# The density property for vector bundles

Riccardo Ugolini  Riccardo Ugolini
Faculty of Mathematics
Ruhr University Bochum
Germany<EMAIL_ADDRESS>and  Jörg Winkelmann  Jörg
Winkelmann
Faculty of Mathematics
Ruhr University Bochum
Germany<EMAIL_ADDRESS>
###### Abstract.

We prove that holomorphic vector bundles over Stein manifolds with the density
property also satisfy the density property, provided that the total space is
holomorphically flexible. We apply this result to provide a new class of Stein
manifolds with the density property.

###### Key words and phrases:

density property, Stein manifold, flexibility, complex automorphism

###### 2020 Mathematics Subject Classification:

Primary: 32M18; Secondary: 32Q28, 32M25

##  1\. Introduction

The class of Stein manifolds with the density property is at the center of the
Andersén-Lempert Theory. The latter began when Andersén and Lempert first
described the group of holomorphic automorphisms of complex affine spaces
${\mathbb{C}}^{n},\ n>1$ [AL92]. Later, Varolin generalized their ideas and
introduced the density property:

######  Definition 1.1 ([Var01]).

A complex manifold $X$ has the density property if the Lie algebra generated
by complete holomorphic vector fields (those whose flows are defined for all
complex times) is dense in the Lie algebra of all holomorphic vector fields in
the compact-open topology.

Note that the flow of a complete vector field at a fixed time
$t\in{\mathbb{C}}$ defines an automorphism of $X$. Under the additional
condition that $X$ is Stein, the density property hence guarantees the
existence of a large number of automorphisms. This has many striking
consequences; we refer the interested reader to the monograph [For17] or the
more recent survey [FK21].

Our main result is the following:

######  Theorem 1.2 (Main theorem).

Let $X$ be a Stein manifold with density property, $\pi:E\to X$ a holomorphic
vector bundle.

Assume that the total space $E$ admits an automorphism which is not compatible
with $\pi$.

Then $E$ satisfies the density property.

The first application of Theorem 1.2 concerns a specific class of flexible
manifolds.

######  Definition 1.3.

A complex algebraic manifold $X$ is flexible at a point $p\in X$ if the
tangent space $T_{p}X$ is spanned by the tangent vectors to the orbits $Hp$ of
one-parameter unipotent algebraic subgroups
$H\subset\mathop{\mathrm{Aut}}_{alg}(X)$ of the group of algebraic
automorphisms. We say that $X$ is flexible if it is flexible at all of its
points.

These one-parameter subgroups induce algebraic actions by the additive group
of ${\mathbb{C}}$, i.e. i.e. they correspond to locally nilpotent derivations
(LNDs). In other words: $X$ is flexible at $p$ if for every $v\in T_{p}X$
there exists an algebraic vector field $\Theta$ such that the induced
derivation $\theta$ of the function ring ${\mathbb{C}}[X]$ satisfies

$\forall f\in{\mathbb{C}}[X]:\exists
n:\underbrace{\theta(\ldots(\theta(}_{\text{$n$-times}}f)\ldots)=0$

These LND proved useful in many aspects of the Andersén-Lempert theory; more
information can be found in the monograph [Fre17].

Denoting by $\mathop{\mathrm{SAut}}(X)$ the group generated by all unipotent
one-parameter subgroups of $\mathop{\mathrm{Aut}}_{alg}(X)$, the main result
about flexible affine manifolds is the following:

######  Theorem 1.4.

[AFK+13, Theorem 0.1] Let $X$ be an affine algebraic manifold of dimension
greater than one. Then, the following are equivalent:

  * •

$X$ is flexible

  * •

the group $\mathop{\mathrm{SAut}}(X)$ acts transitively

  * •

the group $\mathop{\mathrm{SAut}}(X)$ acts $n$-transitively for every
$n\in{\mathbb{N}}$

An action is $n$-transitive if any ordered $n$-tuple of distinct points can be
sent to any other such $n$-tuple by the action. This implies that the
automorphism group of a flexible manifold is infinite dimensional.

An easy example of a flexible variety is ${\mathbb{C}}^{n}$ for $n\geq 2$.

In view of Theorem 1.4, we have the immediate Corollary of our main Theorem:

######  Corollary 1.5.

Let $X$ be a Stein manifold with density property, $E\to X$ a holomorphic
vector bundle such that the total space $E$ is a flexible affine variety.

Then $E$ satisfies the density property.

This raises the question how to check whether $E$ is flexible. It is therefore
useful to know that flexibility of the base manifold $X$ implies the
flexibility of the total space $E$.

######  Theorem 1.6.

Let $E\to X$ be an algebraic vector bundle over a flexible smooth affine
variety $X$.

Then $E$ is a flexible affine variety.

The proof of the main theorem for vector bundles of rank greater than one
consists of two main ingredients: Proposition 7.2 gives that all vector fields
on $E$ that are compatible with the projection map are limits of Lie
combinations of complete vector fields, while Proposition 7.3 grants that
there is no intermediate closed Lie algebra between the Lie algebra of vector
fields compatible with the projection map and the Lie algebra of all vector
fields. The fact that $E$ is flexible is used to guarantee the existence of a
complete vector field which is not compatible with the projection map.

The case of line bundles is handled separately in Section 10. The main
difference is that the fiber does not satisfy the density property, hence a
new approach is required when approximating vertical vector fields.

Using Theorem 1.2, we are able to provide a class of Stein manifolds with the
density property.

######  Corollary 1.7.

Let $X$ be a smooth irreducible affine variety with only constant invertible
functions. Assume a reductive algebraic group $G$ acts on $X$ with an open
orbit and that the unique closed orbit is biholomorphic to neither the complex
plane $\mathbb{C}$ nor a torus $(\mathbb{C}^{*})^{k},\ k\in\mathbb{N}$.

Then $X$ has the density property.

With the exception on the condition about the unique closed orbit, the
hypotheses are the same as in [GS19, Theorem 2] where the authors prove that
such manifolds are flexible.

The fact that there is a unique closed orbit is a consequence of Luna’s slice
Theorem ([Lun73], see also [Sha94, Theorem 6.7]).

###### Proof.

Luna’s slice Theorem gives that $X$ is $G$-biholomorphic to
$G\times_{H}\mathbb{C}^{N}$ for a reductive subgroup $H\subset G$ acting
linearly on $\mathbb{C}^{N}$ for some $N\in\mathbb{N}$. In particular, the
projection $G\times_{H}\mathbb{C}^{N}\to G/H$ onto the first factor is a
holomorphic vector bundle. Since $G$ and $H$ are reductive, $G/H$ is affine
and biholomorphic to the unique closed orbit of the $G$-action on $X$. The
work of Donzelli, Dvorsky, and Kaliman [DDK10] guarantees that $G/H$ has the
density property; the conclusion follows from Theorem 1.2. ∎

Manifolds equipped with a Lie group action admitting an open orbit are called
_quasi-homogeneous_ ; many quasi-homogeneous affine varieties were classified
by Popov [Pop73a, Pop73b] and Popov and Vinberg [VP72].

If we require $G$ to be semisimple instead of reductive, the result holds
without additional hypotheses and is a corollary of the following [AFK+13,
Theorem 5.6]:

######  Theorem 1.8.

Suppose that a connected semisimple algebraic group $G$ acts on a smooth
irreducible affine variety $X$ with an open orbit.

Then $X$ is homogeneous with respect to a connected affine algebraic group
$\tilde{G}\supset G$ without nontrivial characters.

For the sake of completeness, we explicitly state the result for semisimple
groups; we do not claim any contribution to it.

######  Corollary 1.9 (Arzhantsev, Flenner, Kaliman, Kutzschebauch,
Zaidenberg).

Let $X$ be an affine manifold of dimension greater than one and not isomorphic
to a torus $\mathbb{C}^{k},\ k\in\mathbb{N}$. Assume that a connected
semisimple algebraic group $G$ acts on $X$ with an open orbit.

Then $X$ satisfies the density property.

###### Proof.

By Theorem 1.8, $X$ is a homogeneous space. Since it is neither one-
dimensional nor a torus, it satisfies the density property thanks to [KK17,
Theorem 1.3]. ∎

Let us illustrate a new example of Stein manifold with the density property
that can be obtained from our results:

######  Example 1.10.

Let $P$ be a polynomial on ${\mathbb{C}}$ without multiple zeroes. Then

$X=\\{(x,y,z)\in{\mathbb{C}}^{3}:xy=P(z)\\}$

is called a Danielewski surface. Danielewski surfaces are known to be flexible
affine varieties and to satisfy the density property ([KK08]). One may easily
verify that $b_{2}(X)=\deg(P)-1$ for a Danielewski surface (compare [KZ99]).
The classification of homogeneous surfaces (see [HL81]) implies that no
simply-connected complex surface with $b_{2}(X)>1$ is biholomorphic to a
quotient $G/H$ of complex Lie groups. Furthermore, $b_{2}(X)>0$ implies that
there exist non-trivial holomorphic line bundles. Fix such a non-trivial line
bundle over a Danielewski surface and call the total space $Y$. By Theorem 1.6
flexibility of the base manifold $X$ implies that the total space $Y$is also
flexible. Thus $Y$ is a Stein manifold with density property by Corollary 1.5.
To our knowledge, this is a new example of a manifold with density property
that does not appear elsewhere in the literature.

