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% Author Package file for use with AMS-LaTeX 1.2
\controldates{22-JUL-1997,22-JUL-1997,22-JUL-1997,22-JUL-1997}
 
\documentstyle[newlfont,draft]{era-l}
\issueinfo{3}{09}{January}{1997}
\dateposted{July 31, 1997}
\pagespan{63}{71}
\PII{S 1079-6762(97)00025-5}
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%\newtheorem{prop}{Proposition}[section]
\newtheorem{prop}[subsection]{Proposition}

%\newtheorem{thm}{Theorem}[section]
\newtheorem{thm}[subsection]{Theorem}
\newtheorem*{linthm}{Linearization Theorem}

\theoremstyle{remark}
\newtheorem*{stepi}{Step I, {\em the quotient}}
\newtheorem*{stepii}{Step II, {\em reduction of weights}}
\newtheorem{rmk}{Remark}[subsection]
\catcode`\@=11

\theoremstyle{definition}
\newtheorem*{hcase}{Hyperbolic Case}

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\begin{document}
\title{$\Bbb C^*$-actions on $\Bbb C^3$ are linearizable} 
%\markboth{S. KALIMAN, M. KORAS, L. MAKAR-LIMANOV, AND P. RUSSELL}
%{$\C^{\hbox{*}}$-ACTIONS ON $\C^{\hbox{3}}$ ARE LINEARIZABLE}

%\setlength{\parskip}{1mm}
%\begin{center}{\Large\bf $\C^*$-actions on $\C^3$ are linearizable}\\
%by\\
%S. Kaliman 
%\footnote {Partially supported by NSA grant}, 
%M. Koras, L. Makar-Limanov and P. Russell
%\end{center}

\author{S. Kaliman}
\address{Department of Mathematics \& Computer Science,
University of Miami,
Coral Gables, FL 33124}
\email{kaliman@paris-gw.cs.miami.edu}

\author{M. Koras}
\address{Institute of Mathematics, Warsaw University,
Ul. Banacha 2,
Warsaw,
Poland}
\email{koras@mimuw.edu.pl}

\author{L. Makar-Limanov}
\address{Department of Mathematics \& Computer Science,
Bar-Ilan University,
52900 Ramat-Gan,
Israel, and 
Department of Mathematics,
Wayne State University,
Detroit, MI  48202}
\email{lml@bimacs.cs.biu.ac.il; lml@math.wayne.edu}

\author{P. Russell}
\address{Department of Mathematics \& Statistics,
McGill University,
Montreal, QC,
Canada, and 
Centre Interuniversitaire,
en Calcul Math\'ematique,
Alg\'ebrique (CICMA)}
\email{russell@Math.McGill.CA}

\thanks{The first author was partially supported by an NSA grant}
\commby{Hyman Bass}

\date{March 5, 1997}

\subjclass{Primary 14L30}

%\issueinfo{3}{1}{June}{1997}

\copyrightinfo{1997}{American Mathematical Society}

\begin{abstract}
We give the outline of the proof of
the linearization conjecture:
every algebraic $ %{\bf 
\C^*$-action on $ %{\bf 
\C^3$ is linear in 
a suitable coordinate system.
\end{abstract}

\maketitle


%\vspace{1.0cm}\bigskip\begin{center}{\bf 1.\quad Introduction}\end{center}\medskip
\section{Introduction}

The purpose of this note is to outline the main ingredients in a 
proof of the 
following

%\vspace{.1cm}{\bf  
\begin{linthm} %:\quad 
Every algebraic action of the torus 
$T=\C^*$ on 
affine space $X=\C^3$ is linearizable, that is linear in suitably 
chosen coordinates for $X$.
\end{linthm}
%\vspace{.1cm}
It is known that the action has a fixpoint $0\in X$ (\cite{B-B}).  The 
{\em weights} of 
the action are the weights
%$$
\[
a_1,\ a_2,\ a_3
%$$
\]
of the (diagonalized) action on the tangent space $T_0X$.  (They are 
independent of the choice of fixpoint \cite{KbR}.)  We will assume tacitly 
that the action is effective, or, 
equivalently, that $\GCD(a_1,
a_2, a_3)=1$.  Put
%$$
\[
\delta =\dim X//T,\quad \tau = \dim X^T.
%$$
\]
Then $2\ge\delta\ge\tau\ge 0$.

