Extensions of isomorphisms of subvarieties in flexible varieties
Shulim Kaliman
University of Miami, Coral Gables, FL 33124, USA
[email protected]
Abstract.
Let X be an algebraic variety isomorphic to the complement of a closed subvariety of
dimension at most n−3 in Akn. We find some conditions under which an isomorphism of two closed subvarieties of X can be extended
to an automorphism of X. We also study the similar problem for subvarieties of affine quadrics and SL(n,k).
††footnotetext: 2010
Mathematics Subject Classification:
14R10, 14R20, 32M17, 32M25.
Key words: affine
varieties, group actions, one-parameter subgroups, transitivity.
Contents
- 1 Algebraically generated groups of automorphisms
- 2 Flexible varieties
- 3 Relative version of Theorem 2.8
- 4 General projections for flexible varieties. I
- 5 General projections for flexible varieties. II
- 6 General projections for partial quotient morphisms of flexible varieties
- 7 The case of Gromov-Winkelmann flexible varieties
- 8 The case of quadrics
- 9 Comparable morphisms
- 10 Holomorphic extension of θ
- 11 The case of SL(n,k)
Introduction
Let X be a smooth
quasi-affine variety over an algebraically closed field k of characteristic zero
and Y1 and Y2 be closed subvarieties of X.
We study the following extension problem:
Under what restrictions on Yi and TYi an isomorphism Y1→Y2 extends to an automorphism of X?
This question makes sense when X itself possesses a large automorphism
group Aut(X) which leads to the notion of a flexible variety [2].
Recall that it is a quasi-affine algebraic variety of dimension at least 2 on which the group SAut(X) (generated by elements of all one-parameter unipotent subgroups of Aut(X)) acts m-transitively for every m>0.
The simplest example of a flexible variety is X=Akn and the extension problem was studied extensively for such an X.
The starting point (and an inspiration) of that research was, of course, the Abhyankar-Moh-Suzuki theorem [1], [35] which
states that given two plane curves isomorphic to a line one can be transferred to the other by an automorphism of Ak2.
Then, disproving an Abhyankar’s conjecture, Jelonek [18] established
that if one requires
that 4dimY1+2≤n then one gets a positive answer to the extension problem in Akn
for the case of smooth Yi 111 Abhyankar’s conjecture was also disproved slightly earlier by
Craighero [6] but he did not consider the extension problem in the full generality.. In the non-smooth case we have to take into consideration dimTYi
and the more general result established by the author [19] and Vasudevan Srinivas [36] states the following:
Theorem 0.1**.**
Let φ:Y1→Y2 be an isomorphism of two closed subvarieties of Akn such that n≥ED(Y1)+1 where ED(Y1)=max(2dimY1+1,dimTY1).
Then φ extends to an automorphism of Akn.
The first result in the case of a flexible variety different from Akn is due to Van Santen (formerly Stampfli) [33] who proved that given two curves isomorphic to a line
in an algebraic variety X isomorphic to the special linear group SL(n,k) over k
one can be transferred to the other by an automorphism of X provided that n≥3
(in particular, any such curve is an orbit of a Ga-action). This theorem was generalized later
in the paper of Van Santen and Feller [8] where they showed that the same is true if one considers two curves isomorphic to a line in a connected linear algebraic group modulo some exceptions. Then Van Santen jointly with J. Blanc showed (among other facts) that there are closed surfaces isomorphic
to Ak2 in SL(2,k) which cannot be transferred to each other by an
automorphism of SL(2,k) (as an algebraic variety).
In the present paper beside SL(n,k) we study smooth quadrics in Akn
and the case of X equal to the complement to a codimension at least 2 subvariety in Akn
(we call it the Gromov-Winkelmann case since these authors established the flexibility of such an X [37], [14]). The main results of our paper are the following.
Theorem 0.2**.**
Let Z, Y1, and Y2 be closed subvarieties of Akn such that Y1∩Z=Y2∩Z=∅, dimZ≤n−3
and ED(Y1)≤n−2. Let φ:Y1→Y2 be an isomorphism and X=Akn∖Z.
Suppose also that either
(a)* dimZ+dimY1≤n−3, or*
(b)* dimY1=1 and dimZ=n−3.*
Then there exists an automorphism γ∈SAut(X) for which γ∣Y1=φ.222If ED(Z)≤n−1
then there is no need for this theorem. Indeed, one can consider the isomorphism ψ:Y1∪Z→Y2∪Z such
that ψ∣Y1=φ and ψ∣Z=idZ. Then Theorem 0.1 implies that ψ extends to an automorphism
of Akn as soon as ED(Y1)≤n−1.
Theorem 0.3**.**
Let m≥6 and X be a hypersurface in Akm that is a nonzero fiber of a non-degenerate quadratic form. Suppose that
φ:Y1→Y2 is an isomorphism of two closed subvarieties of X.
Let ED(Yi)+dimYi≤m−2.
Then φ extends to an automorphism of X which belongs to SAut(X).
Theorem 0.4**.**
Let X=SL(n,C) and
φ:Y1→Y2 be an isomorphism of two closed subvarieties of X such that either
(i)* ED(Yi)+dimYi≤n−2, Hi(Y1)=0 for i≥3
and H2(Y1) is a free abelian group; or*
(ii)* dimY1 is a curve and ED(Yi)≤n−2, or;*
(iii)* Y1 is a once-punctured curve and ED(Y1)≤2n−3.*
Then there exists a holomorphic automorphism β of X such that β∣Y1=φ.
Theorem 0.5**.**
Let φ:Y1→Y2 be an isomorphism of two closed
subvarieties of X≃SL(n,k) with n≥3 such that Yi is isomorphic to Akk. Suppose that
either k≤3n−1 or k=1.
Then there exists α∈SAut(X) such that α∣Y1=φ.
The paper is organized as follows. In the first six sections we develop some technique which is valid for a wide class of flexible varieties.
More precisely, in Section 1 we consider a morphism κ:X→P
of smooth irreducible varieties
such that X is equipped with an action of a group G⊂Aut(X) which
preserves every fiber κ−1(p),p∈P and acts transitively on it.
We describe some conditions under which one can
find an algebraic family A⊂G of automorphisms of X such that
for two subvarieties Y and Z of X and a general
element α∈A the varieties Y and α(Z) are transversal (this is a relative version of the transversality theorem from [2] that dealt with the case when P was
a singleton). In Section 2 we remind
some facts about flexible varieties and in Section 3 we prove a relative version of a theorem from [2] which yields automorphisms of a given flexible
variety with prescribed jets at a finite number of points.
Sections 4-6 are crucial.
Namely, the proof of Theorem 0.1 is heavily based on the fact that for a general linear projection
θ:Akn→Akn−1 the variety θ(Yi) is closed in Akn−1
and the restriction of θ yields an isomorphism Yi→θ(Yi). Note that
θ can be viewed as the composition of a fixed projection θ0 and a general linear automorphism
of Akn.
Hence in Section 4 we imitate this idea for a morphism ϱ:X→Q over P as above
but with each fiber of κ being a G-flexible variety. We show
that when the morphism ϱ and the variety Q are smooth and
Z is a closed subvariety of X with ED(Z)≤dimQ
then for a general element α of some algebraic family A⊂G the morphism
ϱ∣α(Z):α(Z)→Q is an injection and, furthermore, the induced morphism
Tα(Z)→TQ of the Zariski tangent bundles is also an injection.
However, a priori this map is not proper and in Section 5 we describe some conditions
under which the morphism ϱ∣α(Z) is also proper and, therefore,
ϱ∘α(Z) is closed in Q and ϱ∣α(Z):α(Z)→ϱ∘α(Z)
is an isomorphism. These facts are
already sufficient for the proof of Theorem 0.2 in Section 7. Section 6 is devoted
to the independently interesting case when ϱ
is a partial quotient morphism of a Ga-action on X. In this situation we cannot
claim that ϱ and Q are smooth and cannot guarantee the properness of ϱ∣α(Z).
However, we establish that for any given finite subset S⊂Z and a general α∈A one can find
a neighborhood V′ of ϱ(α(S)) in ϱ(α(Z))⊂Q such that for
V=ϱ−1(V′)∩α(Z) the restriction of ϱ∣α(Z) yields an isomorphism V→V′.
Applications of this result go beyond the present paper (e.g., see [21]).
Section 8 contains Theorem 0.3 and in Sections 9 and 10 we prove some technical results
that enable us to obtain Theorems 0.4 and 0.5 in Section 11
where we also present an example of a topological
obstruction for the extension problem in the case of general flexible varieties.
Acknowledgements. The author is deeply indebted to his referee for catching mistakes in the original version of this manuscript
and for an unusually thorough review which was a great help to the author.
1. Algebraically generated groups of automorphisms
Let X be an irreducible algebraic variety and Aut(X) be the group of its algebraic automorphisms.
Recall the following terminology introduced by Ramanujam [30].
Definition 1.1**.**
(1) Given an irreducible algebraic variety A and
a map φ:A→Aut(X) we say that (A,φ)
is an algebraic family of automorphisms on X if the induced map
A×X→X, (α,x)↦φ(α).x is a morphism.
(2) In the case when A is a connected algebraic group and the induced map
A×X→X is not only a morphism but also an action of A on X we call this family a connected algebraic subgroup of Aut(X).
Definition 1.2**.**
Following [2, Definition 1.1] we call a subgroup G of Aut(X) algebraically generated if it is generated as an abstract group by a family
G of connected algebraic subgroups of Aut(X).
We have the following important fact [2, Theorem 1.15] (which is the analogue of the Kleiman transversality theorem [22]
for algebraically generated groups).
Theorem 1.3**.**
(Transversality Theorem) Let a subgroup G⊆Aut(X) be
algebraically generated by a system G of connected algebraic
subgroups closed under conjugation in G. Suppose that G acts
with an open orbit O⊆X.
Then there exist subgroups H1,…,Hm∈G such that for
any locally closed reduced subschemes Y and Z in O one can
find a Zariski dense open subset U=U(Y,Z)⊆H1×…×Hm such that every element (h1,…,hm)∈U satisfies the following:
- (a)
The translate (h1⋅…⋅hm).Zreg
meets Yreg
transversally. **
2. (b)
dim(Y∩(h1⋅…⋅hm).Z)≤dimY+dimZ−dimX*. *333We put the dimension of empty sets equal to −∞.
In particular Y∩(h1⋅…⋅hm).Z=∅
if dimY+dimZ<dimX.
We need to generalize [2, Theorem 1.15] further.
Theorem 1.4**.**
(Collective Transversality Theorem)* Let X and P be smooth irreducible
algebraic varieties and κ:X→P be a smooth morphism (in particular X×PX is smooth
and dimX×PX=2dimX−dimP).
Let a group G⊆Aut(X) be
algebraically generated by a system G of connected algebraic
subgroups closed under conjugation in G.
Suppose that the G-action transforms every fiber κ−1(p) into
itself and, furthermore, the restriction of the G-action to κ−1(p) is transitive for every p∈P.*
Then there exist subgroups H1,…,Hm∈G such that for
any locally closed reduced subschemes Y and Z in X one can
find a Zariski dense open subset U=U(Y,Z)⊆H1×…×Hm so that every element (h1,…,hm)∈U satisfies the following:
(i)*
dim(Y∩(h1⋅…⋅hm).Z)≤dim(Y×PZ)+dimP−dimX.*
In particular, when dimY×PZ≤dimY+dimZ−dimP one has
(ii)* dim(Y∩(h1⋅…⋅hm).Z)≤dimY+dimZ−dimX.*
Furthermore, suppose that the inequality dimY×PZ≤dimY+dimZ−dimP holds
and also that Z, Y×PZ, and Y×PX are smooth.
Then
(iii)* (h1⋅…⋅hm).Z meets Y transversally.*
The proof of Theorem 1.4 is an adjustment of the proof of [2, Theorem 1.15].
Hence following the latter we establish first three facts which in the case of a singleton P are nothing but Propositions 1.5, 1.8, and 1.16 in [2].
Proposition 1.5**.**
(Analogue of [2, Proposition 1.5])* Let the assumptions of Theorem 1.4 hold.
There are (not necessarily distinct) subgroups
H1,…,Hm∈G such that for every p∈P and each x∈κ−1(p) one has*
[TABLE]
Proof.
Let us
introduce the partial order on the set of sequences in G such that for H=(H1,…,Hm) and H′=(H1′,…Hs′) one has
[TABLE]
Assuming first that κ has a section λ:P→X with S=λ(P)
we consider H.S=⋃x∈SH.x where H.x=H1.(H2.⋯.(Hm.x). Then
such a set H.S is constructible (since it is the image of the algebraic variety H1×…×Hm×S under a morphism). In particular, XH:=X∖H.S is a constructible set.
Furthermore, the following property holds:
if H≽H′, then H′.S⊂H.S and, therefore, XH⊂XH′.
Because of transitivity for every y∈κ−1(p) we can find a sequence H=(H1,…,Hm) and g=h1⋅…⋅hm (where hi∈Hi) for which
y=g.x where x=S∩κ−1(p). Hence y∈H~.S for any sequence H~ of the form H~=(H1,H). In particular, choosing y in any given irreducible component
C of XH1 we guarantee C is not contained in XH~. That is, dimXH~∩C<dimC since XH~⊂XH1.
Thus, enlarging H we can reduce the dimension of XH and continuing this process we can make XH=∅. In particular, for every
p∈P and x=S∩κ−1(p) one has H.x=κ−1(p). Therefore, for every y∈κ−1(p) there exists g=h1⋅…⋅hm as before (with H now
being independent from y) for which y=g.x. Let Ht=(Hm,Hm−1,…,H1) and H~=(H,Ht), i.e., x∈Ht.y.
Then one has H~.y=κ−1(p) for every y∈κ−1(p), i.e we get the desired
conclusion in the presence of a section λ.
In the general case consider an étale neighborhood W of a point p∈P and suppose that V⊂P is the image of W under the natural morphism. Since κ is smooth one can suppose that
for an appropriate choice of W the natural projection τ:X×PW→W has a section. Consider the induced G-action on X×PW.
By the previous argument there is a sequence H such that for every w∈W and z∈τ−1(w) one has H.z=τ−1(w). Applying the natural projection
X×PW→X we see that for every p∈V and every y∈κ−1(p) we have H.y=κ−1(p). Choosing now a finite number of étale neighborhoods that cover X,
we can enlarge H so that it works for each of these neighborhoods. This implies the desired conclusion.
∎
Proposition 1.6**.**
(Analogue of [2, Proposition 1.8])* Let the assumptions of Theorem 1.4 hold.
Assume that the generating family G of
connected algebraic subgroups is closed under conjugation in G,
i.e., gHg−1∈G for all g∈G and H∈G. Then
there is a sequence H=(H1,…,Hm) in G such that
for all p∈P and x∈κ−1(p) the tangent space Txκ−1(p)
is spanned by the tangent spaces*
[TABLE]
to the orbits H1.x,…,Hm.x at x.
Proof.
Let H=(H1,…,Hm) be a sequence in G satisfying the conclusion of Proposition 1.5. Consider the map
[TABLE]
The fiber τ−1(x) of the second projection τ:X×PX→X over x∈X is naturally isomorphic to κ−1(p) where p=κ(x).
Hence, by Proposition 1.5 ΦH is a surjective map while the assumptions of Theorem 1.4 imply that τ is a smooth morphism.
Let us consider the map of relative tangent bundles
[TABLE]
and its restriction to {(e,…,e)}×X≅X (where e is the identity in the group G),
[TABLE]
The set UH of points in X where this map
is surjective is, of course, open. By [2, Proposition 1.8] for every p∈P and x∈κ−1(p) the tangent space Txκ−1(p) is spanned by
the tangent spaces Tx(H.x), where H∈G.
Hence ⋃HUH coincides with X. Since an increasing union of open subsets stabilizes,
we obtain that X=UH for H sufficiently large (with respect to the partial order introduced in Proposition 1.5). This yields the desired conclusion.
∎
Proposition 1.7**.**
(Analogue of [2, Proposition 1.16])*
Let the assumption of Theorem 1.4 hold.
Then there is a sequence H1,…,Hm in G
so that for a suitable open dense subset U⊆Hm×…×H1, the map*
[TABLE]
is surjective and smooth on U×X.
Proof.
By Proposition 1.5 there are subgroups
H1,…,Hm⊆G in G such that
Φm is surjective. Let Um⊂Hm×…×H1×X be the set of points
where Φm is smooth. Then Um is non-empty by [16, Chapter III, Corollary 10.7] and it is open
by [SGA1, Exp. II, Prop. 1.1].
Consider the
complement Am=(Hm×…×H1×X)\Um. Let us study the effect of increasing the number m of factors in the product Hm×…×H1.
Suppose that H is an element
of G and Φ′ plays the same role for the sequence H1,…,Hm,H as Φm for the sequence H1,…,Hm.
Note that for a fixed h∈H the restriction of Φ′ yields the morphism {h}×Hm×…×H1×X⟶X×PX
that is the composition of Φm and the automorphism φ of
X×PX given by (x,y)→(h.x,y) which implies smoothness of Φ′ on H×Um.
Thus, Um+1⊇Hm+1×Um and
Am+1⊆Hm+1×Am. Increasing the
number of factors by Hm+1,…,Hm+k in a suitable way,
we can achieve that
[TABLE]
Indeed, if
(hm,…,h1,x)∈Am and y=(hm⋅…⋅h1).x
then by Proposition
1.6 for suitable Hm+k,…,Hm+1 the map
[TABLE]
is smooth in all points (e,…,e,y) where e is the identity of the group G. In particular, Φm+k is smooth in all points
(e,…,e,hm,…,h1,x) with x∈X, i.e.,
[TABLE]
Now (3) follows.
Thus increasing the number of factors suitably we can achieve
that
[TABLE]
That is, the image of Am under the projection
[TABLE]
is nowhere dense. Hence there is an open dense subset
U⊆Hm×…×H1
such that Φm:U×X→X×PX is smooth.
∎
Remark 1.8**.**
Let Φm and H1,…,Hm be as in Proposition 1.7 and let
H be an element G. Suppose that Φ′ (resp. Φ′′) plays the same role for the sequence H1,…,Hm,H (resp. H,H1,…,Hm) as Φm
for the sequence H1,…,Hm. We showed already in the proof of Proposition 1.7 that
that Φ′ is smooth on H×Um. Similarly, Φ′′ is smooth on the preimage U′′ of Um under the natural
projection Hm×…×H1×H×X→Hm×…×H1×X.
Indeed, the smoothness of Φm∣Um is equivalent to the fact that
it factors locally through an étale morphism Um→Akn×(X×PX) over X×PX or, in other words, that Φm∣Um admits
a local étale section s:X×PX→Um
(where s(x,y)=(s^(x,y),y) with s^(x,y)∈Hm×…×H1). Denoting by φ the automorphism of
X×PX given by (x,y)→(h.x,y) one observes that
(x,y)→(s^∘φ(x,y),h,y)∈Hm×…×H1×h×X
is a local section of Φ′′∣U′′, i.e., Φ′′ is smooth on U′′.
Proof of Theorem 1.4.
By Proposition 1.7 there are subgroups H1,…,Hm in G such
that Φm:U×X→X×PX is smooth for some open
subset U⊆Hm×…×H1.
Let C=Φm∗(Y×PZ)∩(U×X).
Consider first the case in (i) when Y×PZ is smooth.
Then C is smooth.
By [16, Chapter III, Corollary 10.7] the general fibers of the
projection πC:C→U
are smooth as well.
Suppose that πC is dominant (otherwise the general fibers of πC are empty). Shrinking U we may now assume that all fibers of πC are
smooth. Then the dimension of every fiber of Φm is dimU−dimX+dimP.
Thus dimC=dimU+dimP−dimX+dimY×PZ and the dimension of every fiber πC∗(h) of πC is
[TABLE]
Observe now that for a point h=(hm,…,h1)∈U the fiber
πC∗(h) maps bijectively via {h}×X→X,(h,x)→(hm⋅…⋅h1).x onto Y∩(h1⋅…⋅hm).Z which yields (i) (and therefore (ii)) in the case of smooth Y×PZ.
In the general case stratifying Z and Y we can find Zariski dense open subsets Z0⊂Z and Y0⊂Y such that
Y0×PZ0 is smooth.
By Formula (4) we see that for a general h∈U the dimension
of Y0∩(h1⋅…⋅hm).Z0 is at most dimY×PZ+dimP−dimX. Let Z′=Z∖Z0 and Y′=Y∖Y0 and
consider, say, the pair (Y0,Z′). We can suppose that Y0×PZ′ is smooth (otherwise stratify further). Then the same argument with Formula (4)
implies that the dimension of Y0∩(h1⋅…⋅hm).Z′ is at most dimY0×PZ′+dimP−dimX≤dimY×PZ+dimP−dimX. Repeating this procedure for the pairs
(Y′,Z0) and (Y′,Z′) we get (i) and (ii) in full generality.
For (iii) consider Z=U×Z and Y=Φm∗(Y×PX)∩(U×X), i.e., C (as a scheme) is the
intersection of Z and Y. As before, shrinking U we can suppose that all fibers of the
natural projections πY:Y→U and πZ:Z→U
are smooth.
Observe also that
πZ∗(h)={h}×Z⊂{h}×X and πY∗(h)={h}×(h1−1⋅…⋅hm−1).Y⊂{h}×X 444In particular,
the last equality implies that Y is smooth under the assumption of (ii).. It remains to note that if these two smooth
subvarieties of {h}×X do not meet transversely and the dimension of their intersection πC∗(h) is dimY+dimZ−dimX then πC∗(h) cannot be smooth (as a scheme).
Indeed, the absence of transversality implies that at some closed point of x∈πC∗(h)
the dimension of the intersection of the tangent spaces TxπY∗(h) and TxπZ∗(h)
is greater than dimY+dimZ−dimX. Note that this intersection coincides with the intersection of
kernels of the differentials of all functions from the defining ideals of πZ∗(h) and πY∗(h),
and, therefore, from the defining ideal of πC∗(h). However, for a smooth πC∗(h)
this dimension must be equal to dimY+dimZ−dimX.
Hence the smoothness of πC∗(h)
established before yields (iii) which concludes the proof.
∎
Remark 1.9**.**
(1) Suppose that κ(Y) is dense in P and all fibers of κ∣Y are of the same dimension (say, κ∣Y is flat).
That is, the dimension of each of these fibers is dimY−dimP. Then dimY×PZ=dimY+dimZ−dimP
and we are under the assumption of (ii) in Theorem 1.4, i.e., the dimension of Y∩(h1⋅…⋅hm).Z
is at most dimY+dimZ−dimX.
(2) Let us emphasize the following fact the first part of which follows from the argument in the proof of Theorem 1.4.
Proposition 1.10**.**
If a sequence H1,…,Hm∈G satisfies Proposition 1.7 then it satisfies also
Theorem 1.4. Furthermore, for any element H of G
the sequence H1,…,Hm,H (resp. H,H1,…,Hm) satisfies Theorem 1.4 as well.
Proof.
The second statement is true because by Remark 1.8
the sequence H1,…,Hm,H (resp. H,H1,…,Hm) satisfies Proposition 1.7.