Most of the paper is devoted to the proof of Theorem 1.2. Section 11
independently proves Theorem 1.6, while in the last section we discuss the
existence of tame sets in the newly provided Stein manifolds with the density
property; the definition of tame set and a panoramic of the context in which
they arise will be provided there.

##  2\. Main result for rank at least two

######  Theorem 2.1.

Let $X$ be a Stein manifold with the density property, $\pi:E\to X$ a
holomorphic vector bundle of rank greater than one.

Assume that the total space $E$ admits an automorphism which is not compatible
with $\pi$.

Then $E$ satisfies the density property.

###### Proof.

Let $L$ be the Lie algebra of all holomorphic vector fields on the manifold
$E$ and let $P$ denote the subalgebra of those vector fields (whose flows are)
preserving the projection map $\pi$.

Vertical vector fields are approximable due to lemma 7.1. Complete vector
fields on $X$ may be lifted to complete vector fields on $E$ which are
compatible with the projection, by using a holomorphic connection (Proposition
3.2).

Combined, we obtain that every vector field in $P$ is approximable.

Let $H$ denote the Lie sub algebra of $L$ containing all approximable vector
fields.

Then

$P\subsetneq H\subset L$

Now Theorem 7.3 implies that $H=L$. ∎

##  3\. Horizontal vector fields

To find complete vector fields on $E$, we will lift complete vector fields on
$X$ along the projection $E\to X$. In order to do so, we need a holomorphic
connection.

######  Proposition 3.1.

Every holomorphic vector bundle $E$ over a Stein manifold $X$ admits a
holomorphic connection.

###### Proof.

The obstruction to the existence of a holomorphic connection on $E$ is given
by a cohomology class in $H^{1}(X,\Omega^{1}\otimes End(E))$. The sheaf
$\Omega^{1}\otimes End(E)$ is coherent and $X$ is Stein, hence Cartan’s
Theorem B guarantees that the cohomology group is trivial and there is no
obstruction to the existence of a holomorphic connection. ∎

Now that we can lift vector fields, we need to check that completeness is
preserved.

######  Proposition 3.2.

Let $E\to X$ be a holomorphic vector bundle equipped with a holomorphic
connection.

Then for every complete vector field on $X$ the lift to $E$ which is
horizontal with respect to the connection is complete, too.

###### Proof.

Let $V=\sum_{i}f_{i}(x)\frac{\partial}{\partial x_{i}}$ be a complete vector
field on the base with associated flow $\Phi(x,t)$, i.e.,
$\Phi^{\prime}(x,t)=f(\Phi(x,t))$.

Locally, on some neighborhood $U\subset X$ we may trivialize $E$ such that
$D(\sigma)=d\sigma+\alpha\sigma$ where $\alpha$ is a holomorphic function on
$U$ with values in $Mat(n\times n,\mathbb{C})$. The horizontal lift of $V$
amounts to $f(x)\frac{\partial}{\partial
x}+f(x)\alpha(x)(s)\frac{\partial}{\partial s}$.

We are looking for a function $H:U\to\mathbb{C}^{r}$ such that

$H_{i}^{\prime}(x,s,t)=\sum_{i,j,k}\alpha^{i}_{jk}(\Phi(x,t))v_{j}(\Phi(x,t))H_{k}(x,s,t)$

But this is a linear differential equation. Thus there is a global solution
(global on the fiber).

∎

######  Remark 3.3.

Let $X$, $Y$ be complete vector fields on a manifold $M$.

Then $[X,Y]$ can be approximated by linear combinations of complete vector
fields.

###### Proof.

For the benefit of the reader we indicate a proof although this is well known.
Let $\phi_{t}$ be the flow associated to $X$. Then

$[X,Y]={\mathcal{L}}_{X}(Y)=\lim_{t\to\infty}\frac{\phi_{t}^{*}Y-Y}{t}$

where ${\mathcal{L}}$ denotes the Lie derivative. ∎

######  Corollary 3.4.

Let $M$ be a manifold, $D$ the space of all vector fields and $C$ the vector
space generated by all complete vector fields.

Then the closure of $C$ in $D$ is a Lie subalgebra.

We conclude this section with a lemma which will be used later.

######  Lemma 3.5.

Let $E\to X$ be a holomorphic vector bundle equipped with a holomorphic
connection over a Stein manifold with the density property $X$.

Then every horizontal vector field preserving $\pi$ on $E$ can be approximated
by Lie combinations of complete vector fields.

###### Proof.

A horizontal vector field preserving $\pi$ on $E$ is the lift of a vector
field $V$ on $X$. Because of the density property and Remark 3.3, $V$ can be
approximated by a linear combination of complete vector fields. Since lifting
a vector field is a linear operation, the same linear combination of the lifts
of these vector fields will approximate our original vector field on $E$. ∎

##  4\. Euler Vector field

Let $E\to X$ be a complex vector bundle. Then there is a natural
${\mathbb{C}}^{*}$-action on $E$ with a corresponding vector field, the “Euler
vector field” $\Theta$.

In local coordinates: Let $(x_{i})_{i}$ be coordinates on the base space $X$
and let $(y_{j})_{j}$ be fiberwise linear coordinates on $E$. Then

$\Theta=\sum_{j}y_{j}\frac{\partial}{\partial y_{j}}$

The vector bundle structure is completely determined by the Euler vector
field: The zero-section coincides with the vanishing locus of $\Theta$,
$\pi(p)=\pi(q)$ for $p,q\in E$ iff the closures of the
${\mathbb{C}}^{*}$-orbits through $p$ and $q$ intersect. A function $\lambda$
on $E$ is fiber wise linear if and only if $\Theta\lambda=\lambda$.

Let $V$ be a complete vector field on $E$. Then the associated flow

  * •

consists of vector bundle automorphisms if $[\Theta,V]=0$.

  * •

consists of automorphisms which preserve the fiber and are fiberwise given as
translations if $[\Theta,V]=-V$.

##  5\. Vector fields incompatible with the vector bundle structure

Let $\pi:E\to X$ a holomorphic vector bundle and let $\phi$ be an automorphism
of the complex manifold $E$. We call $\phi$ “compatible with the bundle
structure”, or simply “$\pi$-compatible” if $\phi$ maps $\pi$-fibers to
$\pi$-fibers. If $X$ is smooth and therefore normal, being “compatible” is
equivalent to the existence of an automorphism $\psi$ of $X$ such that the
diagram

${E}$${E}$${X}$${X}$$\scriptstyle{\phi}$$\scriptstyle{\pi}$$\scriptstyle{\pi}$$\scriptstyle{\psi}$

is commutative.

Note that we do not require the map $\phi$ to be fiberwise linear. Hence the
notion “$\pi$-compatible automorphism” is weaker than the notion of a vector
bundle automorphism.

A vector field $v$ on $E$ (not necessarily complete) is called
“$\pi$-compatible” if for every $p,q\in E$ with $\pi(p)=\pi(q)$ we have
$\pi_{*}(v_{p})=\pi_{*}(v_{q})$. This is equivalent to the existence of a
vector field $w$ on $X$ with $\pi_{*}(v_{p})=w_{\pi(p)}\ \forall p\in E$.

If a $\pi$-compatible vector field is also complete, the associated flow
consists of $\pi$-compatible automorphisms.

######  Lemma 5.1.

Let $v$ be a $\pi$-compatible vector field on $E$ and let $f$ be a holomorphic
function on $E$. Assume that $v$ is not everywhere tangent to the $\pi$-fibers
(i.e., $\pi_{*}v\not\equiv 0$).

Then $fv$ is $\pi$-compatible if and only if $f$ is constant along the
$\pi$-fibers.

###### Proof.

Assume that $f$ is not constant along the $\pi$-fibers.

The set $\Omega_{1}$ of $p\in E$ where $f\circ\pi$ is not constant along the
$\pi$-fiber is open and dense in $E$. The set $\Omega_{2}$ of points where $v$
is not tangent to the $\pi$-fiber is also open and dense.

Hence the intersection of these two sets $\Omega_{1}\cap\Omega_{2}$ is still
dense and in particular not empty. Let $p$ be a point in
$\Omega_{1}\cap\Omega_{2}$. Choose $q\in\Omega_{1}\cap\Omega_{2}$ with
$\pi(p)=\pi(q)$, but $f(p)\neq f(q)$. (This is possible, since
$p\in\Omega_{1}$.) By construction, we have $\pi_{*}(v_{p})\neq 0$ and
$\pi_{*}(v_{p})=\pi_{*}(v_{p}q)$.

We obtain

$\pi_{*}(fv_{p})=f(p)\underbrace{\pi_{*}(v_{p})}_{\neq 0}\neq
f(q)\pi_{*}(v_{p})=f(q)\pi_{*}(v_{p})=\pi_{*}(fv_{q}).$

Thus $fv$ can not be $\pi$-compatible unless $f$ is constant along the
$\pi$-fibers.

