数理解析研究所講究録1033 巻 1998 年 81-93

81Compact and reductive subgroups of the group ofholomorphic automorphisms of$\mathbb{C}^{n}$

FRANK KUTZSCHEBAUCH
Mathematisches Institut, Rheinsprung 214051 Basel, Switzerland
kutzsche@math.unibas.ch

1. INTRODUCTION
This article is centered around the following question about complex reductive subof holomorphic automorphisms of :groups of the group$\mathbb{C}^{n}$$\mathrm{A}\mathrm{u}\mathrm{t}_{\mathrm{h}\circ}1(\mathbb{C}^{n})$be a complex reducHolomorphic Linearization Problem. Lettive subgroup of the groupof holomorphic automorphisms of . Can
one conjugate this subgroup by a single automorphism into the general linear group, is every action of a complex reductive group on?
$G\mathrm{L}_{arrow}\mathrm{A}\mathrm{u}\mathrm{t}_{\mathrm{h}\circ}1(\mathbb{C}^{n})$$\mathbb{C}^{n}$$\mathrm{A}\mathrm{u}\mathrm{t}_{\mathrm{h}\circ}1(\mathbb{C}^{n})$$\mathbb{C}^{n}$$\mathrm{G}\mathrm{L}_{n}(\mathbb{C})\subset \mathrm{A}\mathrm{u}\mathrm{t}_{\mathrm{h}\circ}1(\mathbb{C}^{n}),\dot{i}.e.$$l_{\dot{i}near}\dot{i}zable$

It is natural to ask the same question for compact groups. These two questions arestrongly related since complex reductive groups are exactly those groups which are.
the universal complexifications of its maximal compact subgroup $K$ , i.e.,
$K$
hence
is a totally real submanifold of half real dimension in
Recall that
$K$ is an identity set infor holomorphic maps. So if we have a non-linearizable
on ,of a complex reductive groupholomorphic action :compact
subgroupmaximal:tothethen by restricting that action$K$ ofwould be conjugate by an
we get a non-linearizable $K$-action. Indeed if$K$automorphism to a linear -representation, then the same automorphism conjugates. The aim of the first section isto the corresponding linear representation of
to explain the equivalence of the holomorphic linearization questions for compact andis a complex
complex reductive groups. In fact we prove that the group
group in the following sense:. Then the
Let $H$ be a real Lie group acting by holomorphic au tomorphisms on.
oncomplexification
action extends to a holomorphic action of the universal$G$$G=K^{\mathbb{C}}$$G=K^{\mathbb{C}}$$G$$\phi$$G=K^{\mathbb{C}}$$G\cross \mathbb{C}^{n}arrow \mathbb{C}^{n}$$\phi|_{K}$$\mathbb{C}^{n}$$K\cross \mathbb{C}^{n}arrow \mathbb{C}^{n}$$G$$\phi|_{K}$$K^{\mathbb{C}}$$\alpha$$\mathrm{A}\mathrm{u}\mathrm{t}_{\mathrm{h}\circ}1(\mathbb{C}^{n})$$\mathbb{C}^{n}$$H^{\mathbb{C}}$$\mathbb{C}^{n}$

Also we give two interesting consequences of this extension result. The first is thatby holomorphic transformations andsome real Lie groups can not act effectively on
invariant under a complete
Fatou-Bieberbach domains inistheholomorphic vectorfield.
In the second section we give a short overview over the positive results known about
holomorphic linearization and in the third section we explain the method to constructcounterexamples to the holomorphic linearization problem found by DERKSEN and theauthor.
$\mathbb{C}^{n}$$\mathbb{C}^{2}$$\overline{\mathrm{s}}\mathrm{e}\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{d}$$\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{C}\mathrm{e}\mathrm{r}\mathrm{n}\mathrm{i}\mathrm{n}.\mathrm{g}$partially supported by SNF (Schweizerischer Nationalfonds)

82
Compact and red uctive subgroups of the group of holomorphic au tomorphisms of

2. COMPLEXIFYING AN ACTION OF A REAL LIE GROUP ON$\mathbb{C}^{n}$$\mathbb{C}^{n}$

We start with some notation.
We say th at a real Lie group acts on a complex space $X$ by holomorphic transforma tions if the action map : $G\cross Xarrow X$ is real analytic and for all $g\in G$ the: $Xarrow X$ is holomorphic (in fact a holomorphic automorphism). If is amap
is holomorphic we say that acts holomorphicallycomplex Lie group and the$G$$\phi$$G$$\phi(g, \cdot)$$G$$m\mathrm{a}p\phi$

on $X$ .

REMARK 2.1. Every continous action of a real Lie group on a complex space by holomorphic automorphisms is already real analytic ([2] 1.6.). So it would be sufficient torequire only continouity of the action map : $G\cross Xarrow X$ in the first part of theabove definition.
If $X$ is a complex manifold the compact-open topology makes the group of holomorphic
of $X$ a topological group. The action map : $G\cross Xarrow X$ isautomorphisms
is continous.
continous iff the corresponding group homomorphism of into
holomorphic
by
on
group
By the above remark we see that an action of a real Lie.
transformations is the same as a continous group homomorphism
a
is
as
which
is the same as a subgroup of
An effective action of on.
topological group isomorphic to
Before we formulate the extension result we recall the definition of the universal
complexification of a real Lie group.
together with a continous (hence real-analytic)
Definition. A complex Lie groupis called the universal complexification of a real Liehomomorphism :group if it has the following property: For any continous group homomorphism: $Garrow H$ of into a complex Lie group $H$ there exists a unique complex Lie:such that the diagram:group homomorphism$\phi$$\phi$$\mathrm{A}\mathrm{u}\mathrm{t}_{\mathrm{h}\circ 1}(x)$$G$$\mathrm{A}\mathrm{u}\mathrm{t}_{\mathrm{h}_{0}1}(X)$$G$$\mathbb{C}^{n}$$\alpha:Garrow \mathrm{A}\mathrm{u}\mathrm{t}_{\mathrm{h}\circ}1(\mathbb{C}^{n})$$G$$\mathrm{A}\mathrm{u}\mathrm{t}_{\mathrm{h}\circ 1()}\mathbb{C}^{n}$$\mathbb{C}^{n}$$G$$G^{\mathbb{C}}$$Garrow G^{\mathbb{C}}$$\dot{i}$

$G$

$G$

$\psi$$\psi^{\mathbb{C}}$$G^{\mathbb{C}}arrow H$$arrow\psi$$G\searrow$$\nearrow_{\psi^{\mathrm{C}}}H$:$G^{\mathbb{C}}$commutes.

can be found in [13] in the case where $G$ is connected. If the
The construction forgroup is not connected the universal complexification can be easily constructed fromthe universal complexification of the component of the identity of (see the proof ofis not injective. For
proposition 2.3). We remark that in general the map :the universal complexification equalsofthe universal covering
is isomorphic to Z. There are also examples
and the kernel of :where the kernel is of positive dimension (see [14]). We call a real Lie group extendable,
is injective. Linear groups and in particular compact
if the homomorphism :groups are extendable.
$G^{\mathbb{C}}$

$G$

$G$

$\dot{i}$$Garrow G^{\mathbb{C}}$$SL_{2}(\mathbb{C})$$SL_{2}(\mathbb{R})$$\underline{\mathrm{g}\mathrm{r}\mathrm{o}}\mathrm{u}\mathrm{p}$$\dot{i}$$SL_{2}(\mathbb{R})arrow SL_{2}(\mathbb{C})$$\dot{i}$$Garrow G^{\mathbb{C}}$of the universal:
Definition. We say that a holomorphic action$X$ is the extension of an
space
complex
on
a
group
Lie
complexification of a realaction $\phi:G\cross Xarrow X$ of on $X$ by holomorphic transformations if$\phi^{\mathbb{C}}$$G^{\mathbb{C}}\cross Xarrow X$

$G$

$G$

$\phi^{\mathbb{C}}(\dot{i}(g), x)=\emptyset(g, x)$$\forall g\in G,$ $x\in

X$

.

