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For the special linear group SL2(C)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{SL}_2(\mathbb {C})$$\end{document} and for the singular quadratic Danielewski surface xy=z2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x y = z^2$$\end{document} we give explicitly a finite number of complete polynomial vector fields that generate the Lie algebra of all polynomial vector fields on them. Moreover, we give three unipotent one-parameter subgroups that generate a subgroup of algebraic automorphisms acting infinitely transitively on xy=z2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x y = z^2$$\end{document}.

Definition 1 Let X be a complex variety and let G be group acting on X through (algebraic or holomorphic) automorphisms, then we call the action of G infinitely transitive if G acts on the regular part X reg m-transitively for any m ∈ N.
In the algebraic category, the situation is slightly different; in particular, flows of complete algebraic vector fields need not be algebraic. Hence, the notion of the density property does not help in the study of algebraic automorphisms. Arzhantsev et al. introduced the notion of flexibility in 2013: Definition 2 [7] A regular point x ∈ X reg is called flexible if the tangent space T x X is spanned by the tangent vectors to the orbits H x of one-parameter unipotent subgroups H ⊆ Aut(X ). A complex variety X is called flexible if every regular point x ∈ X reg is flexible.
One of their main results [ [7], Theorem 0.1] implies that the group of algebraic automorphisms of a flexible variety acts infinitely transitively.
The question whether one can find finitely many one-parameter unipotent subgroups that generate a subgroup of automorphisms acting infinitely transitively was first studied by Arzhantsev, Kuyumzhiyan and Zaidenberg [8] for toric varieties. For C n , n ≥ 2, they showed that three one-parameter unipotent subgroups are sufficient. This was shown independently by the author using a different approach [6]; in the context of the volume density property, it seems natural instead to find finitely many complete vector fields with algebraic flows that generate the Lie algebra of all volume preserving algebraic vector fields. Similarly, one can consider the question of finding finitely many complete vector fields that generate the Lie algebra of all algebraic vector fields. In this case, the flows won't necessarily be algebraic anymore.
In this article, we focus on the question of generating the Lie algebra of all polynomial vector fields on the affine varieties SL 2 (C) and x y = z 2 . The corresponding one-parameter subgroups then generate a group that acts infinitely transitively. In Sect. 2 we provide the necessary tools and the theoretical background.
In Sect. 3 we consider the special linear group SL 2 (C). Theorem 14 gives explicitly four complete vector fields that generate the Lie algebra of all polynomial vector fields on SL 2 (C).
In Sect. 4 we consider the singular quadratic Danielewski surface {(x, y, z) ∈ C 3 : x y = z 2 } which is a normal surface and also a toric variety; it has one isolated singularity at the origin. First, Theorem 19 gives explicitly four complete vector fields that generate a certain Lie sub-algebra which is smaller than the Lie algebra of polynomial vector fields, but its flows nonetheless approximate holomorphic automorphisms that are isotopic to the identity and fix the origin to a certain order. The group generated by these four flows acts infinitely transitively on the regular locus. Second, Theorem 26 gives explicitly five complete vector fields that generate the Lie algebra of all polynomial vector fields on {(x, y, z) ∈ C 3 : x y = z 2 }.
In Sect. 5 we revisit the same surface again, this time focusing on algebraic automorphisms, but no longer on generating the whole polynomial Lie algebra of vector fields. Our motivation is that [ [8], Theorem 5.20] only applies to toric varieties that are smooth in codimension 2, which does not cover the case of the quadric x · y = z 2 . In Theorem 31 we give three unipotent one-parameter subgroups that generate a subgroup of the algebraic automorphisms acting infinitely transitively on x · y = z 2 .
As a side note, we remark that the latter proof involves some tools from analysis such as the implicit function theorem to prove a purely algebraic result.
Note that the defining equation in Question (2) is equivalent to x y = z 2 by a linear change of coordinates if n = 3.