∎
Remark 1.11**.**
(1) Let us consider an application of Proposition 1.10.
Suppose that X and P are smooth irreducible
algebraic varieties and κ:X→P be a morphism which is not in general smooth or even dominant.
Let a group G⊆Aut(X) be
algebraically generated by a system G of connected algebraic
subgroups closed under conjugation in G.
Suppose that the G-action transforms every fiber κ−1(p) into
itself and, furthermore, the restriction of the G-action to κ−1(p) is transitive for every p∈P.
By the Generic Smoothness theorem (e.g., see [16, Chapter III, Corollary 10.7]) we can present κ(X) as a disjoint union ⋃k=1nPk of
smooth varieties such that for Xk=κ−1(Pk) the morphism κ∣Xk:Xk→Pk is smooth.
For Y and Z as in Theorem 1.4 let Yk=Xk∩Y and Zk=Xk∩Z.
Then by Theorem 1.4 there exist subgroups H1k,…,Hmkk∈G such that
one can find a Zariski dense open subset Uk=U(Yk,Zk)⊆H1k×…×Hmkk so that for every element (h1k,…,hmkk)∈Uk we have the inequality
[TABLE]
Consider now a general element α in
[TABLE]
Then Proposition 1.10 implies that one has
[TABLE]
(2) Let us consider now the case when in (1)
all nonempty fibers of the morphism κ∣Y:Y→P are of the same dimension.
By Remark 1.9 (1) we have dim(Yk×PkZk)=dimYk+dimZk−dimPk for every k=1,…,n.
Hence
[TABLE]
Furthermore, if all nonempty fibers of κ∣X:X→P are of the same dimension then
dimYk−dimXk≤dimY−dimX and we have the following.
Proposition 1.12**.**
Let X and P be smooth irreducible
algebraic varieties and κ:X→P be a flat morphism.
Let a group G⊆Aut(X) be
algebraically generated by a system G of connected algebraic subgroups closed under conjugation in G.
Suppose that the G-action transforms every fiber κ−1(p) into
itself and, furthermore, the restriction of the G-action to κ−1(p) is transitive for every p∈P.
Let Y and Z be locally closed reduced subschemes of X such that
all nonempty fibers of the morphis κ∣Y:Y→P are of the same dimension.
Then there exist subgroups H1,…,Hm∈G such that such that for any Y and Z as above
one can find a Zariski dense open subset U=U(Y,Z)⊆H1×…×Hm so that for every element (h1,…,hm)∈Uk we have the inequality
dim(Y∩(h1⋅…⋅hm).Z)≤dimY+dimZ−dimX.
2. Flexible varieties
Definition 2.1**.**
(1) A derivation σ on the ring A of regular functions on a quasi-affine algebraic variety X is called locally nilpotent
if for every 0=a∈A there exists a natural n for which σn(a)=0. For the smallest n with this property one defines
the degree of a with respect to σ as degσa=n−1. This derivation can be viewed as a vector field on X which
we also call locally nilpotent. The flow of this vector field is an algebraic Ga-action on X, i.e., the action of the group (k,+)
which can be viewed as a one-parameter unipotent group U in the group Aut(X) of all algebraic automorphisms of X.
In fact, every Ga-action is generated by a locally nilpotent vector field (e.g, see [13]).
(2) A smooth quasi-affine algebraic variety X of dimension at least 2
is called flexible if for every x∈X the tangent space TxX is spanned by the tangent vectors
to the orbits of one-parameter unipotent subgroups of Aut(X) through x.
(3) The subgroup SAut(X) of AutX generated by all one-parameter unipotent subgroups is called special.
We have the following [2, Theorem 01] and [9, Theorem 2.12].
Theorem 2.2**.**
For every irreducible smooth quasi-affine algebraic variety X of dimension at least 2 the following are equivalent
(i)* the special subgroup SAut(X) acts transitively on X;*
(ii)* the special subgroup SAut(X) acts infinitely transitively on X (i.e., for every natural m
the action is m-transitive 555Recall that a group G acts m-transitively on a space Y if for any two m-tuples (y1,…,ym) and (y1′,…,ym′)
of distinct points in Y there is an element α∈G such that α(yi)=yi′ for every i=1,…,m.);*
(iii)* X is flexible.*
Definition 2.3**.**
(1) For every locally nilpotent vector fields σ and each function f∈Kerσ from its kernel the field
fσ is called a replica of σ. Recall that such a replica is automatically locally nilpotent.
(2) Let N be a set of complete algebraic vector fields on X. We say that a subgroup GN⊂Aut(X) is generated by N if GN is generated
by the elements of all one-parameter groups that are flows of complete vector fields from N.
(3) A collection of locally nilpotent vector fields N is called saturated if N is closed under conjugation by elements in GN
and for every σ∈N each replica of σ is
also contained in N.
Definition 2.4**.**
Let N be a saturated set of locally nilpotent vector fields on X and G=GN be a subgroup of SAut(X)
which is generated by N.
Then X is called G-flexible if for any x∈X the vector space TxX is generated by the values of locally nilpotent vector fields from N at x.
Remark 2.5**.**
A priori the notion of G-flexibility depends on N while N is not determined uniquely by the group G=GN.
However, the following generalization of Theorem 2.2 ( see [9, Theorem 2.12]) shows that
this notion depends on G only.
Theorem 2.6**.**
Let X be an irreducible smooth quasi-affine algebraic variety of dimension at least 2
and G be a subgroup generated by a saturated set N of locally nilpotent vector fields on X.
Then the following are equivalent
(i)* G acts transitively on X;*
(ii)* G acts infinitely transitively on X;*
(iii)* X is G-flexible.*
The next fact is a straightforward consequence of Theorem 2.6.
Proposition 2.7**.**
Let X be a G-flexible variety where G is as in Definition 2.4 and Δ be the diagonal in X×X.
Then the natural action of G on the variety X×X∖Δ is transitive.666Precaution:
this natural action of G on X×X∖Δ is not infinitely transitive.
The following result will be very useful later in this paper.
Theorem 2.8**.**
([2, Theorem 4.14 and Remark 4.16])* Let x1,…,xm be distinct points in a G-flexible variety X of dimX=n where G
is generated by a saturated set N of locally nilpotent vector fields on X.
Then there exists an automorphism α∈G⊂SAut(X) such that it fixes the
points x1,…,xm and for every i the linear map dα∣TxiX coincides
with a prescribed element βi of SL(n,k).
Furthermore, let k∈N and γi be a k-jet of an isomorphism
between two étale neighborhoods of xi in X
which preserves xi and a local volume form at xi. Then α can be chosen so that for every i=1,…,m the k-jet of
α at xi coincides with γi.*
By the Rosenlicht Theorem (e.g., see [27, Theorem
2.3]) for X, A, and U as in Definition 2.1 (1)
one can find a finite set of U-invariant functions a1,…,am∈A,
which separate general U-orbits in X. They generate a morphism ϱ:X→Q into an affine
algebraic variety Q (in particular, dimQ=dimX−1 because general U-orbits are one-dimensional). Note that this set of invariant functions can be chosen so that Q is normal (since X is normal).
Definition 2.9**.**
Such a morphism ϱ:X→Q into a normal Q is called a partial quotient.
In the case when a1,…,am generate the subring AU of U-invariant elements of A
such a morphism is called the categorical quotient.777However, in general AU is not finitely generated by the
Nagata’s example. That is, why, following [9] we prefer to formulate some results for partial quotients.
Proposition 2.10**.**
Let G⊂SAut(X) be generated by a saturated set N of locally nilpotent vector fields on a
smooth quasi-affine algebraic variety X (i.e., X is G-flexible).
Suppose that N0 is a nontrivial subset of N which is also saturated and closed under conjugation by elements of G.
Then X is G0-flexible where the group G0 is generated by N0.
Proof.
By Theorem 2.8 for any point x∈X where σ∈N0 does not vanish we can find α1,…,αn∈G where n=dimX such that αi(x)=x and the values
of the fields α1∗(σ),…,αn∗(σ) at x generate TxX.
That is, TxX is generated by the values of the fields from N0.
Since G acts transitively on X we can guarantee the same for every x∈X
using conjugation by elements in G. Thus,
X is G0-flexible which is the desired conclusion.
∎
Definition 2.11**.**
Let δ1 and δ2 be a pair of locally nilpotent vector fields on X.
Suppose that Kerδ1 and Kerδ2 are finitely generated algebras (in particular,
δi admits the categorical quotient morphism ϱ:X→Qi).
We say that (δ1,δ2) is a compatible pair if
(i) the vector space Span(Kerδ1⋅Kerδ2) generated by elements from
Kerδ1⋅Kerδ2 contains a nonzero ideal in
k[X] (so called associated ideal of the pair) and
(ii) some element a∈Kerδ2 is of degree
1 with respect to δ1, i.e., δ1(a)∈Kerδ1∖{0}.
Remark 2.12**.**
(1) In [20] the vector field δ2 was also allowed to be semi-simple, but we do not consider this case here.
(2) For every locally nilpotent vector field δ on X its nonzero replica fδ has the same kernel
and for each a∈k[X] the degree of a with respect to δ coincides with its degree with respect to fδ.
Hence, for a compatible pair (δ1,δ2)
and fi∈Kerδi∖{0},i=1,2 the pair (f1δ1,f2δ2) is also compatible.
Furthermore, the conjugation by any element of Aut(X) transfers (δ1,δ2) into another compatible pair.
(3) The assumption that Kerδ1 and Kerδ2 are finitely generated was unfortunately missed in [20].
(4) It is worth mentioning that in (ii) the field aδ1 is complete and, furthermore, (ii) holds for any commuting pair of nontrivial non-equivalent888Recall that two
locally nilpotent derivations δ and σ on k[X] are equivalent if Kerδ=Kerσ. locally nilpotent derivations.
Notation 2.13**.**
For every affine algebraic variety X we denote by AVF(X) the space of all algebraic vector fields on X.
If K is an ideal in k[X] then AVFK(X) is the subspace of AVF(X) generated by all fields of
the form fσ where f∈K and σ∈AVF(X). Given a saturated set N of locally nilpotent vector
fields on X and a closed subvariety Z in X we denote by LiealgN(X,Z) the Lie algebra generated by the complete vector fields
vanishing on Z that are of the form bδ where δ∈N and b∈k[X] has degree degδb≤1.
If x∈X then μx is its vanishing maximal ideal in k[X] and for every k[X]-module M
its localization at μx will be denoted by (M)μx.
Lemma 2.14**.**
Let X and Z be as in Notation 2.13 and I⊂k[n] be the vanishing ideal of Z.
Let M be a k[X]-submodule of AVF(X) such that for every x∈X∖Z the localization
(M)μx coincides with the localization of M˘=AVF(X) at μx.
Then there exists k>0 such that M contains AVFIk(X).
Proof.
Since X is affine the k[X]-module M˘ is finitely generated, i.e., we have M˘=∑i=1mM˘i where M˘i=k[X]σi for
some vector field σi∈AVF[X]. Let Mi=M∩M˘i. Note that M˘i has a natural structure of the ring k[X]
and Mi can be viewed is an ideal Ki in k[X]. Since the operations of localization and intersection commute
we have (Mi)μx=(M˘i)μx for every x∈/Z.
Hence, the zero locus of Ki is contained in Z. By Nullstellensatz there exists ki for which Iki⊂Ki and, therefore,
IkiM˘i⊂Mi. Letting k=max{ki∣i=1,…,m} we get the desired conclusion.
∎
Theorem 2.15**.**
Let X, Z and N be as in Notation 2.13 and I⊂k[n] be the vanishing ideal of Z.
Suppose that every vector field in N is tangent to Z (i.e., the flows of these vector fields map Z onto itself).
Let G⊂SAut(X) be the group generated by N
and let X∖Z be G-flexible.
Suppose that X admits a pair of compatible locally nilpotent vector fields δ1 and δ2∈N.
Then for some k≥1 the algebra
LiealgN(X,Z) contains the space AVFIk(X).
Proof.
Let ϱi:X→Qi:=SpecKerδi be the quotient morphism associated with δi, let Zi be the closure of ϱi(Z) in Qi, and let a be as in Definition 2.11.
Choose a nonzero function hi∈Kerδi≃k[Qi] that vanishes on Zi and note that for every fi∈Kerδi,i=1,2 the fields
f1h1δ1,af2h2δ2,af1h1δ1, and f2h2δ2 are contained in LiealgN(X,Z).
Hence
[TABLE]
also belongs to LiealgN(X,Z). By condition (i) in Definition 2.11 Span(Kerδ1⋅Kerδ2)
contains a nontrivial ideal J. Hence, J2:=h1h2J⊂J∩I is also
nontrivial ideal in k[X]. By Formula (5) one has
J2δ2⊂LiealgN(X,Z).
Let N~ be the set {δ′′} of locally nilpotent vector fields δ′′∈N such that for some
δ′∈N the pair (δ′,δ′′) is compatible. By Remark 2.12 (2) this set is saturated and closed under conjugation by elements of G.
By Proposition 2.10 N~ generates a subgroup G~ of SAut(X∖Z) such that X∖Z is G~-flexible.
Let σ1,…,σm∈N~, σ2=δ2
and Ji plays the same role for σi as J2 above for δ2, i.e., Jiσi⊂LiealgN(X,Z).
Applying [2, Proposition 1.8] (with Hi being the one-parameter unipotent group associated with σi)
we can suppose that for every x∈X the values of the fields σ1,…,σm
generate TxX. Set L1=J1⋅…⋅Jm, i.e., L1σi⊂LiealgN(X,Z) for every i and L1 is contained in
I.
Put M1=AVFL1(X).
Then M=∑i=1mL1σi is a k[X]-submodule of M1 such that
for every point x in the complement (in X) to the zero locus W1 of L1 and every nonzero v∈TxX there exists a vector field σ∈M whose value at x is v.
Hence, M/(μxM)=M1/(μxM1) when x∈X∖W1.
In particular, the localizations (M)μx and (M1)μx satisfy the relation (M1)μx=μ(M1)μx+(M)μx and the Nakayama lemma [3, Corollary 2.7] implies that (M)μx=(M1)μx.
Hence by [3, Proposition 3.9] M=M1, i.e., AVFL1(X)⊂LiealgN(X,Z).
Using conjugations by elements of G~, we can transform L1 into a sequence
of ideals L1,…,Lk⊂I
such that for every x∈X∖Z there exists i for which x is not in the zero locus of Li.
Consider the k[X]-module N=∑i=1kAVFLi(X). By construction (N)μx coincides with
the localization of AVF(X) at μx for every x∈X∖Z. Hence, by Lemma 2.14
N contains AVFIk(X) for some k>0. Since AVFLi(X)⊂LiealgN(X,Z) for every i,
we see that N⊂LiealgN(X,Z) which yields the desired conclusion.
∎
Remark 2.16**.**
(1) Let the assumptions of Theorem 2.15 hold with the following execption:
we do not assume that the fields from N are tangent to Z and that X∖Z are G-flexible, but
we suppose that X is G-flexible. Then the conclusion of Theorem 2.15 remains valid.
Indeed, consider the saturated subset NZ of N that consists of all fields that vanish on Z
and let GZ⊂SAut(X) be the group generated by NZ. Then X∖Z is GZ-fleixible
by [9]. Hence, replacing N and G by NZ and GZ respectively we get the assumptions
and, therefore, the conclusion of Theorem 2.15.
(2) If N is the set of all locally nilpotent vector fields on X
and for some k≥1 the algebra LiealgN(X,Z)
contains the space of all algebraic vector fields on X that
vanish on Z with multiplicity at least k then
we say that the pair (X,Z) has the algebraic density property. In this
terminology Theorem 2.15 is a generalization of [20, Theorem 4]
which established the algebraic density property for pairs of the form (Cn,Z)
where Z is a closed subvariety of Cn with dimZ≤n−2.
The algebraic density property in the complex case has some remarkable consequences.
In particular, as in [20]
we get two interesting facts which will not be used in the sections below.
Theorem 2.17**.**
(cf. [11, Theorem 4.10.6])* Let X be a complex affine flexible variety and Z be a closed subvariety of X
whose codimension is at least 2.
Suppose that X admits a pair of compatible vector fields.
Let Φt:Ω0→Ωt=Φt(Ω0)⊂X∖Z(t∈[0,1]) be a
C1-isotropy consisting of injective holomorphic maps between Runge domains999Recall that an open
subset Ω of a Stein manifold Y is called a Runge domain if every holomorphic function on Ω can be approximated
in the compact-open topology by holomorphic functions on Y.
with Φ0=IdΩ0. Suppose also that each Ωt is Stein.
Then Φ1 can be approximated uniformly on compacts of Ω0 by holomorphic automorphisms of X identical on Z.*
Proof.
Consider generators f1,…,fm of the vanishing ideal I⊂k[X] of Z. These functions have no common zeros on any Ωt.
By the weak Nullstellensatz for Stein spaces
(e.g., see [26, Theorem 4.25]) there are holomorphic functions g1t,…,gmt on Ωt
for which ∑i=1mfigit=1 on Ωt. Since Ωt is a Runge domain on every compact Kt⊂Ωt
these functions g1t,…,gmt can be uniformly approximated by global holomorphic functions on X, i.e.,
we get a holomorphic function h on X vanishing on Z which is as close to 1 on Kt as we wish. Furthermore,
replacing h by hk for a given k≥1 we can suppose that h is contained in I~k where I~ is the vanishing ideal of Z
in the algebra Hol(X) of holomorphic functions on X. Hence every holomorphic vector field νt on Ωt
can be approximated in the compact-open topology by holomorphic fields from HVFI~k(X) where HVF(X) is the space
of all holomorphic vector fields on X. Since X is affine every holomorphic function (resp. vector field) on X can be approximated by regular functions
(resp. algebraic vector fields) on X in the compact-open topology
and also I~ is generated by I over Hol(X) (e.g., see [19, Theorem 4]). Hence νt can be approximated in
the compact-open topology by elements of AVFIk(X) and, therefore, by Theorem 2.15 by elements of LiealgN(X,Z).
Let ν~t be an element of LiealgN(X,Z) uniformly close to νt on Kt
and let ψs and ψ~s be the flows of νt and ν~t respectively with s being the time parameter.
Suppose that for some s0>0 both of these flows
are defined for every x∈Kt. Recall that all generators of LiealgN(X,Z) can be chosen as complete
algebraic vector fields that vanish on Z. By [11, Corollary 4.8.4] ψ~s0 can be approximated by compositions
of flows of these generators. Hence we have the following
(*) the element ψs0 of the flow of νt can be uniformly approximated on Kt
by global holomorphic automorphisms of X identical on Z.
To make use of (*) following the proof of [11, Theorem 4.9.2] we consider the trace
Ω~={(t,z):t∈[0,1],z∈Ωt} of the isotropy {Φt} in R×X
and we treat Φt as the flow of the continuous time dependent vector field
[TABLE]
where dot denotes the derivative on t.
The field V is continuous on Ω~ and holomorphic on Ωt for every fixed t∈[0,1].
Divide the interval [0,1] into N subintervals [tk,tk+1] of length 1/N for a given natural N
and consider the locally constant holomorphic vector field V~(t,z) which is equal to V(tk,z) on every interval [tk,tk+1].
Let φt be the flow of V~. Similarly to the estimates in the proof of [11, Theorem 4.8.2] one can check
that as N→+∞ the flow φt converges
to Φt uniformly on compacts in Ω for all t∈[0,1].
Since by (*) φt can be approximated by global holomorphic automorphisms of X identical on Z so does Φt.
This yields the desired conclusion.
∎
Corollary 2.18**.**
Let X be a complex affine flexible variety of dimension n and Z be a closed subvariety of X
whose codimension is at least 2. Suppose that X admits a pair of compatible vector fields.
Then every x∈X∖Z
has a neighborhood Ω in X∖Z that is a Fatou-Bieberbach domain
(i.e., U is biholomorphic to Cn).
Proof.
Since X∖Z is flexible it suffices to prove this statement for some point x0 in X∖Z only.
Choose a dominant morphism φ:X→Cn and choose x0∈X∖Z so that for a ball B0⊂Cn with
center at φ(x0) the component B of the preimage φ−1(B0), containing x0, is naturally biholomorphic to B0.
Taking B small enough we can suppose that there are regular functions g1,…,gm∈C[X] such
that each ∣gi∣ does not exceed 1 on B while it is greater than 1 at each point of φ−1(B0)∖B.
This implies that B0 is Hol(X)-convex. Hence it is a Runge domain in X by the Oka-Weil theorem [11, Theorem 2.2.5].
For an analytic coordinate system (z1,…,zn) on B with the origin at x0 consider the homothety Φ:(z1,…,zn)→(z1/2,…,zn/2).
By Theorem 2.17 Φ can be approximated by a global
holomorphic automorphism F of X identical on Z. Since F(B)⊂B by the Brouwer fixed point theorem F has a fixed point in B.
Without loss of generality we can suppose that this point is x0 and reducing the size of B we can suppose that B contains no other fixed point but x0.
The eigenvalues λ1,…,λn of the map dF
at x0 must be close to those of dΦ which are 1/2. In particular, we can suppose that ∣λ1∣≥∣λ2∣≥…≥∣λn∣ and ∣λ1∣2<∣λn∣. In particular, these eigenvalues satisfy the assumptions of [31, Theorem 9.1]
and we can copy the argument of Rosay and Rudin. Namely, consider the basin Ω of attraction of F, i.e., x∈Ω if there exists
N>0 such that FN(x)∈B. By continuity for every y in a small neighborhood of x one has FN(y)∈B. Thus Ω is open (in
the standard topology). Treating B as a neighborhood of the origin in Tx0X≃Cn and letting A=dF we can consider
an injective map A−M∘FM:F−N(B)→Cn where M≥N. Choose positive constants β>∣λ1∣ and
α<∣λn∣ such that β2<α. Since the local Jacobi matrix of the map A−1∘F at x0 is the identity matrix
the computation in [31, Theorem 9.1]
101010 See the formula immediately after formula (9) in [31, Theorem 9.1].
Formally, this formula is proven for X=Cn but it works in our case as well without change.
implies that for every compact K⊂F−N(B)
there exist some M0>N and a positive real number b such that for every x∈K one has
[TABLE]
where M≥M0. Hence we have a well-defined holomorphic map Φ:Ω→Cn
where Ψ=limM→∞A−M∘FM. Since at every point of F−N(B) the local Jacobian of A−M∘FM
does not vanish we have the following alternative: either the local Jacobian of Ψ does not vanish or it is identically zero.
However, the local Jacobian of Ψ at x0 is 1 and we have the former, i.e., Ψ is an open map. Furthermore, it is
injective since otherwise the maps A−M∘FM are not injective for large M. Note also that Ψ=A−1∘Ψ∘F,
i.e., Ψ and A−1∘Ψ have the same range. Since the linear operator A−1 is an expansion it follows that
Ψ(Ω)=Cn. Thus the basin Ω of attraction of F is biholomorphic to Cn and we have the desired conclusion.∎
Remark 2.19**.**
For X=Cn the question about Fatou-Bieberbach domains in the complement of
subvariety of codimension 2 was posed by Siu and answered by Buzzard and Hubbard [5] (see also [10]).