83
2. Complexifying an action of a real Lie group on$\mathbb{C}^{n}$

The main point for our extension result is that one can complexify every one parameter. This result is due tosubgroup of , i.e. an action of the additive group$X$
is called Liouville if every bounded
FORSTNERIC [6]. Recall that a complex space$X$
plurisubharmonicisfunction on
constant.
from above$G$$(\mathbb{R}, +)$is an action of
on a complex $man\dot{i}foldx$
Theorem 2.2. If :onwhich is Stein and Liouville, then extends to a holomorphic action ofX.
$\mathbb{R}\cross Xarrow X$$\phi$$(\mathbb{R}, +)$$(\mathbb{C}, +)$$\phi$PROOF. We sketch only the main ideas of the proof. For a detailed proof we refer to[6]. By
we denote the globally integrable (real) vectorfield induced by , i.e.,$\phi$$\xi_{\phi}$$\xi_{\phi}(x)=\frac{d}{dt}|t=0\emptyset(t,x)$

and by

on

$X$

$J$

.

we denote the almost complex structure induced from the complex structure

The global action induces a local -action on $X$ which can be constructed as,:follows: Fix a point $x\in X$ and extend the real-analytic map$R_{x}arrow
X$:from some maximalby analytic continuation to a holomorphic map
is schlicht, i.e., a domain inover into $X$ . One shows thatRiemann domain
commuteis an -invariant strip in containing
C. Since the vectorfields and$R_{x}=\{\lambda\in\mathbb{C}:-b(x)<Im(\lambda)<a(x)\}$, where $a(x),$ $b(x)>0$ .
R. So it is of the form
can beClearly $(-b(x), a(x))$ is the maximal interval for which the vectorfield$\Omega:=\{(\lambda,x)\in\mathbb{C}\crossX:\lambda\inR_{x}\}$
integrated starting from the oint . The subset
to the leaf
of $X$ under the map
is just the saturation
spaceof the local -action (for definition see [22], [10]). Hence it is open in $X$ .
This implies that $x\mapsto-a(x)$ and $x\mapsto-b(x)$ are lower semicontinous functions on $X$ .
andThe most important point is to show that if $X$ is Stein, then the$-b$ are plurisubharmonic. If one of the functions $-a$ or $-b$ is not plurisubharmonic
not entirely contained in suchone can find a Hartogs figure
that
$H$ (
)
or
:
$\mathbb{C}$$\mathbb{R}$$\phi_{x}$$\mathbb{R}arrow X$$t\vdasharrow\phi(t, x)$$\tilde{\phi}_{x}$$R_{x}$$R_{x}$$\mathbb{C}$$R_{x}$$J\xi_{\phi}$$\xi_{\phi}$$\mathbb{C}$$\mathbb{R}$$J\xi_{\phi}$$\dot{\mathrm{p}}$$x$$p^{-1}(p(\mathrm{O}\cross X))$

$X^{*}$$p:\mathbb{C}\cross Xarrow X^{*}$$\mathbb{C}$$\mathrm{f}\mathrm{u}\mathrm{n}\mathrm{c}\mathrm{t}\mathrm{i}_{0}\mathrm{n}\mathrm{s}-a$$\Omega$$H:\Delta\cross\Delta_{n}\llcorner_{arrow \mathbb{C}}\cross X$$\{(w_{1},\tilde{w})\in\Delta\cross\Delta_{n}$$|w_{1}|< \frac{1}{2}$$| \tilde{w}|>\frac{1}{2}\}$is contained in thus proving that every holomorphic function defined on can be. Since a Stein manifold admits a closed
extended to the bigger domain
the extension also holds for maps
holomorphic embedding into some linear space
into Stein manifolds. In particular the local -action :could be extended contradicting the maximality in the definition of .
and $-b$ which
Since in addition $X$ is Liouville the plurisubharmonictant on $X$ . This means that the vectorfield
are bounded from above by arecan be integrated in both negative and positive directions for a fixed time independend
on the starting point $x\in X$ . This implies that it is globally integrable hence $-a$ and$\Omega$$\Omega$$\Omega\cup H(\Delta\cross\Delta_{n})$$\mathbb{C}^{N}$$\mathbb{C}$$\tilde{\phi}$$\Omegaarrow X$$(\lambda,x)\mapsto\tilde{\phi}_{x}(\lambda)$$\Omega$$\mathrm{f}\mathrm{u}\mathrm{n}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{s}-a$

$0$

$-b$

are

$\mathrm{i}\mathrm{d}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{i}\mathrm{c}\mathrm{a}\mathrm{u}_{\mathrm{y}-\infty}$$\mathrm{c}\mathrm{o}\mathrm{n}\dot{\mathrm{s}}$$J\xi_{\phi}$.

$\square$Now we are able to prove the extension result
be a real Lie group acting by holomorphic transformations
Proposition 2.3. Let$X$
on a Stein manifold which is Liouville, then the action extends to a holomorphic
on $X$ .
action of the universal complexification$G$$G^{\mathbb{C}}$84
Compact and red uctive subgroups of the group of holomorphic au tomorphisms of$\mathbb{C}^{n}$

PROOF. Let denote the Lie algebra of . First suppose is connected. We shortly
in that case:recall the construction ofand denote the kernel of by:
Take the universal covering groupbe the unique simply connected complex Lie group corresponding to the
F. Letbe the Lie group homomorphism
and :complexified Lie algebrabe the minimal. Let $H$ incorresponding to the inclusion homomorphism:is theand.
Thencomplex normal subgroup containing
group homomorphism determined by . For details we refer to [13]. In the rest of theproof we will use the notation introduced above.
From the -action we get an infinitesimal -transformation group, i.e., a Lie algefrom into the Lie algebra of holomorphic vector
bra homomorphism :$X$
by differentiating the action:
fields $V(X)$ on
$G$

$G$

$\mathfrak{g}$$G^{\mathbb{C}}$$\tilde{G}arrow G$

$\pi$

$\pi$

$\tilde{G}^{\mathbb{C}}$$\tilde{G}arrow\tilde{G}^{\mathbb{C}}$$\mathfrak{g}^{\mathbb{C}}=\mathrm{g}\otimes_{\mathbb{R}}\mathbb{C}$$\dot{i}_{1}$$\tilde{G}^{\mathbb{C}}$$\mathfrak{g}arrow \mathrm{g}^{\mathbb{C}}$$G^{\mathbb{C}}=\tilde{G}^{\mathbb{C}}/H$$\dot{i}_{1}(\Gamma)$$\dot{i}$$Garrow G^{\mathbb{C}}$$\dot{i}_{1}$

$G$

$G$

$\alpha$$\mathrm{g}arrow V(X)$$\mathfrak{g}$$\alpha(v)(x):=\frac{d}{dt,-}|_{t=}0\emptyset(\exp tv, X)$$v\in \mathrm{g},$$x\in X$.