Background and Tools
Definition 3 Let X be a complex variety and let V be a holomorphic vector field on X . We call V complete or C-complete if its flow map exists for all times t ∈ C. We call V R-complete if its flow map exists for all times t ∈ R.
Since the flow satisfies the semi-group property, any time-t map of a Ror C-complete vector field is a holomorphic automorphism.
The density property for complex manifolds was introduced and studied by Varolin in [22,23] around 2000: Definition 4 [22] (1) Let X be a Stein manifold. We say that X has the density property if the Lie algebra generated by the complete holomorphic vector fields on X is dense (in the compact-open topology) is the Lie algebra of all holomorphic vector fields on X . (2) Let X be an affine manifold. We say that X has the algebraic density property, if the Lie algebra generated by the complete algebraic vector fields on X coincides with the Lie algebra of all algebraic vector fields on X .
By a standard application of Cartan's Theorem B and Cartan-Serre's Theorem A, the algebraic density property implies the density property (see e.g. [19], Proposition 6.2 which also covers the singular case). Since flows of algebraic vector fields don't need to be algebraic, there is no direct advantage in proving the algebraic density property over the density property. But polynomial vector fields are much more amenable to algebraic manipulations, and thus the algebraic density property is usually easier to prove directly and can be used a useful tool for establishing the density property.

Example 5
Examples of Stein manifolds with the density property include C n , n ≥ 2 which are a special case of affine homogeneous spaces of linear algebraic groups. The connected components of these homogeneous spaces enjoy the density property except for C and (C * ) n [9,17]. Whether or not (C * ) n has the density property is not known. Other classes of affine manifolds with the density property include smooth Danielewski surfaces {(x, y, z) ∈ C 3 : x y = p(z)} where p is a polynomial with simple zeroes [15]. Moreover, the Koras-Russel cubic threefold [21], Calogero-Moser spaces [5] and a large class of Gizatullin surfaces [3,4] also enjoy the density property. For details and a comprehensive list we refer the reader to the recent survey by Forstnerič and Kutzschebauch [13].
Let X be a complex manifold of complex dimension n. We call a complex differential form of bi-degree (n, 0) on X a volume form if it is nowhere degenerate.
Let X be a complex variety. We denote its group of holomorphic automorphisms by Aut(X ). If X is smooth and if there exists a volume form ω on X , we denote the group of ω-preserving holomorphic automorphisms by Aut ω (X ).
Definition 6 (1) Let X be a Stein manifold with a holomorphic volume form ω. We say that (X , ω) has the volume density property if the Lie algebra generated by the complete ω-preserving holomorphic vector fields on X is dense (in the compactopen topology) in the Lie algebra of all ω-preserving holomorphic vector fields on X . [22] (2) Let X be an affine manifold with an algebraic volume form ω. We say that (X , ω) has the algebraic volume density property if the Lie algebra generated by the complete ω-preserving algebraic vector fields on X coincides with the Lie algebra of all ω-preserving algebraic vector fields on X . [16] Again, the algebraic volume density property implies the volume density property; however, the proof is not straightforward and can be found in [16] by Kaliman and Kutzschebauch. The main result for manifolds with density property is the following theorem which was first stated for star-shaped domains of C n by Andersén and Lempert in 1992, then generalized to Runge domains by Forstnerič and Rosay in 1993 and finally extended to manifolds with the density property by Varolin: Theorem 7 [2,10,11,22] Let X be a Stein manifold with the density property or (X , ω) be a Stein manifold with the volume density property, respectively. Let ⊆ X be an open subset and ϕ : [0, 1] × → X be a C 1 -smooth map such that (1) ϕ 0 : → X is the natural embedding, (2) ϕ t : → X is holomorphic and injective for every t ∈ [0, 1] and, respectively, ω-preserving, and Then for every ε > 0 and for every compact K ⊂ there exists a continuous family Moreover, each of the automorphisms t can be chosen to be compositions of flows of generators of a dense Lie subalgebra in the Lie algebra of all holomorphic vector fields on X .
One of the two main ingredients in the proof of Theorem 7 is the following proposition which has been found by Varolin [22], but is stated best as a stand-alone result in the textbook of Forstneric [12].