3. Relative version of Theorem 2.8
Let us prove first the following analogue of [2, Theorem 3.1].
Theorem 3.1**.**
Let ϱ:X→Q be a dominant morphism of quasi-affine algebraic varieties, Q0 be
a Zariski open dense subset of Q, and X0=ϱ−1(Q0).
Let every fiber ϱ−1(q),q∈Q0 be G-flexible where G⊂SAut(X0)
is a subgroup generated by a saturated set N of locally nilpotent vector fields on X0
which are tangent to the fibers of ϱ.
Suppose that q1,…,qm are distinct points in ϱ(X0)
and αi∈G∣ϱ−1(qi).
Then there exists an automorphism α of X over Q
such that α∣ϱ−1(qi)=αi for every i=1,…,m and α∣X0∈G.
Proof.
Suppose first that Q=Q0. For a locally nilpotent vector field σ denote by exp(tσ)
the element of the one-parameter group associated with σ at time t∈k.
By definition αi is of the form
[TABLE]
where n(i) is a natural number depending on i and tj,i∈k. Choose regular functions fi on Q such that fi(qi)=1 and fi(qj)=0 for every j=i.
Using the natural embedding k[Q]⊂k[X] we treat fi as a function on X. Then by the assumption fi∈Kerσ for every σ∈N,
i.e., the replica fiσ∈N. It remains to put α=
[TABLE]
and we are done in the case of Q=Q0.
In the general case this α is only an automorphism of X0 and its extension to X may have poles on X∖X0.
However, one can choose fi from the above so that they vanish on Q∖Q0 with sufficiently high multiplicity.
That is, we can assume that each fiσj,i is a regular vector field on X that vanishes on X∖X0. Then
the extension of α becomes a regular automorphism on X whose restriction to X∖X0 is the identity map.
∎
Remark 3.2**.**
(1) Let Q be affine. Then Theorem 3.1 remains valid with the same proof when the finite set q1,…,qm is replaced by a collection of m disjoint closed suvarieties of Q which are contained in Q0.
(2) One can consider a more general situation when every αi is the restriction of an element of G to a k-infinitesimal neighborhood Vi
of ϱ−1(qi).111111For every reduced subvariety Y of X with a defining ideal I⊂k[X] one can treat an automorphism of the k-infinitesimal neighborhood of Y
as an automorphism of the ring k[X]/Ik. Then we can still find α∈G
for which α∣Vi=αi,i=1,…,m. For this it suffices to require that each fi vanishes with
multiplicity at least k at qj (where j=i) and takes the value 1 with multiplicity at least k at qi.
Notation 3.3**.**
Further in this section X is a smooth algebraic variety of dimension n, G⊂SAut(X) is a subgroup generated by a saturated set N of locally nilpotent vector fields on X,
Gz⊂G is the isotropy group of a point z∈X, mz is the maximal ideal in the local ring OX,z at z, and Am(X,z)=mz/mzm+1
(in particular, A1(X,z) coincides with the cotangent space Tz∗X).
We consider the set Aut(Am(X,z)) of
k-algebra isomorphisms f:Am(X,z)→Am(X,z) satisfying the following condition:
[TABLE]
Let u1,…,un∈μz be such that they generate the cotangent space. Then we call the n-tuple (u1,…,un) a local coordinate system
at z (indeed, if the ground field k=C then u1,…,un form a local analytic coordinate system in a small neighborhood of z in the standard topology).
In terms of this local coordinate system elements of Aut(Am(X,z)) can be described
as follows. The k-algebra Am is contained in the quotient
A/mAm+1 of the local ring
A=k[[u1,…,un]] of formal power series with respect to the power of its maximal ideal mA. Therefore, we treat any map f∈Aut(Am(X,z)) as an n-tuple of polynomials
(F1,…,Fn)∈(A/mAm+1)n in n variables u1,…,un of degree at most m such that they vanish at the origin and the determinant of the matrix [∂xj∂Fi]i,j is 1
modulo terms of degree higher than m. In particular, each Fi is the sum of homogeneous k-forms where k runs from 1 to m. Let Fi′ be the m-form present is this sum
and θz,m be the linear map from Aut(Am(X,z)) to the space of n-tuples of m-forms given by
[TABLE]
Suppose also that λ(f) is the n-tuple of linear parts of f. In particular, λ(f)∈SL(n,k) (because of the assumption on the Jacobian).
Note that SL(n,k) admits different natural actions on the space θz,m(Aut(Am(X,z))) of n-tuples F(uˉ) of m forms in n variables (i.e., uˉ=(u1,…,un))
for which we use the following notations
[TABLE]
Lemma 3.4**.**
Let Notation 3.3 hold and Autm−1(Am(X,z)) be the subgroup of the group Aut(Am(X,z)) consisting of those automorphisms f for which f≡idmodmzm
(i.e., (f−θz,m(f))(uˉ) coincides with the n-tuple (u1,…,un)). Then we have the following.
(a)* For every f∈Autm−1(Am(X,z)) and g∈Aut(Am(X,z)) one has*
[TABLE]
In particular, if g is also in Autm−1(Am(X,z)) then θz,m(g∘f)=θz,m(f∘g)=θz,m(f)+θz,m(g).
(b)* For m≥2 the set Fz,m:=θz,m(Autm−1(Am(X,z))) is the linear space of n-tuples F(uˉ) of m-forms in n variables of divergence zero and
the SL(n,k)-action on Fz,m given by λ.F(uˉ) is irreducible.*
(c)* There is a natural homomorphism Jz,m:Gz→Aut(Am(X,z)) such that
in the case when X is G-flexible one has Jz,1(Gz)=Aut(A1(X,z))≃SL(n,k)=SL(Tz∗X).*
(d)* If ∂ is a locally
nilpotent vector field on X with a zero of order m≥2 at z
then θz,m(Jz,m(exp(t∂)))=tθz,m(Jz,m(exp(∂))).*
Proof.
Statement (a) is straightforward (see also [2, Lemma 4.12]). The first clause in statement (b) can be found in [2, Lemma 4.13] and the second on in [28, IX.10.2].
Define Jm,z(α) as the operation of taking the m-jet of α∈Gz. Since α∈SAut(X)
we see that the Jacobian J(α)≡1.
Hence Jz,m(Gz)⊂Aut(Am(X,z)). The fact that Jz,1(Gz)≃SL(Tz∗X)
when X is G-flexible follows from [2, Corollary 4.3] which concludes (c).
For (d) note (as in [2, Lemma 4.12]) that exp(∂)∈Gz,m and it induces the map
id+∂^∈Autm−1(Am) where ∂^ denotes the derivation on Am induced by ∂. Hence exp(t∂) induces id+t∂^ which is (d) and we are done.
∎
Notation 3.5**.**
In addition to Notation 3.3 suppose that ϱ:X→Q is a smooth morphism of smooth
quasi-affine algebraic varieties
such that every fiber Y=ϱ−1(q),q∈Q is of dimension at least 2 and G-flexible (i.e., we are
under the assumptions of Theorem 3.1 and G is generated by a saturated set N
of locally nilpotent vector fields such that every δ∈N is tangent to the fibers of ϱ). In particular,
for every z∈X and q=ϱ(z) a local coordinate system can be chosen in the form (uˉ,vˉ):=(u1,…,uk,v1,…,vn−k) where ui and vi are regular functions on X such that v1,…,vn−k are the lifts
of functions on Q
that form a local coordinate system at q∈Q while the restriction (u1,…,uk) yields a local coordinate system at z∈Y.
Note that in such a coordinate system for every α∈Gz its image Jz,m(α)∈Aut(Am(X,z)) is of the form
[TABLE]
where Fi is a polynomial of degree at most m and the determinant of the matrix [∂uj∂Fi]i,j is 1 up to terms of degree higher than m.
Lemma 3.6**.**
Let δ∈N and δq be the restriction of δ to Y=ϱ−1(q) for q∈Q.
Then a partial quotient morphism τ:X→P (resp. τq:Y→Pq) of δ (resp. δq)
can be chosen so that ϱ factors through τ and τ∣Y factors through τq
(i.e., ϱ=θ∘τ and τ∣Y=κq∘τq for some morphisms θ:P→Q
and κq:Pq→P).
Proof.
The quasi-affine variety Q is contained as an open subset in an affine variety Q˘. Under the natural embedding
k[Q˘]↪k[X] generators of k[Q˘] can be treated as elements of Kerδ.
Thus we can choose τ so that k[P] contains these generators which implies that ϱ factors through τ,
i.e., ϱ=θ∘τ.
Similarly, we can choose τq so that k[Pq] contains generators of the ring of regular functions
on an affine variety in which θ−1(q)
is an open subset. This implies that τ∣Y factors through τq.
∎
Lemma 3.7**.**
Let Z be a finite subset of Y=ϱ−1(q) and z∈Z.
Suppose that there exists δ∈N such that δ∣Y=0 and partial quotient morphisms
τ and τq from Lemma 3.6 can be chosen so that
[TABLE]
Then such a field δ and a coordinate system (uˉ,vˉ) as in Notation 3.5 can be chosen so that
the following holds:
(i)* δ induces a trivial derivation on Am(X,w) for every w∈Z∖{z};*
(ii)* ui belongs to the kernel Kerδ for every i≥2;*
(ii)* the derivation σ on Am(X,z)⊂A/mAm+1 (where A=k[[uˉ,vˉ)]]) induced by δ coincides with σ:=∂/∂u1.*
Proof.
Treating δ as a derivation δ:k[X]→k[X] consider its conjugate δ~=g∗∘δ∘(g∗)−1:k[X]→k[X]
for g∈G. Similarly, let δ~q=g∣Y∗∘δq∘(g∣Y∗)−1.
Note that Kerδ~=g∗(Kerδ) and Kerδ~q=g∣Y∗(Kerδq).
Therefore, for τ~=τ∘g:X→P and τ~q=τq∘g∣Y:Y→Pq one has
τ~∗(k[P])⊂Kerδ~ and τ~q∗(k[Pq])⊂Kerδ~q. Hence these morphisms
commute with the Ga-actions associated with δ~ and δ~q respectively. Each fiber of τ~ (resp. τ~q)
is the image of a fiber of τ (resp. τq) under the action of g. This implies that the general fibers of τ and τq are isomorphic
to lines and these morphisms are partial quotient morphisms
for δ~ and δ~q respectively. Furthermore, one can see that ϱ=ϱ∘g=θ∘τ∘g=θ∘τ~ and,
similarly, τ~∣Y=κq∘τ~q. In particular, the assumption (8) holds for the locally nilpotent vector field δ~.
Since N is saturated δ~∈N and we can replace δ by δ~ while trying to achieve (i)-(iii) for the finite set Z.
Note that the validity
of (i)-(iii) for the pair (δ~,Z) is equivalent to the validity of these conditions for the pair (δ,g−1(Z)).
Indeed, (i) is obvious and for the rest let (u1,…,uk,v1,…,vn−k) be a local coordinate system at z such that (i)-(iii) are true for the pair (δ~,Z).
Let (u^1,…,u^k,v1,…,vn−k) be a local coordinate system at g−1(z) such that u^i=ui∘g, i.e.,
the local form of g is
[TABLE]
Since ui∈Kerδ~i for i≥2 and δ~=g∗∘δ∘(g∗)−1 we see that u^i∈Kerδ, i.e., we have (ii) for the pair (δ,g−1(Z)). Condition (iii) is equivalent to
the fact that modulo t2 the exponent exp(tδ~),t∈k has the following local form at z
[TABLE]
where f is a function
vanishing at the origin with multiplicity m+1 or higher. Since exp(tδ~)=g∗∘exp(tδ)∘(g∗)−1 the local form of exp(tδ) is
[TABLE]
and we have (iii) for the pair (δ,g−1(Z)).
Therefore, we replace Z by g(Z) while keeping δ intact. By virtue of infinite transitivity (Theorem 2.6) we can suppose now that
Z consists of general points {wi} of Y with z=w1.
Hence pi′=τq(wi) are distinct
general (and, therefore, smooth) points pi′ of Pq such that for some neighborhood Ui⊂Pq of pi′ one has a natural
Ui-isomorphism τq−1(Ui)≃Ui×Ak.
Furthermore, we can suppose also that δ is nontrivial on τq−1(p1′).
Let pi=κq(pi′).
By (8) we can suppose that {pi} are general points of θ−1(q), i.e., there is a function f∈k[P] that vanishes at each pi,i≥2 with multiplicity at least m but has f(p1)=1
also with multiplicity at least m.
Then replacing δ by its replica fδ∈N we get (i).
Since p1 is a smooth point of θ−1(q) a local coordinate system at p1∈P can be chosen in the form
(u2,…,uk, v1,…,vn−k) where each ui,i≥2 is a regular function on P.
Taking u1 as an appropriate extension to X of a coordinate function on the Ga-orbit τq−1(p1′)≃Ak we can
treat (u1,u2,…,uk,v1,…,vn−k)
as a local coordinate system at z∈X. This is the desired coordinate system with δ being a desired derivation which concludes the proof.
∎
Remark 3.8**.**
Condition (8) is very mild and it is automatically true when q is a general point of Q.
Lemma 3.9**.**
Let z∈Z⊂Y and (uˉ,vˉ) be as in Lemma 3.7. Suppose that GZ=⋂w∈ZGw and GZ,zm is the subgroup
of GZ such that it induces the identity map on the m-th infinitesimal neighborhood of every point w∈Z∖{z}. Then the image Im:=Jz,m(GZ,zm) contains
all automorphisms of Am(X,z) with the following coordinate form
[TABLE]
where every ℓim(vˉ) is an m-form in variables v1,…,vn−k. Furthermore,
for every λ(uˉ)=(λ1(uˉ),…,λk(uˉ)) in SL(Tz∗Y) and each k-tuple
(ℓ11(vˉ),…,ℓk1(vˉ)) of 1-forms the subgroup I1 contains
the following automorphism of Tz∗X
[TABLE]
Proof.
Let σ be as in Lemma 3.7.
Then for every h∈Kerδ the automorphism
exp(hδ)∈GZ,zm induces the automorphism exp(h′σ) on Am(X,z) (where h′ is the image of h in Am(X,z))
of the following coordinate form
[TABLE]
In particular, choosing h so that h′ is equal to an m-form l1m(vˉ) we see that the automorphism
[TABLE]
is contained in Im.
Note the variety (Y∖Z)∪{z} is GZ,zm-flexible by [9]. Therefore,
by Theorem 2.8 for every λ(uˉ)∈SL(Tz∗Y)
the subgroup I1 contains an automorphism as in Formula (10) for certain 1-forms ℓi1(vˉ).
Using notations of Lemma 3.4 we suppose that f and g∈GZ,z are such that
Jz,1(g) is given by Formula (10) and Jz,m(f) is given by Formula (12). In particular, Jz,m(f)∈Autm−1(Am(X,z))
and θz,m(Jz,m(f))=(ℓ1m(vˉ),0,…,0) while
[TABLE]
By Lemma 3.4 (a) Jz,m(g−1∘f∘g)∈Autm−1(Am(X,z))
and θz,m(Jz,m(g−1∘f∘g)) is obtained from θz,m(Jz,m(f)) via conjugation by
θz,1(Jz,m(g)). Using such conjugations
we see that automorphisms of the form
[TABLE]
are also contained in Im. Taking the product of automorphism in Formula (13) with i running from 1 to k we
obtain every automorphism from Formula (9) as an element of Im. This implies in turn that
any automorphism from Formula (10) is contained in I1 regardless of the choice of ℓi1(vˉ)
which concludes the proof.
∎
Now we can formulate the main result of this section.
Theorem 3.10**.**
Let Notation 3.5 hold and V(Am(X,z)) be the subset of Aut(Am(X,z))
which consists of automorphisms as in Formula (7) satisfying the assumption on the determinant of [∂uj∂Fi]i,j in Formula (6).
Let Z be a finite subset of X such that for every
q∈ϱ(Z) there exists δ∈N (where δ may depend on q) for which condition (8) in Lemma 3.7 holds. Suppose that
GZ=⋂z∈ZGz and JZ,m:GZ→∏z∈ZV(Am(X,z)) is the natural homomorphism.
Then JZ,m is surjective
(in brief, one can choose an element α∈G so that for every z∈Z the m-jet of α at z coincides with a prescribed jet from Formula (7)
satisfying the assumption on the determinant).
Proof.
By Theorem 3.1 and Remark 3.2 (2) it suffices to consider the case when ϱ(Z) is a singleton q∈Q.
Furthermore, assume that for every z∈Z we can find an element αz in the subgroup GZ,zm⊂GZ from Lemma 3.9
such that at z this automorphism has a prescribed m-jet from Formula (7)
satisfying the assumption on the determinant. Recall that for every
w∈Z∖{z} the m-jet of αz at w is the identity map in the m-th infinitesimal neighborhood of w.
Hence the composition of such automorphisms αz with z running over Z (in any order) yields an automorphism α∈GZ
such that at every point of Z it has a prescribed
m-jet from Formula (7)
satisfying the assumption on the determinant. Therefore, it is enough to consider the case when Z is a singleton z which we do below.
We shall use induction on m with the case of m=1 provided by Lemma 3.9.
Assume now that the statement is true for m−1.
Let h∈V(Am(X,z)). Then h−θz,m(h)=h~∈V(Am−1(X,z)). By the assumption
there exists α∈GZ for which Jz,m−1(α)=h~. Let JZ,m(α)=g and let λ(g) be as in Notation 3.3. Consider the n-tuple f~=θz,m(h)−θz,m(g) of m-forms and let f∈Autm−1(Am(X,z)) be such that θz,m(f)=λ(g)−1.rf~.
Then by Lemma 3.4 (a) we have f∘g=h. That is, it suffices to show that f belongs to Jz,m(GZ), or, equivalently θz,m(f) is contained in
[TABLE]
Note that since h and g are in V(Am−1(X,z)) the last n−k coordinates of the n-tuple f~ (and, therefore, θz,m(f)) are equal to
zero for m>1.
Note also that any element of I (and, thus, θz,m(f)) is of the form ∑μμpμ(uˉ) where μ is a monomial in coordinates vˉ
(with (uˉ,vˉ) from Notation 3.5) and
pμ(uˉ) is an n-tuple of homogeneous polynomials in uˉ of degree l=m−degμ such that
the last n−k coordinates of this n-tuple are equal to zero and its divergence is also zero (the latter fact follows from the assumption of
the Jacobian, see [2, Lemma 4.13]). Thus applying
Lemma 3.4 (a) again we see that it suffices to show that
μpμ(uˉ) belongs to I. Furthermore, we can suppose that l>0 since the case of l=0 is taken care of by Lemma 3.9.
Recall that δ can be chosen so that conditions (i)-(iii) from Lemma 3.7 are satisfied. In particular, u2 belongs to the kernel
Kerδ as well as every vj and, therefore, μ. Since N is saturated the replica μu2l−1δ belongs to N.
Thus, exp(μu2l−1δ)∈GZ, and
[TABLE]
belongs to Jz,m(GZ)∩Autm−1(Am(X,z)).
Consider a finte set {γi} in GZ and gi=Jz,m(γi).
Then
[TABLE]
and θz,m(ei)=λ(gi).θz,m(e) is the result of the natural action of
λ(gi)∈SL(TzX) from Lemma 3.4 (b).
Consider the projection κ of the space of n-tuples of m-forms in uˉ and vˉ to the space of
k-tuples
forgetting the last n−k coordinates. Note that κ(θz,m(e)) belongs to μθz,l(Autl−1(Al(Y,z))).
Recall also that by Lemma 3.9 we can suppose that λ(gi) is given by
[TABLE]
where (λ1(uˉ),…,λk(uˉ)) is a prescribed
element of SL(TzY)≃SL(k,k). Therefore, κ(θz,m(ei))∈μθz,l(Autl−1(Al(Y,z))) and
by Lemma 3.4 (b) we can suppose that these elements
generate the vector space μFz,l where Fz,l=θz,l(Autl−1(Al(Y,z)) is the space of k-tuples of l-forms in uˉ
whose divergence is zero.
Lemma 3.4 (d) implies that tiλ(gi).θz,m(e) is also
contained in I for every ti∈k.
Applying Lemma 3.4 (a) again we see that every linear combination of the elements λ(gi).θz,m(e) belongs to I
and, therefore, κ(I) contains the space μθz,lFz,l. Hence μp(uˉ)∈I which
concludes the proof.
∎
Corollary 3.11**.**
Let Notation 3.5 hold and condition (8) in Lemma 3.7 be valid for every
q∈Q. Suppose that
Z is a finite subset of X, and m is a natural number.
Let Sz′ and Sz′′ be étale sections of ϱ:X→Q through z∈Z.
Then there exists an automorphism α∈G such that for every z∈Z one has α(z)=z and α(Sz′) is tangent to Sz′′
at z with multiplicity at least m.
4. General projections for flexible varieties. I
Notation 4.1**.**
(1) If κ:X→P is a morphism of algebraic varieties then we denote
by Aut(X/P) (resp. SAut(X/P)) the subgroup of Aut(X) (resp. SAut(X)) that preserves each fiber of κ.
(2) If κ:X→P is a smooth morphism of smooth varieties we denote by T(X/P) the relative tangent bundle
(which is the kernel of the induced map TX→κ∗(TY)). Furthermore, if Z is a subvariety of X then we still
denote by T(Z/P) the intersection of the Zariski tangent bundle TZ with T(X/P). Similarly, for every z∈Z we
let Tz(Z/P)=TzZ∩T(Z/P).
The aim of this section is to describe analogues of general linear projections of Akn for flexible varieties. More precisely, we shall prove the following fact.
Theorem 4.2**.**
Let X and P be smooth algebraic varieties and Q be a normal algebraic variety. Let ϱ:X→Q and τ:Q→P be dominant morphisms such that κ:X→P is smooth for κ=τ∘ϱ.
Suppose that Q0 is a non-empty Zariski open smooth subset of Q so that for X0=ϱ−1(Q0) the morphism ϱ∣X0:X0→Q0 is smooth.
Let G⊂Aut(X/P) be an algebraically generated group acting 2-transitively on each fiber of κ:X→P and
Z be a locally closed reduced subvariety in X.
(i)* Let dimZ×PZ≤2dimZ−dimP and dimQ≥dimZ+m where m≥1. Then there exists an algebraic family A⊂G of automorphisms such that
for a general element α∈A one can find a subvariety R of α(Z)∩X0 of dimension dimR≤dimZ−m for which
ϱ(R)∩ϱ(α(Z)∖R)=∅ and
the morphism ϱ∣(α(Z)∩X0)∖R:(α(Z)∩X0)∖R→Zα′∩Q0
is injective where
Zα′=ϱ∘α(Z). In particular, if dimQ≥dimZ+1 (resp. dimQ≥2dimZ+1)
for a general element α∈A
the morphism ϱ∣α(Z)∩X0:α(Z)∩X0→Zα′∩Q0
is birational (resp. a bijection).*
(ii)* Let dimZ×PZ≤2dimZ−dimP, dimQ≥dimZ+1
and F be a closed subvariety of Z such
dimF×PZ<dimQ−dimP (which is the case when F is a finite set and P is a singleton).