Here holomorphic vectorfield means a real vectorfield inducing holomorphic transformations on $X(V\vdasharrow V-\dot{i}JV$ gives an isomorphism between the holomorphic
vectorfields and the holomorphic sections of the complex tangent bundle $\tau^{(1,0)}(X)$ of$X)$ . Clearly all the vectorfields.are globally integrable.
homomorphismfrom the complexified Lie algebra
We define a complex Lie lgebra.
into the holomorphic vectorfields on $X$ by:
with
are globally integrable. Together
By theorem 2.2 all vectorfields
they form a spanning set (as a vector
vectorfields
the globally integrable
space) of the real Lie algebra underlying . By a theorem of PALAIS (theorem III
are globally integrable and theyof . in [22] all vectorfields in the Lie algebraof the:integrate to a (real analytic, not necessarily effective) action. Clearly this action iscorresponding tosimply connected complex Lie groupholomorphic ([2] p.25). We have$(*)$
, $x\in X,g\in G$$\alpha(v),$$v\in \mathfrak{g}$$\dot{\mathrm{a}}$$\alpha^{\mathbb{C}}(v+\dot{i}w)=\alpha(v)+J\alpha(w)$$\mathfrak{g}^{\mathbb{C}}arrow V(X)$$\alpha^{\mathbb{C}}$$J\alpha(v),$$v\in \mathrm{g}$$\alpha(v),$$v\in \mathrm{g}$$9^{\mathbb{C}}$$\mathrm{c}\mathrm{h}$$4$$\mathfrak{g}^{\mathbb{C}}$$)$$\phi_{1}^{\mathbb{C}}$$\tilde{G}^{\mathbb{C}}$$\tilde{G}^{\mathbb{C}}\cross Xarrow X$$\mathrm{g}^{\mathbb{C}}$$\phi_{1}^{\mathbb{C}}(\dot{i}1(g),x)=\phi 1(g,X)$

since these two -actions have the same infinitesimal generator . This implies thatis contained in the ineffectivity of the action . The ineffecticity is a normal
and it is also a complex Lie subgroup ([2] 1.7. Prop. 1). Hence $I<H$
subgroup of-action . This is the desired extension, thefactors to a-actionand the, $g\in G,$ $x\in X$ follows from $(*)$ and the definition ofproperty. This proves the theorem for connected groups.
from anIf the Lie group is not connected one easily constructs the extension$G_{0}\cross
Xarrow
X$ . Here:of the restricted action:
extension.
is the component of the identity of . For that write in the form$[g, h]\cdot[g_{1}, h_{1}]=[gg_{1}, g_{1}^{-}hg1h_{1}]1$ .
The multiplication on the right hand side is given byis the universal complexification of where the group multiNow
is the unique Lie groupplication is given by $[g, h]\cdot[g_{1}, h_{1}]=[gg_{1}, \beta_{g_{1}}(h)h_{1}]$ . Hereof .
determined by the Lie group automorphism
automorphism ofdefines an e.xtension
It is straightforward to check that$G$$\alpha$$I$$\dot{i}_{1}(\Gamma)$$\phi_{1}^{\mathbb{C}}$$\tilde{G}^{\mathbb{C}}$$G^{\mathbb{C}}$$\tilde{G}^{\mathbb{C}}$$\phi_{1}^{\mathbb{C}}$$\phi^{\mathbb{C}}$$\emptyset^{\mathbb{C}}(\dot{i}(g), X)=\phi(g, x)$$\dot{i}$$G$$\phi^{\mathbb{C}}$$\phi_{1}=\phi|_{G_{0}\mathrm{X}x}$$G_{0}^{\mathbb{C}}\cross Xarrow X$$\phi_{0}^{\mathbb{C}}$$G$$G_{0}$$G$$G\cong G\cross c_{0}G_{0}$$G$$G^{\mathbb{C}}:=G\cross_{G_{0}}G_{0}^{\mathbb{C}}$$\beta_{g_{1}}$$h\mapsto g_{1}^{-1}hg_{1}$$G_{0}^{\mathbb{C}}$$G_{0}$$\phi^{\mathbb{C}}([g, h], X):=\phi(g, \phi_{0}^{\mathbb{C}}(h, X))$$\mathrm{o}\mathrm{f}\phi$.

$\square$is not extendable, then it can not act effecCorollary 2.4. If a real Lie groupby holomorphic transformations. For instance there is no $su\dot{b}group$ of
tively on).
isomorphic as a topological group to$G$$\mathbb{C}^{n}$$S\overline{L_{2}(}\mathbb{R}$$\mathrm{A}\mathrm{u}\mathrm{t}_{\mathrm{h}\circ}1(\mathbb{C}^{n})$85
2. Complexifying an action of a real Lie group on$\mathbb{C}^{n}$is Stein and Liouville (see for example [21] 1.38). If
PROOF. First of all note thatby holomorphic transformations by proposition 2.3 the action extends
acts on. Hence the kernel of :is contained in theto a holomorphic action of
only
a reformulation of the firstineffectivity of the -action. The second statement is) is not extendable.
part and uses the fact that$\mathbb{C}^{n}$$G$$\mathbb{C}^{n}$$G^{\mathbb{C}}$$Garrow G^{\mathbb{C}}$$\dot{i}$$G$$S\overline{L_{2}(}\mathbb{R}$$\square$is compact HEINZNER proved the
REMARK 2.5. In the case where the Lie groupof a Stein space $X$ , i.e., one embedds $X$
existence of a. universal complexification
space
so thatholomorphically and -equivariantly as an open subset into some$X$-space extends uniquely toevery holomorphic -map fromto some complex[9],[10]).
Proposition
-map
(see
2.3 implies that if
from
into
a holomorphic$X$$X$ in addition is Liouville, then one doesn’t have to enlarge, i.e.
.
$G$$X^{\mathbb{C}}$$G$$X^{\mathbb{C}}$$G^{\mathbb{C}}$$G$$G^{\mathbb{C}}$$X^{\mathbb{C}}$$G^{\mathbb{C}}$$\mathrm{Y}$$\mathrm{Y}$$X=X^{\mathbb{C}}$

Corollary 2.6. The holomorphic linearization questions for compact and complex reductive groups are equivalent.

is
PROOF. Suppose every holomorphic action of a complex reductive group on$K$by holomorphic transformations
linearizable. Given an action of a compact groupon
we can extend this action by proposition 2.3 to a holomorphic action of thecomplex reductive group
on C. By assumption this action is conjugate by anautomorphism to a linear action. The same automorphism clearly conjugates the. K-action. The other direction of theaction to the restriction to $K$ of the linear
equivalence is clear (see Introduction).
$\mathbb{C}^{n}$$\mathbb{C}^{n}$$K^{\mathbb{C}}$$K^{\mathbb{C}}$$\square$

For the last corollary of our ex.tension result recall that a Fatou-Bieberbach domain is. It seems to be unknown
which is biholomorphic toa proper open subset
which is invariant under thewhether there exists a Fatou-Bieberbach domain in-action byflow of a globally integrable holomorphic vectorfield, i.e., under a. We prove that it doesn’t exist in the special
holomorphic transformations on
case of a holomorphic vectorfield with periodic flow, i.e., of an action of the circle
group $S^{1}=\{z\in \mathbb{C} : |z|=1\}$ :$\Omega\subset \mathbb{C}^{n}$$\mathbb{C}^{n}$$\mathbb{C}^{2}$$(\mathbb{R}, +)$$\mathbb{C}^{2}([7])$which is biholomorphic to
Corollary 2.7. Let be a domain in. Thenunder an -action by holomorphic transformations on$\mathbb{C}^{2}$$\mathbb{C}^{2}$$\Omega$$S^{1}$$\mathbb{C}^{2}$and invariant$\Omega=\mathbb{C}^{2}$.

