On nice $\mathbb{G}_m$-actions arising from locally nilpotent derivations with slice

Abstract: Let $k$ be an algebraically closed field of characteristic zero and $B$ a finitely generated $k$-domain. Given a locally nilpotent derivation $D$ on $B$ admitting a slice $s$, the derivation $\partial=NsD$ ($N\in\mathbb{Z}\setminus{0}$) is semisimple and defines a regular $\mathbb{G}_m$-action on $\mathrm{Spec}(B)$. We show that this derivation provides a new explicit description of the $\mathbb{G}_m$-action introduced by Freudenburg in terms of the infinitesimal generator $\partial=NsD$. In the nice case ($D^2(x_i)=0$ for all generators), we prove a linearizability criterion: the associated $\mathbb{G}_m$-action is linearizable if and only if $D$ is automorphically conjugate to $\partial/\partial x_n$ and the slice becomes affine-linear in the distinguished variable; moreover, this criterion is independent of the choice of slice.

• The paper establishes an explicit linearizability criterion for ‘nice’ Gₘ-actions constructed from locally nilpotent derivations with slice.
• It demonstrates that slice independence allows conjugation of actions via algebraic automorphisms and provides clear geometric criteria.
• The study fully resolves cases in dimensions 1 to 3 and identifies the Makar-Limanov invariant as an obstruction for linearizability in higher dimensions.

Explicit Description and Linearizability of $\mathbb{G}_m$-Actions arising from Locally Nilpotent Derivations with Slice

Overview

The paper addresses the structure and linearizability of $\mathbb{G}_m$-actions on affine varieties generated from locally nilpotent derivations (LNDs) admitting a slice. Specifically, it elucidates the explicit correspondence between such multiplicative group actions and their infinitesimal generators, provides a refined linearizability criterion for the so-called "nice" case, and offers a thorough resolution in small dimensions. Furthermore, it analyzes the implications of slice independence, supplies geometric criteria, and highlights the connection to the Makar-Limanov invariant for obstructing linearizability in higher dimensions.

Main Results and Theoretical Contributions

Given a finitely generated $k$-domain $B$ with $k$ algebraically closed and of characteristic zero, the core focus is on LNDs $D$ with a slice $s$ ($D(s)=1$). By leveraging the Slice Theorem ($B = \ker(D)[s]$), the paper constructs the semisimple derivation $\partial = NsD$, for integer $\mathbb{G}_m$0, which induces a regular $\mathbb{G}_m$1-action. The paper rigorously identifies this action with the one introduced by Freudenburg, providing an explicit algebraic description in terms of $\mathbb{G}_m$2.

The pivotal result is the linearizability criterion: for the "nice" case ($\mathbb{G}_m$3 for all generators $\mathbb{G}_m$4), the $\mathbb{G}_m$5-action defined by $\mathbb{G}_m$6 with slice $\mathbb{G}_m$7 is linearizable if and only if $\mathbb{G}_m$8 is automorphically conjugate to $\mathbb{G}_m$9, and the slice $k$0 becomes affine-linear in the distinguished variable ($k$1), independent of the choice of slice. This reduces the linearization problem to the study of the automorphism group of $k$2 acting on $k$3 and $k$4, eliminating dependence on the group action's internal structure.

Explicit Algebraic and Geometric Criteria

A sequence of results establishes:

• Slice Independence: For any two slices $k$5 and $k$6 of a given LND, the induced $k$7-actions are conjugate via an explicit automorphism, verifying that the linearizability criterion is robust under variations of slice representatives.
• Geometric Reformulation: The action is linearizable precisely when $k$8 can be mapped onto a coordinate hyperplane ring and the slice mapped to a translate of the complementary variable. This criterion is constructive and tangible for computational purposes.

Resolution in Low Dimensions

For $k$9, the paper provides:

• $B$0: Trivial, as every LND with slice is $B$1.
• $B$2: Utilizing Rentschler's and Jung–van der Kulk theorems, every nice LND with slice on $B$3 is conjugate to $B$4; the corresponding $B$5-action is always linearizable.
• $B$6: Via Wang's algebraic classification and, over $B$7, KKMR's linearization theorem, every nice LND with slice in $B$8 is conjugate to $B$9; all associated $k$0-actions are linearizable.

These results provide complete answers in small dimensions, cementing the structure theory for $k$1-actions arising from LNDs with slice.

Computational Obstructions in Higher Dimensions

In dimensions $k$2, the linearizability problem remains open. The paper emphasizes the utility of the Makar-Limanov invariant ($k$3) as a practical obstruction: if $k$4 contains non-constant elements, the associated $k$5-action cannot be linearizable. This connects the problem with the broader question of distinguishing affine space from exotic structures (e.g., Koras–Russell threefolds), where invariants like $k$6 are computationally accessible but structural classification is subtler.

Implications and Future Directions

Practically, the explicit algebraic criterion and slice independence result facilitate algorithmic checks of linearizability and automorphic conjugacy for regular actions, both in computational algebra and geometric invariant theory. Theoretically, the reduction to automorphism groups and kernel geometry intertwines group actions with deeper aspects of affine algebraic geometry, particularly in the context of exotic structures and invariants.

Looking forward, establishing analogous results or obstructions in higher dimensions will require more refined invariants or structural theorems, possibly extending the methods via the Makar-Limanov invariant or invoking new algebraic or geometric techniques for dealing with polynomial automorphism groups.

Conclusion

The paper provides a precise, explicit, and computationally tractable description of $k$7-actions arising from LNDs with slice, clarifies the conditions under which these actions are linearizable (in the nice case), and establishes the independence of the linearizability criterion from the choice of slice. The explicit resolution for $k$8 positions this framework as a foundation for further exploration in higher dimensions, with the Makar-Limanov invariant serving as a central tool for detecting non-linearizable actions. The synthesis of additive and multiplicative group actions via infinitesimal generators and slices strengthens the conceptual bridge between algebraic group actions and derivation theory in affine algebraic geometry (2604.03471).