Then for a general element α in the family A from (i) and every z∈α(F)∩X0
one has ϱ−1(ϱ(z))∩α(Z)=z.*
(iii)* Suppose that G is generated by a saturated set N of locally nilpotent vector fields on X
(in particular, every fiber of κ is G-flexible).
Let P0=τ(Q0), Z0=Z∩κ−1(P0) and dimT(Z0/P0)≤dimQ−dimP. 121212
In the case of an irreducible Z the above inequality follows from this one: dimTZ≤dimQ−dimP+dimκ(Z).
Indeed, in the presence of the latter the exact sequence 0→Tz(Z0/P0)→TzZ0→Tκ(z)κ(Z0) implies the former.
In particular, if κ(Z) is dense in P one has to require that dimTZ≤dimQ.
Then there exists an algebraic family A⊂G of automorphisms such that
for a general element α∈A,
every z∈α(Z0)∩X0 the induced map ϱ∗:Tzα(Z0)→Tϱ(z)Q of the tangent spaces is injective.*
(iv)* Suppose that G is again generated by a saturated set N of locally nilpotent vector fields on X.
Let dimZ×PZ≤2dimZ−dimP,2dimZ+1≤dimQ and dimT(Z0/P0)≤dimQ−dimP. Then the family A from (i) can be chosen so that
for a general element α∈A the morphism ϱ∣α(Z)∩X0:α(Z)∩X0→Zα′∩Q0 is
injective and it induces an injective map of the Zariski tangent bundle of α(Z)∩X0 into the Zariski tangent bundle of Q0.*
Proof.
For every variety X over P denote by SX the variety SX=(X×PX)∖ΔX where ΔX is the diagonal in X×X.
Then every automorphism in Aut(X/P) can be lifted to an automorphism of SX. In particular, we have a G-action on SX and by the assumption this action is transitive
on every fiber of the projection SX→P.
Consider the subvariety Y⊂SX that is the intersection of X0×Q0X0 and SX in X×PX. The codimension of Y in SX is dimQ−dimP and, because of smoothness, all fibers of the natural morphism Y→P
are of the same dimension. Hence, by Remark 1.9 (1) dimY×PSZ=dimY+dimSZ−dimP. By
Theorem 1.4 (ii) we can choose algebraic subgroups H1,…,Hm of G such that for a general element (h1,…,hm)∈H1×⋯×Hm
one has
[TABLE]
where
W=Y∩α(SZ) for α=h1⋅…⋅hm.
Hence, in case (i) the dimension of W is at most dimZ−m. Let R be the image of W under one of the two natural projections X×QX→X
(in particular, R⊂α(Z)∩X0 and dimR≤dimZ−m).
Note that for z∈α(Z)∩X0 one has ϱ−1(ϱ(z))∩α(Z)=z iff z∈/R.
Hence the restriction of ϱ to (α(Z)∩X0)∖R is injective.
Therefore, letting A=H1×⋯×Hm, we get (i).
In (ii) we let SF,Z=(F×PZ)∩SX.
By the assumption of (ii) we have dimSF,Z<dimQ−dimP.
By Theorem 1.4 (i) for a general element (h1,…,hm)∈H1×⋯×Hm the dimension of the intersection of
α(SF,Z) (where α=h1⋅…⋅hm) with Y
is at most dimSF,Z+dimY−dimSX=dimSF,Z+dimP−dimQ<0,
i.e this intersection is empty. It remains to note that the fact that α(SF,Z)∩Y=∅
is exactly the statement that for every z∈α(F)∩X0
one has ϱ−1(ϱ(z))∩α(Z)=z.
In (iii) for every variety X and a subvariety Y of the tangent bundle TX let Y∗=Y∖S where S
is the zero section of the natural morphism TX→X.
Every automorphism α∈Aut(X/P) generates an automorphism of T(X/P). In particular, G acts on T(X/P)∗
and by Theorem 2.8 this action is transitive on every fiber of T(X/P)∗→X→κP.
Note that the codimension of Y∗=T(X0/Q0)∗ in T(X/P)∗ is equal to dimQ−dimP
and all fibers of the natural projection Y∗→P0 are of the same dimension.
Hence, by Remark 1.9 dimY∗×PT(Z0/P0)∗=dimY∗+dimT(Z0/P0)∗−dimP.
By Theorem 1.4 and the inequality dimT(Z0/P0)≤dimQ−dimP
we can choose one-parameter unipotent algebraic subgroups H~1,…,H~m~ of G such that
for a general element (h~1,…,h~m~)∈H~1×⋯×H~m~ and
Z′′=(h~1⋅…⋅h~m~)(Z0) one has dimY∗∩T(Z′′/P0)∗≤0.
Note that if Y∗∩T(Z′′/P0)∗ contains a point then dimY∗∩T(Z′′/P0) must be at least 1
(since this point is a vector in TZ′′ and then Y∗∩T(Z′′/P0)∗
contains all nonzero vectors proportional to that one).
That is, Y∗∩T(Z′′/P0)∗=∅.
This implies that for every z∈Z′′∩X0 the restriction of ϱ∗ to
Tz(Z′′/P0) is injective. Consequently, the restriction of ϱ∗ to
TzZ′′ is injective i.e., we have (iii).
In the last statement we note that (i) and (iii) in combination with Proposition 1.10 imply that for a general element
α of H1×…×Hm×H~1×…×H~m~
the restriction of ϱ to α(Z)∩X0 is injective and it induces an injective map of the Zariski tangent bundle of
α(Z)∩X0 into the Zariski tangent bundle
of Q0. Thus we have (iv).
∎
Remark 4.3**.**
(1) Since the family A in Theorem 4.2 is the product of connected groups we see that A is irreducible and contains the identity map.
(2) Recall that by Proposition 1.10 one can choose
the family H1×…×Hm in Theorem 1.4
independent of Y and Z. Hence the argument in Theorem 4.2
implies that in each of the claims (i)-(iv) one can suppose that
the family A does not depend on the choice of Z.
Furthermore, Theorem 4.2 remains valid (by the same Proposition 1.10)
if one replaces A with a family H×A (or A×H) where H is any connected algebraic subgroup
of G. In particular, if Ak⊂G (k=1,…,s) is an algebraic family of automorphisms of
the form H1×…×Hm such that some property Pk is satisfied for a general element of
Ak then there exists an algebraic family A⊂G of automorphisms such that for a general
element of A all these properties are true simultaneously.
(3) Let P be a singleton and X be G-flexible. Under the assumption of (iv) we can find α∈A such that for a given z0∈Z one has z:=α(z0)∈X0.
In particular, for a general α the morphism ϱ∣α(Z):α(Z)→Zˉα′ induces the injective map
ϱ∗:Tzα(Z)→Tϱ(z)Q.
Note also that under the assumption of
(iv) we can suppose that for some neighborhood U of z in α(Z) the morphism ϱ∣U:U→Q induces an
injective map of the Zariski tangent bundles.
However, even if ϱ∣α(Z):α(Z)→Q induces an
injective map of the Zariski tangent bundles it may not be
proper (and, in particular, Zα′ may not be closed in Q).
As a counterexample one can consider a bijective
morphism of C∗ onto a polynomial curve which has only one singular point and this singularity is a node.
(4) Note that the natural morphism T(X/P)→P (resp. Y∗→P0) in the proof of Theorem 4.2 (iii) has
equidimensional fibers. Hence by Proposition 1.12 we conclude that the assumption that
ϱ and τ are dominant and κ is smooth can be omitted in the formulation of Theorem 4.2 (iii).
(5) Similarly, if κ is not smooth or even dominant we can suppose as in Remark 1.11
that κ(X) as a disjoint union ⋃k=1nPk of
smooth varieties such that for Xk=κ−1(Pk) the morphism κ∣Xk:Xk→Pk is smooth.
Letting Yk=Xk∩Y and Zk=Xk∩Z and suing further stratification we can assume that
for every k the morphism κ∣Zk:Zk→Pk has all fibers of the same dimension
(and, in particular dimZk×PkZk=2dimZk−dimPk 131313Of course, this does not imply
that dimZ×PZ≤2dimZ−dimP.).
Suppose that for every k and Qk=τ−1(Pk) one has
[TABLE]
By Theorem 4.2 (i) (with X,Q,P and Z replaced by Xk,Qk,Pk and Zk) there exists an algebraic family Ak⊂G of automorphisms such that
for a general element α∈Ak one can find a subvariety Rk of α(Zk)∩X0 of dimension dimRk≤dimZk−m for which
ϱ(Rk)∩ϱ(α(Zk)∖Rk)=∅ and
the morphism ϱ∣(α(Zk)∩X0)∖Rk:(α(Zk)∩X0)∖Rk→ϱ∘α(Zk)∩Q0
is injective. By (2) one can suppose that Ak=A is independent of k. Thus we see that
the conclusion of Theorem 4.2 (i) (and, therefore, (iv)) remains valid even in the case when κ is not dominant provided that
one replaces the assumption that dimZ×PZ≤2dimZ−dimP and dimQ≥dimZ+m
by the inequalities in Formula (14).
Corollary 4.4**.**
Let the assumptions of Theorem 4.2 (iv) hold with Q=Q0 being quasi-affine.
Suppose that Z is a once-punctured curve 141414That is, Z is the complement to a point in a complete curve..
Then there exists an algebraic family A⊂G of automorphisms of X such that for a general α∈A
the set ϱ(α(Z)) is a closed curve in Q and the restriction of ϱ yields an isomorphism between
α(Z) and ϱ(α(Z)).
Proof.
By Theorem 4.2 (iv) there is a family A such that for a general α∈A the restriction of ϱ yields an injective map of the Zariski tangent bundle of
α(Z) into the Zariski tangent bundle of ϱ(α(Z)). Note also that,
being an injective image of Z,
ϱ(α(Z)) is closed in Q
since any once-punctured curve in a quasi-affine algebraic variety is automatically closed.
Therefore, the bijective morphism ϱ∣α(Z):α(Z)→ϱ(α(Z)) is proper.
This yields the finiteness of ϱ∣α(Z).
Furthermore, since the map Tzα(Z)→TqQ is injective,
[19, Proposition 7]) implies that ϱ∣α(Z):α(Z)→ϱ(α(Z)) is an isomorphism
which is the desired conclusion.
∎
We shall need later the following fact.
Proposition 4.5**.**
Let X be a G-flexible variety for a group G⊂SAut(X) generated by a saturated set of locally nilpotent vector fields,
ϱ:X→Q be a dominant morphism into another algebraic variety Q and
Z be a locally closed reduced subvariety in X. Suppose that F is a finite subset of Z such that
dimTz0Z≤dimQ for every z0∈F.
Then one can find an automorphism α0∈G with the following property:
for any irreducible algebraic family A⊂G of automorphisms of X containing α0
and any general element α∈A there is a neighborhood Vα of α(F) in α(Z) such that for every z∈Vα and q=ϱ(z) the induced map ϱ∗:Tzα(Z)→TqQ
of the tangent spaces is injective.
Proof.
By flexibility and Theorem 2.8 there is an automorphism α0 of X such that
for every z0∈F and
z0′=α0(z0) the variety Q (resp. the morphism ϱ) is smooth at (resp. over) q0′=ϱ(z0′) and
the induced map Tz0′α0(Z)→Tq0′Q
of the tangent spaces is injective. In the case when α0 is contained in an irreducible family A of automorphisms
put z1=α(z0), Z1=α(Z) and q1=ϱ(z1) for a general α∈A. Then by continuity
the variety Q (resp. the morphism ϱ) is smooth at (resp. over) q1
and the induced map ϱ∗∣Tz1Z1:Tz1Z1→Tq1Q of the tangent spaces is also injective.
That is, the kernel of ϱ∗∣Tz1Z1 is zero. Hence for every z in some neighborhood of z1 in Z1 the kernel of
the induced map TzZ1→Tϱ(z)Q is also zero.
This yields the desired conclusion.
∎
We shall also need a parametric version of Theorem 4.2 (iii).
Proposition 4.6**.**
Let the assumptions and notations of Theorem 4.2 (iii) hold with Q0=Q and let U be a smooth algebraic variety.
Let Z=U×Z, P=U×P, X=U×X, Q=U×Q, κˇ=(id,κ):X→P
and τˇ=(id,τ):Q→P. Suppose that ϱˇ:X→Q is a smooth morphism such that κˇ=τˇ∘ϱˇ. Consider the G-action
on X such that the natural projection X→X is G-equivariant,
(1)* Then there exists an algebraic family A⊂G of automorphisms of X such that
for a general element α∈A, a general u∈U and
every z∈α(u×Z) the induced map ϱˇ∗:Tzα(u×Z)→Tϱˇ(z)(u×Q) is injective.*
(2)* Suppose that the assumptions of Theorem 4.2 (iv) are satisfied as well. Then the morphism
ϱˇ∣α(u×Z):α(u×Z)→u×Q in (1) is also injective for general α∈A and u∈U.*
(3)* Furthermore, suppose that W is a subvariety of X for which dimW+dimZ<dimX (resp. dimW+dimZ≤dimX)
and W=U×W. Then one can suppose additionally that W∩Xu does not meet α(Zu) (resp.
W∩Xu∩α(Zu) is finite) for a general u∈U.*
Proof.
Let T(X/P)∗ , T(Z/P)∗ and Y∗ be as in the proof of Theorem 4.2 (iii).
Let T∗=U×T(X/P)∗, Y∗=U×Y∗ and E=U×T(Z/P)∗.
Then G acts on T∗
and by Theorem 2.8 this action is transitive on every fiber of T∗→X→κˇP.
Note that dimE∗=dimT(Z/P)+dimU≤dimQ−dimP+dimU (because
dimT(Z/P)≤dimQ−dimP by the assumption).
Since all fibers of the natural projection Y∗→P are of the same dimension,
by Remark 1.9 (1) one has dimY∗×PE∗=dimY∗+dimE∗−dimP.
Since the codimension of Y∗ in T∗ is equal to dimQ−dimP, by Theorem 1.4
we can choose one-parameter unipotent algebraic subgroups H~1,…,H~m~ of G such that
for a general element α=(h~1,…,h~m~)∈H~1×⋯×H~m~ and
E′′=α∗(E∗) one has dimY∗∩E′′≤dimU. Hence for a general u∈U the fiber Eu of E′′∩Y∗ over u is at most finite.
Note that up to the zero vector the set (Tα(u×Z))∩Y∗ is
the kernel of the natural projection Tα(u×Z)→T(u×Q). This implies that Eu is empty and, therefore, the map
(Tα(u×Z))∩E∗→T(u×Q) is injective. Consequently, this yields (1).
The similar argument works in the case of Theorem 4.2 (i) with dimQ≥2dimZ+1. That is,
ϱˇ∣α′(u×Z):α′(u×Z)→u×Q is injective for a general u∈U and
a general automorphism α′ in some algebraic family A′⊂G.
By Remark 4.3 (2) one can suppose that the induced map of the tangent bundles is still injective and we have (2).
For (3) note that Theorem 1.4 implies that there is a family A′⊂G of algebraic automorphisms of X such that for a general α′∈A′ one
has dimW∩α′(Z)<dimU (resp. dimW∩α′(Z)≤dimU).
Hence W∩Xu does not meet α(Zu) (resp.
W∩Xu∩α(Zu) is finite) for a general u∈U. Remark 4.3 (2) implies that one can suppose that A=A′
which yields the desired conclusion.
∎
.
Remark 4.7**.**
By Remark 4.3 (4) one can drop the assumption that κ is dominant in the first statement of
Propostion 4.6. Similarly, this assumption can be dropped in the second statement if one replaces
the inequality dimZ×PZ≤2dimZ−dimP and dimQ≥dimZ+m
by the ones in Formula (14).
5. General projections for flexible varieties. II
Notation 5.1**.**
In this section X and P are smooth algebraic varieties,
P is a closed affine subvariety of Akm, κ:X→P is a surjective morphism, and Z is a closed subvariety of X.
We suppose that P^ is a completion of P, X^ is a completion of X, D^=X^∖X,
and κ^:X^⇢P^ is a rational extension of κ.
Let f^ be the rational extension of a regular function f∈k[X] to X^.
Denote by R(f^) the subvariety of D^ that consists of points which are either indeterminacy points of f^
or points at which f^ is regular and takes finite values.
Given any morphism of the form λ=(f1,…,fN):X→AkN
we let R(λ^)=⋂i=1NR(f^i) where
λ^ is the extension of λ to X^. In particular, since
κ is given by m coordinate functions we can define R(κ^).
The aim of this section is to describe some conditions under which
the morphism κ∣Z:Z→P is proper and, in particular, κ(Z) is closed in P.
Proposition 5.2**.**
Let Z˘ be
the intersection of D^ with the closure of Z in X^. If
Z˘∩R(κ^)=∅ then κ∣Z:Z→P is proper.
Proof.
Using a resolution π:Xˉ→X^ of the indeterminacy points of κ^ one gets a morphism
κˉ=κ^∘π:Xˉ→P^.
Treat X as a subvariety of Xˉ and denote by Zˉ the closure of Z in Xˉ. Note that κ∣Z:Z→P is
proper if and only if (κˉ−1(P)∖X)∩Zˉ=∅. Note also that π(κˉ−1(P)∖X) consists of
all point in D^ where either κ^ has indeterminacy or where κ^ is regular and takes values in P.
Since P is closed in Akm the latter means that all coordinate functions of κ^ take finite values and, therefore,
π(κˉ−1(P)∖X) is contained in R(κ^). Furthermore, if Z^ is the closure of Z in X^ the
properness of π implies that (κˉ−1(P)∖X)∩Zˉ=∅ if and only if
π(κˉ−1(P)∖X)∩Z^=∅. Since π(κˉ−1(P)∖X)∩Z^=Z˘∩R(κ^)
we have the desired conclusion. ∎
Remark 5.3**.**
One may have properness of κ∣Z:Z→P even if
Z˘∩R(κ^)=∅. Indeed, consider X=Ak2⊂Pk2=X^, P=Ak1
and the map X→P given by (x,y)→x for a coordinates system (x,y) on Ak2.
Let Z⊂X be the parabola given by y=x2. Then κ∣Z:Z→P is proper but Zˉ meets D^
at an ideteminacy point of x^.
Corollary 5.4**.**
Let D^ be irreducible and I(f^i) be the set of indeterminacy points of f^i
for κ=(f1,…,fm). Suppose that every fi is non-constant on Z and
Z˘∩⋂i=1mI(f^i)=∅. Then the morphism
κ∣Z:Z→P is proper and κ(Z) is closed in P.
Proof.
Let x∈Z˘.
By the assumption at least for one index i the rational function f^i yields a regular morphism into P1 in a neighborhood of x.
Note the value
of f^i at x is ∞ (indeed, f^i−1(∞)⊂D^ and, therefore, f^i−1(∞) must be equal to D^ since the latter is irreducible).
Thus Z˘∩R(κ^) is empty and Proposition 5.2 yields the desired conclusion.
∎
Theorem 5.5**.**
Let κ^:X^→P^ be regular
and κ~:X~→P be its restriction to X~=κ^−1(P).
Suppose that H is an algebraically generated group acting on X~ so that
it transforms every fiber of κ into itself.
Let D~=D^∩X~ be a finite disjoint union of
irreducible smooth subvarieties {Ti} such that every set κ~(Ti) is a smooth
variety and
κ~∣Ti:Ti→κ~(Ti) is a smooth morphism with each fiber being
an orbit of H.
Let θ:X→P×AkN be a morphism of the form θ=(κ,λ)
and let Si=R(λ^)∩Ti and Zi=Z˘∩Ti where Z˘ from Proposition 5.2.
Suppose that for
every i either Si is empty or
dimZi×PSi+dimκ~(Ti)<dimTi.
Then there exists an algebraic fammily A⊂H of automorphisms of X such that
for a general element h∈A the morphism θ∣h(Z):h(Z)→P×AkN is proper.
Proof.
Since P is closed in Akn the description of X~ implies that for every point x∈D^∖D~
there is a coordinate function f of κ for which f^
has a regular value ∞ at x.
Thus (D^∖D~)∩R(κ^)=∅ and Z˘∩R(θ^)⊂Z˘∩R(λ^)∩D~.
Applying Theorem 1.4 to the morphism
κ~∣Ti:Ti→κ~(Ti) in the case of a non-empty Si
we see that
for a general element h of some algebraic family A⊂H the variety h(Zi)∩Si is empty
when
dimZi×PSi+dimκ~(Ti)<dimTi.
Hence, by Proposition 1.10 h(Z˘)∩R(λ^)∩D~=∅ and we are done by Proposition 5.2.
∎
Corollary 5.6**.**
Let κ~:X~→P,
H, Ti, θ=(κ,λ) and Si be as in Theorem 5.5.
Suppose that for every i either Si is empty or κ~∣Si:Si→κ~(Ti) is a flat morphism
with fibers of dimension li. Let k=dimZ and mi be the dimension of the fibers
of κ~∣Ti:Ti→κ~(Ti). Suppose that
for every i one has li+k−1<mi.
Then for a general element h∈H the morphism θ∣h(Z):h(Z)→P×AkN is proper.
Proof.
Let Zi be as in Theorem 5.5. By the same theorem we can assume that Si is not empty.
Since κ~(Si)=κ~(Ti) Remark 1.9 (1) implies that
Zi×PSi=dimSi+dimZi−dimκ~(Ti), i.e., the desired inequality
from Theorem 5.5 can be rewritten as dimZi+dimSi<dimTi.
By the assumption we have li+dimκ~(Ti)+k−1<mi+dimκ~(Ti).
Note that dimSi=li+dimκ~(Ti), dimTi=mi+dimκ~(Ti),
and dimZ˘=k−1. Hence we have this inequality dimSi+dimZ˘<dimTi which concludes the proof.
∎
Notation 5.7**.**
From now on by ED(X) we denote ED(X)=max(2dimX+1,dimTX).
Corollary 5.8**.**
Let the assumptions of Theorem 5.5 or Corollary 5.6 hold with θ being a dominant morphism.
Suppose also that X is a G-flexible variety where
G⊂SAut(X) is generated by a saturated set of locally nilpotent derivations
and that ED(Z)≤dimP+N.
Then there exists an algebraic family A⊂G of automorphisms such that for a general α∈A
the morphism θ∣α(Z):α(Z)→P×AkN is a closed embedding.
Proof.
Consider Theorem 4.2 (iv) with P in its formulation being a singleton unlike P in the formulation above.
Applying this special case one can see that there exists an algebraic family A0⊂G of automorphisms such that for a general β∈A0 the morphism
θ∣β(Z):β(Z)→P×AkN is an embedding. By Proposition 1.10 this property remains valid
if one replaces A0 by A=H×A.
Hence by Theorem 5.5 for a general h∈H and α=h∘β∈A the morphism θ∣α(Z):α(Z)→P×AkN is
also proper. Repeating now the argument from Corollary 4.4 we conclude that
it is a closed embedding which is the desired conclusion.
∎
Remark 5.9**.**
Suppose that unlike in Notation 5.1 P is not an affine variety
but only a quasi-affine one and let R be an affine variety containing P as an open subset.
Assume additionally that κ(Z) is closed in R. Then the conclusion about properness of
θ∣h(Z):h(Z)→P×AkN in Theorem 5.5 remains valid.