%\vspace{.1cm}
It is known that {\em fixpointed} actions, that is those for which 
all weights have 
the same sign, are linearizable  \cite{KbR}.  This settles the following 
cases:

%\vspace{.1cm}
$\delta =0=\tau$, or three nonzero weights of the same sign;

$\delta =1=\tau$, or one zero weight, two nonzero weights of the 
same sign;

$\delta =2=\tau$, or two zero weights, one nonzero weight.
%\vspace{.1cm}

\noindent The case

%\vspace{.1cm}
$\delta=2, \tau=1$, or one zero weight, two nonzero weights of 
opposite sign,
%\vspace{.1cm}

\noindent was settled in \cite{KR1}.

%\vspace{.1cm}
It remains to consider the 

%{\bf Hyperbolic Case}:\quad
\begin{hcase} $\delta=2, \tau=0$, or three nonzero 
weights, not 
all of the same sign.
\end{hcase}
%\vspace{.2cm}
A program to settle this case was proposed in \cite{KR2}.  It has two 
quite distinct 
components.

%\vspace{.1cm}{\bf Step I}, {\em the quotient}.  
\begin{stepi} Show that $X//T$ is as expected 
for a linear 
action, i.e.
%$$
\[
X//T\simeq T_0X//T.
%$$
\]
\end{stepi}

Let $\omega_\alpha\subset\C^*$ be the group of $\alpha $-roots of 
1.  
Linearizability follows from Step I (see \ref{1:4} below) in the case $\dim 
X^{w_\alpha}\le 1$ for all $ \alpha >1$, or 
equivalently, if the weights are pairwise relatively prime. This 
leads to

%\vspace{.1cm}{\bf Step II}, {\em reduction of weights}. 
\begin{stepii} Reduction of the proof to 
the case of 
pairwise relatively prime weights.
\end{stepii}

%\vspace{.1cm}
If $\alpha >1$ and $\alpha$ divides two weights, then 
$X'=X/\omega_\alpha $ is a 
smooth, affine threefold, but only after linearizability has been 
established is it at all 
clear that $X'\simeq\C^3$. We are therefore led to study more general 
$\C^*$-threefolds.

%\vspace{.2cm}
%{\bf 1.1\quad Standard conditions}: 
\subsection{Standard conditions}\label{1:1} Let $X$ be a $\C^*$-threefold.  
We consider the 
following conditions.

%\vspace{.1cm}
(i) $X$ is smooth and the action of $T=\C^*$ is {\em hyperbolic},
%$$
\[
%\mbox
\text{i.e. there is a unique fixpoint}\ 0\ %\mbox
\text{and}\ \dim X//T=2.
%$$
\]

(ii) $X$ is contractible.

(iii) $\ov\kappa (X)=-\infty $ ($\ov\kappa =$ logarithmic Kodaira 
dimension).

%\vspace{.1cm}
If we have \ref{1:1} (i), the weights of the action are defined as above for 
$X=\C^3$, and we assume 
%$$
\[
a_1 <0,\ a_2>0,\ a_3>0,\qquad \GCD(a_1,a_2,a_3)=1.
%$$
\]
We put
%$$
\[
\alpha _i=\GCD(\{ a_1, a_2, a_3\}-\{ a_i\}).
%$$
\]
Then
%$$
\[
-a_1=a\alpha_2\alpha_3, \quad a_2=b\alpha_1\alpha_3, \quad a_3=c\alpha_1\alpha_2
%$$
\]
with $a,b,c>0$ and {\em reduced} (pairwise relatively prime).
%\vspace{.2cm}{\bf 1.2\quad Proposition} (\cite{KR3}, 2.5):
\begin{prop}[{\cite{KR3}, 2.5}]\label{1:2}
 Let $X$ satisfy \ref{1:1} (i).

%\vspace{.1cm}
(i) Suppose $\alpha_i>1$. Then $\dim X^{\omega_{\alpha_i}}=2$ and
%$$
\[
X'=X/\omega_{\alpha_i}
%$$
\]
satisfies \ref{1:1} (i) for $T'=T/\omega_{\alpha_i}\simeq\C^*$ with weights 
$a_i$ and $a_j/\alpha _i$ for $j\ne i$.

(ii) $X^\# =X/\omega_{\alpha_1\alpha_2\alpha_3}$ satisfies \ref{1:1} (i) 
for $T^\#
=T/\omega_{\alpha_1\alpha_2\alpha_3}\simeq\C^*$ and {\em reduced} 
weights $-a,b,c$.

(iii) $X//T=X'//T'=X^\# //T^\#$.

