Table Of ContentCHOW POINTS OF C-ORBITS
7
0 ANNETTPU¨TTMANN
0
2
Abstract. Weconsiderfreealgebraicactionsoftheadditivegroupofcomplex
n numbers on a complex vector space X embedded in the complex projective
a space. Wefindanexplicitformulaforthemappthatassignstoagenericpoint
J x ∈X the Chow point of the closure of the orbit through x. The properties
3 Hausdorff quotient topology andproper action areequivalently characterized
bytheclosureoftheimageofpintheclosedChowvariety.
]
G
A
h. 1. introduction
t
a In this article we consider free algebraic C-actions, i.e., actions of the additive
m group of complex numbers, on Cn = X. Such an action is given by a locally
[ nilpotent derivation δ : C[X| → C[X]. We have t.f = ∞ tjδjf for t ∈ C and
j=0 j!
1 f ∈C[X],where C×C[X]→C[X], (t,f)7→t.f, denotes the action ofC onC[X].
P
v ClosuresofC-orbitsinacompactificationP CofX areprojectivecurvesinP C.
n n
0 In section 2, Theorem 1, we derive an explicit formula for the map p : U → C
0 d,n
1
from the Zariski-openset U ⊂X of orbits of maximal degree d to the open Chow
1 0
variety C of curves of degree d in P C that assigns to a point x ∈ X the Chow
1 d,n n
0 point of C.x⊂PnC.
7 Since the closed Chow variety C is projective, there is a sequence of blow-ups
d,n
0 π :X˜ →X and an regular map p˜:X˜ →C that lifts p.
d,n
/
h We are interested in properties of the C-action on Cn that are relevant for
t geometric quotients. By definition, the quotient topology is Hausdorff if the orbits
a
m can be separated by continuous functions. Recall that the action of a Lie group G
on a manifold X is called proper if the map G×X →X ×X, (g,x)7→(x,g.x), is
:
v proper.
i
X Insection3weexpresspropernessandHausdorff quotient topology usingproper-
tiesofthecyclesinp˜(X˜). Corollary3statesthatthequotienttopologyisHausdorff
r
a if and only if for all x˜∈X˜ all irreducible components of p˜(x˜) except the closure of
C.π(x˜) are contained in infinity. In Corollary 4 we prove that the action is proper
if and only if it has Hausdorffquotient topology andthe multiplicity of the closure
of C.π(x˜) in p˜(x˜) is 1 for all x˜∈X˜.
WeexploitthefactthattheanalyticspaceassociatedtheChowschemeofcurves
in P C is the Barlet space of compact 1-cycles in P C [1]. This enables us to
n n
describe convergence of Chow points by the behavior of the corresponding cycles.
In section 4 we apply our results to free C-actions of degree two that have
been studied by other authors using different methods. So far our approach does
not describe the geometric quotient in the case where the action is proper. But
Research supported by SFB/TR12 “Symmetrien und Universalita¨t in mesoskopischen Syste-
men”oftheDeutscheForschungsgemeinschaft.
1
2 ANNETTPU¨TTMANN
the map p˜ gives complete information about non-separable orbits if the quotient
topology is not Hausdorff.
The constructionofthe explicit formof the map p:U →C stronglydepends
0 d,n
on the fact that the acting groupis C. Whereas it should be possible to generalize
theresultsofsection3tofreeactionsofothergroupsonmoregeneralaffinevarieties
than Cn.
We would like to thank A. T. Huckleberry for valuable discussions.
2. Chow point of a generic orbit
Fixing a set of coordinates {x ,...,x } ⊂ C[X] on X = Cn defines an embed-
1 n
ding Cn ֒→ P C by x 7→ [1 : x (x) : ... : x (x)]. Furthermore, every deriva-
n 1 n
tion δ can be uniquely written as δ = n δ(x ) ∂ . The C-action on X cor-
j=1 j ∂xj
responding to a locally nilpotent derivation δ on C[X] is completely described by
P
x (−t.x)= d tj(δjx )(x).
k j=0 j! k
ForalocallynilpotentderivationδonC[X]wedefineitsdegreetobethesmallest
P
numberdsuchthatδd+1(x )=0forallj. Ofcourse,this notionofdegreedepends
j
onthechosencoordinates. LetC :=C.x⊂P C betheclosureinP C oftheorbit
x n n
throughthepointx∈X. AlthoughallorbitsofafreeC-actionareisomorphictoC
theirclosuresinP C mayleadto non-isomorphicprojectivecurvesdistinguishable
n
by their degree as projective varieties.