.
PROOF. By proposition 2.3 the -action extends to a holomorphic action ofare linearizable. So we can assume
A result of SUZUKI [25] states that -actions on
is of the form
after a holomorphic change of coordinates that the -action on$S^{1^{\mathbb{C}}}=\mathbb{C}^{\star}$$S^{1}$$\mathbb{C}^{2}$$\mathbb{C}^{\star}$$\mathbb{C}^{2}$$\mathbb{C}^{\star}$$\lambda,$$(z, w)\mapsto\lambda^{a}\cdot z,$$\lambda^{b}\cdot w$,$\lambda\in \mathbb{C}^{\star},$$(z, w)\in \mathbb{C}^{2}$

we see that is notfor some integers and . Applying proposition 2.3 to-invariant. We consider the following three cases:
only -invariant but also
Case 1: a and are both nonzero and have the same sign
has exactly one fixed point, the point $(0,0)$ . Since a differentiable
The action on
always admits a fixed point (see for instance [4] IV 1.5) we conclude-action onthat $(0,0)$ is contained in . Since is open it contains an open neighborhood ofinvariant we$(0,0)$ . Since every
has $(0,0)$ as limit point and is-orbit in.
have
Case 2: one of the numbers or is zero (say $a=0$ , hence $b\neq 0$ )$\Omega\cong \mathbb{C}^{2}$

$b$

$a$

$S^{1}$$\Omega$$\mathbb{C}^{\star}$$\mathrm{b}$$\mathbb{C}^{2}$$S^{1}$$R^{n}$$U$$\Omega$$\Omega$$\mathbb{C}^{2}$$\mathbb{C}^{\star}$$\mathbb{C}^{2}=\mathbb{C}^{\star}\cdot U\subset\Omega$

$a$

$b$

$\Omega$$\mathbb{C}^{\star}$

86
Compact and red uctive subgroups of the group of holomorphic au tomorphisms of$\mathbb{C}^{n}$

Like in case 1 we conclude that
containes a neighborhood of some fixed point$(z_{0},0)$ . Linearization at a fixed point shows that the actions on
andare locallyis linearizable too, in some coordinates
the same. Since the-action on(see also remark 3.2 below). Inon
it is of the same form like that onparticular the fixed point set is biholomorphic to C. Since an injective holomorphic
map from to is an biholomorphism we conclude that contains the fixed point
set
together with some open neighborhood . Again we haveand the desired conclusion like in case 1.
Case 3: a and are both nonzero and have different signs. The fibres of thisConsider the -invariant map :map are the closed -orbits and the cross of axis $\{zw=0\}$ . The composition of theinjection :with is by the big Picard theorem surjective or leaves out one
point. Like in the other cases we know that contains an neighborhood of the fixed
point $(0,0)$ hence the cross of axes. If
is surjective contains points from each-orbit inso we are done. If the map
leaves out one point we conclude{ closed -orbit}. But this contradicts the simply-connectedness of .
$\Omega$$\mathbb{C}^{2}$$\Omega$$\Omega\cong \mathbb{C}^{2}$$\mathbb{C}^{\star}$$\Omega\cong \mathbb{C}^{2}$$\mathbb{C}^{2}$$\mathbb{C}$$\Omega$$\mathbb{C}$$\{(z, 0)\in \mathbb{C}^{2} :$U$z\in \mathbb{C}\}$$\mathbb{C}^{\star}\cdot U=\mathbb{C}^{2}$$\mathrm{b}$$\mathbb{C}^{\star}$$\mathbb{C}^{2}arrow \mathbb{C}(z, w)\mapsto z^{|b|}\cdot w^{|a|}$$\pi$$\mathbb{C}^{\star}$$\dot{i}$$\Omega\llcorner_{arrow \mathbb{C}^{2}}$$\pi$$\Omega$$\pi\circ\dot{i}$$\mathbb{C}^{2}$$\mathbb{C}^{\star}$$\Omega=\mathbb{C}^{2}\backslash$$\Omega$$\pi\circ\dot{i}$$\Omega$$\mathbb{C}^{\star}$$\mathrm{o}\mathrm{n}\mathrm{e}$$\square$

REMARK 2.8. For $n\geq 3$ there are clearly Fatou-Bieberbach domains ininvariant. Just take any Fatou-Bieberbach domain
under an action ofinand. Clearly is a Fatou-Bieberbach domain
consider the domaininin
and it is invariant under any-action of the form $(z, w)\mapsto z,$ $\phi(t, w)$ ,where is some.
-action on$\mathbb{C}^{n}$$(\mathbb{R}, +)$$\Omega_{1}$$\Omega:=\Omega_{1}\cross \mathbb{C}^{n-2}$$\mathbb{C}^{n}$$\Omega$$\mathbb{C}^{n}$$(\mathbb{R}, +)$$t\in \mathbb{R},$ $z\in \mathbb{C}^{2},$$w\in \mathbb{C}^{n-2}$3.

$\mathrm{P}\mathrm{o}\mathrm{S}\mathrm{I}\mathrm{T}\mathrm{I}\mathrm{E}$$t,$$\mathbb{C}^{n-2}$$(\mathbb{R}, +)$$\phi$$\mathbb{C}^{2}$

RESULTS ON HOLOMORPHIC LINEARIZATION

For the convenience of the reader we first recall the notion of categorical quotient foran action $G\cross Xarrow X$ of a Lie group on a complex space $X$ .
$G$Definition. A complex space $X//G$ together with a -invariant holomorphic map: $Xarrow X//G$ is called categorical quotient for the action $G\cross Xarrow X$ if itsatisfies the following universality property:
For every holomorphic -invariant map :from $X$ to some complex-space there exists a unique holomorphic -invariant map :suchthat the diagram$G$$\pi_{X}$

$G$

$G$

$\psi$$Xarrow \mathrm{Y}$$G$$\mathrm{Y}$$\tilde{\psi}$$X//Garrow \mathrm{Y}$$arrow\psi$$\pi_{X}X\searrow$$\nearrow_{\overline{\psi}}\mathrm{Y}$$X//G$

commutes.
The existence of the categorical quotient in the case where is a complex reductive
group acting holomorphically on a Stein space $X$ was proved by SNOW [24] and if
is a compact group acting by holomorphic transformations on a Stein space $X$ by
HEINZNER [9]. As a topological space $X//G$ is just the topological quotient of $X$ withrespect to the equivalence relation associated to the algebraof G-invariant
holomorphic functions on $X$ :
$G$

$G$

$\mathcal{O}^{G}(X)$$\mathrm{R}$$R=\{(x, y)\in X\cross \mathrm{Y},$$f(x)=f(y)$$\forall f\in \mathcal{O}^{G}(x)$87
3. Positive results on holomorphic linearization
If is reductive and $X$ Stein, the map: $Xarrow X//G$ parametrices the closed Gorbits in $X$ , i.e., every-fibre containes exactly one closed -orbit and moreover
every -orbit in the fibre contains in its closure. If is a finite group the categoricalquotient is the same as the orbit quotient $X./G$ . The following easy example will be
used in chapter 3.
$G$$\pi_{X}$$G$$\pi_{X}$

$G$

$O$

EXAMPLE 3.1. Let$S^{1}$$\lambda,$

act on$\mathbb{C}^{2}$

$O$

$G$

by the rule$(z, w)\mapsto\lambda^{a}\cdot z,$$\lambda^{b}\cdot w$,$\lambda\in \mathbb{C}^{\star},$$(z, w)\in \mathbb{C}^{2}$

.