(iv) If $X$ satisfies \ref{1:1} (ii) or (iii), then so do $X'$ and 
$X^\# $.
\end{prop}

%\vspace{.2cm}
Let $X$ satisfy \ref{1:1} (i) and (ii).  We put (\cite{KR3}, 1.4)
%$$
\[
X^+=\{x\in X |\lim_{t\rightarrow 0}t\cdot x=0\}.
%$$
\]

Then $X^+\simeq\C^2$ and $X^+=F^{-1}(0)$, where $F$ is semiinvariant 
of 
weight 
$a_1.\ \omega_{a_1}$ acts on 
%$$
\[
X_1=F^{-1}(1)
%$$
\]
and we have (\cite{KR1}, Lemma 2)
%\vspace{.2cm}{\bf 1.3} \hskip 2truein 
\subsection{} $X//T\simeq X_1/w_{a_1}$.

%\vspace{.2cm}
The reduction of the proof to Steps I and II is now contained in
%\vspace{.2cm}{\bf 1.4 Proposition} ([KR3] ): 
\begin{prop}[{\cite{KR3}, 2.3, 2.8, and 1.10}]\label{1:4} Let $X$ satisfy 
\ref{1:1} (i) and (ii) and suppose the weights are reduced. If
%$$
\[
X//T\simeq T_0 X//T,
%$$
\]
or equivalently
%$$
\[
X_1/\omega_a\simeq\C^2/\omega_a,
%$$
\]
where $\omega_a$ acts diagonally on $\C^2$ with weights $\equiv b,c
\mod a$, then
%$$
\[
X_1\simeq\C^2,
%$$
\]
and
%$$
\[
X\simeq_e\C^3
%$$
\]
($X$ is equivariantly isomorphic to $\C^3=T_0X$).
\end{prop}
%\vglue .5truein\bigskip\bc{\bf 2.\quad The quotient}\ec\medskip
\section{The quotient}
%\vspace{.2cm} {\bf 2.1 Theorem} (\cite{KR4}, 1.2): 
\begin{thm}[{\cite{KR4}, 1.2}]\label{2:1} Suppose $X$ satisfies all conditions 
of \ref{1:1}. Then
%$$
\[
S'=X//\C^*\simeq T_0 X//\C^* .
%$$
\]
\end{thm}

By \ref{1:2}, we may assume the weights are reduced when studying the 
quotient. 
Also, \ref{2:1} is known (\cite{KR2}) when $S'$ is
smooth, or equivalently $a=1$. So we assume $a>1$. Then by \cite{KR4}, 2.4
%\vspace{.2cm}{\bf \ref{2:2}} 
\subsection{}\label{2:2} $S'$ is contractible, $\ov\kappa(S')=-\infty,\ S'$ has a 
unique singular point $q,\ q$ is analytically of the
type of the origin in $\C^2/\omega_a$, and Pic$(S'-q)\simeq\Z/a\Z$.
%\vspace{.2cm}{\bf 2.3 Theorem } (\cite{K}): 
\begin{thm}[{\cite{K}}]\label{2:3} If $S'$ is as in \ref{2:2}, then

%\vspace{.1cm}
(i) if $\ov\kappa(S'-q)=-\infty$, then $S'\simeq\C^2/\omega_a$,

%\vspace{.1cm}
(ii) $\ov\kappa(S'-q)\neq 0,1$.
\end{thm}

%\vspace{.2cm}
It remains to rule out $\ov\kappa(S'-q)=2$ to complete Step I.

%\vspace{.2cm}{\bf 2.4 Theorem} (\cite{KR4}, \ref{1:1}):
\begin{thm}[{\cite{KR4}, 1.1}]\label{2:4} Let
%$$
\[
S'=X//T
%$$
\]
with $X$ satisfying all conditions of \ref{1:1}. Then
%$$
\[
\ov\kappa(S'-q)<2.
%$$
\]
\end{thm}

%\vspace{.2cm}
The proof is rather involved. It relies in a crucial way on the 
theory of open algebraic
surfaces, in particular the inequalities of Miyaoka \cite{M} and Kobayashi 
\cite{Ko} and the results on the existence of affine
rulings of Miyanishi and Tsunoda \cite{MT}.
%\vspace{.2cm}{\bf 2.5 Proposition} (\cite{KR4}, 2.8): 
\begin{prop}[{\cite{KR4}, 2.8}]\label{2:5} Let $S'$ be as in \ref{2:4}. There 
exists a desingularization $S$ of $S'$ admitting an
$\A^1$-ruling with all but one component $E$ of the exceptional locus 
$\hat E$ in fibres. Moreover, $S-\Delta$ is
simply connected, where $\Delta=\hat E-E$.
\end{prop}

%\vspace{.2cm}
The proof of \ref{2:4} proceeds by a detailed analysis of such ``good'' 
rulings under the conditions of \ref{2:2}. 

