C^*- Actions on Stein analytic spaces with isolated singularities

Abstract

Let $V$ be an irreducible complex analytic space of dimension two with normal singularities and $\vr:\mathbb{C^*}\times V\to V$ a holomorphic action of the group ${\mathbb{C}}^{\ast }$ on $V$. Denote by $\fa_\vr$ the foliation on $V$ induced by $\vr$. The leaves of this foliation are the one-dimensional orbits of $\vr$. %and its singularities are the fixed points of $\vr$. We will assume that there exists a \emph{dicritical} singularity $p\in V$ for the $\bc^*$-action, i.e. for some neighborhood $p\in W\subset V$ there are infinitely many leaves of $\mathcal {F}_\vr|_{W}$ accumulating only at $p$. The closure of such a local leaf is an invariant local analytic curve called a \emph{separatrix} of $\mathcal{F}_\vr$ through $p$. In \cite{Orlik} Orlik and Wagreich studied the 2-dimensional affine algebraic varieties embedded in ${\mathbb{C}}^{n+1}$, with an isolated singularity at the origin, that are invariant by an effective action of the form ${\sigma }_Q(t,(z_0,...,z_n))=(t^{q_0}{z}_0,...,t^{q_n}{z}_n)$ where $Q=(q_0,...,q_n)\in {\mathbb{N}}^{n+1}$, i.e. all $q_i$ are positive integers. Such actions are called \emph{good} actions. In particular they classified the algebraic surfaces embedded in ${\mathbb{C}}^3$ endowed with such an action. It is easy to see that any good action on a surface embedded in ${\mathbb{C}}^{n+1}$ has a dicritical singularity at $0\in {\mathbb{C}}^{n+1}$. Conversely, it is the purpose of this paper to show that good actions are the models for analytic ${\mathbb{C}}^{\ast }$-actions on Stein analytic spaces of dimension two with a dicritical singularity.