Lemma1. Ifδ definesafreealgebraic C-actiononCn ofdegreed,thendegC ≤d
x
for all x∈Cn. Furthermore, degC =d iff x∈ n {δd(x )6=0}.
x j=1 j
Proof. For a general hyperplane H = { n a SX = 0} in P C the intersection
j=0 j j n
H ∩C is given by the solutions of a + n a d tj(δjx )(x)=0 which is a
x 0 Pk=1 k j=0 j! k
polynomial in t of degree at most d. (cid:3)
P P
The set U := ∪n {δd(x ) 6= 0} is Zariski-open and C-invariant, because
0 j=1 j
δd+1x = 0 for all j and δdx 6= 0 for at least one j. An orbit in U is called
j j 0
generic orbit. Hence, givena free algebraicC-actionon Cn of degree d and a com-
pactificationCn ⊂P C theclosuresofgenericorbitsareprojectivecurvesofdegree
n
dinP C. ProjectivecurvesofdegreedinP C areparametrizedbytheopenChow
n n
variety C . We will give an explicit description of the map p : U → C that
d,n 0 d,n
assigns to a point x∈X the Chow point of C .
x
We refer to [2] for an introduction to the Chow variety. The Chow variety C
d,n
is a quasiprojective variety in P(Γ(P2(C),Hd)). Here, Γ(P2(C),Hd) is the vector
n n
space of bihomogeneous polynomials of bidegree (d,d) with respect to the two sets
of n + 1 variables α ,...,α and β ,...,β . The Chow point P ∈ C of a
0 n 0 n C d,n
curve C of degree d in P C is the polynomial defining the hypersurface H :=
n C
∪ {(α,β)|x ∈ H ∩H } ⊂ P2(C). Here, the hyperplane H ⊂ P C assigned
x∈C α β n α n
to a point α∈P C is H ={ n α X =0}.
n α j=0 j j
We denote by τ : P2(C) → P2(C), (α,β) 7→ (β,α), the permutation of the
n P n
componentsofP2(C). Notethatγ :=β α −β α areτ-invariantaselements
n j1j2 j1 j2 j2 j1
of P(Γ(P2(C),H1)) for j ,j =0,...,n. For k,l=0,...,d we set
n 1 2
n n n n
(1) f (x):= β (δlx )(x) α (δkx )(x)− α (δlx )(x) β (δkx )(x)
lk j j j j j j j j
j=0 j=0 j=0 j=0
X X X X
CHOW POINTS OF C-ORBITS 3
Note that f (x) = n n γ δl(x )(x)δk(x )(x). Since γ = τ.γ =
kl j1=0 j2=0 j1j2 j1 j2 j1j2 j1j2
−γ , we have f =−f . Let
j2j1 lkP klP
mind−l,k d−l−r−1 r−1
(2) F (x):= f (x) (d−j) (k−j),
lk l+r,k−r
r=0 j=0 j=0
X Y Y
(3) P(x):=det(F (x))
lk l6=0,k6=d
Lemma 2. The polynomial P is a τ-invariant element of P(Γ(P2(C),Hd)).
n
Proof. By construction, P is a homogeneous polynomial of degree d in the terms
γ . It remains to show that P is non-trivial, P 6≡0.
lk
To see this, we examine the coefficients c of the monoms γd . Among the
j0 j0
polynomialsf onlyf andf contributetothecoefficientsc ,becauseδl(x )=
lk 0k l0 j0 0
0 for all l > 0. Thus, the coefficient of γ in F vanishes if d−l < k. Since the
j0 lk
definition of P involves a determinant, c equals the coefficient of γd in fd . By
j0 j0 d0
definition, f = ( n γ δdx + n n γ (δdx )x )(x). Hence, c =
d0 j=1 j0 j j1=1 j2=1 j1j2 j1 j2 j0
(δdx )d(x). Since the action has degree d, we have c 6=0 for at least one j. (cid:3)
j P P P j0
Lemma 3. Let f be elements of a polynomial ring satisfying f = −f for
lk lk kl
all k,l = 0,...,d and t ∈ C. The equations d f tk = 0 are satisfied for all
k=0 lkk!
