Infinite transitivity and polynomial vector fields

Technical Summary: Infinite Transitivity and Polynomial Vector Fields

Problem Statement
The paper investigates the conditions under which groups generated by root subgroups (specifically $\mathbb{G}_a$-subgroups normalized by a torus) act with infinite transitivity on the affine plane $\mathbb{C}^2$. An action is infinitely transitive if, for any $m$, the group can map any $m$-tuple of distinct points to any other $m$-tuple of distinct points. The authors focus on the group $G_{r,s}$ generated by two specific one-parameter unipotent subgroups:
$$L_r: (x + \alpha y^r, y) \quad \text{and} \quad R_s: (x, y + \beta x^s),$$
where $\alpha, \beta \in \mathbb{C}$ and $r, s$ are non-negative integers.

While it was previously established that $G_{r,s}$ acts infinitely transitively on its open orbit only if $rs=2$ (and that $rs=2$ implies infinite transitivity on $\mathbb{C}^2 \setminus \{0\}$), the paper addresses the broader question of generic infinite transitivity. Specifically, it seeks to determine when the diagonal action of $G_{r,s}$ on $(\mathbb{C}^2)^m$ possesses a Zariski open orbit for any $m$.

Methodology
The authors employ a two-pronged approach combining algebraic Lie theory with analytical approximation techniques:

1. Lie Algebraic Analysis: The study centers on the Lie algebra $\mathcal{L}$ of polynomial vector fields on $\mathbb{C}^2$ with zero divergence, which is isomorphic to the Lie algebra of polynomials $\mathbb{C}[x, y]$ equipped with the standard Poisson bracket $\{f, h\} = \frac{\partial f}{\partial x}\frac{\partial h}{\partial y} - \frac{\partial f}{\partial y}\frac{\partial h}{\partial x}$. The authors analyze subalgebras generated by specific monomials, such as $\text{Lie}(x^p, y^q)$, to determine their codimension and intersection properties with subalgebras of univariate polynomials.
2. Approximation of Flows: To bridge the gap between Lie algebras and transformation groups, the authors utilize a result by Varolin regarding the approximation of polynomial flows. This allows them to construct automorphisms by composing flows of locally nilpotent derivations. The proof relies on the Implicit Function Theorem and the ability to approximate analytic flows with algebraic ones, ensuring that the group generated by the flows of specific derivations acts with an open orbit.

Key Contributions and Results

• Structure of Poisson Subalgebras: The paper establishes precise descriptions of the subalgebra $\text{Lie}(x^p, y^q)$ generated by two monomials.

  • Proposition 7: For $\min(p, q) \ge 2$ and $\max(p, q) \ge 3$, the intersection of this subalgebra with univariate polynomials is given by $\text{Lie}(x^p, y^q) \cap \mathbb{C}[x] = x^p \mathbb{C}[x^{pq-p-q}]$ (and similarly for $y$).
  • Corollary 9: The subalgebra $\text{Lie}(x^p, y^q)$ has finite codimension in $\mathbb{C}[x, y]$ if and only if $(p, q) = (2, 3)$ or $(3, 2)$.
  • Theorem 10: Under the same conditions, the subalgebra generated by $x^p - y$ and $y^q$ is the entire algebra $\mathbb{C}[x, y]$.

• 2-Generation of Lie Algebras:

  • Proposition 1: The Lie algebra $\mathcal{L}$ of zero-divergence polynomial vector fields is generated by two locally nilpotent derivations: $\partial_1 = y^2 \frac{\partial}{\partial x}$ and $\partial_2 = \frac{\partial}{\partial x} - x \frac{\partial}{\partial y}$.
  • Proposition 11: The Lie algebra $\tilde{\mathcal{L}}$ of polynomial vector fields with constant divergence (the tangent algebra of $\text{Aut}(\mathbb{C}^2)$) is also 2-generated.

• Main Theorems on Transitivity:

  • Theorem 18 (Main Result): The action of the group $G_{r,s}$ on $\mathbb{C}^2$ is generically infinitely transitive if and only if $rs \ge 2$. This extends previous results which were limited to the case $rs=2$ and the punctured plane.
  • Theorem 17: The group $G_{r,s}$ acts infinitely transitively on an open subset $O \subseteq \mathbb{C}^2$ if and only if $rs=2$. In this case, $O = \mathbb{C}^2 \setminus \{0\}$. The paper provides a new proof of this result using the Lie algebraic framework.

• Generalizations:

  • Proposition 25: The results are extended to Danielewski surfaces $Y_p = \{(x, y, z) \in \mathbb{C}^3 : xy = p(z)\}$, showing that groups generated by specific locally nilpotent derivations act generically infinitely transitively on these surfaces.
  • Corollary 29: The authors construct a specific pair of $\mathbb{G}_a$-subgroups (one of which is not a root subgroup) that generate a group infinitely transitive on $\mathbb{C}^2$, utilizing the 2-generation of the zero-divergence Lie algebra.

Significance and Claims
The paper claims that its primary significance lies in connecting the property of infinite transitivity to the structure of the underlying tangent Lie algebra. By proving that the action is generically infinitely transitive if and only if the associated Lie algebra subalgebra has finite codimension (specifically when $rs \ge 2$), the authors provide a structural criterion for transitivity.

The authors note that while the Lie algebraic results are purely algebraic and valid over any field of characteristic zero, the transitivity results for group actions rely on analytical methods (specifically the Implicit Function Theorem and flow approximation). They explicitly state that obtaining purely algebraic criteria for infinite transitivity remains an open and interesting problem. The work also serves as a proof of concept for applying these Lie algebraic techniques to other affine surfaces, such as Danielewski surfaces.