Proposition 8 [[12], Corollary 4.8.4]
Let V 1 , . . . , V m be R-complete holomorphic vector fields on a complex manifold X . Denote by g the Lie subalgebra generated by the vector fields {V 1 , . . . , V m } and let V ∈ g. Assume that K is a compact set in X and t 0 > 0 is such that the flow ϕ t (x) of V exists for every x ∈ K and for all t ∈ [0, t 0 ]. Then ϕ t 0 is a uniform limit on K of a sequence of compositions of time-forward maps of the vector fields V 1 , . . . , V m .
For the proof of Theorem 31 where we can't make use of the density property, we will need to use Proposition 8 directly.
As one of many standard applications of Theorem 7 we obtain the following. It is implicit in the paper of Varolin [23], but can also be found with a detailed proof in [[6], Lemma 7 and Corollary 8].
Proposition 9 Let X be a Stein manifold with the density property resp. (X , ω) be a Stein manifold with the volume density property with dim C X ≥ 2. Let g be a Lie algebra that is dense in the Lie algebra of all holomorphic vector fields on X resp. in the Lie algebra of all ω-preserving holomorphic vector fields on X . Then the group of holomorphic automorphisms generated by the flows of completely integrable generators of g acts infinitely transitively on X .
The following lemma can be found in the proof of [ [14], Corollary 2.2] by Kaliman and Kutzschebauch. Its proof is a straightforward calculation.
Lemma 10 (Kaliman-Kutzschebauch formula) Let and be holomorphic vector fields and f , g, h be holomorphic functions on a complex space. Then the following holds: The power of this formula lies in the observation that if all the vector fields on the l.h.s. are in a certain Lie algebra and if in addition (h) = 0, then we found a new vector field on the r.h.s. that is a multiple of and lies the same Lie algebra.
The notion of the density property was extended to singular varities by Kutzschebauch, Liendo and Leuenberger [19]. Following them, we introduce these notations: Let X be a normal reduced Stein space and let X sing be its singular locus. Let A ⊆ X be a closed analytic subvariety containing X sing and let Let Lie hol (X , A) be the Lie algebra generated by all the complete vector fields in VF hol (X , A). Definition 11 [[19], Definition 6.1] Let X be a normal reduced Stein space and let A ⊂ X be a closed subvariety. We say that X has the (strong) density property relative to A if the Lie algebra generated by the complete holomorphic vector fields vanishing in A is dense in the Lie algebra of all holomorphic vector fields vanishing on A. Furthermore, we say that X has the weak density property relative to A if there exists ≥ 0 such that I A · VF hol (X , A) ⊆ Lie hol (X , A).

Theorem 12 [[19], Theorem 6.3] Let X be a normal reduced Stein space and let A ⊂ X be a closed analytic subvariety that contains the singularity locus of X .
Assume that X has the weak relative density property with respect to A for some ≥ 0. Let ⊆ X be an open subset and ϕ : [0, 1] × → X be a C 1 -smooth map such that Then for every ε > 0 and for every compact K ⊂ there exists a continuous family Moreover, these automorphisms can be chosen to be compositions of flows of generators of a dense Lie subalgebra in the Lie algebra of all holomorphic vector fields on X , see Varolin [22].

The Special Linear Group SL 2 (C)
Throughout this section, we will use the coordinates a, b, c, d for SL 2 (C) in the following way: The vector fields corresponding to left-multiplication by 1 t 0 1 and 1 0 t 1 , respectively, are: We have the commutation relations [H , V ] = 2V and [H , W ] = −2W . Similar to the case of the singular quadratic Danielewski surface, we obtain the following: Using the above, we find the following lemma. Proof yields a H and cH. We get the other terms in front of H by proceeding symmetrically with dW : The missing term in front of V then follow from: The following higher powers are easily obtained: By taking linear combinations, we obtain c k+1 H . Analogously, we treat a k+1 H , b k+1 H and d k+1 H .
Recall that It is now easy to see that Lie brackets of the form

Corollary 15 The group of holomorphic automorphisms generated by the flows of V , W , (b + c)W and d H acts infinitely transitively on SL 2 (C).
Proof Theorem 14 implies the (algebraic) density property for SL 2 (C). This corollary then follows from Proposition 9.

