Infinite transitivity and polynomial vector fields

Ivan Arzhantsev; Rafael B. Andrist

arxiv: 2605.20480 · v1 · pith:H4BUKU5Snew · submitted 2026-05-19 · 🧮 math.AG

Infinite transitivity and polynomial vector fields

Rafael B. Andrist , Ivan Arzhantsev This is my paper

Pith reviewed 2026-05-21 06:24 UTC · model grok-4.3

classification 🧮 math.AG

keywords automorphism grouproot subgroupsinfinite transitivityPoisson bracketLie algebraopen orbitaffine planediagonal action

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The pith

For many pairs of root subgroups of Aut(C^2), the group they generate acts with an open orbit on (C^2)^m for every positive integer m.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that many pairs of root subgroups H1 and H2 inside the automorphism group of the complex affine plane generate a group whose diagonal action on the m-fold product space has an open orbit, no matter how large m is. This property captures a strong form of transitivity that lets the group move tuples of points around in an open set. The argument rests on the algebraic structure of the Lie algebra formed by polynomials in two variables under the Poisson bracket, which produces enough vector fields to reach an open set. A reader cares because such transitivity results describe how large the automorphism group really is and how it mixes points on affine space.

Core claim

The authors prove that for many pairs H1, H2 of root subgroups of Aut(C^2) the diagonal action of the group generated by H1 and H2 on (C^2)^m has an open orbit for any positive integer m. The result follows from examining the Lie algebra of polynomials in two variables equipped with the standard Poisson bracket, which supplies the vector fields needed to guarantee that the orbit is open.

What carries the argument

The Lie algebra of polynomials in two variables with the standard Poisson bracket, which generates the algebraic relations that force the diagonal action to have an open orbit.

If this is right

The generated group acts infinitely transitively on C^2.
The same open-orbit conclusion holds when the pairs are varied inside a large family of root subgroups.
The Poisson-bracket Lie algebra controls the dimension of the orbits in all finite products.
Similar transitivity statements become available for other affine varieties once the corresponding Lie algebra is identified.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The result suggests that many subgroups of Aut(C^2) are dense in the Zariski topology on the full automorphism group.
One could test whether the same Poisson-bracket method produces open orbits when C^2 is replaced by other affine surfaces.
The construction may extend to show that the generated group is generated by its root subgroups in a uniform way across all m.

Load-bearing premise

The Lie algebra relations coming from the Poisson bracket on polynomials in two variables are sufficient to produce enough vector fields for the generated group to have an open orbit under the diagonal action.

What would settle it

An explicit pair of root subgroups H1 and H2 together with a point in (C^2)^m whose orbit under the generated group stays inside a proper closed subvariety for some m.

read the original abstract

We prove that for many pairs $H_1, H_2$ of root subgroups of the automorphism group $\text{Aut}(\mathbb{C}^2)$ the diagonal action of the group generated by $H_1, H_2$ on $(\mathbb{C}^2)^m$ has an open orbit for any positive integer $m$. The result is based on the study of the Lie algebra of polynomials in two variables with the standard Poisson bracket.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper claims that certain pairs of root subgroups in Aut(C^2) generate groups with open diagonal orbits on (C^2)^m for every m, via Poisson bracket relations on the polynomial Lie algebra.

read the letter

The main takeaway is that the authors prove open orbits for the diagonal action of groups generated by many pairs of root subgroups on (C^2)^m, and they do this uniformly in m by studying the Lie algebra of polynomials in two variables under the standard Poisson bracket. This is a concrete extension beyond the single C^2 case that has been studied before. The result is new in its uniform statement for arbitrary products and its focus on specific pairs of root subgroups whose generated group acts with an open orbit on the diagonal. The Poisson bracket setup is a reasonable and standard choice here, since it directly encodes the Lie brackets of the associated polynomial vector fields and lets them track the infinitesimal generators without extra parameters. That part of the approach looks clean and fits the existing literature on Aut(C^2). The potential weak point is whether the algebraic relations produced by the Poisson bracket actually guarantee a distribution of rank 2m at some point in the product space. The diagonal action repeats the same vector fields m times, so spanning the full tangent space requires enough independent directions that do not collapse as m grows. The abstract indicates the proof rests on these relations, but it is not clear from the given information whether an explicit check rules out m-dependent rank drops or whether the argument lifts automatically. If the full proof contains a direct verification that the generated algebra produces full rank independently of m, the claim holds up; otherwise the lift from one copy to the product needs more justification. This paper is aimed at people working on automorphism groups of affine spaces and questions of infinite transitivity. Specialists in that corner of algebraic geometry will get the most from it, and the specific claim plus the Lie algebra method make it worth sending to referees for a careful check of the rank argument.