Since
the -invariant holomorphic functions on
are -invariant whereacts by the same rule. Hence they are constant on-orbits. If and havethe same sign all-orbits contain the point $(0,0)$ in its closure. So the value of a-invariant holomorphic function at some point $(z, w)$ is the same as the value at$(0,0)$ . So the invariant functions are constant and{one point}.
If and have different signs (and no common divisor to make the action effective)
all invariant holomorphic functions on
are functions in one variable of. So:
is given by.
$S^{1\mathbb{C}}=\mathbb{C}^{\star}$$S^{1}$$\mathbb{C}^{2}$$\mathbb{C}^{\star}$$\mathbb{C}^{\star}$$\mathbb{C}^{\star}$

$a$

$b$

$\mathbb{C}^{\star}$$S^{1}$$c^{2}//s^{1}=\mathbb{C}^{2}//C^{\star}=$

$a$

$b$

$z^{|b|}\cdot w|a|$$\mathbb{C}^{2}$$\pi_{\mathbb{C}^{2}}$$(z, w)\mapsto z^{|b|}\cdot w^{1}a|$$\mathbb{C}^{2}arrow \mathbb{C}^{2}//S^{1}=\mathbb{C}^{2}//\mathbb{C}^{\star}\cong \mathbb{C}$The categorical quotient $X//G$ is a normal complex space if $X$ is normal and contractible if $X$ is contractible [11]. So if
has dimension one it is necessarilysmooth and contractible and admits no bounded holomorphic functions, hence is biholomorphically to C.
Now we formulate some positive results about holomorphic linearization.
First of all we remark that a local linearization of an action of a compact group $K$
on a manifold $X$ in a $K$-invariant neighborhood of a fixed point is easyly achievedby averaging over the group $K$ (with respect to the Haar measure) a suitable local
biholomorpism from a neighborhood of in $X$ to a neighborhood of in the tangent
space $T_{x}X$ (see for instance [2] 2.2). For this note that differentiating the action map: $K\cross Xarrow X$ with respect to the first variable one gets a $K$-action on the tangent
bundle $TX$ hence a linear representation of $K$ on the tangent space $T_{x}X$ at the fixed
point . This implies the following useful observation$\mathbb{C}^{n}//G$

$x$

$U$

$0$

$x$

$\phi$$x$

REMARK 3.2. If an action of a compact group on
by holomorphic transformations
is linearizable, it is conjugate to the tangent representation at some fixed point.
$\mathbb{C}^{n}$Moreover if the reductive groupacts
is Stein one can extend this lo$K$
cal -equivariant biholomorphism to a-equivariant biholomorphism between
invariant neighborhoods, which are saturated with respect to the categorical quotient
maps. This is a special case of the holomorphic version of LUNA’S slice theorem ([9]$\mathrm{a}\mathrm{n}\dot{\mathrm{d}}X$$K^{\mathbb{C}}$$K^{\mathbb{C}}$$K^{\mathbb{C}_{-}}$

5.5).
From this one easy deduces the classical fact that holomorphic actions of a reductive
group onwith zero dimensional categorical quotient are linearizable, i.e., we are
speaking about the case thatis just one point or equivalently all G-invariant
holomorphic functions on
are constant.
The first remarkable result is due to SUZUKI [25] who gave a classification of theproper holomorphic-actions on. Here proper means that the limit set ofeach -orbit is finite.
-actions on a Stein space are proper in this sense.
$G$$\mathbb{C}^{n}$$\mathbb{C}^{n}//G$$\mathbb{C}^{n}$$(\mathbb{C}, +)$$\mathbb{C}$$\mathbb{C}^{2}$$\mathbb{C}^{\star}$

Theorem 3.3. Holomorphic actions of$\mathbb{C}^{\star}$on$\mathbb{C}^{2}$are linearizable.

88Compact an red uctive subgroups of the group of holomorphic automorphisms of$\mathbb{C}^{n}$$d$

By proposition 2.3 the same holds for -actions by holomorphic transformations on
for all. In chapter 3 we will give examples of nonlinearizable -actions on
is open.
on$n\geq 4$ . The linearization question for
The complex algebraic analogue of the linearization question for reductive groups
is studied quite well. We refer the interested reader to the papers by KRAFT [18], [17].
is open in general,
We only remark that algebraic linearization of -actions on$n\leq3[16]$. The first non-linearizable algebraic actions
but has a positive solution for
were found by SCHWARZ [23]. They come from non -trival $G$ -vectorbundles over(e.g.
representations. More concretely he proved that for some reductive groupsof the form
there exist algebraic actions on$S^{1}$$\mathbb{C}^{n}$$\mathbb{C}^{*}$$\mathbb{C}^{2}$$\mathbb{C}^{3}$$\mathbb{C}^{\star}$$\mathbb{C}^{n}$$\mathbb{C}^{\star}$

$G$

$G$

$\mathbb{C}^{n}$$O_{2}(\mathbb{C}))$$\varphi(g, Z)\cdot w$$g,$ $(z, w)\mapsto\alpha(g)\cdot z,$,$g\in G,$ $(Z, w)\in \mathbb{C}^{n}1\mathrm{x}\mathbb{C}^{n_{2}}=\mathbb{C}^{n}$,(1)

is a linear represenwhich are not algebraically linearizable. Here :is an algebraic map satisfying tho two conditions
tation and :$h\in G$ to make (1) an$\phi(e, z)=\dot{i}d$ and $\phi(g, \alpha(h)\cdot Z)\cdot\phi(h, Z)=\phi(gh, z)$ ,action. All examples of SCHWARZ have one-dimensional categorical quotient. His original aim together with KRAFT was to prove that algebraic actions with one dimensional
categorical quotient are algebraically linearizable. This turned out to be false, the first
counterexamples of the form (1) found by SCHWARZ have one dimensional quotient.
But they proved that algebraic actions with one dimensional categorical quotient are
holomorphically linearizable [20]. This result was generalized to the natural setting by
JIANG [15].
with one dimensional
Theorem 3.4. Holomorphic actions of reductive groups on
categorical quotient are linearizable.
$\alpha$$\phi$$Garrow GL_{n_{1}}(\mathbb{C})$$G\cross \mathbb{C}^{n_{1}}arrow GL_{n_{2}}(\mathbb{C})$$z\in \mathbb{C}^{n_{1}},$$g,$$\mathbb{C}^{n}$Note that this result generalizes theorem 3.3.
Oka principle implies that all
An application of an equivariant version ofholomorphic actions of reductive groups of the above form (1) are holomorphicallylinearizable. So the only up to now known method to construct counterexamples tocomplex algebraic linearization does not work in the holomorphic setting. To explain
this we introduce the following setting and notations:
Let $X$ be a complex space equipped with a holomorphic action of a reductive group. A holomorphic fibre bundle : $Earrow X$ over $X$ with structure group $H$ and typicalacts holomorphically on thefibre $F$ is called a holomorphic -bundle over $X$ if$E$total space such that the projection is equivariant and the automorphisms of theinduced from the -action are given by biholomorphic maps which
fibresbelong to $H$ . We call such bundles holomorphically -isomorphic if there exists a-equivariant holomorphic bundle isomorphism between them. Let $K$ be a maximal. The following theorem and corollary are due
compact subgroup of , i.e.,
to HEINZNER and the author [12]$\mathrm{G}\mathrm{r}\mathrm{a}\mathrm{u}\mathrm{e}\mathrm{r}\mathrm{t}‘ \mathrm{s}$