The opposite direction is obvious. ∎

######  Proposition 5.2.

Let $E\to X$ be a vector bundle over a Stein manifold $X$.

If there exists a $\pi$-incompatible automorphism $\phi$, there also exists a
$\pi$-incompatible complete vector field $v$.

###### Proof.

Let $\Theta$ denote the Euler vector field. If $\phi_{*}\Theta$ is
$\pi$-incompatible, we are done. Thus we may assume that $\phi_{*}E$ is
$\pi$-compatible. Let $F$ be a $\pi$-fiber such that $\phi^{-1}(F)$ is not a
fiber. Then we choose a holomorphic function $f$ on $X$ such that $f\circ\pi$
is not constant along $\phi^{-1}(F)$. This implies that $f\circ\pi\circ\phi$
is not constant on $F$. Now the lemma implies that
$\phi_{*}\left((f\circ\pi)\Theta\right)$ is a complete vector field on $E$
which is incompatible with $\pi$. ∎

######  Corollary 5.3.

Let $\pi:E\to X$ be an algebraic vector bundle and assume that $E$ is a
flexible smooth affine variety.

Then there exists a $\pi$-incompatible complete holomorphic vector field $v$
on $E$.

###### Proof.

The automorphism group of every flexible smooth affine variety is
$n$-transitive for every $n\in{\mathbb{N}}$. Choose $p,q_{1},q_{2}\in E$ with
$\pi(p)=\pi(q_{1})\neq\pi(q_{2})$ and $p\neq q_{1}$. Since
$\mathop{\mathrm{Aut}}(E)$ (the group of variety automorphisms of $E$) is
$2$-transitive, there exists an automorphism $\phi$ of the variety $E$ such
that $\phi(p)=p$, $\phi(q_{1})=\phi(q_{2})$. Now
$\pi(\phi(p))\neq\pi(\phi(q_{1}))$ although $\pi(p)=\pi(q_{1})$. Thus $\phi$
is $\pi$-incompatible and the statement follows from Proposition 5.2. ∎

##  6\. Preparations

###  6.1. Existence of certain vector fields

######  Lemma 6.1.

Let $X$ be a Stein manifold, $p\in X$, $u\in T_{p}X$ and let $W$ be a
holomorphic vector field with $W(p)\neq 0$.

Then there exists a holomorphic vector field $V$ on $X$ with $[V,W](p)=u$.

###### Proof.

In suitable chosen local coordinates in some neighborhood of $p$ we have

$W=\frac{\partial}{\partial x_{1}},\quad
u=(u_{1},\ldots,u_{n})=\sum_{j}u_{j}\frac{\partial}{\partial
x_{j}}\quad(u_{j}\in{\mathbb{C}})$

We choose $V$ such that

$V=-\sum_{j}u_{j}x_{1}\frac{\partial}{\partial x_{j}}+\text{ terms vanishing
of order $\geq 2$ in $p$}$

(This is possible, because $X$ is Stein.) Then

$[V,W]=\sum_{j}u_{j}\frac{\partial}{\partial x_{j}}+\text{ terms vanishing in
$p$}$

∎

###  6.2. Runge Theory

######  Proposition 6.2.

Let $\pi:E\to X$ be a holomorphic vector bundle over a Stein manifold $X$ and
let $\Omega\subset X$ be a Runge domain (i.e. every holomorphic function on
$\Omega$ may be approximated by holomorphic functions on $X$).

Then every section $s\in\Gamma(\Omega,E)$ may be approximated by sections on
$X$. (We assume that $E$ is equipped with some fixed hermitian metric.)

###### Proof.

This is a consequence of [For17, Theorem 5.4.4]. ∎

######  Theorem 6.3.

Let $\pi:E\to X$ be a vector bundle on a Stein manifold $X$. Let
$(\sigma_{j})_{j\in I}$ be a family of sections in $E$ which span $E$
everywhere.

Then the $\mathcal{O}_{X}$-module generated by the $\sigma_{i}$ is dense in
the space of global sections of $E$.

######  Remark 6.4.

Density is the best one may hope for, equality is not always satisfied. For
example, let $X={\mathbb{C}}$, $E$ the trivial line bundle and consider

$\sigma_{k}=\frac{e^{2\pi iz}}{z-k}$

for $k\in{\mathbb{Z}}$. Then every holomorphic function ($=$ section in the
trivial line bundle) which is contained in the $\mathcal{O}_{X}$-module
generated by the $(\sigma_{k})_{k\in{\mathbb{Z}}}$ must vanish on almost all
integers, hence this $\mathcal{O}_{X}$-module does not equal
$\mathcal{O}({\mathbb{C}})$, although it is dense.

###### Proof.

Fix a Runge exhaustion $U_{k}$, i.e., we choose open subsets $U_{k}$ of $X$
such that $U_{k}$ is relatively compact in $U_{k+1}$, $U_{k}$ is Runge in $X$
and $X$ equals the union of all $U_{k}$.

Fix $p\in X$ and $k\in{\mathbb{N}}$. Choose $\sigma_{1},\ldots,\sigma_{r}$
within the given family such that $\sigma_{1}(p),\ldots,\sigma_{r}(p)$
generate $E_{p}$. Then these sections generate $E$ outside an analytic subset
$S_{1}$ dimension at least $1$.

Now we may proceed by recursion: Choosing one point on each irreducible
component of $S_{1}\cap U_{k}$ and then enough sections such that they
generate at these points. Since $U_{k}$ is relatively compact, there are only
finitely many such irreducible components. Hence finitely many additional
sections are enough. In this way we may increase the codimension of the
singular set step by step, finally verifying: For every $k\in{\mathbb{N}}$
there exists a finite subfamily $\\{\sigma_{j}:j\in I_{k}\\}$ such that these
sections generate $E_{p}$ for every $p\in U_{k}$.

Let $N_{k}=\\#I_{k}$. Let $I_{k}=\\{\sigma_{1},\ldots,\sigma_{N_{k}}\\}$. We
consider the sheaf morphism $\mathcal{O}^{N_{k}}\to{\mathcal{E}}$ given by

$(f_{1},\ldots,f_{N})\mapsto\sum_{j=1}^{N_{k}}f_{j}\sigma_{j}$

Let ${\mathcal{K}}$ denote the kernel sheaf. We obtain a long exact sequence

$\to\Gamma(X,\mathcal{O}^{N_{k}})\longrightarrow\Gamma(X,E)\longrightarrow
H^{1}(X,{\mathcal{K}})\to\ldots$

Since $U_{k}$ is Stein, the cohomology group $H^{1}(U_{k},{\mathcal{K}})$ must
be trivial. Thus $\Gamma(U_{k},\mathcal{O}^{N})\to\Gamma(U_{k},E)$ is
surjective. Hence every section of $E|_{U_{k}}$ is contained in the
$\mathcal{O}_{U_{k}}$-module generated by the $\sigma_{i}$.

Let $s\in\Gamma(X,E)$. For every $k\in{\mathbb{N}}$ the section $s$ is
contained in the $\mathcal{O}_{U_{k}}$-module generated by the $\sigma_{j}$.

Fix a hermitian metric on $D$. Since $U_{k}$ is Runge in $X$, we may find
$s_{k}\in\Gamma(X,E)$ such that

$||s-s_{k}||_{\overline{U_{k-1}}}\leq\frac{1}{k}$

Then $s=\lim_{k\to\infty}s_{k}$ for the topology of locally uniform
convergence. Therefore $\Gamma(X,E)$ equals the closure of the
$\mathcal{O}_{X}$-module generated by the $\sigma_{j}$. ∎

###  6.3. Fréchet Spaces

######  Theorem 6.5 (Peter-Weyl-Theorem).

Let $V$ be a Fréchet space and let $G$ be a reductive group acting linearly
and continuously on $V$. An element $v\in V$ is called $G$-finite if the
$G$-orbit through $v$ spans a finite-dimensional vector subspace of $V$.

Then the set of all $G$-finite elements is dense in $V$.

###### Proof.

See [PW27]. ∎

######  Corollary 6.6.

Let $\pi:E\to X$ be a vector bundle, $\Theta$ the Euler vector field and let
$H$ be a closed subspace of $\Gamma(E,TE)$ which is invariant under

$ad(\Theta):v\mapsto[\Theta,v]$

Let

$H_{k}=\\{v\in H:[\Theta,v]=kv\\}$

Then

$\oplus_{k\in{\mathbb{N}}_{0}}H_{k-1}$

is dense in $H$.

##  7\. Vertical vector fields

As mentioned in the introduction, we will prove that vector fields preserving
the projection can be approximated by Lie combinations of complete vector
fields. We already proved that horizontal vector fields preserving the
projection can be approximated by Lie combinations of complete vector field.
We will now show the same result for vertical vector fields. These aspects
will be put together in Proposition 7.2.

######  Lemma 7.1.

Let $\pi:E\to X$ be a holomorphic vector bundle over a Stein manifold $X$.

