[Paper Review] Characterization of affine surfaces with a torus action by their automorphism groups

Alvaro Liendo, Andriy Regeta+1 more|arXiv (Cornell University)|May 10, 2018

Algebraic Geometry and Number Theory4 citations

TL;DR

This paper establishes that normal affine surfaces with a torus action are uniquely determined by the abstract group structure of their regular automorphism groups. It proves that isomorphisms between automorphism groups preserve algebraic group actions and dynamical types of $
\mathbb{G}_{\mathrm{m}}$-actions, and shows that complex affine toric surfaces are classified up to isomorphism by their automorphism groups, with the exception of the algebraic torus itself.

ABSTRACT

In this paper we prove that if two normal affine surfaces $S$ and $S'$ have isomorphic automorphism groups, then every connected algebraic group acting regularly and faithfully on $S$ acts also regularly and faithfully on $S'$. Moreover, if $S$ is non-toric, we show that the dynamical type of a 1-torus action is preserved in presence of an additive group action. We also show that complex affine toric surfaces are determined by the abstract group structure of their regular automorphism groups in the category of complex normal affine surfaces using properties of the Cremona group. As a generalization to arbitrary dimensions, we show that complex affine toric varieties, with the exception of the algebraic torus, are uniquely determined in the category of complex affine normal varieties by their automorphism groups seen as ind-groups.

Motivation & Objective

• To determine whether the automorphism group of a normal affine surface uniquely determines the surface up to isomorphism.
• To investigate how algebraic group actions and dynamical types of $\mathbb{G}_{\mathrm{m}}$-actions are encoded in the abstract group structure of the automorphism group.
• To extend the characterization to higher-dimensional affine toric varieties and identify necessary exceptions.
• To clarify the role of unipotent and non-unipotent subgroups in the automorphism group structure.

Proposed method

• Use of ind-group structures and the Zariski topology on the Cremona group to analyze algebraic subgroups of $\operatorname{Aut}(X)$.
• Classification of root subgroups in affine toric and non-toric $\mathbb{G}_{\mathrm{m}}$-surfaces to detect dynamical behavior.
• Application of Kleiman’s criterion and equivariant projective completions to study fixed points and orbit structures.
• Construction of explicit counterexamples to show the necessity of conditions such as non-toricity and presence of $\mathbb{G}_{\mathrm{a}}$-actions.
• Use of invariant fibrations and morphisms (e.g., $\pi: S \to C$) to constrain automorphism groups via invariance under automorphisms.
• Proof of isomorphism preservation via group-theoretic techniques, particularly analyzing the image of algebraic subgroups under abstract group isomorphisms.

Experimental results

Research questions

• RQ1 Can the automorphism group of a normal affine surface, as an abstract group, determine the surface up to isomorphism?
• RQ2 Does an abstract group isomorphism between automorphism groups of two surfaces preserve the structure of algebraic subgroups, especially non-unipotent ones?
• RQ3 Is the dynamical type of a $\mathbb{G}_{\mathrm{m}}$-action on a non-toric $\mathbb{G}_{\mathrm{m}}$-surface invariant under automorphism group isomorphisms?
• RQ4 Are complex affine toric surfaces uniquely determined by their automorphism groups in the category of normal affine surfaces?
• RQ5 What is the role of $\mathbb{G}_{\mathrm{a}}$-actions in preserving dynamical types under automorphism group isomorphisms?

Key findings

• Abstract group isomorphisms between automorphism groups of normal affine surfaces preserve non-unipotent algebraic subgroups, mapping them to isomorphic algebraic subgroups in the target group.
• For non-toric $\mathbb{G}_{\mathrm{m}}$-surfaces with a non-trivial $\mathbb{G}_{\mathrm{a}}$-action, the dynamical type of the $\mathbb{G}_{\mathrm{m}}$-action is preserved under automorphism group isomorphisms.
• Complex affine toric surfaces are uniquely determined by their automorphism groups as abstract groups, with the exception of the algebraic torus.
• The automorphism group of a non-toric $\mathbb{G}_{\mathrm{m}}$-surface with a $\mathbb{G}_{\mathrm{a}}$-action cannot be isomorphic to that of a surface of different dynamical type.
• In higher dimensions, complex affine toric varieties (except the algebraic torus) are uniquely determined by their automorphism groups seen as ind-groups.
• The algebraic torus is an exception: $\operatorname{Aut}(T) \cong \operatorname{Aut}(C \times T)$ as ind-groups for certain curves $C$, showing the necessity of excluding the torus in the characterization.