Quadratic Singular Danielewski Surface
Definition 16 We consider the following singular quadratic Danielewski surface M = {(x, y, z) ∈ C 3 : x y = z 2 } and the following complete vector fields Note that [H , ] = 2 and [H , ] = −2 , i.e. and form a sl 2 -pair. However, the singular surface M is not a homogeneous space of SL 2 (C). We will need the following equations where we act with and , respectively, from the left. [ Note that if not sending it directly to zero, none of , and H can reduce the polynomial degree. We therefore need and (or a linear combination thereof) for sure, since they can't be produced otherwise. Also note that the r.h.s. of the equations (2) and (6) only differ by a sign. The above computation shows that , and x can generate y . Similarly, , and y can generate x .

Lemma 17 The Lie algebra generated by , , x contains
H , x k , y k for all k ∈ N Proof By induction in k ∈ N we obtain all terms of the form x k : From equations (1)-(4) we obtain y and proceed with the calculation analogous to the above, where we reverse the roles of and as well as x and y, respectively.
Note that the flows of , , x are all algebraic. If we also allow holomorphic flows of complete algebraic vector fields, then we can generate a larger Lie algebra containing the above, if we take , , z , z H as generators:

Lemma 18
The Lie algebra generated by , , z H, z contains all the vector fields with all polynomial coefficients of degree one in front of , , H.

Proof
We proceed step by step as follows: After obtaining y , we use the equations from (5) to (8) to obtain x . By taking linear combinations of (10) and (5), we obtain y H and z . Equation (1) gives us also x H. Finally, using z H and equation (9) we obtain y . Analogous to (9) we have and thus obtain x .

Theorem 19
The four complete vector fields , , z , z H generate the Lie algebra

Proof of Theorem 19
(1) Using Lemma 17 and Lemma 18 we obtain the following (actually complete) vector fields for all k ∈ N 0 : (2) Next, we obtain all vector fields of the form z k . We proceed by induction in k. However, the induction step breaks down when passing from z 2 to z 3 .
(3) Similarly, we can obtain all vector fields of the form z k .
, h(z) be any polynomial in y, x, z, respectively. We apply the Kaliman-Kutzschebauch formula (KK) twice: Similarly, we can also obtain By taking linear combinations, we obtain all polynomial coefficients in front of H . Similarly, we obtain a result for , while the calculation for would be completely analogous to : Thus, we have obtained all polynomial coefficients in front of , and H .

Lemma 21
The vector fields , , H together span the tangent space of {(x, y, z) ∈ C 3 : x y = z 2 } in each point except the origin.
Proof Let us recall the definition of these vector fields: On each point of {x = 0} ∩ {y = 0} the vector fields and are linearly independent in C 3 and hence must span the 2-dimensional tangent space. On {x = 0} ∩ {y = 0} the vector fields and H span, and on {y = 0} ∩ {x = 0} the vector fields and H span, by the same argument.

Remark 22
One might also decide to work with , and some of their pullbacks instead: Pulling back with the flow of for fixed time t -and vice versa, we obtain new vector fields t , t . A direction calculation yields: which shows that these pullbacks are already in the span of , and H . This can also be compared to a result on algebraic ellipticity in the monograph of Alarcón, Forstnerič and López [[1], Proposition 1.15.3] where vector fields V 1 , V 2 , V 3 with polynomial flows are given explicitly in coordinates (z 1 , z 2 , z 3 ) ∈ C 3 on the quadric z 2 1 + z 2 2 + z 2 3 = 0; they are related by By a global linear change of coordinates, we can map our +i 2 and −i 2 to those V 1 and V 2 , respectively:

Corollary 23
The group of the holomorphic automorphisms generated by the flows of , , z and z H acts infinitely transitively on the regular locus of {(x, y, z) ∈ C 3 : x y = z 2 }.
Proof Theorem 19 implies the weak algebraic relative density property for {(x, y, z) ∈ C 3 : x y = z 2 } using the flows of the generators above. This corollary then follows from the same proof as in Proposition 9, but we instead have to use relative version with respect to the singularity in the origin.