If $rank(E)\geq 2$, then every vertical vector field on $E$ can be
approximated by Lie combinations of complete vertical vector fields.

###### Proof.

Let $r$ denote the rank of the vector bundle $E\to X$. Fix a point $p\in X$.
Since $X$ is Stein, $E$ admits global sections $\sigma_{1},\ldots,\sigma_{r}$
such that the vector space $\pi^{-1}(p)=E_{p}\simeq\mathbb{C}^{r}$ is
generated by the sections $\sigma_{i}(p)$. Then

$\left(h_{1},\ldots,h_{r}\right)\mapsto\sum_{i}h_{i}\sigma_{i}$

defines a morphism $\tilde{\phi}_{0}$ of coherent sheaves from
$\mathcal{O}^{r}$ to $\mathcal{E}$ (the sheaf of sections in $E$). This
morphism $\tilde{\phi}$ is an isomorphism in a neighborhood of $p$, because
the $\sigma_{i}(p)$ form a basis of $E_{p}$.

Let $C$ be the zero set of $\sigma_{1}\wedge\ldots\wedge\sigma_{r}$ and let
$f$ be a holomorphic function on $X$ which vanishes on $C$, but not in $p$.
Define $\phi=f\phi_{0}$. Due to the choice of $f$, for every $q\in X$ the map
$\phi_{q}:{\mathbb{C}}^{r}\to E_{q}$ is either the zero map or an isomorphism.

By abuse of language, $\phi$ also denotes the corresponding holomorphic map
from $X\times{\mathbb{C}}^{r}$ to $E$.

Claim: For every complete vertical vector field $w$ on the trivial vector
bundles $X\times{\mathbb{C}}^{r}$, the push-forward $\phi_{*}w$ is a complete
vertical vector field on $E$.

First we verify that the push-forward exists. The push-forward clearly exists
where $\phi$ is an isomorphism. If $q\in X$ is a point where $\phi$ is the
zero map, we set $\phi_{*}w=0$.

Similarily we check completeness: If $\phi_{q}$ is an isomorphism over a point
$q\in X$, then obviously completeness of $w$ implies completeness of
$\phi_{*}w$. But if $\phi_{q}$ is the zero map, then $\phi_{*}w$ vanishes on
the fiber of $E$ over $q$ which trivially implies completeness.

This proves the claim.

We observe that $\phi_{*}$ is continuous for the topology of locally uniform
convergence.

As a consequence of the claim we obtain:

Given any complete vector field $w=\sum_{i}P(z)\frac{\partial}{\partial
z_{i}}$ on $\mathbb{C}^{r}$, we obtain a complete vector field on $E$ as

$\tilde{w}=\phi_{*}w=\sum_{i}P(z)=\sum_{i}\sigma_{i}\otimes
P_{i}(z)\frac{\partial}{\partial z_{i}}$

In other words: For every complete holomorphic vector field $w$ on a fiber
$E_{p}$ of the vector bundle $E\to B$ there exists a complete vertical vector
field $\hat{w}$ on $E$ such that $\hat{w}|_{E_{p}}=w$.

We recall that ${\mathbb{C}}^{r}$ enjoys the density property for $r\geq 2$.
Thus every vector field $w$ on $E_{p}$ may be realized as the limit of a
sequence of complete vector fields $w_{k}$ on $E_{p}$. It follows that every
vector field $w$ on a given fiber $E_{p}$ extends to a global vector field
$\hat{w}$ on $E$ which is the limit of a sequence of complete vector fields on
$E$, i.e., $\hat{w}$ is approximable.

Let $\Theta$ denote the Euler vector field on $E$ resp. the trivial vector
bundle, i.e., $\Theta$ is the fundamental vector field of the natural
${\mathbb{C}}^{*}$-action on a vector bundle. A vertical vector field $w$ is
called to be “of degree $d$” (with $d\in\mathbb{N}$) if $[\Theta,w]=(d-1)\cdot
w$. If $(x_{i})$ resp. $(y_{j})$ are local coordinates on $X$ resp. linear
coordinates in the fiber direction, then a vertical vector field

$w=\sum_{j}c_{j}(x,y)\frac{\partial}{\partial y_{j}}$

is of degree $d$ if the functions $y\mapsto c_{j}(x,y)$ are homogeneous
polynomials of degree $d$.

Consider the vector space $W_{d}$ of vector fields of degree $d$ on
$\mathbb{C}^{r}$.

As we have seen, for every $p\in X$ and every isomorphism
$E_{p}\simeq{\mathbb{C}}^{r}$ every vector field on $E_{p}$ (in particular,
every such vector field of degree $\leq d$) extends to a vertical vector field
on $E$ which is the limit of a sequence of complete vertical vector fields.

We claim: For fixed $d\in{\mathbb{N}}$, there are finitely many approximable
vertical vector fields $A_{1},\ldots,A_{N}$ on $E$ such that
$A_{1}(p),\ldots,A_{N}(p)$ generate the vector space of vector fields of
degree $d$ for every $p\in E$.

Indeed, first we choose approximable vertical vector fields
$A_{1},\ldots,A_{M_{1}}$ such they generate at one given point $p$. Then they
generate outside an analytic subset $S_{1}$ dimension at least $1$.

Now we may proceed by recursion: Choosing one point on each irreducible
component of $S_{1}$ and then enough vector fields such that they generate at
these points, we may increase the codimension of the singular set step by
step, finally verifying the claim.

Let ${\mathcal{V}}_{d}$ denote the sheaf of all vertical vector fields of
degree $d$. This is a coherent analytic sheaf on $X$.

We consider the sheaf morphism $\mathcal{O}^{N}\to{\mathcal{V}}_{d}$ given by

$(f_{1},\ldots,f_{N})\mapsto\sum_{j}f_{j}A_{j}$

As we have seen, this is a sheaf epimorphism. Let ${\mathcal{K}}$ denote the
kernel sheaf. We obtain a long exact sequence

$\to\Gamma(X,\mathcal{O}^{N})\longrightarrow\Gamma(X,{\mathcal{V}}_{d})\longrightarrow
H^{1}(X,{\mathcal{K}})\to\ldots$

Since $X$ is Stein, the cohomology group $H^{1}(X,{\mathcal{K}})$ must be
trivial. Thus $\Gamma(X,\mathcal{O}^{N})\to\Gamma(X,{\mathcal{V}}_{d})$ is
surjective. Therefore every $w\in\Gamma(X,{\mathcal{V}}_{d})$ may be written
as $\sum f_{j}A_{j}$ for some holomorphic functions $f_{j}$ on $X$. It follows
that every such vertical vector field $w$ is approximable. ∎

Now that we established these facts, let us proceed with one of the two main
ingredients of the proof of Theorem 2.1.

######  Proposition 7.2.

Let $E\xrightarrow[]{\pi}X$ be a holomorphic vector bundle of rank $\geq 2$
over a Stein manifold $X$ with the density property.

Then every $\pi$-compatible vector field on $E$ can be approximated by Lie
combinations of complete vector fields.

###### Proof.

A vector field $W$ on $E$ preserving $\pi$ can be decomposed as the sum of a
vertical vector field $V$ and a horizontal vector field $H$ preserving $\pi$.

The conclusion is obtained by applying lemma 7.1 and lemma 3.5 to $V$ and $H$,
respectively. ∎

######  Theorem 7.3.

Let $\pi:E\to X$ be a holomorphic vector bundle over a Stein manifold $X$.

Let $L$ be the Lie algebra of all holomorphic vector fields on the manifold
$E$ and let $P$ denote the subalgebra of those vector fields (whose flows are)
preserving the projection map $\pi$.

Then there does not exist any intermediate closed Lie algebra, i.e., if $H$ is
a closed Lie subalgebra of $L$ with $P\subset H$, then either $H=P$ or $H=L$.

###### Proof.

Fix $p\in X$. In local coordinates near $p$, vector fields on $E$ may be
written as

(1) $v=\sum_{\alpha,k}f_{\alpha,k}(x)y^{\alpha}\frac{\partial}{\partial
x_{k}}+\sum_{j}g_{j}(x,y)\frac{\partial}{\partial y_{j}}$

where $x=(x_{1},\ldots,x_{n})$ are local coordinates on the base and
$y=(y_{1},\ldots,y_{r})$ local coordinates in the fiber direction with
$\alpha=(\alpha_{1},\ldots,a_{k})$ being a multi-index. As usual,
$|\alpha|=\sum_{j}\alpha_{j}$.

Note that $v\in P$ if and only if $f_{\alpha,k}\equiv 0$ for $|\alpha|>0$.