For a more general setting we need the following.
Notation 5.10**.**
Let ϱ:X→P be an affine morphism of smooth quasi-affine varieties
and P=⋃j=1mPj, where each Pj is an affine Zariski dense open subset of P.
Let Xj=ϱi−1(Pj), ϱj=ϱ∣Xj:Xj→Pj and
ϱ~j:X~j→Pj be an extension of ϱj to a proper morphism, where X~j is a smooth variety
and D~j=X~j∖Xj. Suppose that for every j=1,…,m there is a set Nj of locally nilpotent vector fields on Xj
tangent to the fibers of ϱj and extendable to complete vector fields on X~j. Let Hj be the group of automorphisms of Xj (and X~j)
generated by the elements of the flows of the vector fields from Nj.
Let λ:X→AkN be a morphism, Θ=(ϱ,λ):X→P×AkN,
Z be a closed subvariety of X and Zj=Z∩Xj. Suppose also that for every j=1,…,m
the assumptions (and, therefore, the conclusion) of Theorem 5.5 are true if κ:X→P, κ~:X~→P, Z, D~, H and θ in
the formulation of that theorem are replaced with ϱj:Xj→Pj, ϱ~j:X~j→Pj, Zj D~j, Hj and θj=Θ∣Xj.
Theorem 5.11**.**
Let Notation 5.10 hold.
Then there exists an algebraic family A of automorphisms of X over P such that
for a general element h∈A the morphism Θ∣h(Z):h(Z)→P×AkN is proper.
Proof.
Let Ij⊂k[P] be the defining ideal of the variety P∖Pj.
For every δ∈Nj one can find f∈Ij such that fδ is a locally nilpotent vector field on X tangent to
the fibers of ϱ. Furthermore, for a given point p∈P such f can be chosen so that f(p)=1.
Consider the set Nj′ of all locally nilpotent vector fields of this form fδ.
The elements of their flows generate a group Hj′⊂SAut(X/P).
Without loss of generality we can suppose that for every σ∈Nj and each r∈k one has rσ∈Nj.
Then for every point p∈Pj the restrictions of Nj and Nj′ to ϱj−1(p) (resp. ϱ~j−1(p)) coincide.
Hence, the restriction of the Hj′-action to ϱj−1(p) (resp. ϱ~j−1(p)) is the same as the Hj-action.
Thus, we can replace Hj in Notation 5.10 by Hj′.
In particular, from the beginning we can assume that Hj⊂SAut(X/P). By Theorem 5.5 there exists an algebraic family
Aj⊂Hj⊂SAut(X/P) such that for a general element h∈Aj the morphism Θ∣h(Zj):h(Zj)→Pj×AkN is proper.
Arguing as in Remark 4.3 (2) we can find an algebraic family A⊂SAut(X/P) such that for a general element α∈A
the morphism Θ∣α(Zj):α(Zj)→Pj×AkN is proper for every j which yields the desired conclusion.
∎
6. General projections for partial quotient morphisms of flexible varieties
In the case of partial quotients of Ga-actions one can get local properness under much milder assumptions than in Theorem 5.5. It is reflected in the following fact
which plays an important role in [21].
Theorem 6.1**.**
Let X be a smooth quasi-affine algebraic variety, N be a saturated set of locally nilpotent vector fields on X,
and G⊂SAut(X) be the group generated by N. Suppose also that X is G-flexible.
Let ϱ0:X→Q be a partial quotient morphism
associated with a nontrivial δ0∈N, Z be a
locally closed reduced subvariety of X
of codimension at least 2 and F be a finite subset of Z such that dimTz0Z≤dimQ
for every z0∈F.
Then there exists a connected algebraic family A⊂G of automorphisms
such that for a general element α∈A
and the closure Zˉα′ of Zα′=ϱ0∘α(Z) in Q
one can find a neighborhood V0′ of ϱ0(α(F)) in Zˉα′ such that for V0=ϱ0−1(V0′)∩α(Z) the morphism ϱ0∣V0:V0→V0′
is an isomorphism.
The proof consists mostly of reminding some results from [9].
Proposition 6.2**.**
(Proposition 2.15 in [9])* Let X, G, and N be as in Theorem 6.1.
Then for any locally nilpotent derivation
δ0∈N one
can find another one
δ1∈N such that the subgroup H⊂G
generated by δ0, δ1 and all their replicas acts with an open orbit on X.*
Remark 6.3**.**
In fact we have more. It follows from the proof of [9, Proposition 2.15]) that δ1 can be chosen so that the open orbit of H contains a given finite subset of
X.
Notation 6.4**.**
(a)
Let δ0 and δ1 be as in Proposition 6.2 and Ui be the one-parameter unipotent subgroup of SAut(X) associated with δi.
Any function f∈kerδ0\kerδ1 yields the one-parameter group Uf0 associated with the replica
fδ0, and similarly
g∈kerδ1\kerδ0 yields the one-parameter group Ug1 associated with the replica
gδ1.
(b) To any sequence of invariant functions
[TABLE]
we associate an algebraic family
of automorphisms defined by the product
[TABLE]
More generally, given a tuple
κ=(ki,li)i=1,…,s∈N2s the product
[TABLE]
is as well an algebraic family of automorphisms.
Proposition 6.5**.**
(Corollary 5.4 in [9])There is a finite collection of invariant functions F as in (15)
such that for any sequence
κ=(ki,li)i=1,…,s∈N2s the algebraic family of automorphisms
Uκ as in (17)
has an open orbit O(Uκ) that coincides with O(H) and so
does not depend on the choice of κ∈N2s.
Notation 6.6**.**
We keep the notation and assumptions from 6.4(a).
(a) Let ϱ0:X→Q0 and ϱ1:X→Q1 be partial quotient morphisms with respect
to the unipotent subgroups U0 and U1, respectively.
It is proven in [9, Lemma 3.3] that there are open embeddings X↪Xˉ, Q0↪Qˉ0, and Q1↪Qˉ1
into normal projective varieties such that the following conditions
are satisfied.
- (i)
ϱ0 and ϱ1 extend to morphisms
ϱˉ0:Xˉ→Qˉ0 and ϱˉ1:Xˉ→Qˉ1.
2. (ii)
the unique “horizontal” divisors D0⊂Xˉ∖X and D1⊂Xˉ∖X, that
are mapped birationally (by ϱˉ0 and ϱˉ1) onto
Qˉ0 and Qˉ1 respectively, are smooth.
3. (iii)
the completion Xˉ satisfies some other conditions which we assume to be true but omit because they are not needed for the formulation of
Proposition 6.7 below.
(b) Given a closed subscheme Y⊆X of codimension at least 2 we call
[TABLE]
the partial boundaries.
(c)
For a one-parameter group U we let U∗=U\{id}
and for Uκ=Ufsks1⋅Ugsls0⋅…⋅Uf1k11⋅Ug1l10 as in (17) we let
[TABLE]
Proposition 6.7**.**
(Proposition 5.11 in [9])*
Let (Yα)α∈A be a flat family of proper closed subsets of X.
Assume that the partial boundaries
∂iYα are contained in Eα,i,
where the (Eα,i)α∈A, i=0,1,
form flat families of proper closed subsets of Di.
Then one can find an open dense subset Ao of A, flat families of proper, closed subsets
(Eα,io)α∈Ao of Di (i=0,1), and a sequence
κ=(k1,l1,…,ks,ls)∈N2s such that for any element h∈Uκ∗ we have*
[TABLE]
Proof of Theorem 6.1.
Let Q0 be a smooth Zariski open dense subset of Q such that
for X0=ϱ0−1(Q0) the morphism ϱ0∣X0:X0→Q0 is smooth.
Since the G-action on X is m-transitive for every m replacing Z by g(Z) for some g∈G
we can suppose that F⊂X0∩O(H) where O(H) is the open orbit
of H from Proposition 6.2.
Let Zˉ be the closure of Z in X.
By Theorem 4.2 (i), (ii) and Remark LABEL:gp1.r1_(4) there exists an irreducible family A⊂G of algebraic
automorphisms of X such that for a general α∈A one has
(a) ϱ0∣α(Zˉ):α(Zˉ)∩X0→ϱ0∘α(Zˉ) is birational and
ϱ0−1(ϱ0(α(F))∩α(Zˉ)=α(F) (here we use the fact that
α(F)⊂X0∩O(H) since F⊂X0∩O(H)).
Recall that by Remark 4.3 (1) we can suppose that A is irreducible and it contains the identity map e.
By the definition of SAut(X) any automorphism α0∈SAut(X) is an element of an algebraic family
H1×…×Hm of automorphisms on X as in Theorem 1.3 where every Hi is a unipotent one-parameter group. Using the way to enlarge A as in Remark 4.3 (2) we can replace A by H1×…×Hm×A where the last family is still irreducible and contains e
but it contains also α0 now.
Hence, since dimTz0Z≤dimTQ for every z0∈F,
choosing α0 as in Proposition 4.5
we also have
(b) a neigborhood V~α of α(F) in α(Z) such that for every z∈V~α the induced map Tzα(Z)→Tϱ0(z)Q
of the tangent spaces is injective.
Let Uκ be from Proposition 6.7
and let β=h∘α be a general element of the family Uκ⋅A.
By Proposition 1.10 and Remark 4.3 (1) we still have conditions (a) and (b) for this bigger family.
By Proposition 6.7 the partial boundary ∂0(β(Zˉ))=∂0(h.α(Zˉ))⊆Eα,0o where
Eα,0o is a proper subvariety of D0 from Notation 6.6.
This implies that the morphism ϱ0∣β(Zˉ):β(Zˉ)→Q is proper over
Q∖ϱˉ0(Eα,0o) where ϱˉ0 is again from Notation 6.6.
On the other hand, by Proposition 6.5
β(z0)=h.α(z0) runs over the open set O(H) when h runs over Uκ. Hence
ϱ0∘β(F) does not meet ϱˉ0(Eα,0o) for a general β. Therefore,
the morphism ϱ0∣β(Zˉ):β(Zˉ)→Q is proper over an neighborhood of
F′=ϱ0∘β(F) in Q.
Assume that there is an irreducible subvariety Y′⊂ϱ0∘β(Z)∩Q0 such
that dimY′<dimY where Y is an irreducible component of ϱ0−1(Y′)∩β(Z) with
some z0′=ϱ0(β(z0)) from F′ contained in the closure of ϱ0(Y).
Then by the Chevalley’s theorem [15, 13.1.3] dimϱ0−1(z0′)∩Y≥1 (this fiber cannot be empty because of properness) contrary to the fact that by (a)
ϱ0−1(z0′)∩β(Z)=β(z0).
Hence we can suppose that ϱ0∣β(Z) is not only birational but also quasi-finite over a neighborhood
V0′⊂ϱ0∘β(Z) of F′ which is combination with properness implies finiteness by the Grothendieck theorem [15, Theorem 8.11.1].
If for a sufficiently small V0′ and V0=ϱ0−1(V0′)∩β(Z) the morphism ϱ0∣V0 is not injective
then for some curve C′∈V0 through a point z0′∈F′ and C=ϱ0−1(C′)∩β(Z)
the finite morphism ϱ0∣C:C→C′ must be ramified at β(z0). However, this is contrary to the fact that the induced map Tβ(z0)T(β(Z))→Tz0′Q
of the tangent spaces is injective by (b). That is, we can suppose that ϱ0∣V0:V0→V0′ is injective and proper. Since it is also finite condition (b) implies that
this map is an embedding (e.g., see [19, Proposition 7]) which yields the desired conclusion.
∎
Remark 6.8**.**
(1) Note that by Remark 4.3 (1) and construction the family A from Theorem 6.1 is a Zariski open subset in a larger family of automorphisms which contains the identity map.
(2) Similar to Remark 4.3 (2) we also note that the family A from Theorem 6.1 does not depend on the choice of the subvariety Z.
(3) Let X, Z, F, N and G be as in Theorem 6.1, S={δ1,…,δs}⊂N
and ϱi:X→Qi be a partial quotient morphism associated with δi for i=1,…s. Then an easy adjustment of
the proof yields the following generalization of Theorem 6.1.
There exists a connected algebraic family A⊂G of automorphisms such that for a general element α∈A,
every i=1,…,s, and the closure Zαi:=ϱi∘α(Z) of ϱi∘α(Z) in Qi
one can find a neighborhood Vi′ of ϱi(α(F)) in Zαi such that for Vi=ϱi−1(Vi′)∩α(Z) the morphism
ϱi∣Vi:Vi→Vi′ is an isomorphism.
7. The case of Gromov-Winkelmann flexible varieties
The aim of this section is the following fact.
Theorem 7.1**.**
Let Z, Y1, and Y2 be closed subvarieties of Akn such that Y1∩Z=Y2∩Z=∅, dimZ≤n−3
and ED(Y1)≤n−2 (where ED(Y1) is as in Notation 5.7). Let φ:Y1→Y2 be an isomorphism and X=Akn∖Z.
Suppose also that either
(a)* dimZ+dimY1≤n−3, or*
(b)* dimY1=1 and dimZ=n−3.*
Then there exists an automorphism γ∈SAut(X) for which γ∣Y1=φ.
The proof requires some preparations.
Notation 7.2**.**
Further in this section we write An instead of Akn and the symbol
Au1,…,unn means that An is equipped with a fixed coordinate system
uˉ=(u1,…,un). In particular, this system induces an embedding An↪Pn
into a projective space.
We also suppose that
Z, Y1 and Y2 are closed subvarieties of Au1,…,unn such that dimZ≤n−3, ED(Y1)≤n−2,
Y1 and Y2 are disjoint from Z and there exists an isomorphism φ:Y1→Y2.
The following result will be important this section: in the case when Z is of codimension at least 2 Gromov [14] observed that An∖Z is a flexible variety
and Winkelmann [37]
showed that SAut(An∖Z) acts transitively on An∖Z which is equivalent by virtue of Theorem 2.2.
(In particular, for a finite Y1 Theorem 7.1 is valid even when dimZ=n−2.)
The next fact is well-known (e.g., see [19]).
Proposition 7.3**.**
Let ED(Z)≤k≤n and Lin(An,Ak) be the affine variety of surjective linear maps An→Ak.
Then for a general element ϱ∈Lin(An,Ak) the restriction ϱ∣Z:Z→Ak is a closed embedding.
Proposition 7.4**.**
Let ϱ:An→Ak be a general element of Lin(An,Ak).
(1)* If k≤dimZ then ϱ∣Z:Z→Ak is surjective
and for every w∈Ak the fiber F=ϱ−1(w)∩Z is of dimension dimZ−k.*
(2)* If k=dimZ then ϱ∣Z:Z→Ak is finite.*
(3)* Let T(Z/Ak) be as in Notation 4.1. Then for every k one has dimT(Z/Ak)≤dimTZ−k.*
Proof.
Let ϱ=(f1,…,fk) be the coordinate form of ϱ and ϱˉ=(fˉ1,…,fˉk) be the rational extension of ϱ to Pn.
Let R(ϱˉ)=⋂i=1kR(fˉi) be the same as in Notation 5.1. Since ϱ is surjective
(i.e., f1,…,fk are linearly independent) R(ϱˉ) is of codimension k in D=Pn∖An.
The natural action of SL(n,k) on An extends to an action to Pn whose restriction to D
is transitive. In particular, replacing ϱ by ϱ∘h where h is a general element of SL(n,k), by virtue of Theorem 1.3 we can assure that
the intersection Zˉ∩R(ϱˉ) is of dimension dimZ−1−k where Zˉ is the closure Zˉ of Z in Pn
(and, therefore, Zˉ∩R(ϱˉ)=∅ when dimZ=k).
Assume that there exists a fiber F of ϱ∣Z with dimF>dimZ−k. Then the closure Fˉ of F meets D along a subvariety of dimension at least dimZ−k.
However, Fˉ∩D⊂Zˉ∩R(ϱˉ) which yields a contradiction.
Thus dimF=dimZ−k (since it cannot be less than dimZ−k [32, Chap. 1, Sec. 6, Theorem 7]).
In particular, ϱ∣Z:Z→Ak is quasi-finite when dimZ=k. Since Zˉ∩R(ϱˉ)=∅ in the latter case
the morphism ϱ∣Z:Z→Ak is proper by Proposition 5.2 and, therefore, it is finite by [15, Theorem 8.11.1].
Hence, we have (1) and, in particular, this map is surjective. This implies that when k<dimZ
we have also surjectivity for a general ϱ which is (2).
In (3) treat every point x∈An as a vector and consider the map ψ:TZ→An that sends each v∈TzZ,z∈Z
to z+v. Denote by V (resp. V0) the closure of ψ(TZ) (resp. T(Z/Ak)) in Pn. Note that dimV does not exceed dimTZ.
Hence, for a general ϱ and R(ϱˉ) as above the dimension of V∩R(ϱˉ) does not exceed dimTZ−k−1.
It remains to note that V0∩D is contained in V∩R(ϱˉ) (indeed, if v∈Kerϱ∗ then f1(v)=…=fk(v)=0
and the closure
of every line {z+tv∣t∈k} in Pn meets D at a point of R(ϱˉ)). That is, dimV0∩R(ϱˉ)≤dimTZ−k−1
and, thus, dimV0≤dimTZ−k which concludes the proof.
∎
Notation 7.5**.**
For every group G acting on an algebraic variety X and every subscheme W of X
we denote by GW the subgroup of G such that the action of every element of GW yields the identity map on W.
Lemma 7.6**.**
Let X=P×AN+1, ϱ:X→P be the natural projection and m>0.
Suppose that G=SAut(X/P) and Xp:=ϱ−1(p),p∈P. Let Lin(AN+1,AN) be the
variety of surjective linear maps AN+1→AN and Z be a closed subvariety of X such that for every p∈P
one has dimZ∩Xp≤N−1
and for a general
λ∈Lin(AN+1,AN) and θ=(ϱ,λ) the morphism θ∣Z:Z→P×AN is proper.
Then each variety Xp∖Z is GZm-flexible where Zm is the m-th infinitesimal neighborhood
of Z in X and GZm has the same meaning as in Notation 7.5.
Proof.
Let ψ:P×AN→P be the natural projection. Since θ∣Z is proper one has
θ(Z∩Xp)=θ(Z)∩ψ−1(p).
Let x be any point in Xp∖Z. Since dimZ∩Xp≤N−1 we see that θ(x)∈/θ(Z∩Xp)
for a general λ and hence θ(x)∈/θ(Z).
Note also that θ can be treated as a quotient morphism of a locally nilpotent vector δ field on X~ whose value at x
is a general vector ν in TxXp (since λ is general).
Note also that the kernel of δ coincides with k(P×AN) where we treat k(P×AN) as a subring of k(X) under the natural embedding.
Hence for every regular function f on P×AN that vanishes on θ(Z) but not at θ(x)
(which exists since θ(Z) is closed in P×AN) the vector field fmδ is locally nilpotent
and its value at x coincides with this general vector ν up to a nonzero factor. Furthermore, the flow of fmδ is a one-parameter group in GZm.
Hence Xp∖Z is GZm flexible by the definition of flexibility and we are done.
∎
Notation 7.7**.**
Further in this section by X we denote X=Au1,…,unn∖Z.
Lemma 7.8**.**
Let κ:An→As be the natural projection
(u1,…,un)→(u1,…,us) where s≤ED(Y1). Let κ=κ∘φ, dimZ≤n−s−2
and F⊂Y1 be a finite set for which φ(y)=y for every y∈F.
Then for every m>0 there exists an automorphism β of X⊂An over As such that for each y∈F
the m-jet of φ at y coincides with the restriction of the m-jet of β.
Proof.
By Lemma 7.6 every fiber of κ∣X is GZ-flexible.
By Theorem 3.10 we can suppose now that the tangent spaces of Y1 and Y2
at every y∈F coincide and the isomorphism of these tangent spaces induced by φ is the identity map.
Without loss of generality we can also suppose that TyY1 is contained in {uk+1=…=un=0} where k≥s
and y is the origin 0ˉ of An.
Then the m-jet φm of φ at y can be chosen in the
following form (u1,…,un)→(u1+h1,…,un+hn)
where h1,…,hn are polynomials in u1,…,uk without free and linear terms.
Let h~n=∂u1∂h1+…+∂uk∂hk.
Then the m-jet (u1,…,un−1,un)→(u1+h1,…,un−1+hn−1,un−unh~n) is of divergence 1 and, hence,
it satisfies the assumption on
the Jacobian in Notation 3.5 (e.g., see the proof of [2, Lemma 4.13]).
By Theorem 3.10 we can find an automorphism β1∈GZ of X over As with such precise jets at every point of F.
Hence φm is a composition of β1 with an m-jet ψm of the
form (u1,…,un−1,un)→(u1,…,un−1,un+h) where a priori h depends on u1,…,un.
However, restricting ψm to the étale germ of Y1 at y we can replace h with a function of u1,…,uk, i.e.,
we can view ψm as the restriction of the m-jet of an automorphism β2∈GZ. Letting β=β2∘β1 we get
the desired conclusion.
∎
Lemma 7.9**.**
Let ϱ:An→Ak be the natural projection (u1,…,un)→(u1,…,uk)
with k≥max(dimZ+1,ED(Y1)). Assume that for the closure Z′ of ϱ(Z) in Ak
the set Y1∩ϱ−1(Z′) is finite.
Let ϱ∣Y1:Y1→Ak be a closed embedding and
φ(y)=y for every y∈Y1∩ϱ−1(Z′). Then there exists m>0 such that whenever
the m-th jets of φ are equal to the jets of the identity map for all y∈Y1∩ϱ−1(Z′), one has
an automorphism α of An over Ak for which the following holds
(a)* α(z)=z for every z∈Z (in particular, α is an automorphism of X=An∖Z) and*
(b)* after replacing Y1 by α(Y1) and φ by φ∘α−1 one has ui∣Y1=ui∘φ
for every ui with i≥k+1.*
Proof.
Let I′⊂k[Ak] be the vanishing ideal of Z′, Y1′=ϱ(Y1), I⊂k[Y1′] be the image of I under the
natural morphism k[Ak]→k[Y1′] and J⊂k[Y1′] be the vanishing ideal of Y1′∩Z′. By Nullstellensatz one has
Jm⊂I for some m>0.
Let gi=ui∘φ−ui∣Y1 for i≥k+1. Since and Y1 and Y1′ are isomorphic we can consider gi as a function on Y1′.
Note that it always belongs to J and, furthermore, under the assumption it belongs to Jm.
Thus gi is the restriction of a polynomial g~i on Ak which vanishes on Z′.
Consider the automorphism ψi of X given by ui→ui+g~i(u1,…,uk)
and uj→uj for j=i.
Then the composition ψk+1∘…∘ψn yields the desired automorphism α.
∎
Remark 7.10**.**
Since m in the proof above can be chosen as large as we wish one can suppose that the extension g~i vanishes not only on
Z′ but also on the m-th infinitesimal neigborhood of Z′.
This implies that the automorphisms {ψi} in the proof are elements of the flows of locally nilpotent vector fields that vanish on Z with multiplicity m.