%\vglue .5truein\newpage\bigskip\bc{\bf 3.\quad
\section{Reduction of weights and ``exotic affine spaces''} %\ec\medskip

%\vspace{.2cm} 
Step II, the reduction of weights, is achieved in a roundabout way. 
In \cite{KR3}, an explicit construction is given of a
class of smooth, contractible $\C^*$-threefolds that encompasses, in 
the equivariant sense, all possible
counterexamples to linearization. It is then shown in \cite{KM-L} that 
only the ``obviously'' equivariantly trivial threefolds
in the class are isomorphic to $\C^3$ (without reference to the 
$\C^*$-action). The others are in themselves
interesting examples of ``exotic affine
spaces'' (algebraic varieties homeomorphic to $\C^3$). They include 
the threefolds described in \cite{D}, 4.36.

%\vspace{.2cm}{\bf 3.1 Theorem} (\cite{KR3}, 4.1): 
\begin{thm}[{\cite{KR3}, 4.1}]\label{3:1} The threefolds
%$$
\[
X=%\mbox
\Spec A
%$$
\]
satisfying \ref{1:1} (i) and (ii) and
%$$
\[
X//\C^*\simeq T_0 X//\C^*
%$$
\]
are precisely the ones obtained as follows.

%\vspace{.1cm}
(1) Let
%$$
\[
-a=a'_1,\ b=a'_2,\ c=a'_3
%$$
\]
be a triple of reduced weights with $a,b,c>0$. (These define a 
hyperbolic $\C^*$-action on
%$$
\[
W=%\mbox
\Spec B\simeq\C^3
%$$
\]
with $B=\C[\eta,\xi,\zeta]$ and $\eta,\xi,\zeta$ homogeneous of 
weight $-a,b,c$).

%\vspace{.1cm}
(2) Let
%$$
\[
\alpha_1,\alpha_2,\alpha_3
%$$
\]
be a reduced triple of positive integers with $\GCD(\alpha_i, 
a'_i)=1,\ i=1,2,3$.

%\vspace{.1cm}
(3) Let $C_2$ and $C_3$ be $\omega_a$-homogeneous ``lines'' (curves 
isomorphic to $\C$) in $W_1=%\mbox
\Spec
k[\xi,\zeta]\simeq\C^2$, identified with $\eta^{-1}(1)\subset W$, 
such that

%\vspace{.2cm}
(i) $C_2$ and $C_3$ meet normally in $r\geq 1$ points, including the 
origin,

%\vspace{.1cm}
(ii) $U_i=\overline{\C^*\cdot C_i}\subset W$ is smooth, $i=2,3$.

%\vspace{.2cm}
(4) Let $U_1=W^+=\eta^{-1}(0)$.

%\vspace{.2cm}
Then $X$ is the ``tri-cyclic'' cover of $W$ ramified to order 
$\alpha_i$ over $U_i,\
i=1,2,3$, that is,
%$$
\[
A=B[z_1,z_2,z_3],
%$$
\]
where $z^{\alpha_i}_i=u_i$ with $u_1=\eta$ and for $i=2,3,\ u_i$ is 
an equation for $U_i$ and uniquely
determined by
%$$
\[
s^{-a'_i}f_i(\xi s^{a'_2},\zeta s^{a'_3})=u_i(s^{-a'_1},\xi,\zeta),
%$$
\]
where $f_i$ is an equation for $C_i\subset W_1$.

%\vspace{.2cm}
Moreover,
%$$
\[
B=\C[u_1,u_2,u^*_3]=\C[u_1,u^*_2,u_3],
%$$
\]
with $u_i$ and $u^*_i$ homogeneous of weight $a'_i$ and if
\[ %\begin{array}{rl}
u_2=G_2(u_1,u^*_2,u_3)\quad%\mbox
\text{and}\quad
u_3=G_3(u_1,u_2,u^*_3),
\]
then the equations
\[ %\begin{array}{l}
z^{\alpha_2}_2=G_2(z^{\alpha_1}_1,z^*_2,z^{\alpha_3}_2)
\quad%\mbox
\text{and} \quad
z^{\alpha_3}_3=G_3(z^{\alpha_1}_1,z^{\alpha_2}_2,z^*_3)
\]
describe $X$ (in two ways) as a hypersurface in $\C^4$.
\end{thm}
%\vspace{.2cm}{\bf 3.1.1 Remark} (i):
\begin{rmk} 1) (3)(ii) imposes a rather mild restriction 
that can be 
made quite explicit (\cite{KR3}, 1.11.1).