$G$

$G$

$\pi$

$G$

$G$

$\pi$

$G$

$E_{x}arrow E_{g\cdot x}$

$G$

$G$

$G$

$G=K^{\mathbb{C}}$bundles with a complex Lie group as structure
Theorem 3.5. Two holomorphic
group are holomorphically -isomorphic iff they are topologically K-isomorphic.
$G$

$G$

Corollary 3.6. Holomorphic actions of the form (1) are linearizable.

PROOF. The action (1) is ancomplex Lie group$G$-action on the vector bundle (structure group is theover the Stein manifold$GL_{n}(\mathbb{C}))E=\mathbb{C}^{n_{1}}\cross \mathbb{C}^{n_{2}}arrow C^{n_{1}}=X$

89
4. Non-linearizable actions
with linear action . Since the base is $K$ -equivariantly retractible to the
fixed point , an equivariant version of the covering homotopy theorem for compact
group actions implies that the bundle $E$ is topologically $K$-isomorphic to the bundle
equipped with the action$X=\mathbb{C}^{n_{1}}$$\alpha$$0$$F=\mathbb{C}^{n_{1}}\cross \mathbb{C}^{n_{2}}arrow C^{n_{1}}=X$$\varphi(g, 0)\cdot w$$g,$ $(z, w)\mapsto\alpha(g)\cdot z,$

,

(2).

$g\in G,$ $(Z, w)\in \mathbb{C}^{n}1\mathrm{x}\mathbb{C}^{n_{2}}=\mathbb{C}^{n}$linear representation of on the fibre over the fixed point. By theorem 3.5 there exists a holomorphic -equivarant biholomorphism
between the two bundles $E$ and $F$ . This is the automorphism of the total space
which conjugates the action (1) to the linear action (2).
Note that$G$$g-\varphi(g, 0)\mathrm{i}\mathrm{s}\cdot \mathrm{t}\mathrm{h}\mathrm{e}$$G$$0\in \mathbb{C}^{n_{1}}$$\mathbb{C}^{n}$$\square$is the subgroup of
For the last result recall that the group of overshears
generated by affine automorphisms and automorphisms of the form$\mathrm{S}\mathrm{h}_{\mathrm{n}}$$\mathrm{A}\mathrm{u}\mathrm{t}_{\mathrm{h}\circ}1(\mathbb{C}^{n})$$(z_{1}, \ldots, z_{n})\mapsto(a(z2, \ldots, Zn)z_{1}+b(_{Z}2, \ldots, Zn), z2, \ldots, Zn)$

and is invertible. By a theoare arbitrary holomorphic functions on
where
subgroup of
proper)(but
ANDERS\’EN
and LEMPERT this is a dense
rem of$n\geq 2[3]$ . The following theorem was proved by AHERN and RUDIN in the case offinite cyclic groups [1] and generalized by KRAFT and the author to the present rm
[19].
$a,$

$\mathbb{C}^{n-1}$

$b$

$a$

$\mathrm{A}\mathrm{u}\mathrm{t}_{\mathrm{h}\circ}1(\mathbb{C}^{n})$$\dot{\mathrm{f}}\mathrm{o}$Theorem 3.7. Every holomorphic action of a compact goup $K$ on
is linearizable.

$\mathbb{C}^{2}$by elements of$Sh_{2}$

In view of theorem 3.4 this result is only interesting for finite groups. By the methods explained in chapter 3 one can for instance prove that there are non-linearizable
for $m\geq n+2([5])$ . The
onholomorphic actions of finite cyclic groups
smallest dimension in which non-linearizable holomorphic actions are known is 4. So
is open.
the linearization question for finite groups on$C^{m}$$\mathbb{Z}/\mathbb{Z}_{n}$$\mathbb{C}^{2}$

4. NoN-LINEARIZABLE ACTIONS
The first counterexamples to the holomorphic linearization problem were constructedby DERKSEN and the author in [5]. There the following theorem is proved:(except the trivial group)
Theorem 4.1. For every complex reductive Lie groupsuch that for all $n\geq N_{G}$ there exists an effective
there exists a natural number.
non-linearizable holomorphic action of on$G$$N_{G}$$G$$\mathbb{C}^{n}$

We will not prove this theorem here in full generality. Instead we explain the main
. The construction uses theidea-restricting ourselves to the interesting casewhich are not equivalent to theexistence of holomorphic embeddings of into
standard embedding proved by FORSTNERIC, GLOBEVNIK and ROSAY in [8],$G=\mathbb{C}^{\star}$$\mathbb{C}^{2}$$\mathbb{C}$Definition. A proper holomorphic embedding
such that
there exists a$\varphi$:$\mathbb{C}^{k}arrow \mathbb{C}^{n}$is called straightenable if$\beta\in \mathrm{A}\mathrm{u}\mathrm{t}_{\mathrm{h}\circ 1()}\mathbb{C}^{n}$$\beta\circ\varphi(z1, \ldots, Zk)=(z_{1}, \ldots, z_{k}, 0, \ldots, 0)$

Since every automorphism ofis equivalent to the existence of$\mathbb{C}^{k}\cross\{0\}\subset \mathbb{C}^{n}$.

extends to an automorphism ofwith$\beta\in \mathrm{A}\mathrm{u}\mathrm{t}_{\mathrm{h}\circ 1()}\mathbb{C}^{n}$$\mathbb{C}^{n}$this$\beta(\varphi(\mathbb{C}^{k}))=\mathbb{C}^{k}\cross\{0\}\subset \mathbb{C}^{n}$.

90
Compact and red uctive$s\mathrm{u}$bgroups of the group of holomorphic au tomorphisms of$\mathbb{C}^{n}$

We will use the following special version of non-straightenable embeddings (see [5]Cor.2.4)
Theorem 4.2. There exists a non-straightenable proper holomorphic embedding::
such that in addition all embeddings
defined$\varphi_{k}(Z,y)=(\varphi(Z),
y)$
$k>0$
.
integeralltoonon-straightenable
are
by
for
$\mathbb{C}\cross \mathbb{C}^{k}arrow \mathbb{C}^{2}\cross \mathbb{C}^{k}$$\mathbb{C}^{\mathrm{L}}arrow \mathbb{C}^{2}$$\varphi_{k}$$\varphi$

In the proof of proposition 4.4 we will need the fact that if the image of an embedding
is contained in an linear subspace of high enough codimension this:embedding is straightenable which is made precise in the following lemma. For a proof
we refer to [5] (Lemma 2.5).
$\mathbb{C}^{k}\mathrm{L}arrow \mathbb{C}^{n}$$\varphi$