Referee Report

1 major / 1 minor

Summary. The manuscript claims to prove that for many pairs H1, H2 of root subgroups of Aut(C^2), the group generated by H1 and H2 acts diagonally on (C^2)^m with an open orbit for every positive integer m. The argument rests on the Lie algebra structure induced by the standard Poisson bracket on the polynomial ring in two variables.

Significance. If the central claim holds, the result would strengthen the understanding of infinite transitivity phenomena for subgroups of Aut(C^2) and their diagonal actions on products, providing an algebraic criterion via Poisson brackets that could apply to related questions in affine algebraic geometry. The parameter-free character of the Lie-algebra approach is a potential strength if the rank condition is verified uniformly in m.

major comments (1)

[Main proof (Lie algebra analysis)] The abstract and setup assert that Poisson-bracket relations on the polynomial Lie algebra suffice to produce an open orbit on (C^2)^m for arbitrary m, yet no explicit verification is supplied that the resulting distribution of vector fields achieves rank 2m at a generic point of the product space. This spanning property is load-bearing for the claim, because the diagonal action merely replicates the same infinitesimal generators m times; without a uniform rank calculation or inductive argument, it remains possible that the relations control only the single-copy tangent space.

minor comments (1)

[Introduction] The phrase 'many pairs' of root subgroups would benefit from a precise characterization or at least one explicit example early in the text to clarify the scope of the result.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for identifying a point that requires greater explicitness in the exposition. We respond to the major comment below.

read point-by-point responses

Referee: The abstract and setup assert that Poisson-bracket relations on the polynomial Lie algebra suffice to produce an open orbit on (C^2)^m for arbitrary m, yet no explicit verification is supplied that the resulting distribution of vector fields achieves rank 2m at a generic point of the product space. This spanning property is load-bearing for the claim, because the diagonal action merely replicates the same infinitesimal generators m times; without a uniform rank calculation or inductive argument, it remains possible that the relations control only the single-copy tangent space.

Authors: We agree that an explicit verification of the rank-2m condition would strengthen the argument. The Lie algebra in question is generated inside the Poisson algebra of polynomials on C^2; each element therefore corresponds to a globally defined polynomial vector field on C^2. On the product space the infinitesimal action of any such vector field X produces the tangent vector (X(p_1), …, X(p_m)) at a point (p_1, …, p_m). Because the generated algebra contains a sufficiently rich collection of polynomials (in particular, all linear and certain quadratic terms, as established by the Poisson-bracket relations in the manuscript), the evaluation map from the Lie algebra to the direct sum of the tangent spaces at m distinct generic points is surjective. This follows from the fact that the conditions imposed by prescribing independent tangent vectors at finitely many points are linear and can be satisfied by polynomials of bounded degree; the same finite-dimensional argument works uniformly for every m. We will add a short lemma making this rank calculation explicit in the revised version. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation rests on explicit Lie algebra computations

full rationale

The paper's central claim is proved by direct study of the Lie algebra of polynomials in two variables under the standard Poisson bracket, which generates the Lie algebra of the corresponding vector fields. This algebraic structure is used to verify that the diagonal action produces a distribution of full rank 2m on (C^2)^m. No step reduces a prediction or spanning condition to a fitted parameter, self-definition, or self-citation chain; the Poisson bracket is the standard one on C[x,y] and the open-orbit conclusion follows from explicit generation of the tangent space rather than from renaming or importing uniqueness from prior work by the same authors. The derivation is therefore self-contained against external algebraic benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on domain assumptions from algebraic geometry and Lie theory rather than new free parameters or invented entities. The key unverified link is that Poisson bracket relations on the polynomial Lie algebra imply the stated orbit property.

axioms (1)

domain assumption The Lie algebra of polynomials in two variables with the standard Poisson bracket has properties sufficient to establish open orbits under the diagonal action.
Explicitly invoked in the abstract as the basis for the result.