2) Possibilities for $f_2,f_3$, and hence for $G_2,G_3$, can be 
worked out explicitly with the help of the {\em
epimorphism theorem} of Abhyankar, Moh and Suzuki \cite{AM}, \cite{S}.
\end{rmk}
%\vspace{.2cm}
The key to \ref{3:1} is the following observation.

%\vspace{.2cm}{\bf 3.2 Proposition} (\cite{KR3}, 2.6, 2.7): 
\begin{prop}[{\cite{KR3}, 2.6, 2.7}]\label{3:2} Suppose $X$ is as in \ref{3:1} and 
$\alpha_2=1,\ \alpha_3>1$. Then
%$$
\[
X/\omega_{\alpha_3}\simeq_e\C^3\ %\mbox
\text{implies}\ X\simeq_e\C^3 .
%$$
\]
A similar result holds if $\alpha_2>1,\ \alpha_3=1$. Also,
%$$
\[
X/\omega_{\alpha_1}\simeq_e\C^3\ %\mbox
\text{implies}\ X\simeq_e\C^3.
%$$
\]
\end{prop}

In view of \ref{1:4} we obtain a commutative diagram
%\newpage
\[\begin{array}{ccccc}
 & &X& & \\
 & & & & \\
 &\swarrow& &\searrow& \\
 & & & & \\
\C^3\simeq_eX/\omega_{\alpha_2}& & & 
&X/\omega_{\alpha_3}\simeq_e\C^3\\
 & & & & \\
 &\searrow& &\swarrow& \\
 & & & & \\
 & &X/\omega_{\alpha_2\alpha_3}\simeq_e\C^3& & \\
 & & & & \\
 & &\downarrow& & \\
 & &X/\omega_{\alpha_1\alpha_2\alpha_3}\simeq_e\C^3& &
\end{array}\]

\ref{3:1} is an elaboration of the possibilities for such a diagram. It is 
not difficult to decide when $X$ is
equivariantly isomorphic to $\C^3$ (see \ref{3:4}). The question of just 
isomorphism with $\C^3$, on the other hand,
proved to be much more elusive.
%\vspace{.2cm}
%{\bf 3.3} 
\subsection{} Let us for instance choose $a=b=c=1,\ 
\alpha_2=2$ and $\alpha_3=3$ and a
parabola and straight line for $C_2$ and $C_3$. Then in suitable 
coordinates $X$ is defined in $\C^4$ by
%$$
\[
x+x^2y+z^2+t^3=0.
%$$
\]

$X$ is dominated birationally by $\C^3$ and there exists a surjective 
quasi-finite map $\C^3\rightarrow X$ (\cite{KR3},
7.7 and 7.8).
It is shown in \cite{M-L1}
that, nevertheless, $X$ is not isomorphic to $\C^3$.
The proof is based implicitly on the computation of
the following invariant:
%$$
\[ AK(X) = \bigcap_{\partial \in LND(X)} %{\rm Ker}
\ker \partial %$$
\]
where $LND(X)$ is the set of locally nilpotent derivations
on the ring $\C [X]$ of regular functions on $X$. 
For this hypersurface $AK(X) \ne \C$, but clearly $AK(\C^3 ) = \C$.\\

%\vspace{.2cm}
Let $X$ be as in \ref{3:1}. We define
%$$
\[
\varp=(r-1)(\alpha_2-1)(\alpha_3-1)
%$$
\]
($\varp=$ rank $\pi_2(X-X^+)$ is an invariant of the higher-dimensional 
knot $(X,X^+)$ (\cite{KR3}, 4.8)).
%\vspace{.2cm}{\bf 3.4 Theorem} (\cite{KR3}, remark following 5.1):
\begin{thm}[{\cite{KR3}, remark following 5.1}]\label{3:4}
 Let $X$ be as in 
\ref{3:1}. Then $X\simeq_e\C^3$ if and only if
$\varp=0$.
\end{thm}
%\vspace{.2cm}{\bf 3.5 Theorem} (\cite{KM-L}):
\begin{thm}[{\cite{KM-L}}]\label{3:5} Let $X$ be as in \ref{3:1}. If $\varp>0$, then 
$X\not\simeq\C^3$. %\\
\end{thm}

%\vspace{.2cm}
If now $X$ is $\C^3$ with a hyperbolic $\C^*$-action, then by \ref{2:1} it 
is one of the $X$ in \ref{3:1} and hence
$X\simeq_e\C^3$ by \ref{3:4} and \ref{3:5}.