l=0,...,d iff d−1F tk =0 for all l =1,...,d, where F is defined as in (2).
k=0 lkk! P lk
Proof. ConsidePr the following recursion: f(0) :=f ,
lk lk
f(s) l6=d−s−1
f(s+1) := lk
lk (fd(s−)s,k−1k+fd−s−1,k sj=0(d−j) l=d−s−1
We will prove by induction on s that f(s) =Q0 for l ≥d−s and
ld
f l<d−s
f(s) = lk
lk ( rm=in0d−l,kfl+r,k−r dj=−0l−r−1(d−j) rj=−01(k−j) l≥d−s
If s=0, thenPf(0) =f = 0 Qf . In parQticular, f(0) =0. Now,
dk dk r=0 d+0,k−0 dd
mins,k−1 P s−r−1 r s
f(s+1) =k f (d−j) (k−j)+f (d−j)
d−s−1,k d−s+r,k−1−r d−s−1,k
r=0 j=0 j=1 j=0
X Y Y Y
mins+1,k s−r r−1 s
= f (d−j) (k−j)+f (d−j)
d−s−1+r,k−r d−s−1,k
r=1 j=0 j=0 j=0
X Y Y Y
mins+1,k s+1−r−1 r−1
= f (d−j) (k−j)
d−s−1+r,k−r
r=0 j=0 j=0
X Y Y
Furthermore, f(s+1) = s+1f s+1−r−1(d − j) r−1(d − j) = 0,
d−s−1,d r=0 d−s−1+r,d−r j=0 j=0
since f =−f .
d−(s+1−r),d−r d−r,d−(s+1−r)
P Q Q
Multiplying the equation d f(s) tk =0 by t6=0 gives an equivalent equa-
k=0 d−s,kk!
tion d−1f(s) tk+1 = d f(s) ktk = 0, since f(s) = 0. Consequently,
k=0 d−s,k k! k=P0 d−s,k−1 k! d−s,d
the systems d tkf(s) =0 for all l=0,...,d are equivalent for all s.
P k=0 k! lk P
P
4 ANNETTPU¨TTMANN
Note that F = f(d). Recall that F = 0 for all l. Since f = −f , we
lk lk ld r,k−r k−r,r
have for all k
k d−r−1 r−1 k
F = f (d−j) (k−j)= f d! k! =0.
0k r,k−r r,k−rr!(k−r)!
r=0 j=0 j=0 r=0
X Y Y X
(cid:3)
Theorem 1. Let δ be a locally nilpotent derivation of C[x ,...,x ] that defines a
1 n
free C-action on Cn of degree d and x∈Cn. If C.x a generic orbit, then the Chow
point p(x) of C is P(x) as defined by equation (3).
x
Proof. Except for one point, the point at infinity, the curve C is parametrized
x
by the C-action. Therefore, H = {(α,β)|∃t∈C :t.x∈H ∩H }. To find the
Cx α β
defining polynomial of of H , we can assume that if (α,β) ∈ H then there is
t ∈ C such that n α x (C−xt.x) = 0 and n β x (−t.x) = 0.CxSince for all j
j=0 j j j=0 j j
t.x = d tkδkx , we obtain the equations
j k=0 k! Pj P
P d n
(4) tkA =0, A := α δkx (x)
k! k k j j
k=0 j=0
X X
d n
(5) tkB =0, B := β δkx (x)
k! k k j j
k=0 j=0
X X
The terms A and B are polynomials of bidegree (1,0) and (0,1), respectively.
k k
The combination of the equations (4) and (5) leads to a system of d+1 equations
d tk
(6) (BA −A B )=0, l =0,...,d.
l k l k
k!
k=0
X
Note that f =BA −AB . By Lemma 3, the system (6) is equivalent to
lk l k l k
d−1tk
F =0, l=1,...,d
lk
k!