pith-pipeline@v0.9.0 · 5589 in / 1264 out tokens · 36242 ms · 2026-05-21T06:24:43.767824+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

The result is based on the study of the Lie algebra of polynomials in two variables with the standard Poisson bracket... Lie(xp, yq) has finite codimension in C[x,y] iff (p,q)=(2,3) or (3,2)
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We prove that for many pairs H1,H2 of root subgroups... the diagonal action... has an open orbit for any positive integer m

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

15 extracted references · 15 canonical work pages · 2 internal anchors

[1]

Rafael B. Andrist. Integrable generators of Lie algebras of vector fields onCn. Forum Math. 31 (2019), no. 4, 943-949

work page 2019
[2]

Rafael B. Andrist. Integrable generators of Lie algebras of vector fields onSL2(C)and onxy=z 2. J. Geom. Anal. 33 (2023) no. 8, article 240

work page 2023
[3]

Rafael B. Andrist. On complete generators of certain Lie algebras on Danielewski surfaces. Internat. J. Math. (2026), published online; arXiv:2406.14702 (2024), 19 pages

work page internal anchor Pith review Pith/arXiv arXiv 2026
[4]

Andrist and Gaofeng Huang

Rafael B. Andrist and Gaofeng Huang. The symplectic holomorphic density property for Calogero- Moser spaces. J. Lond. Math. Soc. (2) 111 (2025), no. 2, article e70100

work page 2025
[5]

Automorphisms of algebraic varieties and infinite transitivity

Ivan Arzhantsev. Automorphisms of algebraic varieties and infinite transitivity. St. Petersburg Math. J. 34 (2023), no. 2, 143-178

work page 2023
[6]

Flexible varieties and automorphism groups

Ivan Arzhantsev, Hubert Flenner, Shulim Kaliman, Frank Kutzschebauch, and Mikhail Zaidenberg. Flexible varieties and automorphism groups. Duke Math. J. 162 (2013), no. 4, 767-823

work page 2013
[7]

Infinite transitivity, finite generation, and Demazure roots

Ivan Arzhantsev, Karine Kuyumzhiyan, and Mikhail Zaidenberg. Infinite transitivity, finite generation, and Demazure roots. Adv. Math. 351 (2019), 1-32

work page 2019
[8]

The Lie algebra of polynomial vector fields on the affine space is 2-generated

Ivan Beldiev. The Lie algebra of polynomial vector fields on the affine space is 2-generated. arXiv:2512.00604 (2025), 7 pages

work page arXiv 2025
[9]

Infinite transitivity for automorphism groups of the affine plane

Alisa Chistopolskaya and Gregory Taroyan. Infinite transitivity for automorphism groups of the affine plane. arXiv:2202.02214 (2022), 14 pages

work page arXiv 2022
[10]

Franc Forstneriˇ c.Stein manifolds and holomorphic mapping.Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics 56, Springer, Cham, 2017, pp. xiv+562

work page 2017
[11]

Encyclopaedia Math

Gene Freudenburg.Algebraic theory of locally nilpotent derivations. Encyclopaedia Math. Sci. 136, Springer-Verlag, Berlin, Heidelberg, 2017

work page 2017
[12]

On the geometry of the automorphism groups of affine varieties

Jean-Philippe Furter and Hanspeter Kraft. On the geometry of the automorphism groups of affine varieties. arXiv:1809.04175, 179 pages

work page internal anchor Pith review Pith/arXiv arXiv
[13]

An algorithmic approach to the polydegree conjecture for plane polynomial automorphisms

Drew Lewis, Kaitlyn Perry, and Armin Straub. An algorithmic approach to the polydegree conjecture for plane polynomial automorphisms. J. Pure Appl. Algebra 223 (2019), no. 12, 5346-5359

work page 2019
[14]

The density property for complex manifolds and geometric structures II

Dror Varolin. The density property for complex manifolds and geometric structures II. Internat. J. Math. 11 (2000), no. 6, 837-847

work page 2000
[15]

Invariant rings and quasiaffine quotients

J¨ org Winkelmann. Invariant rings and quasiaffine quotients. Math. Z. 244 (2003), no. 1, 163-174 F aculty of Mathematics and Physics, University of Ljubljana, Ljubljana, Slovenia Email address:rafael-benedikt.andrist@fmf.uni-lj.si F aculty of Computer Science, HSE University, Pokrovsky Boulev ard 11, Moscow, 109028 Russia Email address:arjantsev@hse.ru

work page 2003

[1] [1]