Since α from Lemma 7.9 is a composition of such automorphisms we can suppose that
the restriction of α to the m-th infinitesimal neigborhood Zm of Z is the identity map. The same is true for β from
Lemma 7.8 since by Lemma 7.6 Xp∖Z is not only GZ-flexible
but also GZm-flexible.
Proof of Theorem 7.1 (a).
Let H=SL(n,k) act naturally on An,
ϱk:An→Ak be the natural projection given by (u1,…,un)→(u1,…,uk) and
ϱk,h be the composition ϱk∘h where h∈H.
By Propositions 7.3 and 7.4 we have the following for a general h∈H.
(1) For k=dimZ the morphism ϱk,h∣Z:Z→Ak is finite
while the morphism ϱn−2,h∣Z:Z→An−2 is a closed embedding.
(2) For every k<dimY2 the morphism ϱσ,k∣Y2:Y2→Ak is surjective
with the dimension of each fiber equal to dimY2−k, for k=dimY2
the morphism ϱk,h∣Y2:Y2→Am is finite and
for every k≤dimY2 one has dimT(Y2/Ak)≤dimTY2−k.
Let us show inductively that for every k≤dimY1 and a general h in H one can also assume the following.
(3) ϱk,h∘φ=ϱk,h∣Y1 and, in particular, by (2)
if k<dimY1 then ϱσ,k∣Y1:Y1→Ak is surjective
with the dimension of each fiber equal to dimY1−k,
if k=dimY1 then ϱk,h∣Y1:Y1→Am is finite and in any case
dimT(Y1/Ak)≤dimTY1−k.
(4) ϱn−2,h∣Y1:Y1→An−2 is a closed embedding.
(5) ϱn−2,h(Y1) does not meet ϱn−2,h(Z).
For k=0 the assumption (3) is automatically true while (4) and (5) follow from Proposition 7.3 applied to the variety Z∪Y1.
Let us assume that (3)-(5) are valid
for k−1 and let us establish that this implies their validity for k.
Since (4) and (5) hold for k−1, by Lemma 7.9 we obtained an automorphism β∈SAut(X) over An−2 such that after replacing
Y1 by β(Y1) one has ui∣Y1=ui∘φ for i=n−1 and i=n.
Consider the element σ of H such that σ∗(uk)=un,σ∗(un)=−uk while the rest coordinates remain intact.
Note that since h is general we can suppose that σ∘ϱk,h is still of the form ϱk,h1 where h1 is
a general element of H. Hence, replacing ϱk,h with σ∘ϱk,h we get (3) while keeping (1) and (2).
Treat now ϱk,h:X→Ak (resp. ϱn−2,h:X→An−2) as κ:X→P (resp. ϱ:X→Q)) in Theorem 4.2.
Note the assumptions of Theorem 4.2 are valid for the group G=SAut(X/P) and the subvariety Y1 of X (indeed, the
fibers of κ is G-flexible by Lemma 7.6 while dimY1×PY1≤2dimY1−dimP by Remark 1.9
and the fact that the fibers of κ∣y1:Y1→P are equidimensional because of (3)).
By Theorem 4.2 (i) and (iii) we can find an algebraic family A1⊂G (a priori depending on h) such that for a general element α1∈A1
the morphism ϱ∣α1(Y1):α1(Y1)→Q is injective and generates an injective map of the Zariski tangent bundles.
Similarly, since ϱ−1(ϱ(Z)) is of dimension dimZ+2<dimX−dimY1 by Theorem 1.4 we can find
an algebraic family A2⊂G (also depending on h) such that for a general element α2∈A2 the variety
α2(Y1) does not meet ϱ−1(ϱ(Z)) which is equivalent to (5). By Remark 4.3 (2) we can suppose that
A1=A2.
Hence, in order to prove (4) and (5) one needs to show that the morphism ϱ∣α(Y1):α(Y1)→Q can be made proper.
Let H′ be the subgroup of H consisting of all elements whose action on the first summand of
An=Au1,…,ukk⊕Auk+1,…,un−2n−k−2⊕Aun−1,un2 is the identity map.
Consider Proposition 4.6 with U=H′ and ϱˇ:H′×X=X→Q=H×Q given by (h′,x)→(h′,ϱn−2,h′h(x)).
Then Proposition 4.6 implies that we can find a family A1 as above which provides the injectivity condition for general h′ in H′
simultaneously. Note that the composition of the H′-action
with the projection An→Auk+1,…,un−2 can be viewed as the variety Lin(An,An−k−2) of surjective
linear maps ϱn−k−2,h′h′:An→An−k−2 and, hence, Proposition 7.4 implies
that ϱn−k−2,h′h′∣Y1:Y1→An−k−2 is proper for a general h′ because dimY1<n−k−2
by the assumption of Theorem 7.1. Since for a general h′ the element h′h
is still general in H, replacing h by h′h we get (4) without ruining (1)-(3) while the second statement of Proposition 4.6
provides us with (5).
Now we shall establish by induction conditions (3)-(5) for a general h and k=k1+1,…,n where k1=dimY1.
Since ϱh,k1∣Y1:Y1→Ak1 is finite by (3) we can suppose now every ϱh,k∣Y1:Y1→Ak is proper.
Assume that (4) and (5) hold for k−1. Applying Lemma 7.9 as above we can achieve (3) for k without ruining this properness
(but may be violating (4) and (5)).
Using notation κ:X→P and ϱ:X→Q in the same meaning as before we observe that though
the assumption dimY1×PY1≤2dimY1−dimP does not hold we can still apply Theorem 4.2 (i) and (iv)
by virtue of Remark 4.3 (5).
Hence the same argument as above yields an algebraic family A⊂G of automorphisms of X over Ak
such that for a general α∈A
the morphism ϱn−2,h∣α(Y1):α(Y1)→Q is injective and generates an injective map of the Zariski tangent bundles,
and, furthermore, α(Y1) does not meet ϱn−2,h−1(ϱn−2,h(Z)). The latter fact is (5) and in combination with the properness
the former one yields (4).
Note that this process of replacing Y1 by its automorphic image in X in the case of k=n yields the equality Y1=Y2 and, hence,
the desired conclusion.∎
Lemma 7.11**.**
Let m>0 and L be a finite collection of 3m distinct lines in P2
such that the intersection of any three of them is empty. Then for every finite subset F⊂P2 of cardinality m there is a line L∈L such that L∩F=∅.
Proof.
Let us use induction by m. For m=1 the claim follows from the definition of L. Suppose that
the statement is true for m−1 and L is a collection of 3m lines with the desired property.
Present L as a disjoint union L1∪L2∪L3 where each collection Li consists of 3m−1 lines.
Let F=F0∪{x}⊂P2 be of cardinality m, i.e., F0 is of cardinality m−1.
By the assumption we can find lines L1∈L1, L2∈L2 and L3∈L3 such that Li∩F0=∅ for every i.
Since L1∩L2∩L3=∅ one of these lines does not contain x and we are done.
∎
Proof of Theorem 7.1 (b).
Let uˉ=(u1,…,un) and ϱk,h:An→Ak have the same meaning as in the proof of Theorem 7.1 (a) where h∈H=SL(n,k).
We can suppose again that (1) as in the proof of Theorem 7.1 (a) holds.
Let us show inductively that for k≤n−3 and some general h (depending on k)
we can also assume
(2) ϱk,h∘φ=ϱk,h∣Y1
(3) ϱn−2,h∣Y1:Y1→An−2 is a closed embedding.
(4) ϱn−2,h(Y1) meets ϱn−2,h(Z) at a finite number of points and, furthermore, these points are the images of smooth points of Y1.
(5) for every singular point y of Y1 the image of TyY1 in TAk under the map induced by ϱk,h is of
dimension min(k,dimTyY1) (which implies that dimT(Y1/Ak)≤dimTY1−k) and the cardinalities
of SingY1 and ϱn−2,h(SingY1) are the same.
Step 1 (condition (5) and the case of k=0). Let k=0 and h be a general element of H. Then the assumption (2) is automatic and (3) follows from Proposition 7.3.
Denote by E the set of singular points of Y1. Since X is SAut(X)-flexible, replacing Y2 by its automorphic image in X
we can suppose by Theorem 2.6 that E is also the set of singular points of Y2 and φ(y)=y for every y∈E.
Furthermore, by Theorem 2.8 we can suppose that for every y∈E and every m≥0 the image of TyY2 in TAm under the map induced by
ϱm,h is of dimension min(m,dimTyY2). Applying now an automorphism of X to Y1 we can suppose
also that φ∗:TyY1→TyY2 is the identity map. Let G be the subgroup of SAut(X) whose restriction to
the second infinitesimal neighborhood E2 of E is the identity map. From now on we are going to use replacements of
Y1 and Y2 by their images under the actions of some elements of G. Note that this quarantees (5) for all m≥0.
Returning to the case of k=0 we observe that since h is general by Proposition 7.3 the morphism ϱn−2,h∣E∪Z is injective (and, in particular,
the sets ϱn−2,h(E) and ϱn−2,h(Z) are disjoint) and also dimY1+dimϱn−2,h−1(ϱn−2,h(Z))=dimX.
Hence, since X′=X∖E is G-flexible, applying to Y1 an element of G we get (4) by Theorem 1.4 which concludes Step 1.
Step 2. Now let us assume that (1)-(4) are valid for k−1 and proceed with
the case of k≤n−3.
Let F1=Y1∩ϱn−2,h−1(ϱn−2,h(Z)) and F2=φ(F1).
Since ϱk−1,h∘φ=ϱk−1,h∣Y1 we see that the sets ϱk−1,h−1(w)∩F1 and ϱk−1,h−1(w)∩F2
are of the same cardinality for every point w∈Ak−1. Since by Lemma 7.6 every fiber of ϱk−1,h∣X′ is
G-flexible by Theorems 2.8 and 3.1 we can find an automorphism α1∈G
such that replacing Y2 by α1(Y2) one has F1=F2 and φ∣F1=id∣F1.
Suppose that m is as in Lemma 7.9 (with ϱ replaced by ϱn−2,h). Choose β
as in Lemma 7.8 and replace Y2 by β(Y2), i.e., the m-jet of φ at any point of F1 is now
the m-jet of the identity map151515By flexibility such β can be chosen as an element of G
since the finite sets E and F1 are disjoint.. By Lemma 7.9 we can find α2∈G such that after replacement of Y1 by
α2(Y1) one has un∣Y1=un∘φ.
Consider σ∈H such that σ∗(uk)=un,σ∗(un)=−uk while the rest coordinates remain intact.
Since we started with a general h one has σ∘ϱk,h=ϱk,h1 where h1 is
a general element of H. Thus, replacement of h by h1 leave (1) valid while providing us with (2).
If we want to apply Theorem 4.2 further we need to guarantee the condition (14) in Remark 4.3 (5)
with ϱk,h as κ. For k=1 we can get this condition by
choosing a couple of points y1 and y2 in each irreducible component of Y2 and requiring that for α1
as above one has additionally ϱ1,h(α1(y1))=ϱ1,h(α1(y2)). Then ϱ1,h∣Y2:Y2→A1 is a quasi-finite morphism
(after the replacement of Y2 by α1(Y2))
and we have (14) for Y2. By (2) we have it also for Y1.
Furthermore, this condition will survive for k>1 provided that in further replacements of h by another general h~∈H one has ϱ2,h=ϱ2,h~.
As in the previous proof we treat ϱk,h:X′→Ak (resp. ϱn−2,h:X′→An−2) as κ:X→P (resp. ϱ:X→Q)) in Theorem 4.2.
By Theorem 4.2 (i) and (iii) we can find an algebraic family A3⊂G such that for a general element α3∈A3
the morphism ϱ∣α3(Y1∖E):α3(Y1∖E)→Q is injective and generates an injective map of the Zariski tangent bundles.
By (5) this is also true for the morphism ϱ∣α3(Y1):α3(Y1)→Q.
Similarly, since ϱ−1(ϱ(Z)) is of dimension dimZ+2≤dimX−dimY1 by Theorem 1.4 we can find
an algebraic family A4⊂G such that for a general element α4∈A4 the variety
α4(Y1) meet ϱ−1(ϱ(Z)) at a finite number of points which is equivalent to (4). By Remark 4.3 (2) we can suppose that
A3=A4. We need to show that the morphism ϱ∣α(Y1):α(Y1)→Q can be made proper for which
we need to change our general element h∈H.
Let H′ be the subgroup of H as in the proof of Theorem 7.1 (a). Applying the same argument as in that proof (i.e., using Proposition 4.6 (1))
we see that for a general h′∈H′ the replacement of h by h~=h′h provides us
with properness of the morphism
ϱn−k−2,h′h′∣Y1:Y1→An−k−2 as soon as n−k−2≥dimY1=1 (i.e., k≤n−3) while preserving
the injectivity of these morphism and injectivity of the induced morphism of the Zariski tangent bundles
(as well as (1), (2), and (5)). This yields (3) and, similarly, by Proposition 4.6 (3) we get (4).
Thus we have ϱn−3,h∘φ=ϱn−3,h∣Y1 which concludes Step 1.
Step 3. Let us show that replacing Y1 and Y2 with their automorphic images in X
and replacing h by another general element of H we can suppose that
ϱn−2,h∘φ=ϱn−2,h∣Y1 while keeping the morphism ϱn−2,h∣Y1:Y1→An−2 finite.
Consider the embedding An=P×Aun−2,un−1,un3⊂P×P3=:W
where P=Au1,…,un−3n−3.
Let D~=W∖An=P×P2 and for every subset Y of An denote by Y˘ the intersection of
D~ with the closure of Y in W. Note that if Y is a curve then Y˘ consists of no more than l points where l
is the number of punctures of Y (that are the points in the complement to a normalization of Y in a smooth completion of this normalization).
Furthermore, let V⊂An be given by un=0. Then V˘=P×L where L is a line in P2
and the natural morphism Y→Aun1 is finite if and only if Y˘ does not meet P×L. Consider the natural action
of H′′=SL(3,k) on Aun−2,un−1,un3 and extend this action to the action on An=An−3⊕A3 such that
it is trivial on the first summand (this enables us to treat this H′′ as a subgroup of H=SL(n,k) acting naturally on An).
Consider general elements h1′′,…,h3l′′ of H′′ where l is the same as above but for Y=Y1.
Let Vi be the zero locus of un∘ϱhi′′h. Then V˘i=P×Li where L1,…,L3l are general lines
in P2 and, hence, the intersection of any three of them is empty.
Consider F1i=Y1∩ϱn−2,hi′′h−1(ϱn−2,hi′′h(Z)) and F2i=φ(F1i). As before
replacing Y2 by its automorphic image one has F1i=F2i and φ∣F1i=id∣F1i for every i.
Furthermore, for every point in ⋃F1i we can suppose that the m-jet of φ coincides with the m-jet of
the identity map where m is as in Lemma 7.9. By Lemma 7.11 for one of the lines, say L1, the intersection
Y˘2 with P×L1 is empty and thus for un′′=un∘(h1′′h) the morphism Y2→Aun′′1 is finite.
Applying Lemma 7.9 we can replace Y1 by its automorphic image such that un′′∣Y1=un′′∘φ.
Exchanging the role of un−2 and un we can suppose that ϱn−2,h1′′h∘φ=ϱn−2,h1′′h∣Y1.
Since h1′′h is a general element of H, after the replacement of h by h1′′h we still have (1) and (5). The finiteness of the morphism
Y2→Aun−2′′1 implies
also that the morphism ϱn−2,h1′′h∣Y1:Y1→An−2 is finite which concludes Step 3.
Step 4. To finish the proof let us treat
ϱn−2,h:X′→An−2 (resp. ϱn−1,h:X′→An−1) as κ:X→P (resp. ϱ:X→Q)) in Theorem 4.2.
By Theorem 4.2 (i) and (iii) (in combination with (5)) we can find an algebraic family A5⊂G such that for a general element α5∈A5
the morphism ϱ∣α5(Y1):α5(Y1)→Q is injective and generates an injective map of the Zariski tangent bundles.
Being also finite (since κ∣Y1:Y1→P is finite and α5 is in Aut(X/P))
it is a closed embedding. Furthermore, by Theorem 1.4 we can suppose that
α5(Y1) does not meet ϱ−1(ϱ(Z)) since the sum of the dimensions of these varieties
is less than dimX. Consider the natural action
of H′′′=SL(2,k) on Aun−1,un2 and extend this action to the action on An=An−2⊕A2 such that
it is trivial on the first summand (i.e., H′′′ is again a subgroup of H acting naturally on An).
By Proposition 4.6 for a general h′′′∈H′′′ the morphism
ϱn−1,h′′′h∣α5(Y1):α5(Y1)→Q is still a closed embedding with α5(Y1) not meeting
ϱn−1,h′′′h−1(ϱn−1,h′′′h(Z)). Since h′′′ is general the same is true for the replacement of h by σ′′′h′′′h where
σ′′′∈H′′′ is up to a sign the transposition of coordinates un−1 and un.
By Lemma 7.9 (with k=n−1) replacing Y1 by its automorphic image we can suppose that for un′′′=un∘(h′′′h)
one has un′′′∣Y1=un′′′∘φ. Exchanging the role of un−1 and un
we can suppose that ϱn−1,σ′′′h′′′h∘φ=ϱn−1,σ′′′h′′′h∣Y1.
The assumptions of Lemma 7.9 are true with ϱ in that lemma equal to ϱn−1,σ′′′h′′′h.
Thus, after additional replacement of Y1 by its automorphic image we can make Y1=Y2 which is the desired conclusion.
∎
Remark 7.12**.**
(1) The assumption that Y1 and Y2 are closed in An (and not in An∖Z) cannot be dropped.
Indeed, consider Z that does not admit nontrivial automorphisms, and let L1≃A and L2≃A be disjoint curves in An
each of which meets Z at one point only. Then Y1=L1∖Z and Y2=L2∖Z are isomorphic but there is no way to extend
this isomorphism to an automorphism of An∖Z.
(2) We constructed an automorphism γ of X such that γ∣Y1=φ as a composition of elements of GZ.
However, Remark 7.10 implies that we can choose γ as a composition of elements of GZm.
More precisely, the proof of Theorem 7.1 yields the following fact.
Proposition 7.13**.**
For every m>0 the automorphism γ from Theorem 7.1
can be chosen as a composition of elements of the flows of locally nilpotent vector fields that vanish on Z with multiplicity m.
In particular, the restriction of γ to the m-th infinitesimal neighborhood Zm of Z is the identity map.
It is interesting to understand how sharp is Theorem 7.1. Therefore, we would like to pose the following.
Question. Let Z be a closed subvariety in An of codimension 2, and let Y1 and Y2 be two smooth closed
isomorphic curves in An disjoint from Z. Can one always find an automorphism of An∖Z
transforming Y1 onto Y2?
8. The case of quadrics
Notation 8.1**.**
In this section m≥6 and
X is a hypersurface in Akm that is a nonzero fiber of a non-degenerate quadratic form.
That is, when m=2n (resp. m=2n+1) we can suppose that X is given by u1v1+…+unvn=1
(resp. u02+u1v1+…+unvn=1) in a suitable coordinate system (uˉ,vˉ)=(u1,…,un,v1,…,vn)(resp. (u0,uˉ,vˉ)).
Recall that such an X is a homogeneous space of a special orthogonal group G:=SO(m,k) acting linearly on Akm.
In particular, it is flexible since homogeneous spaces of any semi-simple group are flexible [2]. Furthermore,
the coordinate system determines an embedding Akm↪Pkm such that the action of G
extends to Pkm. The closure Xˉ of X in Pkm yields a completion of X for which Xˉ∩H is a quadric
in H=Pkm∖Akm≃Pkm−1.
Lemma 8.2**.**
Let Notation 8.1 hold. Then G acts transitively on Xˉ∩H.
Proof.
Making a linear coordinate change we can suppose that the coordinate system (w1,…,wm) on Akm
is such that X is given by w12+…+wm2=1. Then Pkm is equipped with the coordinate
system (w~0:w~1:…:w~m) so that wi=w~i/w~0 for i=1,…,m
and H is given by w~0=0. Hence the equation of Xˉ∩H in H is
w~12+…+w~m2=0.
Let Zi=Xˉ∩H∩{w~i=0} and Zi′=(Xˉ∩H)∖Zi.
Note that Zi′ is isomorphic to a nonzero fiber of a non-degenerate quadratic form on Akm−1 and, therefore,
the subgroup Gi≃SO(m−1,k) of G that preserves the coordinate wi acts transitively on Zi′.
On the other hand
for a point z∈Zm at least one of the coordinates w~1:…:w~m−1 is nonzero (say w~1).
Furthermore, applying an element of Gm we can make w~2 also different from zero.
Hence the subgroup G2 can transform z into
a point of Zm′ since the action of G2 on Z2′ can switch coordinates w~1 and w~m
(and change the sign of one of them). This yields the desired statement about transitivity because
Xˉ∩H=Zm∪Zm′.
∎
The aim of this section is the following theorem.
Theorem 8.3**.**
Let Notation 8.1 hold and
φ:Y1→Y2 be an isomorphism of two closed subvarieties of X. Let ED(Yi)+dimYi≤m−2.
Then φ extends to an automorphism of X which belongs to SAut(X).
Proof.
Consider the case of m=2n, i.e., X⊂Akm is given by u1v1+…+unvn=1.
Note that for m≥6 the assumption that ED(Yi)+dimYi≤m−2 implies that
k:=dimYi+2≤n.
Let (u1,…,un,vk+1,…,vn) be a coordinate system on Akm−k and
ϱ:Akm→Akm−k be the natural projection. Suppose that
ϱˉ:Xˉ⇢Akm−k is the rational
extension of ϱ∣X. Then R(ϱˉ) as in Notation 5.1 has dimension k−1.
Note also that if Yˉj is the closure of Yj in Xˉ then dimYˉi∩H<m−k−2 since
ED(Yi)≤m−k. Hence dimYˉj∩H+dimR(ϱˉ)<dimXˉ∩H.
Lemma 8.2 implies that Corollary 5.8 is applicable and
for a general element α of some algebraic family A⊂SAut(X)
of automorphisms ϱ∣α(Yj):α(Yj)→Akm−k is a closed embedding. Furthermore,
by Theorem 1.3 we can suppose that α(Yj) does not meet the subvariety F of X
given by u2=…=uk=0 since dimYj<k−1=codimXF. Hence replacing each Yj
by its automorphic image we suppose that ϱ∣Yj:Yj→Akm−k is a closed embedding
and Yj∩F=∅. This implies that Yj′:=ϱ(Yj) is isomorphic to Yj and it
does not meet the subspace of Akm−k
given by the same system of equations u2=…=uk=0. In particular, we can treat v1∣Yj
as the lift of a function fj on Yj′ and by the Nullstellensatz one has 1−fj=∑i=2kuigj,i for some regular
functions gj,2,…,gj,k on Yj′.
Claim. For every Yj as in the statement of the Theorem there exists an automorphism of X
that sends Yj onto a subvariety of the hypersurface S in X given by v1=1 (in particular, S is isomorphic to Akm−2).
Indeed, let δi,i=2,…,k be the locally nilpotent
vector fields on X given by δi=ui∂v1∂−u1∂vi∂.