Thus any element $w$ in $P$ may locally be written in the form

(2) $w=\sum_{k}f_{0,k}(x)\frac{\partial}{\partial
x_{k}}+\sum_{j}g_{j}(x,y)\frac{\partial}{\partial y_{j}}$

Consequently, elements in $L/P$ may be represented as

(3) $\sum_{\begin{subarray}{c}\alpha,k\\\ \alpha\neq
0\end{subarray}}f_{\alpha,k}(x)y^{\alpha}\frac{\partial}{\partial x_{k}}$

Assume that $H\neq P$. We consider the adjoint action of the Euler vector
field

$\Theta=\sum_{j}y_{j}\frac{\partial}{\partial y_{j}}$

For $v$ as in (1) we have

$[\Theta,v]=\sum_{\alpha,k}|\alpha|f_{\alpha,k}(x)y^{\alpha}\frac{\partial}{\partial
x_{k}}+\sum_{j}(\theta(g_{j})-g_{j})\frac{\partial}{\partial y_{j}}$

Hence $[\Theta,v]\in P$ for $v\in P$ and we obtain an induced action
$ad(\Theta)$ on $L/P$ via $v\mapsto[\Theta,v]$. This action is given as

$ad(\Theta)(w)=\sum_{\begin{subarray}{c}\alpha,k\\\ \alpha\neq
0\end{subarray}}|\alpha|f_{\alpha,k}(x)y^{\alpha}\frac{\partial}{\partial
x_{k}}\text{ for }w=\sum_{\begin{subarray}{c}\alpha,k\\\ \alpha\neq
0\end{subarray}}f_{\alpha,k}(x)y^{\alpha}\frac{\partial}{\partial x_{k}}$

Note that $\Theta$ is complete and yields a ${\mathbb{C}}^{*}$-action on $L$
given as

${\mathbb{C}}^{*}\ni\lambda:w\mapsto\sum_{\begin{subarray}{c}\alpha,k\\\
\alpha\neq
0\end{subarray}}\lambda^{|\alpha|}f_{\alpha,k}(x)y^{\alpha}\frac{\partial}{\partial
x_{k}}$

with $w$ as in (3). ${\mathbb{C}}^{*}$ is a reductive group and $L$ is a
Fréchet space. Hence we may apply the Peter-Weyl Theorem (Theorem 6.5). This
theorem implies that, given a ${\mathbb{C}}^{*}$-action $\mu$ on a Fréchet
space $V$, the subspace $\oplus_{d\in{\mathbb{Z}}}V_{d}$ is dense in $V$ where

$V_{d}=\\{v\in V:\mu(\lambda)(v)=\lambda^{d}v\\}.$

Since $L$ is a Fréchet space and $P$ and $H$ are closed invariant subspaces,
the quotient space $H/P$ is a Fréchet space on which ${\mathbb{C}}^{*}$ acts.

It follows that for every $w\in H/P$ as in (3) we have

$\sum_{|\alpha|=c}f_{\alpha,k}(x)y^{\alpha}\frac{\partial}{\partial x_{k}}\in
H/P\ \forall c\in{\mathbb{N}}$

In other words, if $H\neq P$, then $H/P$ contains a non-zero vector field $w$
which is homogeneous in the $y$-graduation, i.e., $\exists
m\in{\mathbb{N}}\setminus\\{0\\}$:

$H\ni w=\sum_{\begin{subarray}{c}\alpha,k\\\
|\alpha|=m\end{subarray}}f_{\alpha,k}(x)y^{\alpha}\frac{\partial}{\partial
x_{k}}\quad\text{ modulo $P$}$

The ${\mathbb{C}}^{*}$-action induced by the Euler vector field yields
gradings on $L$, $H$, $P$ and the algebra of vertical vector fields $V$.

For any $d\in{\mathbb{N}}$ the vector spaces $L_{d}$, $P_{d}$ and $V_{d}$ can
be realized as spaces of global section on $X$ of coherent sheaves
${\mathcal{L}}_{d}$, ${\mathcal{P}}_{d}$, ${\mathcal{V}}_{d}$, e.g., for any
open subset $U\subset X$, we define ${\mathcal{P}}_{d}(U)$ as the set of all
vector fields $w$ on $\pi^{-1}(U)\subset E$ which are in the form (2) and
satisfy $[\Theta,w]=dw$.

Similarily the quotient space $(L/P)_{d}=L_{d}/P_{d}$ arises as space of
global sections for a coherent sheaf ${\mathcal{S}}_{d}$ ($d\in{\mathbb{N}}$).

Using this notions, $H\neq P$ implies that there exists a number
$d\in{\mathbb{N}}$ such that $(L/P)_{d}$ has non zero intersection with $H/P$.

Let us now consider vector fields which locally may be written as

$u=\sum_{j}c_{j}(x)\frac{\partial}{\partial y_{j}}$

The family of these vector fields define a coherent sheaf
$\mathcal{A}={\mathcal{V}}_{1}$ on $X$. Since $X$ is Stein, this sheaf is
spanned by global sections. Fix a non-zero such vector field $u$.

The adjoint action $ad(u):v\mapsto[u,v]$ preserves $P$ and defines a morphism
of the graded Lie algebra $L/P$ of degree $-1$, i.e.,

$ad(u):(L/P)_{k+1}\mapsto(L/P)_{k}$

Observe that $u\in P$, because $v$ is a vertical vector field. Therefore

$ad(u):(H/P)\mapsto(H/P)$

Let

$v=\sum_{\begin{subarray}{c}\alpha,k\\\
|\alpha|=d\end{subarray}}f_{\alpha,k}(x)y^{\alpha}\frac{\partial}{\partial
x_{k}}$

represent a non-zero homogeneous element of $H/P$.

Since $\mathcal{A}$ is spanned by global sections, there is a vector field
$u\in P$ as above with

$\sum_{\begin{subarray}{c}\alpha,k\\\
|\alpha|=d\end{subarray}}f_{\alpha,k}(x)u\left(y^{\alpha}\right)\frac{\partial}{\partial
x_{k}}\not\equiv 0$

It follows: If $H/P$ contains a non-zero homogeneous element of degree $d+1$,
it also contains a non-zero homogeneous element of degree $d$.

By repeated application of this argument we deduce that $H/P$ contains a non-
zero homogeneous element of degree $1$.

In other words, for the grading of $L/P=\oplus_{d}(L/P)_{d}$ defined by the
eigen space decomposition with respect to $ad(\Theta)$, we have $H/P$ has non
zero intersection with $(L/P)_{1}$.

Next we claim: There is such a homogeneous element $v$ of degree $1$ in $H/P$
which may locally be written as

$v=\sum_{\begin{subarray}{c}\alpha,k\\\
|\alpha|=1\end{subarray}}f_{\alpha,k}(x)y^{\alpha}\frac{\partial}{\partial
x_{k}}$

such that there exists an index $\alpha$ (with $|\alpha|=1$) such that
$f_{\alpha,k}(p)\neq 0$.

For any $v$ as in the claim we call the minimum of the vanishing orders in $p$
of the components $f_{\alpha,k}$ ($|\alpha|=1$) simply the order of $v$. Thus
our goal is to show that $H$ contains an element $v$ of degree $1$ and order
$0$. Let $N$ denote the minimal number $N\in{\mathbb{N}}_{0}$ for which $H$
contains an element of degree $1$ and order $N$ and fix such an element $v$.

Next let $w$ be a vector field on $X$ which does not vanish in $p$. In
appropriately chosen local coordinates near $p$ on $X$ we have

$w=\frac{\partial}{\partial x_{1}}$

and

$v=\sum_{\begin{subarray}{c}\alpha,k\\\
|\alpha|=1\end{subarray}}f_{\alpha,k}(x)y^{\alpha}\frac{\partial}{\partial
x_{k}}=\sum_{j,k}f_{j,k}(x)y_{j}\frac{\partial}{\partial x_{k}}$

with $N$ being the minimal vanishing order of the $f_{\alpha,k}$.

Next let $\tilde{w}$ be the lift of $w$ with respect to some holomorphic
connection on the vector bundle $E\to X$. Then

$\tilde{w}=\frac{\partial}{\partial
x_{1}}+\sum_{j,l}A_{j,l}(x)y_{j}\frac{\partial}{\partial y_{l}}$

where the $A_{j,l}$ are holomorphic functions which depend on the choice of
the holomorphic connection.

As a horizontal vector field, $\tilde{w}\in P$. Hence $[\tilde{w},v]\in H$.
Now

$[\tilde{w},v]=\sum_{j,k}\frac{\partial
f_{j,k}}{\partial{x_{1}}}y_{j}\frac{\partial}{\partial x_{k}}\text{ $+$ terms
vanishing of order $\geq N$}$

By the minimality of $N$, it follows that all the $f_{j,k}$ vanish of order at
least $N+1$ along the integral curve of $w$ (i.e. in the $x_{1}$-direction).

But $w$ was arbitrary vector field on $X$ (with $w(p)\neq 0$). Hence we
obtain: If $N>0$, then for every integral curve of every vector field not
vanishing in $p$ the restriction of every $f_{j,k}$ to this curve vanishes of
order at least $N+1$ in $p$. The vanishing order of a holomorphic function
equals the minimum of the vanishing order of the restriction to every such
integral curve. Hence the assumption $N>0$ yields a contradiction to the
minimality assumption. This proves the claim, i.e., $H/P$ admits an element
$v\neq 0$ of degree $1$ and order $0$.