k=0
X
Since t0 = 1, it follows that P = det(F ) = 0 for all (α,β) ∈ H . But
lk l6=0,k6=d Cx
P is a homogeneous polynomial of bidegree (d,d) and τ-invariant as an element of
P(Γ(P2(C),Hd)). (cid:3)
n
A local slice of an action of a Lie group G on a manifold X through a point
x∈ X is a locally closed submanifold x ∈ S ⊂ X such that G.S is open in X and
every orbit through S intersects S in exactly one point. For a free action a local
slice establishes a local trivialization of the action, i.e., G×S →G.S, (g,s)7→g.s,
is a diffeomorphism. The properness of a Lie group action implies the existence of
local slices. We include the proof of a special case of this general fact.
Lemma 4. A free action of a Lie group G on a manifold M is proper if and only
if there are local slices through every point of M. A proper action of a Lie group G
on a manifold Mhas Hausdorff quotient topology.
Proof. TheactionofaLiegroupGonamanifoldM isproperiffforanyconvergent
sequence {x } ⊂ M and any sequence {g } ⊂ G the convergence of the sequence
n n
{g .x }⊂M implies the existence of a convergentsubsequence of {g }⊂G.
n n n
CHOW POINTS OF C-ORBITS 5
If there is a local slice through every point in M, then M can be covered by
localtrivializations of the action. Then a local trivializationaroundy :=limg .x
n n
shows that {g } is convergent.
n
Assume that G acts properly on M. Let x∈M be a point. Choose any locally
closed submanifold S′ through x that is transversal to the G-orbit G.x. Such a
S′ can be shrunk to S such that G×S → G.S, (g,s) 7→ g.s, is a diffeomorphism.
Otherwisetwoconvergentsequences{s }→x,{s′ =g .s }→xwouldexist. The
n n n n
properness would imply that {g } is convergentto the identity e contradicting the
n
fact that G×S →G.S is locally diffeomorphic at (e,x).
If {x } and {g .x } are convergent sequences in M, then the properness of
n n n
the action implies that g tends to an element g ∈ G. Then g.lim x =
n n→∞ n
lim g .x . (cid:3)
n→∞ n n
Lemma 5. There are local slices through every x∈U .
0
Proof. There are local trivializations {δdx 6= 0} → C ×{δdx 6= 0,δd−1x = 0}
j j j
given by x7→(t(x),−t(x).x) with t(x)=(δd−1x )(x)/(δdx )(x), because
j j
d (−t(x))k
δd−1x (−t(x).x)= δkδd−1x (x)=δd−1x (x)−t(x)δdx (x)=0.
j j j j
k!
k=0
X
(cid:3)
Corollary 1. The restriction of the C-action on U is a proper action.
0
Note that U = X if d = 1. Hence, a free algebraic C-action of degree one on
0
Cn is proper.
3. Limit cycles
The regular map p :U →C , x7→P(x) defines a rational map p :X →C .
0 d,n d,n
Since the open Chow variety C is quasiprojective, there is a finite sequence of
d,n
blow-ups π : X˜ = X → X → ... → X = X along Y ⊂ X and a regular
m m−1 0 j j
morphism p˜:X˜ → C into the projective Chow variety C such that p˜=p◦π.
d,n d,n
s
A point in C is a cycle C = n C which consists of irreducible curves
d,n j=1 j j
C ⊂P C with multiplicities n ∈N such that d= s n degC .
j n j P j=1 j j
For x˜ ∈ X˜ \π−1(U ) we call p˜(x˜) ∈ C a limit cycle. A cycle p˜(x˜) can be
0 d,n P
uniquely written as p˜(x˜) = n C +Z with n ∈ N, C = C.π(x˜), and a cycle
x˜ x˜ x˜ x˜ x˜
Z ⊂P C ofdegreed−n degC thatdoes notcontainC . Note that n =1 and
x˜ n x˜ x˜ x˜ x˜
Z =0 if π(x˜)∈U .
x˜ 0
InthesequelweusthefactthattheanalyticspaceassociatedtotheclosedChow
variety is contained in the Barlet space.
Theorem 2. If Y is a projective variety, then the Barlet space of Y coincides with
the analytic space associated with the Chow scheme of Y.