Let ψj,i be the flow of gj,iδi at time t=1. Note that the automorphism ψj,2∘…∘ψj,k
transforms Yj into a subvariety of X on which the restriction of v1 is identically 1. This concludes the proof
of the Claim.
Thus we can suppose from the beginning that Y1 and Y2 are contained in S.
By Theorem 0.1 there is an automorphism β of S that transforms Y1 into Y2.
Since X∖{v1=0}≃Ak∗×S we can suppose that β is the restriction of an
automorphism of X∖{v1=0}. Thus by Theorem 3.1 β can be extended to an automorphism
of X which concludes the case of m=2n.
If m=2n+1 then we treat (u0,u1,…,un,vk+1,…,vn) as a coordinate system on Akm−k
and consider the natural projection ϱ:Akm→Akm−k.
The rest of the proof works without change and we are done for the case of m=2n+1 as well.
∎
The assumption on ED(Yi)
implies that in the smooth case (i.e., in the case when ED(Yi)=2dimYi+1) we have the following.
Corollary 8.4**.**
Let X be a nonzero fiber of a non-degenerate quadratic form in Akm.
Suppose that φ:Y1→Y2 is an isomorphism of two closed smooth subvarieties of X
such that dimYi does not exceed 3m−1. Then φ extends to an automorphism of X.
9. Comparable morphisms
Definition 9.1**.**
(1) Let X (resp. X′) be a smooth algebraic variety and ϱ:X→X′ be
a morphism. Consider a family F of closed subvarieties of X.
Suppose that, given Y1 and Y2∈F
isomorphic over X′ and such that ϱ∣Yi:Yi→X′ is a closed embedding,
there is an automorphism α∈Aut(X/X′) of X over X′ that transforms Y1
onto Y2. In this case we say that ϱ is comparable on F.
When the ground field k is C we also say that ϱ
is holomorphically comparable on F if one can find a holomorphic automorphism of X over X′
that transforms Y1 onto Y2.
(2) Let X (resp. X′) be a smooth algebraic variety with a group G⊂Aut(X) (resp. G′⊂Aut(X′)) acting on it.
We say that a morphism ϱ:X→X′ is (G,G′)-comparable if for every g′∈G′ there exists g∈G such that ϱ∘g=g′∘ϱ.
The next fact follows from Definition 9.1.
Proposition 9.2**.**
Let ϱ:X→X′ be (G,G′)-comparable. Let Y1 and Y2 be closed subvarieties of X and φ:Y1→Y2
be an isomorphism. Suppose that each morphism ϱ∣Yi:Yi→X′ is a closed embedding, Yi′=ϱ(Yi), and φ′:Y1′→Y2′ is the isomorphism for
which φ′∘ϱ∣Y1=ϱ∣Y2∘φ. Let φ′ extend to an automorphism g′ of X′ which is an element of G′. Then there exists an element g∈G such that g(Y1) is naturally isomorphic
to Y2 over X′.
Furthermore, if ϱ is also comparable (resp. holomorphically comparable) on a family F containing Y1 and Y2
then there exists an algebraic (resp. holomorphic) automorphism α of X
for which α∣Y1=φ.
Notation 9.3**.**
Given a smooth algebraic variety X and an algebraic group H consider the sheaf
generated by the presheaf whose elements are morphisms from Zariski open subsets of X to H.
Denote by H1(X,H)
the first Cˇech cohomology of X with coefficient in this sheaf. That is, for a Zariski open cover {Ui} of X a one-cocycle
associated with this cover is given by morphisms ψij:Ui∩Uj→H.
Proposition 9.4**.**
Let H be an algebraic group and H′ be a closed algebraic subgroup of H.
Suppose that H acts freely on a smooth algebraic variety X~ so that the quotient map
ϱ~:X~→X~/H=:X′ (resp. τ:X~→X~/H′=:X) is a principal H-bundle (resp. H′-bundle)
over a smooth algebraic variety (and, hence,
for a Zariski locally trivial fiber bundle
ϱ:X→X′ one has ϱ~=ϱ∘τ).
Let h.x be the natural action of h∈H on x∈X and φ:Y1→Y2 be an isomorphism of closed subvarieties of X over X′ such that ϱ∣Yk:Yk→X′ is
a closed embedding. Suppose that Y′=ϱ(Yk) and the group H1(Y′,H′) is trivial (which is
the case when H′ itself is trivial). Then there exists a morphism θ:Y′→H such that
(a)* φ(y1)=θ(ϱ(y1)).y1 for every y1∈Y1 and*
(b)* φ extends to an automorphism of X over X′ if θ extends
to a morphism X′→H.*
Furthermore, if the ground field k is C then
(c)* φ extends to a holomorphic automorphism of X over X′ if θ extends
to a holomorphic map X′→H.*
Proof.
Let y2=φ(y1), y′=ϱ(yk) and y~k∈τ−1(yk). Since y1 and y2 are in ϱ−1(y′)≃H/H′
we have an element h∈H for which y2=h.y1 where h is determined up to an element of H′. However, h becomes unique
if we require additionally that h transfer y~1 to y~2. Since τ:X~→X is a principal H′-bundle
we can choose local sections of Y1 (resp. Y2) in X~ which determines a local choice of y~k as above.
Hence, for a Zariski open cover {Ui} of Y′ we can choose morphisms ψi:Ui→H for which
one has ψi(y′).y1=y2 as soon as y′∈Ui. If y′∈Ui∩Uj then letting ψij(y′)=ψj(y′)∘ψi(y′)−1
we get an element of H1(Y′,H′). Since the latter group is trivial we can construct θ:Y′→H as in (a).
If θ extends to a morphism (reps. holomorphic map) Θ:X′→H then the desired extension of φ
in (b) (resp. (c)) is given by Θ(ϱ(x)).x for x∈X.
∎
Corollary 9.5**.**
Let ϱ:X→X′, H and H′ be as in Proposition 9.4 and X′ be quasi-affine.
(i)* Suppose that either H′ is trivial or H′≃SL(m,k) for some m>0.
Let Faff be the family of closed subvarieties of X isomorphic to Akk
with k=1,2,….
Then ϱ is comparable on Faff.*
(ii)* Suppose that H′ is trivial and H is isomorphic as an algebraic variety to Akn.
Consider the family F of closed subvarieties Z of X such that
ϱ(Z) is closed in an affine variety X′′ containing X′ as an open subvariety. Then ϱ is comparable on F.*
Proof.
Let Y1 and Y2∈Faff be as in Definition 9.1 (1), i.e., Y′=ϱ(Yi)≃Akk is closed in X′.
Note that by the Quillen-Suslin theorem [29], [34] the group H1(Y′,H′) is trivial even in the case of H′≃SL(m,k)161616Formally, the Quillen-Suslin theorem implies the group H1(Akn,GL(m,k)) is trivial. However, each Čech
cocycle on Akn with coefficients in GL(m,k) differs from a cocycle with coefficients in SL(m,k) by multiplication of, say, the first
coordinate of Akn by invertible regular functions. Since the first cohomology of Akn with coefficients in invertible regular functions is trivial
by the Serre Theorem B we see that H1(Akn,SL(m,k)) is also trivial.,
i.e., we are under the assumption of Proposition 9.4.
Consider a morphism g:X′→Akk whose restriction to Y′ is the identity map (such a g exists since X′
is quasi-affine). Let θ be as in Proposition 9.4. Then θ∘g yields an extension of θ to X′
which by Proposition 9.4 concludes (i).
In (ii) note that since Z′:=ϱ(Z),Z∈F is closed in the affine variety X′′ this morphism θ extends to
a morphism Θ:X′′→Akn≃H and again Proposition 9.4 yields (ii).
∎
Remark 9.6**.**
Let ϱ:X→X′ be an affine morphism of smooth varieties
and G⊂Aut(X) (resp. G′⊂Aut(X′)).
(1) Note that if ϱ is (G,G′)-comparable
and G′ acts transitively on X′ then all fibers of ϱ are isomorphic and, furthermore, ϱ:X→X′
is a locally trivial fiber bundle
in étale topology (it follows from the fact that any affine morphism W→V of algebraic varieties with pairwise
isomorphic general fibers is locally trivial over an étale neighborhood of a general point of V [23]).
(2) Let G (resp. G′) be generated by a set N (resp. N′) of complete algebraic vector fields on X (resp. X′).
In order for ϱ to be (G,G′)-comparable it suffices to require that for every δ′∈N′ there exists δ∈N such that for every x∈X and x′=ϱ(x)
one has
[TABLE]
where δx (resp. δx′′) is the value of the field δ at x (resp. δx′′ at x′).
Definition 9.7**.**
We call a pair δ,δ′ of vector fields (on X and X′ respectively) comparable if they satisfy Formula (18).
Similarly, we call the pair (N,N′) from Remark 9.6 (2) comparable
if for every δ′∈N′ there exists δ∈N so that the pair δ,δ′ is comparable.
Example 9.8**.**
Let X=SL(n,k), thus dimX=n2−1.
Let A=[ai,j]i,j=1n be a matrix from SL(n,k)
and let k≤n−1 and m≤n. Consider the (m×k)-matrix A′ obtained from A by removing all rows starting with (m+1)-st and all columns
starting with (k+1)-st.
Then one has the natural morphism of ϱ:X→X′,A→A′ into the space X′≃Akkm of (k×m)-matrices.
Let 1≤i=j≤n and
[TABLE]
i.e., δij is a locally nilpotent vector field on X whose flow is the addition of multiples of the i-th column in A to the j-th column.
Note that
[TABLE]
is a locally nilpotent vector field on X′ for which Formula (18) is valid.
Similarly, the locally nilpotent vector fields
σij=∑l=1nai,laj,l∂ and σij′=∑l=1kai,laj,l∂ on X and X′ respectively also satisfy
Formula (18).
Notation 9.9**.**
Suppose that ϱ:X→X′ is a smooth morphism of smooth algebraic varieties.
Let N (resp. N′) be a set of locally nilpotent vector fields on X (resp. X′) such that the pair (N,N′) is comparable.
Furthermore, we suppose that X is a closed subvariety of X′×Akm with ϱ being the restriction of the natural projection ϱ^:X′×Akm→X′.
That is, for every δ and δ′ satisfying Formula (18) we can write δ=δ′+δ′′ where δ′′ is tangent to each fiber ϱ^−1(x′)≃Akm of ϱ^
(where by abuse of notation we identify δ′ with its natural lift to X′×Akm).
Recall that a vector field on Akm (with a fixed coordinate system uˉ=(u1,…,um))
is a linear vector field if it is of the form x˙=Ax+b where x and b are vectors in Akm and A is a square (m×m)-matrix (i.e., it is a non-homogeneous system of linear differential equations).
Proposition 9.10**.**
Let Notation 9.9 hold. Suppose that for every comparable pair (δ,δ′)∈N×N′
the following condition is true:
(A)* the restriction of the field δ′′=δ−δ′ to every fiber ϱ^−1(x′)≃Akm is a linear vector field (depending on x′∈X′).*
Let N~′ be the smallest saturated set of locally nilpotent vector fields on X′
such that it contains N′. Suppose that L′ is the Lie algebra generated by the fields of the form
aδ′ where δ′∈N~′ and a∈k[X′].
Then there exists a Lie algebra L of vector fields on X such that for every complete
(resp. locally nilpotent) vector field σ′∈L′ there exists a complete (resp. locally nilpotent)
vector field σ∈L such that the pair (σ,σ′) is comparable.
Proof.
Note that for every a∈k[X′] and every comparable
pair (aδ,aδ′) is comparable and satisfies condition (A).
Furthermore, let (δi,δi′),i=1,2,…s be a collection of comparable pairs satisfying condition (A) and let ℓ be a linear form in s variables.
Let κ′=ℓ(δ1′,…,δs′) (resp. κ=ℓ(δ1,…,δs)). Then the pair (κ,κ′) is also comparable and satisfies
condition (A).
For the Lie brackets δ0=[δ1,δ2] and δ0′=[δ1′,δ2′] the pair (δ0,δ0′) is, of course, comparable. Let us check condition (A) for this pair.
Note that
[TABLE]
where the first term belongs to L′ and the restriction of the second one to every fiber of ϱ^ is a linear vector field. Consider, say, the third term [δ1′,δ2′′].
Since the restriction of δ2′′ to ϱ^−1(x′)≃Akm is of the form x˙=Ax+b we see that
[δ1′,δ2′′] is of the form δ1′(A)x+δ1′(b) and, in particular, it
is again a vector field whose restriction to ϱ^−1(x′) is linear. Thus the pair (δ0,δ0′) satisfies condition (A).
Now, a comparable pair (δ,δ′) of complete vector fields induces the
flows φt:X→X and φt′:X′→X′ (where t∈k) so that ϱ∘φt=φt′∘ϱ.
Hence φt tranforms a fiber ϱ−1(x0′)≃Akm to the fiber ϱ−1(φt′(x0′))≃Akm.
When condition (A) is satisfied this map of fibers is an element of the flow of a linear non-autonomous vector field
which is automatically an affine map. Hence for every comparable pair
(σ,σ′) of complete vector fields the conjugation by φt and φt′ yields a comparable pair
(σ~,σ~′) of complete vector fields such that condition (A) is satisfied.
Applying these operations of taking Lie brackets, conjugations, and linear combinations we construct
the desired saturated set N~′ so that the pair (N~,N~′) is comparable for some set N~ of locally nilpotent vector fields on X.
Let L′ (resp. L) be the Lie algebra generated by the fields of the form
aδ′ where δ′∈N~′ and a∈k[X′] (resp. aδ where δ∈N~ and a∈k[X′]⊂k[X]).
By construction for every
element σ′∈L′ there is an element of σ∈L so that the pair (σ,σ′) is also comparable and satisfies condition (A).
Suppose now that σ′ is complete and O′ is an integral curve of this field (in particular, O′ is isomorphic to either Ak or Ak∗).
To show that σ is complete
it suffices to prove that O′ admits a lift to an integral curve O⊂ϱ^−1(O′)≃O′×Akm
of the field σ=σ′+σ′′. The restriction of σ′′ to O′×Akm is of the form x˙=Ax+b where the matrix A and the vector b
depend on the parameter t∈O′. That is, we are dealing with a non-autonomous system of linear equations.
Such a system has a solution for all values of t which yields the desired lift of O′ to an integral curve of σ.
If σ′ is locally nilpotent then in order to show that σ=σ′+σ′′ is locally nilpotent one needs to prove
that for every b∈k[X] there exists n for which σn(b)=0. This is true when b∈k[X′]⊂k[X]
since in this case σ(b)=σ′(b). Since k[X] is generated over k[X′] by the coordinate functions on Akm
it suffices to prove that if b is a coordinate on Akm then σn(b)=0 for n>>0.
Note that in this case σ(b)=σ′′(b) and by condition (A) one has σ′′(b)∈k[X′]. This yields the desired conclusion.
∎
10. Holomorphic extension of θ
In this section the ground field k is C and its aim is to describe conditions under which θ:Y′→H from Proposition 9.4
admits a holomorphic extension Θ:X′→H.
Proposition 10.1**.**
Let the assumption of Proposition 9.4 hold and X′ be Stein.
Suppose that the map θ is homotopy equivalent to a constant map
from Y′ to H. Then θ admits a holomorphic extension Θ:X′→H.
Proof.
By the Oka-Grauert principle [11, Theorem 5.4.4] it suffices to construct a continuous extension of θ.
Let θ~(t):Y→H,t∈[0,1] be a homotopy of θ to a constant map, i.e., θ~(∗,0)=θ and
θ(∗,1)
sends Y′ to a point h0∈H.
The argument is rather simple in the case of a smooth Y′ since
one can consider its closed tubular neighborhood U. Note that U admits a continuous map g:U→[0,1]
such that g−1(0)=Y′ and g−1(1) is the boundary ∂U of U. Define a continuous extension θ′:U→H
of θ via the formula θ′(u)=θ~(p(u),g(u)) where u∈U and p:U→Y′ is
the natural projection. Then we can extend θ′ further to θ′′:X′→H
by putting Θ(x′)=h0 for x′∈X′∖U.
In the general case we recall that complex algebraic varieties admit triangulation [24].
That is, one can view Y′ as a subcomplex in a complex X′. Then the second baricentric derived neighborhood
U of Y′ is regular (e.g., see [17]). This means, that Y′ is a deformation retract of U and
the map U→Y′ (which is identity on Y′)
can be constructed via a sequence of collapses Ui−1↘Ui,i=1,…,k where each Ui is a subcomplex of
the second baricentric subdivision, U0=U, Uk coincides with the second baricentric subdivision of Y′,
and Ui−1∖Ui={σi,τi) where
σi is an l-dimensional simplex
and τi is the only (l−1)-dimensional face of σi that is not contained
in Ui (viewed as a complex). In particular, the union ⋃i=1kτi
contains the boundary of U0. Let us establish the following.
Claim. A continuous map
θi:Ui→H, admitting a homotopy θ~i to the constant map h0 and such that θ(∂τi)=h0, can be extended to a similar continuous map
θi−1:Ui−1→H with a homotopy θ~i−1 to the constant map h0 such that θ~i−1(τi)=h0.
Let ∂σi (resp. ∂τi) be the boundary of σi (resp. τi).
Then there is a homotopy g:σi×[0,1]→σi such
that g(∗,1)=id∣σi, Img(∗,0)=∂σi∩Ui−1, and
for every t∈[0,1] the restriction g(∗,t)∣∂σi∩Ui−1 is the identity
map on ∂σi∩Ui−1.
In particular,
each collapse Ui−1↘Ui
induces a strong deformation retract fi:Ui−1→Ui.
Furthermore, g can be chosen so that it yields
a natural homeomorphism between σi∖∂τi and τ˙i×[0,1]
where τi˙=τi∖∂τi is the interior of τi.
171717In order to see this, treat σi as
a closed l-dimensional ball, τi as the upper (l−1)-dimensional semi-sphere in its boundary,
∂σi∩Ui−1 as the lower semi-sphere and ∂τi as its equator.
Hence we can now define the extension θi−1 of θi so that every u∈τ˙i and t∈[0,1]
one has θi−1(g(u,t))=θ~i(fi(u),t). By construction θi−1(τi)=h0 and the extension is
homotopic to a constant map (indeed, define the homotopy θ~i−1(g(u,s)) on Ui−1∖∂τi via
θ~i(fi(u),t+s(1−t))). This yields the the Claim.
It remains to show that for such extensions one has θ~i∣∂U=h0 as soon as Ui∩∂U=∅. First
note that if σi is one-dimensional and meets Y′ then τi is a singleton (i.e., ∂τi=∅) and, therefore, θ~i(τi)=h0.
Let S be the collection of σi such that each of them has a face contained in Ui. For every σi∈S
there is the only one vertex not in Ui. Suppose that T is the set of such vertices. Since each of σi∈S collapses before
the one-dimensional simplexes mentioned above we can suppose that
θ~i∣T=h0. Let Sm be the collection of m-dimensional simplexes σj∈/S that have faces in ⋃σi∈Sσi.
For every σj∈S2 we have a one-dimensional τj such that ∂τj∈T. Thus by the Claim
we can suppose that the restriction of θ~j to ⋃σj∈S2τj is the constant map to h0.
Similarly, for every σk∈S3 we have ∂τk in ⋃σj∈S2τj. Hence, proceeding by induction
we see that the restriction of every θ~i to ⋃j≥2⋃σk∈Sjτk is h0.
Consequently, for every σi∈/S∪⋃j≥2Sj we have θ~i(τi)=h0.
Since we deal with the second baricentric subdivision we have ⋃σi∈Sτi∩∂U=∅ which
yields the desired conlusion.
∎
Lemma 10.2**.**
Let m≥2 and Z be a complex affine algebraic variety such that Hi(Z)=0 for
i≥m+1. Then Z can be embedded into a contractible topological space Z^
such that Z^∖Z is a union of components each of which is homeomorphic to a ball B
whose real dimension is between 2 and m+2.
Proof.
Since any algebraic variety is a finite CW-complex, Z has a finitely generated fundamental group.
Gluing Z with discs along generators of this group we obtain a topological space Z1⊃Z such that
Z1 is simply connected. Furthermore, the Mayer-Vietoris sequence implies that the embedding
Z↪Z1 induces an isomorphism Hi(Z)≃Hi(Z1) for i≥3.
By the Hurewicz theorem π2(Z1) is naturally isomorphic
to H2(Z1). Thus, we can choose a nontrivial element ω of H2(Z1) presented by
a two-dimensional cell in the CW-complex which is homeomorphic to a two-sphere.
Glue Z1 with a three-dimensional
ball along this sphere and let V1=Z1∪B1 be the resulting simply connected topological space.
Consider the Mayer-Vietoris sequence
[TABLE]
Note that if ω generates a free group then the map H2(Z1∩B1)→H2(Z1) is a monomorpism and,
therefore, d3 sends H3(V1) to zero. That is, after such a gluing Hi(V1)=Hi(Z1) for i≥3
while the rank of H2(V1) is less than the rank of H2(Z1) since ω induces the zero element in H2(V1).
Continuing this procedure we can embed Z1 into a simply connected V2 such that Hi(V2)=Hi(Z1) for i≥3
while H2(V2) is finite. Let V3 be the result of gluing a three-ball to V2 along ω as before and
V3=V2∪B2 be the resulting simply connected topological space.
In this case ω is of finite order l and the kernel of H2(V2∩B2)→H2(V2) in the associated
Mayer-Vietoris sequence is isomorphic
to lZ⊂Z≃H2(V2∩B2). In particular, the image of the
map Hk+1(V3)→Hk(V2∩B2)≃Z is lZ. Since the latter is a free Z-module the map Hk+1(V3)→lZ
has a right inverse. Thus it follows from the Mayer-Vietoris sequence
that H3(V3) is naturally isomorphic to H3(V2)⊕Z while Hi(V3)=Hi(V2) for i≥4.
Since the number of elements in H2(V3) is less than the number of elements in H2(V2) continuing this
procedure we embed Z1 into a simply connected Z2 such that H2(Z2)=0,
Hi(Z2)=Hi(Z1)=Hi(Z) for i≥4, and H3(Z2) is naturally isomorphic
to the direct sum of H3(Z)
and a finitely generated free Z-module.
In the same manner we can embed Z into a simply connected Zm such that Hi(Zm)=0 for i≤m,
Hi(Zm)=Hi(Z)=0 for i≥m+2, and Hm+1(Zm) is naturally isomorphic to the direct sum of Hm+1(Z)
and a finitely generated free Z-module. That is, Hm+1(Zm) is a finitely generated free Z-module
since Hm+1(Z)=0. Gluing Zm with balls of real dimension m+2 we can obtain
a simply connected Z^ with Hi(Z^)=0 for i≤m+1. Furthermore, as we saw before,
since Hm+1(Zm) is free,
this procedure does not affect the i-homology groups with i≥m+2.
Hence Hi(Z^)=0 for i≥1 and
by the Hurewicz theorem πi(Z^)=0 for i≥1. By the Whitehead theorem Z^ is contractible which
yields the desired conclusion.
∎
Remark 10.3**.**
It follows from the proof that if in Lemma 10.2 the group Hm(Z) is free then in order to construct Z^
one needs only balls of real dimension at most m+1.