Next claim: $H$ defines a sub ${\mathcal{O}}_{X}$-module sheaf $H_{1}$:
Indeed, if $h$ is a holomorphic function on $X$, then $[v,h\Theta]=hv$ for any
$v\in L/P$ (note that $\Theta\in P$). $H$ is assumed to be a Lie sub algebra,
and $\Theta\in P\subset H$, hence $[v,h\Theta]\in H$ for every $v\in H$.

Thus we have

$H\ni v=\sum_{j,k}f_{j,k}(x)y_{j}\frac{\partial}{\partial x_{k}}$

such that $\exists j,k:f_{j,k}=1$. Wlog $f_{1,1}=1$.

Fix $m$. Since $X$ is Stein and the tangent sheaf is coherent, we can find a
global vector field $w$ on $S$ such that locally

$w=w_{m}=\sum_{j,l}h_{j,l}(x)x_{j}\frac{\partial}{\partial x_{l}}$

with

$h_{j,l}(p)=\begin{cases}1&\text{ if $j=1,l=m$}\\\ 0&\text{ else. }\\\
\end{cases}$

We lift $w_{m}$ to a horizontal vector field $\tilde{w}_{m}$ on $E$ (with
respect to some holomorphic connection).

$\tilde{w}_{m}=\sum_{j,l}h_{j,l}(x)x_{j}\left(\frac{\partial}{\partial
x_{l}}+\sum_{i,k}A_{i,k,l}(x)y_{i}\frac{\partial}{\partial y_{k}}\right).$

We recall: A holomorphic connection on a vector bundle is unique only up to
adding a section in $End(E)\otimes\Omega^{1}$. Since $V$ is a Stein manifold,
the vector bundle $End(E)\otimes\Omega^{1}$ is spanned by global sections. It
follows that by an appropriate choice of the holomorphic connection we may
assume that $A_{i,k,l}(p)=0$.

Then

$[v,\tilde{w}_{m}]=\sum_{\beta,r}g_{\beta,r}(x)y^{\beta}\frac{\partial}{\partial
x_{r}}\text{ modulo vertical vector fields and terms vanishing in $p$}$

with

$g_{\beta,r}=\begin{cases}1&\text{ if $\beta=e_{1},r=m$}\\\ 0&\text{ else.
}\\\ \end{cases}$

In other words, there is (for every $m$) an element $u=[v,\tilde{w}_{m}]\in H$
with

$u(p)=y_{1}\frac{\partial}{\partial y_{m}}.$

Recall that ${\mathcal{V}}_{d}$ is a coherent sheaf on the Stein manifold $X$
(for every $d\in{\mathbb{N}}$). Fix a multi-index $\beta$. Then there exists a
vector field $z$ with

$z=z_{\beta}=\sum_{\gamma,l}b_{\gamma,l}(x)y^{\gamma}\frac{\partial}{\partial
y_{l}}$

with

$b_{\gamma,l}(p)=\begin{cases}1&\text{ if $\beta=\gamma,l=1$}\\\ 0&\text{ else
}\\\ \end{cases}$

Now

$[z,u](p)=y^{\beta}\frac{\partial}{\partial x_{l}}$

Since $\beta$ and $l$ are arbitrary, we see that elements in $H$ span the
vector bundle corresponding to the sheaf ${\mathcal{L}}_{d}$ in $p$ for every
$d$. Note that $p$ was also arbitrary. Combined with the fact that $H$ is the
set of sections in a ${\mathcal{O}}_{X}$-module sheaf ${\mathcal{H}}$ we may
deduce that ${\mathcal{H}}_{d}={\mathcal{L}}_{d}$ for all $d$, which implies
that $H_{d}=L_{d}$.

Finally, recall that $H$ is assumed to be closed in $L$. Therefore
$L_{k}\subset H\ \forall k$ implies $H=L$. ∎

##  8\. Supporting Lemmas

The present section and the one immediately following it establish a few facts
needed in the proof of the line bundle case.

######  Lemma 8.1.

Let $V,W,Y$ be vector fields on a manifold, and $\phi$ a function.

Assume that $V\phi=0$, $W\phi=0$, $[W,Y]=0$ and $[W,V]=V$.

Define $\zeta=VY\phi$.

Then $W\zeta=\zeta$ and $\zeta W$ is contained in the Lie algebra generated by
$\phi W$, $Y$, $\phi Y$ and $V$.

###### Proof.

$\displaystyle\phantom{=}[\phi W,[Y,V]]-[\phi Y,V]$ $\displaystyle=-[Y,[V,\phi
W]]-[V,[\phi W,Y]]-\phi[Y,V]+(V\phi)Y$
$\displaystyle=-[Y,\phi\underbrace{[V,W]}_{=-V}]-[Y,(V\phi)W]-[V,\phi\underbrace{[W,Y]}_{=0}]+[V,(Y\phi)W]-\phi[Y,V]+(V\phi)Y$
$\displaystyle=[Y,\phi
V]-(YV\phi)W-(V\phi)\underbrace{[Y,W]}_{=0}-0+(VY\phi)W+(Y\phi)\underbrace{[V,W]}_{=-V}-\phi[Y,V]+(V\phi)Y$
$\displaystyle=\hbox{\pagecolor{yellow}$\phi[Y,V]$}+\hbox{\pagecolor{pink}$(Y\phi)V$}+(YV\phi)W+0+(VY\phi)W-\hbox{\pagecolor{pink}$(Y\phi)V$}-\hbox{\pagecolor{yellow}$\phi[Y,V]$}+(V\phi)Y$
$\displaystyle=\left((YV+VY)\phi\right)W+(\underbrace{V\phi}_{=0})Y$
$\displaystyle=(Y\underbrace{V\phi}_{=0})W+\underbrace{(VY\phi)}_{\zeta}W$
$\displaystyle=\zeta W$

Thus $\zeta W$ is in the Lie algebra generated by $\phi W$, $Y$ and $V$ if
$\zeta=VY\phi$.

On the other hand

$\displaystyle W\zeta$ $\displaystyle=WVY\phi$
$\displaystyle=WY\underbrace{V\phi}_{=0}+W[V,Y]\phi$
$\displaystyle=0+[V,Y]\underbrace{W\phi}_{=0}+[W,[V,Y]]\phi$
$\displaystyle=-\left([V,\underbrace{[Y,W]}_{=0}]+[Y,\underbrace{[W,V]}_{=V}]\right)\phi\quad\text{\sl
by Jacobi-Identity}$
$\displaystyle=-[Y,V]\phi=-\left(Y\underbrace{V\phi}_{=0}-VY\phi\right)=\zeta$

∎

######  Lemma 8.2.

Let $X$ be a smooth affine variety of dimension at least two, $Y$ a LND on
$X$. Then there exists an algebraic vector field $V$ and a regular function
$f$ on $X$ such that $Vf\equiv 0$ and $VYf\not\equiv 0$.

###### Proof.

Let $f$ be a non-constant function which is invariant under the flow of $Y$.
Let $q\in X$ be any point such that $(df)_{q}\neq 0$. Then we may chosen local
holomorphic coordinates near $q$ such that $V=\frac{\partial}{\partial x_{1}}$
and

$\frac{\partial f}{\partial x_{2}}(q)\neq 1$

Now we choose $Y$ such that near $q$ we have

$Y=x_{1}\frac{\partial}{\partial x_{2}}+\text{terms vanishing with
multiplicity $\geq 2$ in $q$}$

(This choice is possible, because $X$ is affine.) Our choice of $Y$ implies
$(VYf)(q)=1\neq 0$. ∎

######  Proposition 8.3.

Let $\pi:E\to X$ be a holomorphic vector bundle over a flexible smooth affine
variety, $\dim(X)\geq 2$. Let $\Theta$ denote the Euler vector field on $E$.
Let $k\geq 1$. Let $D_{k}$ denote the space of vector fields on $L$ of weight
$k$ (i.e., $[\Theta,V]=kV$ for $V\in D_{k}$). Let $V_{k}\subset D_{k}$ denote
the subspace of vertical ones.

Let $A$ be a Lie subalgebra of $\Gamma(L,TL)$ which contains all vector fields
of weight $\leq 0$.

Assume that

(4) $(A\cap D_{k})+V_{k}=D_{k}$

Then $A\cap V_{k}\neq\\{0\\}$.

###### Proof.

Using lemma 8.2 we choose vector fields $Y_{0}$, $V_{0}$ and a regular
function $\phi_{0}$ on $X$ such that $V_{0}\phi_{0}\equiv 0$ and
$V_{0}Y_{0}\phi_{0}\not\equiv 0$.

We lift $Y_{0}$ and $V_{0}$ to horizontal vector fields $Y$ resp. $V_{1}$ on
$E$ using some holomorphic connection and define $\phi_{0}=\phi\circ\pi$. Note
that $[\Theta,Y]=0=[\Theta,V_{1}]=\Theta\phi$.

Next we choose a section $\lambda\in\Gamma(X,E^{*})$ and observe that
$\lambda$ defines a fiberwise linear function on $E$. We define
$V=\lambda^{k}V_{1}$. Note that $V\in D_{k}$. From $[\Theta,V_{1}]=0$ it
follows that $[\Theta,V]=(E\lambda^{k})V_{1}=kV$. Due to (4), $V-H\in V_{k}$
for some $H\in A$.