Lemma 4.3. If$\varphi_{1}$:$\varphi$:$\mathbb{C}^{k}arrow \mathbb{C}^{n}$$\mathbb{C}^{k}arrow \mathbb{C}^{n+k}=\mathbb{C}^{n}\cross \mathbb{C}^{k}$is a proper holomorphic embedding, then the embedding
defined by $\varphi_{1}(z)=(\varphi(z), 0)$ is straightenable.

like in the preceeding
From now on we fix a proper holomorphic embedding :precisely on
vanishes
which
holomorphic
function
theorem. Also we fix ainand the gradient of does not vanish when
the closed submanifold. Such a functionofgenerates the ideal
vanishes, in other words
exists by the solution of the multiplicative Cousin problem.
Define$X=\{(x, y, Z, w)\in \mathbb{C}^{4} : f(x, y)=w\cdot z\}$ .
$\mathbb{C}arrow \mathbb{C}^{2}$$\varphi$$f\in \mathcal{O}(\mathbb{C}^{2})$

$f$

$f$

$\mathbb{C}^{2}$$\varphi(\mathbb{C})$$f$$\varphi(\mathbb{C})$$I_{\varphi(\mathbb{C})(\mathbb{C}^{2})}$Since the gradient of does not vanish on $\{f(x, y)=0\}=\varphi(\mathbb{C})$ the set $X$ is a smoothsubmanifold of . The crucial point of our construction is the following$f$$\mathbb{C}^{4}$, whose zero set is biholoProposition 4.4. If is any holomorphic function onmorphic to , and with non-vanishing gradient on its zero set, then the manifold.
is biholomorphic to$\mathbb{C}^{2}$$f$$\mathbb{C}$$\mathbb{C}^{4}$$X\cross \mathbb{C}$PROOF. We have
$X\cross \mathbb{C}=\{(X, y, Z, w, u)\in \mathbb{C}^{5} :The map $x,$ $y,$ $w,$ $u\mapsto x,$ $y,$is the submanifold of
where
$z,$

$A_{1}$$w\cdot u,$ $z,$ $w,$ $u$$\mathbb{C}^{6}$

f(x, y)=w\cdot z\}$.

gives a biholomorphism$\tau_{1}$:$X\cross \mathbb{C}arrow A_{1}$defined by$A_{1}:=\{(x, y, v, z, w, u)\in \mathbb{C}^{6} :f(x, y)=w\cdot z, v=u\cdot z\}$is given by $(x, y, v, z, w, u)\mapsto(x, y, z, w, u)$ .
The inverse mappingwith
By Lemma 4.3 there exists an automorphism$\tau_{1}^{-1}$$\alpha\in \mathrm{A}\mathrm{u}\mathrm{t}_{\mathrm{h}\circ}1(\mathbb{C}^{3})$$\alpha(\varphi(\xi), 0)=(\xi, 0,0)$.

$(\star)$defined by $(x, y, v, z,w, u)\mapsto\alpha(x, y, v),$ $w,$ , i.e., in the
The automorphism ofafirst 3 coordinates and identity in the other 3 coordinates gives if restricted toby:definedsubmanifold
of
from
to
the
biholomorphism$\mathbb{C}^{6}$

$z,$

$u$

$A_{1}$$\alpha$$\tau_{2}$$A_{2}$$A_{1}$$A_{2}:=\{(a, b, C, Z, w,u)\in \mathbb{C}6 :$\mathbb{C}^{6}$(\alpha^{-}1)^{\star}f(a, b, c)=w\cdot z, (\alpha^{-}1)^{\star}v(a, b, C)=u\cdot z\}$generated by the two functions and is thethe Ideal in
Because of, namely the Ideal
andsame as the Ideal generated by the functions$(\star)$$\mathrm{b}$$\mathcal{O}(\mathbb{C}^{3})$$\mathrm{c}$$(\alpha^{-1})^{\star}v$$(\alpha^{-1})^{\star}f$

91
4. Non-linearizable actionsof the first coordinate axis. We can express one set of generators through the$\dot{i},j=1,2$ withother, that means we find holomorphic functions$I_{\mathbb{C}\cross\{\}}0$$a_{ij},$$b_{ij}\in \mathcal{O}(\mathbb{C}^{3})$,$c=a_{21}\cdot(\alpha^{-1})^{\star}f+a_{22}\cdot(\alpha^{-1})^{\star}v$,$(\alpha^{-1})^{\star}v=b_{21}\cdot b+b_{22}\cdot c$$A_{3}$of the holomorphic map$b=a_{11}\cdot(\alpha^{-1})^{\star}f+a_{12}\cdot(\alpha^{-1})^{\star}v$

(1)

and
$(\alpha^{-1})^{\star}f=b_{11}\cdot b+b_{12}\cdot c$

We claim that the restriction
defined byto$\tau_{3}$.

(2)$S_{1}$:$\mathbb{C}^{6}arrow \mathbb{C}^{6}$$S_{1}(a, b, C, z, w, u)=a,$ $b,$ $c,$ $z,$ $a_{1}1(a, b, c)\cdot w+a_{12}(a, b, c)\cdot u,$ $a21(a, b, C)\cdot w+a_{22}(a, b, c)\cdot u$

gives a biholomorphism$\tau_{3}$from$A_{2}$

to the submanifold$A_{3}:=\{(a, b, c, z, w, u)\in \mathbb{C}^{6} :$S_{2}$

We will prove that the inverse
defined by:$\tau_{3}^{-1}$$A_{3}$

ofdefined by$\mathbb{C}^{6}$

b=w\cdot z, c=u\cdot z\}$.

: $A_{3}arrow A_{2}$ is given by the restriction of$\mathbb{C}^{6}arrow \mathbb{C}^{6}$$S_{2}(a, b, c, Z, w, u)=a,$ $b,$ $C,$ $z,$ $b11(a, b, C)\cdot w+b_{12}(a, b, C)\cdot u,$ $b21(a, b, C)\cdot w+b_{22}(a, b, c)\cdot u$to$A_{3}$. For that consider the holomorphic maps$\psi_{1}:\mathbb{C}^{3_{\mathrm{X}\mathbb{C}}}\stararrow \mathbb{C}^{6}$,$a,$ $b,$ $c,$ $z\mapsto a,$ $b,$ $c,$ $z,$$\frac{(\alpha^{-1})^{\star}f}{z},$$\frac{(\alpha^{-1})^{\star}v}{z}$and$\psi_{2}:\mathbb{C}^{3}\cross \mathbb{C}^{\star}arrow \mathbb{C}^{6}$,$a,$ $b,$ $c,$ $z\mapsto a,b,$ $c,$ $z,$$\frac{b}{z},$$\frac{c}{z}$.