%\vglue .5truein\bigskip\bc{\bf 4.\quad The computation of $AK(X)$}\ec\medskip
\section{The computation of $AK(X)$}
%\vspace{.2cm} {\bf 4.1} 
\subsection{}\label{4:1} Theorem \ref{3:5} is again the 
consequence of the fact that $AK(X) \ne \C$ \cite{KM-L}.
More precisely, $AK(X) = \C [X]$ unless $X$ is isomorphic to
a hypersurface in $\C^4$ given by one of the following equations:
\[ \begin{array}{rr}
%\mbox
\text{(i)}& \, \, \, \, \, x+x^ky+z^{\alpha_2}+t^{\alpha_3}=0\quad 
%{\rm
\text{or}\\
\qquad \\
%\mbox
\text{(ii)}& \, \, \, \, \, x+y(x^k+z^{\alpha_2})^l+t^{\alpha_3}=0 
\end{array} \]
where $k \geq 2, l \geq 1,\ %{\rm 
\text{and in the second equation}\ (kl, 
\alpha_3)=1$.
In case (i) $AK(X)$ is the restriction of $\C [x]$ to $X$
and in case (ii) $AK(X)$ is the restriction of $\C [x,z]$ to $X$.

%\vspace{.2cm}{\bf 4.2} 
\subsection{}\label{4:2} The scheme of the computation of $AK(X)$ is
discussed below.
Every $X$ from \ref{3:1} is the hypersurface $P(x,y,z,t)=0$
where 
\[ %\begin{array}{rl}
(x,y,z,t)=(z_3^*,z_1,z_2,z_3) \quad %{\rm 
\text{and}\quad
P(x,y,z,t)=t^{\alpha_3}-G_3(y^{\alpha_1},z^{\alpha_2},x).
\]
The polynomials from \ref{4:1} (i) and (ii) are examples of
such $P$. A derivation $\partial$ on $\C [X]$ is said to be of
{\em Jacobian type} if $\partial (f)$ coincides with the restriction
of $J_{x,y,z,t}(P,\varphi_1,\varphi_2,\varphi )$ to $X$
where $\varphi_1,\varphi_2 \in \C [x,y,z,t]$ are fixed
and the restriction of $\varphi \in \C [x,y,z,t]$ to $X$ coincides
with $f \in \C [X]$.
%\vspace{.2cm}{\bf 4.3 Proposition} (\cite{KM-L}): 
\begin{prop}[{\cite{KM-L}}]\label{4:3}
Let $\delta \in 
LND(X)$ be nontrivial and let
$\varphi_1,\varphi_2$ be such that
$\varphi_1|_X,\varphi_2|_X\in %{\rm Ker
\ker %} 
\delta$ and $P, 
\varphi_1,\varphi_2$ are algebraically independent.
Then $\partial$ 
has the same kernel as $\delta$. 
\end{prop}
Furthermore, since the transcendence degree of the field of
fractions of $%{\rm Ker} 
\ker\delta$ is $2$ \cite{M-L1}, one can always find
$\varphi_1,\varphi_2,$ and therefore $\partial$ as above.

%\vspace{.2cm}{\bf \ref{4:4}} 
\subsection{}\label{4:4} We consider degree functions 
$L$ on $\C[x,y,z,t]$ obtained 
by assigning real weights to the variables. The
$L$-quasi-leading part $\varphi^L$ of a nonzero polynomial $\varphi$ 
is the sum of the terms from $\varphi$ whose
$L$-degree coincides with $L(\varphi)$. Suppose, given $\varphi_1$, 
there exists a degree function $L_1$ with
positive values such that for any other degree function $L_2$ with 
positive values each nonzero monomial from
$\varphi^{L_2}$ is also present in $\varphi^{L_1}$. We then call
%$$
\[
\hat\varphi :=\varphi^{L_1}
%$$
\]
the quasi-leading part of $\varphi$. In cases \ref{4:1} (i) and (ii) $\hat 
{P}$ coincides with
$x^ky+z^{\alpha_2}+t^{\alpha_3}$ and 
$y(x^k+z^{\alpha_2})^l+t^{\alpha_3}$ respectively.
In all other cases $\hat {P}$ also exists and can be computed 
explicitly by virtue of
the Abhyankar-Moh-Suzuki theorem (see 3.1.1 (ii)). 
Consider further only those degree functions (may be with negative 
values)
which satisfy the condition
%$$
\[
P^L = \hat {P}.
%$$
\]
%{\bf 4.5 Proposition} (\cite{KM-L}): 
\begin{prop}[{\cite{KM-L}}]\label{4:5} Let $\partial$ be a
nontrivial locally nilpotent derivation of
Jacobian type on $\C [X]$. 
Then polynomials $\varphi_1,\varphi_2$ can be chosen so that 
$\varphi_1|_X,\varphi_2|_X\in\ %{\rm Ker}
\ker \partial$
and $\hat {P} , \varphi_1^L,\varphi_2^L$
are algebraically independent.
\end{prop}
%\vspace{.2cm}{\bf 4.6}
\subsection{}\label{4:6} With $\partial$ as in \ref{4:5}, suppose that $\hat {X}$ is the 
hypersurface
$\hat {P} (x,y,z,t)=0 $ in $\C^4$
and that $\partial^L$ is the derivation on $\C [{\hat X}]$
such that $\partial^L (f)$ coincides with restriction
of $J_{x,y,z,t}(\hat {P},\varphi_1^L,\varphi_2^L,\varphi )$ 
to $\hat {X}$, where
the restriction of $\varphi \in \C [x,y,z,t]$ to
$\hat {X}$ coincides 
with $f \in \C [\hat {X}]$. Then $\partial^L$ is nontrivial and also
locally nilpotent \cite{M-L1}.