We observe that $(Y_{0}\phi_{0})\circ\pi=Y(\phi_{0}\circ\pi)=Y(\phi)$ (as may
be verified easily calculating in local coordinates).

Similarily it is noted that

$H(Y\phi)=V(Y\phi)=\left(V_{0}Y_{0}\phi_{0}\right)\circ\pi\neq 0$

and

$H(\phi)=V(\phi)=(V_{0}\phi_{0})\circ\pi\equiv 0$

Now we may conclude from lemma 8.1 that

(5) $[\phi\Theta,[Y,H]]-[\phi Y,H]=(HY\phi)\Theta$

We claim that $\phi\Theta,Y,H,\phi Y\in A$. $H\in A$ by the choice of $H$.
Recall that $[\Theta,Y]=0=\Theta\phi$, implying

$[\Theta,\phi
Y]\underbrace{(\Theta\phi)}_{=0}Y+\phi\underbrace{[\Theta,Y]}_{=0}=0.\quad[\Theta,\phi\Theta]=\underbrace{(\Theta\phi)}_{=0}\Theta+\phi\underbrace{[\Theta,\Theta]}_{=0}=0.$

Thus $\Theta$, $\phi\Theta$, $Y$ and $\phi Y$ are all of weight $0$. By
assumption, $A$ contains all vector fields of weight $0$.

Therefore the left hand side of (5) is contained in $A$.

On the other hand, by construction the function $\phi$ is constant along the
$\pi$-fibers and $V-H$ is a vertical vector field. Hence $(V-H)\phi=0$ which
implies that the right hand side of (5) equals $(VY\phi)\Theta$. This implies
the assertion, since $V$ has weight $k$ (implying that $(VY\phi)\Theta\in
V_{k}$) and $VY\phi\not\equiv 0$. ∎

######  Remark 8.4.

The assumption $\dim(X)\geq 2$ in Proposition 8.3 is not seriously restrictive
for the following reason: Let $\pi:E\to X$ be a holomorphic vector bundle over
a non-compact Riemann surface. Then $E$ is biholomorphic to
${\mathbb{C}}^{N}\times X$ and therefore satisfies the density property
[Var01].

##  9\. Generalities

Let $\pi:E\to X$ be a holomorphic vector bundle on a Stein manifold.

We decompose the space $D$ of all vector fields with respect to the weights of
the Euler vector field, i.e., let

$D_{k}=\\{v\in D:[\Theta,v]=kv\\}$

From the Jacobi identity we obtain:

$\forall k,m:\forall v\in D_{k},w\in D_{m}:[v,w]\in D_{k+m}$

We consider the subspace of vertical vector fields

$V_{k}=\\{v\in D_{k}:(D\pi)_{p}(v)=0\ \forall p\in E\\}$

and the quotient

$Q_{k}=D_{k}/V_{k}$

We observe that (locally) the flows corresponding to vector fields in
$D_{-1}\oplus D_{0}$ preserve the vector bundle structure and that those in
$D_{-1}\oplus D_{0}\oplus\bigoplus_{k}V_{k}$ are $\pi$-compatible.

Every vector field $v\in V_{-1}\oplus V_{0}$ is complete, because the
associated flow is by fiber wise affine-linear transformations.

For $k=-1$ we have

$D_{-1}=V_{-1}\simeq\Gamma(X,E)$

A vector field $v\in D_{-1}$ may be written in local coordinates as

$\sum_{j}f_{j}(x)\frac{\partial}{\partial y_{j}}$

and corresponds to the section

$x\mapsto(x;f_{1}(x),\ldots,f_{r}(x))$

######  Lemma 9.1.

Let $A\subset D$ be a vector subspace which is stable under $ad(w)$ for all
$w\in D_{-1}$. Let $k>0$ and assume that $(A\cap D_{k})\setminus
V_{k}\neq\\{0\\}$.

Then $(A\cap D_{j})\setminus V_{j}\neq\\{0\\}$ for all $j\leq k$.

###### Proof.

Let $v\in A\cap D_{k}\setminus V_{k}$. In local coordinates around some point
$p$:

$v=\sum_{\alpha,j}f_{\alpha,j}(x)y^{\alpha}\frac{\partial}{\partial
x_{j}}+\text{vertical terms}$

Fix $\beta$ such that $\exists j:f_{\beta,j}\neq 0$. Observe that elements in
$D_{-1}$ are locally written as

$w=\sum_{h}g_{h}(x)\frac{\partial}{\partial y_{h}}$

Now we choose an element $w\in D_{-1}$ with $w(y^{\beta})\not\equiv 0$. Then

$[w,v]=\sum_{\alpha,j}f_{\alpha,j}(x)w\left(y^{\alpha}\right)\frac{\partial}{\partial
x_{j}}+\text{ vertical terms }$

Therefore $[w,v]\not\in V_{k-1}$ and consequently

$A\ni[v,w]=D_{k-1}\setminus V_{k-1}$

Iterating this process yields the assertion. ∎

Now for every function $\phi$ on $X$, $ad(\phi\Theta):v\mapsto[\phi\Theta,v]$
acts naturally on $D_{k}$; on $V_{k}$ and $Q_{k}$ it acts by multiplication
with $\phi k$:

$[\phi\Theta,v]=\underbrace{(v\phi)}_{=0}\Theta+\phi\underbrace{[\Theta,v]}_{=kv}=\phi
kv\text{ if $v\in V_{k}$}$

Note that $(v\phi)\Theta\in V_{k}$ for every $v\in D_{k}$. Hence

$[\phi\Theta,v]\equiv\phi kv\quad\text{ modulo $V_{k}$ $\forall v\in D_{k}$}$

As a consequence we obtain:

######  Proposition 9.2.

Let $A$ be a vector subspace of $D$ which is invariant under $ad(\phi\Theta)$
for every holomorphic function $\phi$ on $X$.

Then $A\cap V_{k}$ and $(A\cap D_{k})/(A\cap V_{k})$ are $\mathcal{O}_{X}$-sub
modules of $V_{k}$ resp. $Q_{k}$.

######  Lemma 9.3.

Assume $rank(E)=1$, i.e., $E$ is a line bundle. Assume in addition that a
group $G$ of holomorphic automorphisms acts on $E$ such that $\pi:E\to X$ is
equivariant and such that the induced action on the base manifold $X$ is
transitive.

Let $A\subset D$ be a closed $G$-invariant vector subspace. Assume that $A$ is
invariant under $ad(\phi\Theta)$ for every holomorphic function $\phi$ on $X$.
Then (for any $k\in{\mathbb{N}}$):

If $A\cap V_{k}\neq\\{0\\}$, then $V_{k}\subset A$.

###### Proof.

Since $rank(E)=1$, $V_{k}$ is the space of global sections of a line bundle.

Recall that $V_{k}\cap A$ is a $\mathcal{\mathcal{O}}_{X}$-module (lemma 9.2).
Hence, using Theorem 6.3, it suffices to show that for every $p\in X$ there
exists an element $s\in V\cap A_{k}$ with $s(p)\neq 0$. However, this is
obvious, because $G$ acts transitively on $X$ and $A\cap V_{k}\neq\\{0\\}$. ∎

######  Lemma 9.4.

Let $E\to X$ be a holomorphic line bundle over a Stein manifold $X$. Assume
that a Lie group $G$ acts transitively on $X$ such that the action can be
lifted to an action by vector bundles automorphisms on $E$.

Let $k\in{\mathbb{N}}$, and let $A$ be a closed $G$-invariant Lie subalgebra
of $D$ with $D_{0}\subset A$.

Then:

If $(A\cap D_{k})/(A\cap V_{k})\neq\\{0\\}$, then $D_{k}\subset V_{k}+A$.

###### Proof.

The elements in $D_{k}/V_{k}$ may be regarded as global sections in the
corresponding vector bundle. This vector bundle is isomorphic to
$TX\otimes(E^{*})^{k}$.

Let $v\in(A\cap D_{k})\setminus V_{k}$ and let $p\in X$ such that $v(p)$ is a
non-zero element in $T_{p}X\otimes(E_{p}^{*})^{k}$.

In local coordinates on a neighorhood $\Omega$ of $p$ ($x_{1},\ldots,x_{n}$ on
the base, $y$ on the fiber):

$v=v_{0}y^{k},\quad v_{0}\in\Gamma(\Omega,TX)$

Let $u\in T_{p}X$. Using lemma 6.1 we may choose a vector field $w_{0}$ on $X$
with $[w_{0},v_{0}](p)=u$. Using a holomorphic connection we may lift $w_{0}$
to an element $\tilde{w}\in D_{0}$ with

$\tilde{w}=w_{0}+h(x)y\frac{\partial}{\partial y}$

in local coordinates. We choose a global holomorphic function
$g\in\mathcal{O}(X)$ with $g(p)=h(p)$ and define $f(x)=h(x)-g(x)$ and

$w\stackrel{{\scriptstyle
def}}{{=}}\tilde{w}-g(x)\Theta=w_{0}+\underbrace{(h(x)-g(x))}_{f(x)}y\frac{\partial}{\partial
y}$

Note that by construction $f(p)=0$ and $w\in D_{0}$. Now

$[w,v]=[w_{0}+f(x)y\frac{\partial}{\partial
y},v_{0}y^{k}]=[w_{0},v_{0}]y^{k}+kf(x)y^{k}v_{0}-y^{k+1}(v_{0}(f))\frac{\partial}{\partial
y}$

Hence

$[w,v](p)=\underbrace{[w_{0},v_{0}]}_{=u}y^{k}+k\underbrace{f(p)}_{=0}y^{k}v_{0}-\underbrace{y^{k+1}(v_{0}(f))\frac{\partial}{\partial
y}}_{\in V_{k}}$

Thus $A\cap D_{k}$ contains an element $r$ with $r(p)\equiv u\ \text{mod}\
V_{k}$.