The submanifold
is the topological closure (and therefore the holomorphic closure, i.e.,the smallest analytic set containing) of the image ofin . Alsois thein .
closure of the image of$\psi_{2}=S_{1}\circ\psi_{1}$and (2) implies $\psi_{1}=S_{2}\circ\psi_{2}$ . So we have
From (1) follows. This means thatis the identity on the image ofhence. Analogously follows that
it is the identity on the closure of the image of , onis the identity on .
, $(a, b, c, z, w, u)\mapsto a,$ $w,$ is a biholomorphism.
Finally the map:
provides the desired biholomorphism from $X$ to.
The composition$A_{2}$$\psi_{1}$$\psi_{2}$$A_{3}$$\mathbb{C}^{6}$$S_{1}\circ S_{2}$$\psi_{1}=s_{2^{\mathrm{O}}}s_{1^{\mathrm{O}\psi 1}}$$\psi_{1}$$A_{2}$$\psi_{1}$$S_{2}\circ S_{1}$$\mathbb{C}^{6}$$A_{3}$$\tau_{4}$$A_{3}arrow \mathbb{C}^{4}$

$u$

$z,$

$\mathbb{C}^{4}$$\tau_{4^{\circ \mathcal{T}}3}\circ\tau_{2^{\circ}}\mathcal{T}_{1}$$\square$. We do not know whether the manifold(ifitself is biholomorphic tois straightenable then this is clearly true). If for some non-straightenable$\{f=0\}\cong \mathbb{C}$ the manifold $X$ would be biholomorphic tothen we would have a(see the proof of proposition 4.7 below), if it isnon-linearizable -action onthen this would be a counterexample to the following open
not biholomorphic toproblem.
$\mathrm{R}\mathrm{E}\mathrm{M}\overline{\mathrm{A}}\mathrm{R}\mathrm{K}4.5$$X$$\mathbb{C}^{3}$$\{f=0\}\cong \mathbb{C}$$\mathbb{C}^{3}$$X\cong \mathbb{C}^{3}$$\mathbb{C}^{\star}$$\mathbb{C}^{3}$

Problem 4.6 (Holomorphic Zariski Cancellation Problem). Let $Z$ be a complex mani?
. Does it follow thatis biholomorphic tofold such that
$Z\cross \mathbb{C}$$\mathbb{C}^{n+1}(n\geq 2)$$Z\cong \mathbb{C}^{n}$92
Compact and reductive$s\mathrm{u}$bgro$\mathrm{u}ps$of the group of holomorphic au tomorphisms of

Now we come to the non-linearizable$k\geq 1$ a biholomorphism:
stable under the linear -action on$\mathbb{C}^{\star}$-actions. Since. The submanifold
given by$\mathbb{C}^{\star}$

we have for each$X\cross \mathbb{C}\cong \mathbb{C}^{4}$$X\cross \mathbb{C}^{k}\cong \mathbb{C}^{k+3}$$\alpha_{k}$$\mathbb{C}^{n}$$X\cross \mathbb{C}^{k}\subset \mathbb{C}^{4}\cross \mathbb{C}^{k}$is$\mathbb{C}^{4}\cross \mathbb{C}^{k}$$\mathbb{C}^{\star}\cross \mathbb{C}^{4_{\mathrm{X}\mathbb{C}}k}arrow \mathbb{C}^{4}\cross \mathbb{C}^{k}$$\lambda\cdot(x, y, z, w, t_{1}, \ldots,t_{k})=(x, y, \lambda Z, \lambda^{-}1t_{1}w,, \ldots, t_{k})$The restriction of this action to$\sigma_{k}$:$X\cross \mathbb{C}^{k}$.

$\mathbb{C}^{\star}\cross \mathbb{C}^{k+3}arrow \mathbb{C}^{k+3}$

induces via$\alpha_{k}$.

a holomorphic$\mathbb{C}^{\star}$-action

Proposition 4.7. The action
is not linearizable $(k\geq 1)$ . So for all $l\geq 4$ there
action on .
exists a non-linearizable$\sigma_{k}$$\mathbb{C}^{l}$$\mathbb{C}^{\star}$:is a biholomorphic -equivariant map, where
PROOF. Suppose. This representation of on
must be isomorphic to theacts linearly on(see remark
representation of
on the tangent space of some fixed point of
3.2). With respect to some coordinates, this action is given by$X\cross \mathbb{C}^{k}arrow \mathbb{C}^{3+k}\underline{\simeq}$$\mathbb{C}^{\star}$$\alpha_{k}$$\mathbb{C}^{3+k}$$\mathbb{C}^{\star}$$\mathbb{C}^{k+3}$$\mathbb{C}^{\star}$$X\cross \mathbb{C}^{k}$$\mathbb{C}^{\star}$$\lambda\cdot(_{Z,w,u_{\iota},\ldots,u_{k+1}})=(\lambda z, \lambda^{-1}w, u1, \ldots, uk+1)$.

of a -invariant closed subspace of a SteinThe categorical quotient:-space $Z$ is the restriction of the categorical quotient: $Zarrow Z//G$ . This follows
from the fact that all -invariant functions on extend to -invariant holomorphic
functions on $Z$ (first extend to some holomorphic function, then average over the). Using this together with example 3.1 one
maximal compact subgroup $K$ of,:sees that the categorical quotient ofis given by$\pi_{\mathrm{Y}}$$G$$\mathrm{Y}arrow \mathrm{Y}//G$$\mathrm{Y}$$G$$\pi_{Z}$

$G$

$G$

$\mathrm{Y}$$G=K^{\mathbb{C}}$$X\cross \mathbb{C}^{k}arrow \mathbb{C}^{2+k}$$X\cross \mathbb{C}^{k}$$\pi_{X\cross \mathbb{C}^{k}}$$\pi_{X\cross \mathbb{C}^{k}}(x, y, z, w, t_{1}, \ldots, t_{k})=(x, y, t_{1}, \ldots, tk)$

and the categorical quotient of$\mathbb{C}^{3+k}$

is given by$\pi_{\mathbb{C}^{\mathrm{s}+k}}$:$\pi_{\mathbb{C}^{3+k}}(Z, w, u_{1}, \ldots, uk+1)=(Zw, u1, \ldots, uk+1)$

The fixed point set$(X\cross \mathbb{C}^{k})^{\mathbb{C}^{*}}$.

is$\{(x, y, z, w, t1, \ldots,t_{k})\in X\cross \mathbb{C}^{k}|f(x, y)=z=w=0\}$

Its image under$\pi_{X\cross \mathbb{C}^{k}}$is,$\mathbb{C}^{3+k}arrow \mathbb{C}^{2+k}$$\varphi(\mathbb{C})\cross \mathbb{C}^{k}\subset \mathbb{C}^{2}\cross \mathbb{C}^{k}$. On the other hand.

$(\mathbb{C}^{3+k})^{\mathbb{C}^{*}}$

is$\{(z, w, u_{1}, \ldots, uk+1)\in \mathbb{C}^{3+}k|z=w=0\}$and its image underis. Since equivariant maps map thefixed point set into the fixed point set, the -equivariant biholomorphism:
induces a biholomorphism :of the categoricalquotients such that the image of the fixed point set underis mapped onto the, i.e.,. Thisimage of the other fixed point set underaccording to theorem 4.2.
contradicts the choice of$\pi_{\mathbb{C}^{s+k}}$$\{0\}\cross \mathbb{C}^{1+k}\subset \mathbb{C}^{2+k}$$\mathbb{C}^{\star}$$X\cross \mathbb{C}^{k}arrow \mathbb{C}^{3+k}\underline{\simeq}$$\mathbb{C}^{2+k}arrow \mathbb{C}^{2+k}\underline{\simeq}$$\alpha_{k}$$\gamma$$\pi_{X\mathrm{x}\mathbb{C}^{k}}$$\pi_{\mathbb{C}^{\mathrm{s}+k}}$$\varphi(\mathbb{C})$$\gamma(\varphi(\mathbb{C})\cross \mathbb{C}^{k})=\{0\}\cross \mathbb{C}^{k+1}$$\square$93
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$de\iota A_{-}x$