%\vspace{.2cm}{\bf 4.7} 
\subsection{}\label{4:7} Since $\hat {P}$ is known explicitly,
we can find all nontrivial locally nilpotent derivations of Jacobian
type on $\C [{\hat {X}}]$. If $P$ is not as in \ref{4:1} (i) or (ii)
there are no such derivations.
By \ref{4:3} and \ref{4:6}
there is no nontrivial locally nilpotent derivation
on $\C [{X}]$, that is, $AK(X)= \C [X]$.

%\vspace{.2cm}{\bf 4.8} 
\subsection{}\label{4:8} In case \ref{4:1} (ii) the kernel of
any nontrivial locally nilpotent derivation
$\partial^L$ on
$\C [{\hat {X}}]$ is contained in $\C [x,z] |_{\hat {X}}$.
Since this is true for every $L$ satisfying
condition \ref{4:4}, it follows (\cite{KM-L}, Theorem 8.4) that the kernel
of the corresponding nontrivial locally nilpotent
derivation $\partial$ on $\C [X]$ is contained in $\C [x,z]|_X$.
The transcendence degree of the field of fractions
of $\ker\partial$ is 2 and $\ker\partial$ is algebraically
closed in $\C [X]$ \cite{M-L1}.
Hence $\ker\partial = \C [x,z]$. 
In case \ref{4:1} (ii) nontrivial locally nilpotent derivations on $\C [X]$ 
exist, for instance
$J_{x,y,z,t}(P,x,z,\varphi)|_X$. This yields $AK(X)=\C [x,z]|_X$.

%\vspace{.2cm}{\bf 4.9} 
\subsection{}\label{4:9} In case \ref{4:1} (i) nontrivial 
locally nilpotent derivations
on $\C [X]$ exist as well. Examples are 
$J_{x,y,z,t}(P,x,z,\varphi)|_X$
and $J_{x,y,z,t}(P,x,t,\varphi)|_X$. The intersection of the kernels 
of these locally
nilpotent derivations is $\C [x]|_X$, whence it
suffices to show that $x \in %{\rm Ker}
\ker \partial$
for every nontrivial $\partial \in LND(X)$. 
It can be shown that $ %{\rm Ker}
\ker \partial^L \subset \C [x,z,t]|_{\hat 
{X}}$
\cite{KM-L}. Varying $L$ under the condition \ref{4:4} we prove that
$ %{\rm Ker} 
\ker\partial \subset \C [x,z,t]|_X$. From this we deduce that
$\partial (\C [x,z,t]|_X) \subset x^k \C [x,z,t]|_X$ with $k$ as in \ref{4:1}.
Since $\C [x,z,t]|_X \not\subset %{\rm Ker} 
\ker\partial$,
there exists $f \in %{\rm Ker}
\ker \partial \setminus 0$ which
is divisible by $x$, and then $x\in %{\rm Ker}
\ker\partial$
by \cite{FLN}, that is, $AK(X)=\C [x]$.
%Let $g \in %{\rm Ker}\ker \partial^2 \setminus %{\rm Ker}\ker \partial$.
%For every $h \in \C [X]$ there exist 
%$f,f_0,f_1, \ldots ,f_n \in %{\rm Ker} 
%\ker\partial$
%such that $f \ne 0$ and $fh=\sum_{i=0}^n f_ig^i$ [M-L1].
%Using this equality for $h=y$ we see that
%$y=\psi /f$ on $X$ where both $f,\psi \in \C [x,z,t]|_X$.
%On the other hand $y=-(x+z^{\alpha_2}+t^{\alpha_3})x^{-k}$
%by 3.6(i). Hence $f$ is divisible by $x$, and then
%$x \in %{\rm Ker}\ker \partial$ by [M-L1], that is, $AK(X)=\C [x]$.\\

