyclosed
subgroup U ⊆ U(X) isomorphic to k+ the image ϕ(U) ⊆ U(Y) is closed and
ϕ| : U → ϕ(U) is a homomorphism of algebraic groups. We say that U(X) and
U
U(Y) are algebraically isomorphic, U(X)≃U(Y), if there exists a bijective homo-
morphism ϕ: U(X)→U(Y) such that ϕ and ϕ−1 are both algebraic.
Lemma 4.2. Let ϕ: U(X)→U(Y) be an algebraic homomorphism. Then, for any
algebraic subgroup G ⊆ U(X) generated by unipotent elements the image ϕ(G) ⊆
U(Y) is closed and ϕ| : G→ϕ(G) is a homomorphism of algebraic groups.
G
Proof. There exist closed subgroups U ,...,U ⊆ G isomorphic to k+ such that
1 m
the multiplication map µ: U ×U ×···×U → G is surjective. This gives the
1 2 m
following commutative diagram
µ
U ×U ×···×U −−−−→ G
1 2 m
ϕ˜:=ϕ|U1×···×ϕ|Um ϕ|G
 
y µ¯ y
ϕ(U )×ϕ(U )×···×ϕ(U ) −−−−→ ϕ(G)
1 2 m
whereallmapsaresurjective.Itfollowsthatϕ(G)⊆Aut(Y)isa(closed)algebraic
subgroup,andthusϕ(G)=ϕ(G),becauseϕ(G)isconstructible.Itremainstoshow
that ϕ| is a morphism. This follows from the next lemma, because G is normal,
G
and µ and the composition ϕ| ◦µ=µ¯◦ϕ˜ are both morphisms. (cid:3)
G
Lemma 4.3. Let X,Y,Z be irreducible affine varieties where Y is normal. Let
µ: X → Y be a surjective morphism and ϕ: Y → Z an arbitrary map. If the
composition ϕ◦µ is a morphism, then ϕ is a morphism.
Proof. We have the following commutative diagram of maps
⊆
Γ −−−−→ X ×Z
ϕ◦µ
µ¯ µ×id
 
y ⊆ y
Γ −−−−→ Y ×Z
ϕ
p pr
 Y
 
y y
Y Y
whereΓ andΓ denotethe graphsofthe correspondingmaps.We havetoshow
ϕ◦µ ϕ
thatΓ ⊆Y×Z isclosedandthatpisanisomorphism.Thediagramshowsthatµ¯is
ϕ
10 HANSPETERKRAFT
surjective,henceΓ isconstructible,andpisbijective.Thus,theinducedmorphism
ϕ
p¯: Γ → Y is birational and surjective, hence an isomorphism since Y is normal
ϕ
(see [Igu73, Lemma 4, page 379]). Since p is bijective, we finally get Γ =Γ . (cid:3)
ϕ ϕ
The proof of the following theorem will be given in the next section.
Theorem 4.4. Let X be a connected affine variety. If U(X) is algebraically iso-
morphic to U(An), then X is isomorphic to An.
A first consequence is our Theorem 1.1.
Corollary 4.5. Let X be a connected affine variety. If Aut(X) ≃ Aut(An) as
ind-groups, then X ≃An.
Proof. It is clear from the definition that an isomorphism Aut(X) −∼→ Aut(An)
induces an algebraic isomorphism U(X)−∼→U(An). (cid:3)
Finally, we define the following closed subgroups of Aut(X):
Autalg(X):=hG|G⊆Aut(X) connected algebraici,
SAutalg(X):=hU |U ⊆Aut(X) unipotent algebraici.
We have SAutalg(X)= U(X)⊆Autalg(X)⊆Aut(X). Now the same argument as
above gives the next result.
Corollary 4.6. Let X be a connected affine variety. If SAutalg(X) is isomorphic
to SAutalg(An) as ind-groups, then X is isomorphic to An, and the same holds if
we replace SAutalg by Autalg.
5. Root subgroups and modifications
Let G be an ind-group, and let T ⊆G be a torus.
Definition 5.1. A closed subgroup U ⊆ G isomorphic to k+ and normalized by
T is called a root subgroup with respect to T. The character of T on LieU ≃ k is
called the weight of U.
Let X be an affine variety and consider a nontrivial action of k+ on X, given
by λ: k+ →Aut(X). If f ∈O(X) is k+-invariant, then we define the modification
f ·λ of λ in the following way (see [FK14, section 8.3]):
(f ·λ)(s)x:=λ(f(x)s)x for s∈k and x∈X.
It is easy to see that this is again a k+-action. In fact, if the corresponding lo-
cally nilpotent vector field is δ , the fδ is again locally nilpotent, because f is
λ λ
λ-invariant, and defines the modified k+-action f ·λ. This modified action f ·λ is
trivial if and only if f vanishes on every irreducible component X of X where the
i
action λ is nontrivial. It is clear that the orbits of f ·λ are contained in the orbits
of λ and that they are equal on the open subset X ⊆ X. In particular, if X is
f
irreducible and f 6=0, then λ and f ·λ have the same invariants.
If U ⊆ Aut(X) is isomorphic to k+ and if f ∈ O(X)U is a U-invariant, then
we define in a similar way the modification f · U of U. Choose an isomorphism
λ: k+ −∼→ U and set f ·U := (f ·λ)(k+), the image of the modified action. Note
that Lie(f · U) = fLieU ⊆ LieAut(X) ⊆ Vec(X) where we use the fact that
Vec(X) is a O(X)-module.