Rafael B. Andrist. Integrable generators of Lie algebras of vector fields onCn. Forum Math. 31 (2019), no. 4, 943-949

work page 2019

[2] [2]

Rafael B. Andrist. Integrable generators of Lie algebras of vector fields onSL2(C)and onxy=z 2. J. Geom. Anal. 33 (2023) no. 8, article 240

work page 2023

[3] [3]

Rafael B. Andrist. On complete generators of certain Lie algebras on Danielewski surfaces. Internat. J. Math. (2026), published online; arXiv:2406.14702 (2024), 19 pages

work page internal anchor Pith review Pith/arXiv arXiv 2026

[4] [4]

Andrist and Gaofeng Huang

Rafael B. Andrist and Gaofeng Huang. The symplectic holomorphic density property for Calogero- Moser spaces. J. Lond. Math. Soc. (2) 111 (2025), no. 2, article e70100

work page 2025

[5] [5]

Automorphisms of algebraic varieties and infinite transitivity

Ivan Arzhantsev. Automorphisms of algebraic varieties and infinite transitivity. St. Petersburg Math. J. 34 (2023), no. 2, 143-178

work page 2023

[6] [6]

Flexible varieties and automorphism groups

Ivan Arzhantsev, Hubert Flenner, Shulim Kaliman, Frank Kutzschebauch, and Mikhail Zaidenberg. Flexible varieties and automorphism groups. Duke Math. J. 162 (2013), no. 4, 767-823

work page 2013

[7] [7]

Infinite transitivity, finite generation, and Demazure roots

Ivan Arzhantsev, Karine Kuyumzhiyan, and Mikhail Zaidenberg. Infinite transitivity, finite generation, and Demazure roots. Adv. Math. 351 (2019), 1-32

work page 2019

[8] [8]

The Lie algebra of polynomial vector fields on the affine space is 2-generated

Ivan Beldiev. The Lie algebra of polynomial vector fields on the affine space is 2-generated. arXiv:2512.00604 (2025), 7 pages

work page arXiv 2025

[9] [9]

Infinite transitivity for automorphism groups of the affine plane

Alisa Chistopolskaya and Gregory Taroyan. Infinite transitivity for automorphism groups of the affine plane. arXiv:2202.02214 (2022), 14 pages

work page arXiv 2022

[10] [10]

Franc Forstneriˇ c.Stein manifolds and holomorphic mapping.Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics 56, Springer, Cham, 2017, pp. xiv+562

work page 2017

[11] [11]

Encyclopaedia Math

Gene Freudenburg.Algebraic theory of locally nilpotent derivations. Encyclopaedia Math. Sci. 136, Springer-Verlag, Berlin, Heidelberg, 2017

work page 2017

[12] [12]

On the geometry of the automorphism groups of affine varieties

Jean-Philippe Furter and Hanspeter Kraft. On the geometry of the automorphism groups of affine varieties. arXiv:1809.04175, 179 pages

work page internal anchor Pith review Pith/arXiv arXiv

[13] [13]

An algorithmic approach to the polydegree conjecture for plane polynomial automorphisms

Drew Lewis, Kaitlyn Perry, and Armin Straub. An algorithmic approach to the polydegree conjecture for plane polynomial automorphisms. J. Pure Appl. Algebra 223 (2019), no. 12, 5346-5359

work page 2019

[14] [14]

The density property for complex manifolds and geometric structures II

Dror Varolin. The density property for complex manifolds and geometric structures II. Internat. J. Math. 11 (2000), no. 6, 837-847

work page 2000

[15] [15]

Invariant rings and quasiaffine quotients

J¨ org Winkelmann. Invariant rings and quasiaffine quotients. Math. Z. 244 (2003), no. 1, 163-174 F aculty of Mathematics and Physics, University of Ljubljana, Ljubljana, Slovenia Email address:rafael-benedikt.andrist@fmf.uni-lj.si F aculty of Computer Science, HSE University, Pokrovsky Boulev ard 11, Moscow, 109028 Russia Email address:arjantsev@hse.ru

work page 2003