%\vglue .5truein\bigskip\bc{\bf 5.\quad Further results}\ec\medskip
\section{Further results}

Once \ref{2:5} and \ref{2:2} are established, the fact that $S'=X//T$  can be 
forgotten in the proof of \ref{2:4}. In special cases, a
geometric characterization of $\C^2/\omega_a$ is obtained.

%\vspace{.2cm}{\bf 5.1 Theorem} (\cite{KR4}, 10.1): 
\begin{thm}[{\cite{KR4}, 10.1}]\label{5:1} Suppose $S'$ is as in \ref{2:2}. If either
$q$ is an ordinary $a$-fold point, that is $b\equiv c\mod a$, or 
the minimal resolution of $q$ is a single $(-a)$-curve,
or
$q$ is a rational double point, that is, $b\equiv -c \mod a$, or the 
minimal resolution of $q$ is a chain of $a-1\
(-2)$-curves,
then
%$$
\[
S'\simeq\C^2/\omega_a .
%$$
\]
\end{thm}

We do not know whether the restriction on the analytic type of $q$ is 
needed in \ref{5:1}.


%\vspace{.1cm}
Extending the arguments of \cite{KP}, Popov \cite{P} recently proved that any 
effective action of a noncommutative, connected
reductive group on $\C^3$ is linearizable. Since effective actions of 
$(\C^*)^r,\ r>1$, are linearizable by \cite{B-B}, we
obtain

%\vspace{.2cm}{\bf 5.2 Theorem}: 
\begin{thm}\label{5:2} Any action of a connected, reductive group $G$ on 
$\C^3$ is linearizable.
\end{thm}

%\vspace{.1cm}
It is an open question whether the connectedness assumption in \ref{5:2} 
can be removed, and in particular, whether finite
        group actions on $\C^3$ are linearizable.

%\vspace{.1cm}
It is reasonable to expect that our methods and results will shed 
some light in general on codimension 2 torus actions
on $\C^n$. As an illustration, consider the possibility that 
$X\times\C\simeq\C^4$, where $X$ is a $\C^*$-threefold.
For linearizability of the obvious $(\C^*)^2$-action the following 
{\em weak cancellation conjecture} is required: {\em
Let $X$ be an affine threefold such that $X\times\C\simeq\C^4$. Then 
$X\simeq\C^3$ or $X$ does not admit
an effective $\C^*$-action}.

This is known for nonhyperbolic actions. For hyperbolic actions, one 
would have to show that
$X\times\C\not\simeq\C^4$ for the threefolds in \ref{3:1}.
This is true in the case when $X$ is not isomorphic to
a hypersurface of the form \ref{4:1} (i) or (ii) 
since $AK(Y \times \C)= \C [Y]$ for every algebraic
manifold $Y$ with $AK(Y) = \C [Y]$ \cite{M-L2}.

%\vspace{.2cm}
We remark that linearizability of $\G_m$-actions on $\A^3$ in 
positive characteristic is an open question, even in
certain nonhyperbolic cases.
%\bc
%{\bf References}
%\ec
\begin{thebibliography}{GGGG}

%\vglue .25truein
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\MR{94e:14045}
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%\end{itemize}

\end{thebibliography}

%\vglue .5truein
%\begin{tabular}{ll}
%Sh. Kaliman& M. Koras\\
%Department of Mathematics&Institute of Mathematics\\ 
%\& Computer Science&Warsaw University\\
%University of Miami&Ul. Banacha 2\\
%Coral Gables, FL 33124&Warsaw\\
%U.S.A.&Poland\\[3ex]
%L. Makar-Limanov&K.P. Russell\\
%Department of Mathematics&Department of Mathematics \\
%\& Computer Science&\& Statistics\\
%Bar-Ilan University&McGill University\\
%52900 Ramat-Gan&Montreal, QC\\
%Israel;&Canada;\\
%Department of Mathematics&Centre Interuniversitaire\\
%Wayne State University&en calcul Math\'ematique\\
%Detroit, MI  48202&Alg\'ebrique (CICMA)\\
%U.S.A.& \quad
%\end{tabular}

\end{document}    


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