Proof. See [1] and [4]. (cid:3)
This allows us to discuss convergence of Chow points p˜(x ) in terms of the
n
corresponding projective curves C .
xn
Lemma 6. Let x˜ ∈ X˜. For every open neighborhood W ⊂ P C of p˜(x˜), there is
n
an open neighborhood U ⊂X˜ of x˜ such that p˜(y)⊂W for all y ∈U.
6 ANNETTPU¨TTMANN
Proof. The family of compact 1-cycles parametrized by the points in the Barlet
space is an analytic family of compact 1-cycles, see [1] and [4]. (cid:3)
Corollary 2. For all x˜ ∈ X˜ the curve C is a component of the limit cycle p˜(x˜),
x˜
i.e., n ≥1.
x˜
Proof. Let x˜ ∈ X˜. We take a sequence {x˜ } ⊂ π−1(U ) that tends to x˜ in X˜.
n 0
Now, p˜(x˜ )→p˜(x˜), since p˜is continuous. But p˜(x˜ )=p(π(x˜ )) and the sequence
n n n
{π(x˜ )}tends toπ(x˜). Bylemma6,C isacomponentofp˜(x˜),becauset.π(x˜ )→
n x˜ n
t.π(x˜) for all t∈C. (cid:3)
Corollary 3. The C-action on X has Hausdorff quotient topology if and only if
Z ∩X =∅ for all x˜∈X˜.
x˜
Proof. Two distinct orbits C.x and C.x′ in X cannot be separatedby C-invariant
open subsets if and only if for any pair of open neighborhoods U of x and U′
of x′ there exists a t ∈ C such that t.U ∩ U′ 6= ∅. This means that there are
convergent sequences {x˜ } → x˜, {x˜′ } → x˜′ in X˜ and a sequence {t } ⊂ C such
n n n
that π(x˜′ )=t .π(x˜ ), π(x˜ ) tends to x, and π(x˜′ ) tends to x′. By lemma 6 both
n n n n n
limit cycles, p˜(x˜) and p˜(x˜′), contain the irreducible components Cx˜ and Cx˜′. (cid:3)
The condition Z ∩X = ∅ means that Z contains only points at infinity, i.e.,
x˜ x˜
Z is contained in the hyperplane {X = 0} ⊂ P C. Hence, Z ∩ X = ∅ iff
x˜ 0 n x˜
p˜(x˜) = p(π(x˜))nx˜r where r is a polynomial in the set of variables {γlk : l,k 6= 0}
with coefficients in C[X˜].
Lemma 7. Let x˜ ∈ X˜ and U ⊂ X be a neighborhood of π(x˜) with coordinates
{ξ ,...,ξ } satisfying U ∩C ={ξ =...=ξ =0} and ξ (π(x˜))=0 ∀j.
1 n x˜ 2 n j
If Z ∩ X = ∅, then there exists a neighborhood U˜ ⊂ X˜ of x˜ such that the
x˜
restriction of ξ to p˜(y˜)∩U is an unramified covering of degree n for all y ∈U˜.
1 x˜
Proof. The local coordinates define a scale, that is adapted to p˜(x˜), since U ⊂ X
and X ∩ p˜(x˜) = n C ([1], [4]). The resulting covering is unramified, because
x˜ x˜
distinct C-orbits do not intersect. (cid:3)
Corollary 4. The C-action on X is proper if and only if Z ∩X =∅ and n =1
x˜ x˜
for all x˜∈X˜.
Proof. If the C-action on X is proper, Z ∩X =∅, because the quotient topology
x˜
of the action is Hausdorff. Furthermore, there is a local slice S through π(x˜).
By construction we can assume that S is a locally closed complex submanifold of
dimension n−1. Choosing local coordinates {ξ ,...ξ } on S around π(x˜) and a
2 n
coordinate ξ on C the biholomorphic map Φ : C ×S → C.S, (t,s) 7→ (t.s), gives
1
localcoordinatesonasmallneighborhoodU ofπ(x˜)withthepropertiesrequiredin
Lemma7. SinceΦisalocaltrivializationoftheaction,itthenfollowsthatn =1.
x˜
Now, let us assume Z ∩X = ∅ and n = 1. Choosing coordinates {ξ ,...,ξ }
x˜ x˜ 1 n
asdescribedinLemma7thelocallycompactcomplexsubmanifoldπ(U˜)∩{ξ =0}
1
is a local slice through π(x˜). (cid:3)
4. Examples
In this section we apply the explicit formula for the map p :→ C and the
d,n
conditions derived in Corollary 3 and Corollary 4 to free C-actions of degree two.