Proposition 10.4**.**
Let ϱ:X→X′ be a principal bundle for some algebraic group H and
Y′ be a closed subvariety of X′.
Let X′ be Stein and H
simply connected with Hi(H)=0 for i≤m where m≥2.
Suppose also that Hi(Y′)=0 for i≥m+1 and
Hm(Y′) is free. Then any morphism θ:Y′→H admits a holomorphic extension Θ:X′→H.
Proof.
By Lemma 10.2 we can embed Y′ into a contractible topological space Y.
Furthermore, by Remark 10.3 we can suppose that each component of Y∖Y′
is a ball B whose real dimension k is at most m+1
and whose boundary is an image of a (k−1)-dimensional sphere
in Y′. Composition with θ yields a continuous map of this sphere to H.
Since πk−1(H)=0 we can extend any continuous map of
a (k−1)-dimensional sphere in H to a map from a closed k-ball. Hence
we can extend the morphism θ:Y′→H to a continuous
map θ^:Y→H. Let ψ:Y×[0,1]→Y be a contraction of Y, i.e., ψ(∗,0)=id∣Y
and ψ(∗,1)=yo∈Y. Then θ~=θ^∘ψ is a homotopy of θ to
a constant map. Hence we are done by Proposition 10.1.
∎
11. The case of SL(n,k)
Notation 11.1**.**
Let us fix notation for the rest of the paper. From now on X will be always an affine algebraic variety isomorphic to SL(n,k) with n≥3, i.e., dimX=n2−1.
We treat points in X as matrices A=[ai,j]i,j=1n from SL(n,k) and denote by {Aij} the cofactors of this matrix.
For m≤n let Pm be the space of (m×n)-matrices,
Pm0 be its subvariety consisting of matrices of rank m and
ϱm:X→Pm be the natural projection that sends A to the matrix consisting of the first m rows of A.
Consider also the natural projection ϱ:X→Q where Q is the quadric given by the equation
[TABLE]
in the affine space Ak2n equipped with coordinates (a1,1,…,a1,n,A1,1,…,A1,n).
Suppose that
the group H=SL(n,k)×SL(n,k) acts on X so that (B,C)∈SL(n,k)×SL(n,k)
sends A∈X to BAC. Note that H contains the subgroup Hm=SL(m,k)×SL(n,k)
that acts naturally on Pm, while H1 acts naturally on Q.
Under these action the morphism ϱm (resp. ϱ) is Hm-equivariant (resp. H1-equivariant).
Lemma 11.2**.**
Let Notation 11.1 hold. Then the Hm-action (resp. H1-action) on Pm0 (resp. Q)
is transitive.
Proof.
Since any finite sequence of column and row operations on a matrix D∈Pm coincides with
the action of some element of Hm on this matrix we see that for D∈Pm0
there is an element h∈Hm for which h.D=[di,j]
where di,i=1 for every i=1,…,m, while di,j=0 for i=j. This yields the transitivity
of the Hm-action on Pm0.
Similarly, an elementary column operation induced by an element of H1 yields an automorphism
of Q such that
[TABLE]
(where t∈k is a multiple),
while it keeps the rest of coordinates the same.
Since a1,1,…,a1,n cannot vanish simultaneously we can send an arbitrary point q in Q via such automorphism to a point q1 with
[TABLE]
In this case A1,1=1 because of Formula (19).
Let us use now the similar automorphisms which send a1,1→a1,1+ta1,j
and A1,j→A1,j−tA1,1, while preserving the rest of coordinates on Q.
Then we can send q1 to a point q0 with A1,2=…=A1,n=0,
while keeping Formula (21) valid.
That is, we can send an arbitrary point q to this given point q0 which yields the transitivity of the H1-action on Q
and the desired conclusion. ∎
Notation 11.3**.**
Let Ik be the identity (k×k)-matrix for k≥1. Consider the following subgroups of block matrices
in SL(n,k).
[TABLE]
where M is the set of (n−m)×m matrices.
Then we have the subgroup Hn−m′=Fn−m′×In of H acting on X and the subgroup
H′′=F′′×In acting on X.
Proposition 11.4**.**
Let Notations 11.1 and 11.3 hold and n−m≥2.
Then
(i)* ϱm:X→Pm0 is a principal Hn−m′-bundle over Pm0;*
(ii)* ϱ:X→Q is a principal H′′-bundle over Q.*
Proof.
Note that the action of Hn−m′ on X is free and it preserves every fiber of the morphism ϱm.
Therefore, the fibers of ϱm are of dimension at least
dimHn−m′=m(n−m)+(n−m)2−1=n(n−m)−1. Furthermore, the action of Hm transforms each orbit of Hn−m′ into
another orbit.
Hence, the fibers of ϱm are of the same dimension because ϱm
is Hm-equivariant and the Hm-action on Pm0 is transitive.
Since dimPm0=nm and dimX=n2−1,
observing the equality and n2−1=nm+n(n−m)−1 we see that the fibers of
ϱm are nothing but the orbits of Hn−m′′ which shows that
ϱm:X→Pm0 is a principal Hn−m′-bundle. Thus we have (i).
Similarly, the H′′-action on Q is free, it preserves every fiber of the morphism ϱ and it commutes with the H1-action from Notation 11.1.
Hence, the fibers of ϱ are of dimension at least
dimSL(n−1,k)=(n−1)2−1. All these fibers are again of the same dimension because ϱ
is H1-equivariant and the H1-action on Q is transitive by Lemma 11.2.
On the other hand, the dimension of Q is 2n−1.
Observing the equality dimSL(n,k)=n2−1=(n−1)2−1+(2n−1) we conclude that the dimension of each fiber is (n−1)2−1.
That is, the fibers of ϱ
are nothing but the orbits of the H′′-action which yields (ii) and the desired conclusion.
∎
Notation 11.5**.**
(1) Let N={δi,j∣1≤i=j≤n}⋃{σi,j∣1≤i=j≤n} be the set of locally nilpotent vector fields on X given by
[TABLE]
Nm′={δi,j′∣1≤i=j≤n}⋃{σi,j′∣1≤i=j≤m} be the set of locally nilpotent vector fields on Pm given by
[TABLE]
and N′′={δi,j′′∣1≤i=j≤n} be the set of locally nilpotent vector fields on Q given by
[TABLE]
(2) Let N~m′ (resp. N~′′) be the smallest saturated set of locally nilpotent vector fields on Pm containing N′
(resp. on Q containing N′′).
In particular, N~m′ generates a group Gm′ of automorphisms of of Pm and N~′′ generates a
group G′′ of automorphisms of Q.
Lemma 11.6**.**
Let Notation 11.5 hold, n≥3 and 1≤m≤n−2.
(1)* The pair (N,Nm′) (resp. (N,N′′)) is comparable
for the morphism ϱm (resp. ϱ).*
(2)* Every element of (N,Nm′) (resp. (N,N′′)) satisfies Condition (A) in
Proposition 9.10.*
(3)* The variety Pm0 is Gm′-flexible and the variety Q is G′′-flexible.*
(4)* Let Lm′ be the Lie algebra of vector fields on Pm generated
by all fields of the form a′δ′ where δ′∈N~m′ and degδ′a′≤1.
Then Lm′ contains the space of all algebraic vector fields on Pm that vanish at Z=Pm∖Pm0
with some multiplicity s>0.*
(5)* Let L′′ be the Lie algebra of vector fields on Q generated
by all fields of the form a′′δ′′ where δ′′∈N~′′ and degδ′′a′′≤1.
Then L′′ coincides with the space of all algebraic vector fields on Q.*
Proof.
The first and second statements follow from Notation 11.5 and Definition 9.7.
For (3) note that all elementary row and column operations on Pm can be viewed as elements of flows
of fields from Nm′. Since such operations generate the action of the group Hm we see that Gm′ contains
Hm. By the similar reason G′′ contains H′′. Since by Lemma 11.2 Hm acts transitively on Pm0 (resp.
H1 acts transitively on Q) we have (3) by Theorem 2.6.
The vector fields δ1,2′ and δ3,2′ commute,
i.e., we have a compatible pair of locally nilpotent vector fields in N~m′. Since Pm0=Pm∖Z
is Gm′-flexible Theorem 2.15 implies that
the Lie algebra Lm′ contains all algebraic vector fields AVF(Pm) on Pm
vanishing on Z with some multiplicity s>0, i.e., we have (4).
The same argument and the commutativity of δ1,2′′ and δ3,2′′ imply (5).
∎
Proposition 11.7**.**
Let G~=SAut(X), G~′′=SAut(Q) and let
G~m′=SAutZs(Pm) be the subgroup of SAut(Pm) generated by the elements of the flows of locally nilpotent vector fields
on Pm whose restriction to the s-infinitesimal neighborhood Zs is zero, where s is as in Lemma 11.6.
Then ϱm:X→Pm0 is a (G~,G~m′)-comparable morphism and ϱ:X→Q
is a (G~,G~′′)-comparable morphism.
Proof.
By Lemma 11.6 (4) the group G~m′ is generated by elements of the flows of locally nilpotent
vector fields from Lm′. Since Conditions (A) is satisfied by Lemma 11.6 (2) we see that
ϱm is (G~,G~m′)-comparable by Proposition 9.10. Similarly, by Lemma 11.6
the group G~′′ is generated by elements of the flows of locally nilpotent vector fields from L′′ and
and the same reasoning yields the second statement.
∎
Lemma 11.8**.**
Let Notation 11.1 hold. Then the codimension of the subvariety Z=Pm∖Pm0 in Pm is n−m+1.
Proof.
A matrix A∈Pm does not belong to Pm0 if and only if its rank k is at most m−1.
Applying some element of Hm to A=[ai,j] with k≤m−1 we can suppose that A=[aij] has
ai,j=1 where i=j and i≤k, while ai,j=0 in all other cases. Note that the isotropy subgroup F of A in Hm
consists of all elements (B,C) such that B is a block (m×m)-matrix of the form
[TABLE]
and C is a block (n×n)-matrix of the form
[TABLE]
where A′∈GL(k,k), D′∈GL(m−k,k), E′∈GL(n−k,k) with detD′=detA′1 and detE′=detA′, while B′ (resp. C′) is an aribrary
matrix of size k×(m−k) (resp. (n−k)×k). Hence, the dimension of F
is k(m−k)+(m−k)2+(n−k)2+k(n−k)+k2−1. Since the subvariety of matrices of rank k in Pm is isomorphic to Hm/F
we see that its dimension is dimHm−dimF=m2−1+n2−1−(k(m−k)+(m−k)2+(n−k)2+k(n−k)+k2−1)=m(m−k)+k(n−k)−k2−1. For k running from 1 to m−1
the maximum of latter expression is achieved for k=m−1 and it is equal to
m+(m−1)(n−m+1)−(m−1)2−1=(n+1)(m−1)=nm−n+m−1. Since dimPm=nm we see that the codimension of Z in Pm is n−m+1 which is
the desired conclusion.
∎
Lemma 11.9**.**
Let Notation 11.1 hold, 0<m<n and
Y be a closed subvariety of X of dimension at most m.
Suppose that H˘≃SL(n,k) acts on X via left multiplications.
Then for a general element h∈H˘ the morphism ϱm∣h(Y):h(Y)→Pm is proper.
Proof.
Suppose that X=Akn2 is equipped with the coordinates system (a1,1,…,an,n) and, thus, X contains X
as a closed subvariety.
Present X as X=∏i=1nXi0 where Xi0=Akn has the coordinate system (a1,i,…,an,i).
Let Xˉi≃Pn be a completion of Xi0, i.e., Xˉi=Xi0⊔Xi1 (where Xi1≃Pn−1)
and Xˉ=∏i=1nXˉi is a completion of X.
Note that the H˘-action on X extends naturally to an H˘-action on X (resp. Xˉ) and we have also
the transitive H˘-action on each Xi0 with an extension to an action on Xˉi for which Xi1 is an H˘-orbit and
the natural projection X→Xi (resp. ψi:Xˉ→Xˉi) is H˘-equivariant. Let J be the set {0,1}n without
the element (0,…,0). Note that the boundary Xˉ=Xˉ∖X can be presented as
⋃jˉ∈JCjˉ where Cjˉ=∏i=1nXiji and jˉ=(j1,…,jn)∈J.
In particular, for the closure Yˉ of Y in Xˉ one has Yˉ∖Y=⋃Y˘jˉ where Y˘jˉ=(Yˉ∖Y)∩Cjˉ. Let, say, j1=1 for some jˉ′∈J and, hence, ψ1(Cjˉ′)⊂Xˉ11.
Since Yˉ∖Y is of dimension m−1 we see that ψ1(Y˘jˉ′) is of dimension at most m−1.
Note that the set R(a1,1,…,am,1) of common indeterminacy points of the functions a1,1,…,am,1 on Xˉ1
is a subset of X11 of codimension m. By Theorem 1.3 for a general h∈H˘ the intersection
of h(ψ1(Y˘jˉ′)) with R(a1,1,…,am,1) is empty. Since ψ1 is H-equivariant we have
h(Y˘jˉ′)∩ψ1−1(R(a11,…,am,1))=∅ where ψ1−1(R(a1,1,…,am,1))
is the intersection of Cjˉ′ with the set of common indeterminacy points of a11,…,am,1 in Xˉ.
Note that the similar
claims are true for all jˉ∈J. Hence, for a general h∈H˘ the variety h(Yˉ) does not meet
the set of common indeterminacy points of the functions a1,1,…,am1,a1,2,…,am,n.
Furthermore, the boundary X~∖X does not contain points where these functions are regular
and take finite values.
Therefore, the desired conclusion follows now from Proposition 5.2.
∎
Proposition 11.10**.**
Let the assumptions of Lemma 11.9 hold and ED(Y)≤m where m≤n−2. Then
for some element β∈SAut(X) the morphism ϱm∣β(Y):β(Y)→Pm is a closed embedding.
Proof.
By Proposition 11.4 we have X/Hn−m′≃Pm0.
Note that the Hn−m′-action on X is generated by elements of the flows of some set S of locally nilpotent vector fields
on X. Let S~ be the smallest saturated set of locally nilpotent
vector fields on X containing S and tangent to the fibers of ϱm.
Then S~ generates a group F of automorphisms of X over Pm such that it contains Hn−m′
and every fiber of ϱm is F-flexible. By Theorem 4.2 (iv)
there exists an algebraic family A⊂F such that for a general element α∈A
the morphism ϱm∘α:Y→Pm is injective and it induces an injective map
of tangent bundles. By Proposition 4.6 the same is true for a morphism (ϱm∘h)∘α:Y→Pm
where h is a general element of H˘. By Lemma 11.9 the latter morphism is proper. Thus, letting β=h∘α we get
the desired conclusion.
∎
Theorem 11.11**.**
Let X=SL(n,C) and
φ:Y1→Y2 be an isomorphism of two closed subvarieties of X such that either
(i)* ED(Yi)+dimYi≤n−2, Hi(Y1)=0 for i≥3
and H2(Y1) is a free abelian group; or*
(ii)* dimY1 is a curve and ED(Yi)≤n−2, or;*
(iii)* Y1 is a once-punctured curve and ED(Y1)≤2n−3.*
Then there exists a holomorphic automorphism β of X such that β∣Y1=φ.
Proof.
Let m=ED(Yi). By
Proposition 11.10 we can suppose that ϱm∣Yi:Yi→Pm is a closed embedding.
Let φ′:Y1′→Y2′ be the isomorphism
induced by φ, where Yi′=ϱm(Yi).
By Lemma 11.8 one has dimZ=nm−(n−m+1), where Z=Pm∖Pm0. Therefore,
the assumption m+dimYi≤n−2 implies that
dimYi′+dimZ≤dimPm−3 in case (i).
Hence, by Theorem 7.1 (a) and Proposition 7.13 φ′ can be extended
to an automorphism α′∈SAutZs(Pm) for any s>1. By Proposition 11.7 s can be chosen so that
ϱm is (SAut(X),SAutZs(Pm))-comparable and, hence, by Proposition 9.2
we can suppose that Y1′=Y2′ and φ′ is the identity map.
By Proposition 9.2 it suffices to establish now that ϱm is holomorphically comparable
on a family of algebraic varieties containing Yi.
By Proposition 11.4 ϱm is a principal Hn−m′-bundle.
Consider the morphism θ:Y1′→Hn−m′ as in Proposition 9.4 (with H and Y′ replaced by
Hn−m′ and Y1′).
In order to
prove holomorphic comparability it suffices to show that θ extends to a holomorphic map Θ:Pm→Hn−m′.
Note that as an affine variety Hn−m′ is isomorphic to the direct product of Cm(n−m) and SL(n−m,C).
Hence π1(Hn−m′)=π2(Hn−m′)=0 and the existence of an extension Θ is provided by Proposition 10.4
which yields the desired conclusion in (i).
For (ii) exactly the same argument works
with Theorem 7.1 (a) replaced by Theorem 7.1 (b).
For (iii) we recall that ϱ:X→Q is a principal H′′-bundle by Proposition 11.4.
Arguing as in the proof of Proposition 11.10 we can find a subgroup F⊂SAut(X/Q)
containing H′′ such that every fiber of ϱ is F-flexible. Hence, by Corollary 4.4
we can suppose that each ϱ∣Yi:Yi→Q is a closed embedding. Let Yi′′=ϱ(Yi)
and φ′′:Y1′′→Y2′′ be the isomorphism induced by φ. Since ED(Y1)≤2n−3
we see that φ′′ extends to an automorphism α′′∈SAut(Q) by Theorem 8.3.
By Proposition 11.7 ϱ is (SAut(X),SAut(Q))-comparable. Hence, by Proposition 9.2
we can suppose that Y1′′=Y2′′ and φ′′ is the identity map.
As before we have a morphism θ′′:Y1′′→H′′ as in Proposition 9.4 which extends
to a holomorphic map Θ′′:Q→H′′. Thus we have holomorphic comparability of ϱ
which yields (iii) and concludes the proof.
∎
Corollary 11.12**.**
Let X=SL(n,C) and
φ:Y1→Y2 be an isomorphism of two smooth closed subvarieties of X such that
dimYi≤3n−1, Hi(Y1)=0 for i≥3
and H2(Y1) is a free abelian group.
Then there exists a holomorphic automorphism β of X such that β∣Y1=φ.
Proof.
Since dimYi≤3n−1 the smoothness assumption implies that
ED(Yi)+dimYi≤n−2. Hence, Theorem 11.11 implies the desired conclusion.
∎
Theorem 11.13**.**
Let φ:Y1→Y2 be an isomorphism of two closed
subvarieties of X≃SL(n,k) with n≥3 such that Yi is isomorphic to Akk. Suppose that
either k≤3n−1 or k=1.
Then there exists α∈SAut(X) such that α∣Y1=φ.181818The case of k=1 is, of course,
the theorem of Van Santen [33] which we prove by other means.
Proof.
Let k≤3n−1, m=ED(Y1) and let Yi′=ϱm(Yi),i=1,2. Repeating the argument in the proof of Theorem 11.11 we can suppose
that ϱm∣Yi:Yi→Yi′ is a closed embedding, Y1′=Y2′ and the induced isomorphism φ′:Y1′→Y2′
is the identity map.
Since ϱm is a principal Hn−m′-bundle,
ϱm is comparable on the family Faff as in Corollary 9.5.
The desired conclusion follows now from Proposition 9.2.
Similarly, if k=1 then as in Theorem 11.11 (iii) we can suppose that ϱ∣Yi:Yi→Q is a closed embedding
and for Yi′′=ϱ(Yi) the induced isomorphism φ′′:Y1′′→Y2′′ is the identity map.
Since ϱ is a principal H′′-bundle,
ϱ is comparable on the family Faff.
The desired conclusion follows again from Proposition 9.2.
∎
The following necessary condition for the positive solution of the extension problem is straightforward.
Proposition 11.14**.**
Let a group G⊂Aut(W) act on an algebraic variety W
and φ:Y1→Y2 be an isomorphism of
closed subvarieties of W. Suppose that ιk:Yk↪W is the natural embedding.
Suppose also that for every α∈G there exists k≥1
such that the homomorphism πk(Y1)→πk(W) induced by α∘ι1
is different from the similar homomorphism induced by ι2.
Then φ cannot be extended to an automorphism from G.
Remark 11.15**.**
In the complex case note that if ι1:Y1↪W induces
a trivial homomorphism πk(Y1)→πk(X) for some k
while the similar homomorphism induced by ι2:Y2↪W is nontrivial
then φ cannot be extended even to a homeomorphism of W.
Furthermore, if G is contained in the connected component of identity in Aut(X)
then the extension problem does not have a positive solution for the group G if there is no homotopy
of φ to the identity map via closed embeddings of Y1 into W.
In the rest of this section we present a concrete (and more or less obvious) example illustrating Proposition 11.14 in the case of X≃SL(n,C).
Lemma 11.16**.**
There is a closed embedding of Y≃SL(2,C) into X≃SL(n,C)
such that it generates an isomorphism π3(Y)≃π3(X) of the homotopy groups.
Proof.
Let ϱ:X→Q be as in Notation 11.1, i.e., by Proposition 11.4
it is a principal H′′-bundle with fiber F≃SL(n−1,C). Since Q is a complexification of a real (2n−1)-dimensional sphere
it has a homotopy type of this sphere191919Actually, it is well-known that Q is diffeomorphic to the tangent bundle of
the sphere..
Hence πk(Q)=0 for 1≤k≤2n−2 and π2n−1(Q)=Z.
Now the exact homotopy sequence for
the fiber bundle ϱ:X→Q implies that the natural embedding of
SL(n−1,C)≃F↪X≃SL(n,C)
induces an isomorphism πk(X)≃πk(F)
for k<2n−2. Choosing a natural sequence
SL(2,C)⊂SL(3,C)⊂…⊂SL(n−1,C)⊂SL(n,C)
of closed embeddings we get the desired conclusion.
∎
Theorem 11.17**.**
Let X be an affine algebraic variety isomorphic to SL(n,C) with n≥3.
There are two closed subvarieties Y1 and Y2 in X isomorphic to SL(2,C) and such that
there is no automorphism α of X for which α(Y1)=Y2.
Proof.
By Lemma 11.16 we can suppose that the natural embedding Y1↪X
generates an isomorphism π3(Y1)≃π3(X). By Proposition 11.14 and Remark 11.15
in order to prove Theorem 11.17 it suffices to present an embedding Y2↪X such that the induced homomorphism
sends π3(Y2) into the zero element of π3(X).
Treat a point in X as a matrix A=[ai,j]. Let A′ be the (n−1)×(n−1) matrix
obtained from A by removing the first row and the first column.
Consider the subvariety X0 of X that consists of matrices A such that A′ is the identity matrix.
Note that X0 is naturally isomorphic to C2n−1 with coordinates a1,2,…a1n,a21,…,an1
since a1,1 can be expressed as function of these coordinates because of the equation detA=1.
Choose in X0 the quadric Y2≃SL(2,C) given by a1,2a1,3−a2,1a3,1=1 and a1,k=ak,1=0 for all k≥4.
Then the embedding Y2↪X induces the zero map π3(Y2)→π3(X)
since it factors through Y2↪X0 and X0 is contractible. This yields the desired conclusion.
∎