[Paper Review] C^*- Actions on Stein analytic spaces with isolated singularities

César Camacho, Hossein Movasati+1 more|ArXiv.org|Sep 4, 2007

Advanced Operator Algebra Research7 references3 citations

TL;DR

This paper establishes that any holomorphic ℂ*-action on a normal, two-dimensional Stein analytic space with a dicritical singularity is analytically conjugate to a good action on an affine algebraic variety in ℂⁿ⁺¹. The key result is a GAGA-type theorem showing such spaces are biholomorphic to quasi-homogeneous algebraic surfaces, with a complete moduli classification via resolution data including genus, line bundle degree, and continued fraction invariants.

ABSTRACT

Let $V$ be an irreducible complex analytic space of dimension two with normal singularities and $\vr:\mathbb{C^*} imes V o V$ a holomorphic action of the group $\mathbb{C^*}$ 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 $σ_Q(t,(z_{0},...,z_{n}))=(t^{q_{0}}z_{0},..., t^{q_{n}}z_{n})$ 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 $\mathbb{C^*}$-actions on Stein analytic spaces of dimension two with a dicritical singularity.

Motivation & Objective

• To classify ℂ*-actions on normal, two-dimensional Stein analytic spaces with dicritical singularities.
• To prove that such actions are analytically conjugate to good algebraic actions on affine algebraic varieties in ℂⁿ⁺¹.
• To establish a GAGA-type principle for ℂ*-actions on Stein surfaces.
• To provide a complete moduli space description of such pairs (V, φ) in terms of geometric and arithmetic invariants.

Proposed method

• Use resolution of singularities to decompose the space into a union of compact Riemann surfaces, with one special Riemann surface σ₀ on which ℂ* acts transversely.
• Apply local linearization of the ℂ*-action near σ₀, independent of its self-intersection number.
• Construct a linear model for the resolution using the data of σ₀, the line bundle with negative degree, and continued fraction sequences from the resolution graph.
• Prove that the basin of attraction of the dicritical singularity is the entire space V, implying global conjugacy to the linear model.
• Use the Stein property and vanishing of H¹ to show that the basin is simply connected, enabling global biholomorphism.
• Establish a one-to-one correspondence between the moduli data and the resulting linear models, proving the classification.

Experimental results

Research questions

• RQ1 Can every ℂ*-action on a normal, two-dimensional Stein space with a dicritical singularity be realized as a good algebraic action on an affine algebraic variety in ℂⁿ⁺¹?
• RQ2 What is the complete moduli space of such ℂ*-actions, up to biholomorphism?
• RQ3 How do the invariants of the resolution of the foliation induced by the action classify the underlying space and action?
• RQ4 Is the basin of attraction of the dicritical singularity always the entire space?
• RQ5 Are there any other dicritical singularities besides the one at p in such spaces?

Key findings

• Every ℂ*-action on a normal, two-dimensional Stein space with a dicritical singularity is analytically conjugate to a good action on an affine algebraic variety in ℂⁿ⁺¹.
• The basin of attraction of the dicritical singularity is the entire space V, implying global linearization.
• The moduli space of such pairs (V, φ) is parametrized by a Riemann surface σ₀ of genus g with s marked points, a line bundle of degree −k ≤ −1, and continued fraction sequences satisfying a convergence inequality.
• The resolution of the foliation at the singularity determines the moduli data, and the classification is complete and unique up to isomorphism.
• If V is smooth, then V is biholomorphic to ℂ², confirming a special case of the main theorem.
• The singularity at p is necessarily a quasi-homogeneous surface singularity, as defined in the literature.