CHOW POINTS OF C-ORBITS 7
These examples havebeen examinedby different authorsusing other methods. An
expansionofthedeterminantthatdefinesP inequation(3)usingf =−f yields
lk kl
P =2f f −f2 if d=2.
10 21 20
Example 1. We consider the free C-action on C3 that is defined by the derivation
δ = −x2 ∂ + (1 − x x ) ∂ . It is of degree two and U = {x 6= 0}, since
1∂x2 1 2 ∂x3 0 1
δ2 =x3 ∂ . We know that this action, which is induced by a free affine C2-action
1∂x3
onC4,isproperandadmitsaglobalalgebraicslice[3],[5]. Applyingformula(1)we
obtainf = 3 ((1−x x )γ −γ x2)x ,f = 2 x3x γ andf =−x5γ .
10 j=0 1 2 3j 2j 1 j 20 j=0 1 j 3j 21 1 32
Then by equation (3)
P P
3 2
p(x)=−x5(2γ ((1−x x )γ −γ x2)x +x ( x γ )2).
1 32 1 2 3j 2j 1 j 1 j 3j
j=0 j=0
X X
The mapp extends to aregularmaponX with p| =γ (γ x +γ x ). For
{x1=0} 32 30 0 32 2
x˜ ∈ X \U the first factor corresponds the component Z , which lies at infinity,
0 x˜
and the second factor represents the component C , i.e., n =1.
x˜ x˜
Example 2. The derivation δ = (x −x x ) ∂ +x ∂ +(x +1) ∂ defines a
1 2 5 ∂x3 2∂x4 1 ∂x5
free C-action on C5. Since δ2 = −x (x +1) ∂ , we have U = {x (x +1) 6= 0}
2 1 ∂x3 0 2 1
and d = 2. We know that this action is proper and admits Zariski-open local
trivializations [6]. Using the equations (3) and (1) we get
5
p(x)=2((x +1)γ +x γ )( ((x −x x )γ +x γ +(x +1)γ )x )+x (x +1)r
1 35 2 34 1 2 5 3j 2 4j 1 5j j 2 1
j=0
X
with r = ( 4 γ x )2. The map p : U → C extends to a regular map on
j=0 3j j 0 2,5
{x 6=0}∪{x +16=0} and
2 1
P
5
p| =−x γ ( ((1+x x )γ −x γ )x )
{x1+1=0} 2 34 2 5 3j 2 4j j
j=0
X
5
p| =(x +1)γ ( (x γ +(x +1)γ )x )
{x2=0} 1 35 1 3j 1 5j j
j=0
X
Hence,Z ∩X =∅andn =1forallx∈{x 6=0}∪{x +16=0}. Tofindp˜wehave
x x 2 1
toconstructtheblow-upπ :X˜ →X ofX alongY :={x =x +1=0}. Thereisa
2 1
coveringofX˜ by U ={ξ 6=0,x +1=x ξ1}andU ={ξ 6=0,x =(x +1)ξ2}.
1 2 1 2ξ2 2 1 2 1 ξ1
A direct calculation shows that the rational map p : X → C lifts to a regular
2,5
map p˜:X˜ →C given by
2,5
5
p˜| =2(ξγ +γ ) ((x −x x )γ +x γ +(x +1)γ )x +x r
U1 35 34 1 2 5 3j 2 4j 1 5j j 2
j=0
X
5
p˜| =2(γ +ξγ ) ((x −x x )γ +x γ +(x +1)γ )x +(x +1)r.