Remark 24
The strong algebraic density property of x y = z 2 relative to the origin is known (but not with finitely many generators) due to [[19], Corollary 5.5] with d = 2, e = 1. We can now give another proof of this fact using finitely many generators:

Lemma 25 The Lie algebra of the polynomial vector fields on the singular quadratic
Proof Let be any polynomial vector field on M. Since M has an isolated singularity in the origin, must vanish there as well. In particular, each sumand of dz( ) must be divisible by at least one of the monomials x, y or z. We now observe that dz( ) = x, dz( ) = y and dz( ) = z. Hence, we find f , g, k ∈ C[x, y, z] such that y, z]. The requirement to be tangent to M is then equivalent to for some q ∈ C[x, y, z]. This implies that x | a, y | b and x y | q. Writing a = xa and b = yb , we obtain that a = −b mod (x y − z 2 ). Thus, We conclude that Finally, we observe that the following identities hold on M: This means that the polynomial k can in fact be chosen to be a constant.
Combining the preceding lemma with Theorem 19 above, we obtain the following result:

Quadratic Singular Danielewski Surface with Unipotent Subgroups
In this section we discuss how to find finitely many unipotent one-parameter subgroups that generate a subgroup of the algebraic automorphisms that acts infinitely transitively on M := {(x, y, z) ∈ C 3 : x y = z 2 }. We first consider two obvious approaches that do not quite work.

Remark 27
Outside the singularity in the origin, M is equipped with an algebraic volume form Recall that by Lemma 17 the Lie algebra generated by , , x contains Each of the vector fields , , x is a locally nilpotent derivation that preserves the algebraic volume form ω. However, it is not clear how other volume preserving vector fields of the form z k H could be obtained as Lie combinations or even be approximated. Thus, unlike in [6] any kind of transitivity result can't follow from an application of the volume density property. All the polynomial shears (or, in the terminology of [7]: replicas) of the locally nilpotent derivations and are of the form f (x) and g(y) , respectively. Let G be the group generated by their flows. Then we can't expect that the conjugates of f (x) and g(y) by group elements in G can be obtained by Lie combinations of f (x) and g(y) . Hence, the saturation condition of [[7], Theorem 2.5] is not satisfied.

Remark 28
In the following, let S := {x = 0}∪{y = 0} be the set where and are not spanning the tangent space.
Lemma 29 Let B 1 , . . . , B m ⊂ C 3 be balls of radius ε > 0 centered in p 1 , . . . , p m ∈ M\S respectively, with pairwise different projections to the x-axis and with pairwise different projections to the y-axis, and let v 0 ∈ T p m . For sufficiently small ε > 0 and for any δ > 0 there exist polynomials f , g ∈ C[z] such that the following hold for the vector field V := f (x) + g(y) .
Proof For small enough ε > 0, the projection of the union of B 1 , . . . , B m ⊂ C 3 to the x-axis and to the y-axis is Runge. By the Runge approximation theorem, we find holomorphic functions f , g : C → C such that f (x) + g(y) satisfies the desired approximation with an estimate of δ/2 instead of δ. For point (2) observe that and are spanning the tangent space in M \ S. Finally, Taylor expand f and g inside a large enough disk P ⊂ C that contains the projections of B 1 , . . . , B m , such that for their respective Taylor polynomials f and g we have that f − f P < δ/2 and g − g P < δ/2. Then f and g are the desired polynomials.  For a sufficiently close approximation, this map is submersive in 0 ∈ (C 2 ) m . Now by the implicit function theorem there exists a neighborhood U 1 × · · · × U m of ( p 1 , . . . , p m ) ∈ (M\S) m and a neighborhood V 1 × · · · × V m of (0, . . . , 0) ∈ (C 2 ) m such that : V 1 × · · · × V m → U 1 × · · · × U m is a surjective (in fact, bijective) holomorphic map. In particular, for each r ∈ U m =: U we find (t 1,x , t 1,y ) ∈ V 1 , . . . , (t m,x , t m,y ) ∈ V m s.t.
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