U2 35 34 1 2 5 3j 2 4j 1 5j j 1
j=0
X
with r =( 4 γ x )2. Now,
j=0 3j j
p1|U1∩π−P1(Y) =(ξγ35+γ34) γ3jxj, p2|U2∩π−1(Y) =(ξγ34+γ35) γ3jxj,
j6=2,3 j6=2,3
X X
8 ANNETTPU¨TTMANN
where the second factors correspond to C and the first factors to Z .
x˜ x˜
Example 3. The derivationδ =x ∂ +x ∂ +(x2−2x x −1) ∂ defines a free
1∂x2 2∂x3 2 1 3 ∂x4
C-action on C4 of degree two with δ2 =x ∂ and U ={x 6=0}. We know that
1∂x3 0 1
this action has non-Hausdorff quotient topology [6]. Evaluating the map p˜we can
determineallinseparableorbits. Asintheexamplesabovewefindtheexplicitform
of p:U →C using the equations (3) and (1):
0 2,4
4
p=2(x γ +(x2−2x x −1)γ )( (x γ +x γ +(x2−2x x −1)γ )x )−x r
1 32 2 1 3 34 1 2j 2 3j 2 1 3 4j j 1
j=0
X
withr =( 4 γ x )2. ThismapiswelldefinedoutsideofY :={x =x2−1=0}
j=0 3j j 1 2
and
P 4
p| =(x2−1)γ ( (x γ +(x2−1)γ )x )
{x1=0} 2 34 2 3j 2 4j j
j=0
X
which means that n = 1 and Z = (x2 −1)γ for x ∈ {x = 0,x2 6= 1}. The
x x 2 34 1 2
blow-up π :X˜ →X of X along Y is coveredby the charts U+ ={ξ 6=0,x +1=
1 1 2
x ξ2}, U+ = {ξ 6= 0,x = (x + 1)ξ1}, U− = {ξ 6= 0,x −1 = x ξ2}, and
1ξ1 2 2 1 2 ξ2 1 1 2 1ξ1
U− = {ξ 6=0,x =(x −1)ξ1}. It can be checked by an explicit calculation that
2 2 1 2 ξ2
the rational map p:X →C lifts to regular map p˜:X˜ →C and
2,4 2,4
p˜| =(γ −γ +x γ )(γ +γ +(x −4x −4ξ)γ ),
U+∩π−1(Y) 30 32 4 34 30 32 4 3 34
1
p˜| =(γ −γ +x γ )(ξγ +ξγ +(ξx −4ξx −4)γ ),
U+∩π−1(Y) 30 32 4 34 30 32 4 3 34
2
p˜|U−∩π−1(Y) =(γ30+γ32+x4γ34)(γ30−γ32+(x4+4x3−4ξ)γ34),
1
p˜|U−∩π−1(Y) =(γ30+γ32+x4γ34)(ξγ30−ξγ32+(ξx4+4ξx3−4)γ34).
2
Ifx˜∈π−1(Y)andξ(x˜)6=0,thenp˜(x˜)containstwodistinctC-orbits. Inparticular,
orbits in {x =x −1=0} cannot be separatedfromorbits in {x =x +1=0}.
1 2 1 2
References
[1] D. Barlet, Espace analytique r´eduit des cycles analytiques complexes compacts d’un es-
paceanalytiquecomplexededimensionfinie.Fonctionsdeplusieursvariablescomplexes,
II (S´em. Francois Norguet, 1974–1975), pp. 1–158. Lecture Notes in Math., Vol. 482,
Springer,Berlin,1975
[2] J. Harris, Algebraic geometry. A first course. Graduate Texts in Mathematics, 133.
Springer-Verlag,NewYork,1995.
[3] S. Kaliman, Free C+-actions on C3 are translations, Invent. Math. 156 (2004), no. 1,
163–173.
[4] J.Magnusson,LecturesonCycleSpaces,SchriftenreihedesGraduiertenkollegsGeometrie
undMathematische Physik,2005
[5] M. D.Snow, Stein Quotients of Connected Complex Lie Groups, manuscripta math. 50
(1985), 185–214.
[6] J.Winkelmann,OnfreeholomorphicC-actionsonCnandhomogeneousSteinmanifolds,
Math.Ann.286(1990), no.1-3,593–612.
Fakulta¨t fu¨rMathematik,Ruhr-Universita¨tBochum,44780Bochum
E-mail address: [email protected]