Recall that $u$ was arbitrary. Recall further that $A$ is $G$-invariant and
that $G$ acts transitively on $X$. It follows that the elements in $(A\cap
D_{k})/(A\cap V_{k})$ generate the vector bundle
$\Omega_{X}\otimes(E^{*})^{k}$ everywhere.

Due to lemma 9.2, $(A\cap D_{k})/(A\cap V_{k})$ is a $\mathcal{O}_{X}$-sub
module of $D_{k}/V_{k}$. From Theorem 6.3 it follows that $D_{k}\subset
A+V_{k}$. ∎

######  Lemma 9.5.

If $V_{1}\subset A$ and $(A\cap D_{k})/(A\cap V_{k})\neq\\{0\\}$, then $(A\cap
D_{k+1})/(A\cap V_{k+1})\neq\\{0\\}$.

###### Proof.

Let $v\in V_{1}\setminus\\{0\\}$. Then $v=\phi\Theta$ for some non-zero
holomorphic section $\phi$ of $E^{*}$, where $\Theta$ denotes the Euler vector
field.

Let $w\in(A\cap D_{k})\setminus V_{k}$. In local coordinates on some open
neighborhood $\Omega$ of $p$ in $X$:

$v=g(x)y^{2}\frac{\partial}{\partial y}\quad\text{ $\phi=g(x)y$}$

and

$w=w_{0}y^{k}+f(x)y^{k+1}\frac{\partial}{\partial y}\quad\text{ with
$w_{0}\in\Gamma(\Omega,T\Omega)$}$

Then

$[v,w]=[g(x)y^{2}\frac{\partial}{\partial
y},w_{0}y^{k}+f(x)y^{k+1}\frac{\partial}{\partial y}]=ky^{k+1}g(x)w_{0}+\text{
vertical terms}$

∎

##  10\. The Line Bundle Case

######  Theorem 10.1.

Let $\pi:L\to X$ be a holomorphic line bundle over a Stein manifold with the
density property. Assume that there exists a holomorphic automorphism of $L$
which is not $\pi$-compatible.

Then the total space $L$ is a Stein manifold with the density property, too.

###### Proof.

Since $X$ satisfies the density property, $\dim(X)\geq 2$.

Let $A$ denote the closure of the subset of complete vector fields on $L$ in
the space $D$ of all vector fields on $L$.

Recall that $D_{-1}\oplus D_{0}\subset A$.

We assumed that there is a holomorphic automorphism of $L$ which is not
$\pi$-compatible. Due to Proposition 5.2 it follows that there is a complete
vector field $v$ which is not $\pi$-compatible. The vector subspace of all
$\pi$-compatible vector fields equals

$C\stackrel{{\scriptstyle def}}{{=}}D_{0}\oplus
D_{-1}\oplus\bigoplus_{k>0}V_{k}$

The Euler vector field $\Theta$ is an element of $D_{0}$ and contained in the
Lie algebra $A$. Applying Theorem 6.5 to the Fréchet space $A/(A\cap C)$ for
the ${\mathbb{C}}^{*}$-action corresponding to $\Theta$, we conclude that

$\exists k>0:(A\cap D_{k})/(A\cap C\cap D_{k})\neq\\{0\\}\ \iff\ \exists
k>0:(A\cap D_{k})\setminus V_{k}\neq\\{0\\}$

With lemma 9.1 this in turn implies

$(A\cap D_{1})\setminus V_{1}\neq\\{0\\}$

Next $D_{1}\subset V_{1}+A$ due to lemma 9.4. By Proposition 8.3, $A\cap
V_{1}\neq\\{0\\}$. By lemma 9.3, $V_{1}\subset A$. In combination with
$D_{1}\subset V_{1}+A$, this yields $D_{1}\subset A$.

We have established that $V_{1}\subset A$ and $D_{1}\subset A$. Repeated
application of lemma 9.5 implies that

$(A\cap D_{k})\setminus V_{k}\neq\\{0\\}\ \forall k\in{\mathbb{N}}$

This in turn (using once again lemma 9.4, Proposition 8.3 and lemma 9.3)
yields $D_{k}\subset A\ \forall k$. Since $A$ is closed and $\oplus_{k}D_{k}$
is dense in $D$ (Theorem 6.5), we obtain $A=D$. ∎

##  11\. Flexibility

This Section aims to prove that the total space of an algebraic vector bundle
over a flexible affine variety is flexible, too.

######  Theorem 11.1.

Let $E\to B$ be an algebraic vector bundle over a smooth affine variety $B$.

Let $v$ be a locally nilpotent derivation of $k[B]$.

Then the vector bundle $E$ is invariant under the associated flow, i.e.,
$\mu_{t}^{*}E\simeq E$ where $\mu_{t}$ denotes the automorphism of $B$
corresponding to the $k$-algebra automorphism

$f\mapsto\sum_{k=0}^{\infty}\frac{t^{k}v^{k}(f)}{k!}$

###### Proof.

In ([Lin82], see also [Lam06]) Lindel proved (partially) the Bass-Quillen
conjecture. Lindels result implies that, given a smooth affine variety $V$,
every vector bundle on ${\mathbb{A}}^{k}\times V$ is given as a pull-back of a
vector bundle on $V$ via the projection map ${\mathbb{A}}^{k}\times V\to V$.

Let $\mu:G_{a}\times B\to B$ be the map defining the group action induced by
$v$, i.e., $\mu(t,x)=\mu_{t}(x)$. and let $p:G_{a}\times B\to B$ be the
projection on the second factor. By Lindens result there is a vector bundle
$E_{0}$ on $B$ such that $\mu^{*}E\simeq p^{*}E_{0}$. We observe that $\mu=p$
on $\\{0\\}\times B$. Hence $E_{0}\simeq E$. It follows that

$\mu_{t}^{*}E\simeq\mu^{*}E|_{\\{t\\}\times B}\simeq
p^{*}E_{0}|_{\\{t\\}\times B}\simeq p^{*}E|_{\\{t\\}\times B}\simeq E$

∎

######  Lemma 11.2.

Let

(6) $1\to N\to G\to U\to 1$

be a short exact sequence of linear algebraic groups with $U\simeq G_{a}$.

Then the sequence splits.

###### Proof.

Let $S_{1}$ be a maximal connected reductive subgroup of $N$ and let $S$ be a
maximal connected reductive subgroup of $G$ with $S_{1}\subset S$. Then
$S/S_{1}$ is a reductive group of dimension at most one which admits a non-
trivial morphism of algebraic groups into the additive group $G_{a}$. Thus
$S/S_{1}$ is trivial, i.e., $S_{1}=S$. Let $V_{1}$ resp. $V$ denote the
unipotent radical (i.e. the maximal normal unipotent subgroup) of $N$ resp.
$G$. Since $\dim G/N=1$ and $S=S_{1}$, we have $\dim(V/V_{1})=1$. In
particular $V\not\subset N$. We choose a one-dimensional vector subspace $L$
of $Lie(V)$ such that $L\not\subset Lie(N)$. Now $\exp(L)$ is a one-
dimensional unipotent subgroup of $G$ which admits a non-trivial morphism of
algebraic groups $\exp(L)\to U$. Since $\exp(L)$ and $U$ are both one-
dimensional and unipotent, this morphism is an isomorphism. Therefore it admit
an inverse which gives us the desired splitting of (6). ∎

######  Proposition 11.3.

Let $E\to B$ be an algebraic vector bundle over a smooth affine variety $B$.

Then every action of the additive group $G_{a}$ on $B$ lifts to an action on
$E$.

###### Proof.

For each $t\in G_{a}$, let $\mu(t)$ denote the corresponding automorphism of
$B$. Since $\mu(t)^{*}E\simeq E$ (Theorem 11.1) we may lift $\mu(t)$ to an
automorphism of $E$.

Let $G$ denote the group of all vector bundle automorphisms of $E$ such that
the respective automorphism of the base manifold $B$ is contained in
$\mu(G_{a})$. In other words $G$ consists of all variety automorphisms $\phi$
of $E$ which are fiberwise linear and such that there exists a $t$ with
$\pi(\phi(x))=\mu(t)(\pi(x))$ for all $x\in E$.

Now there is a natural surjective morphism of algebraic groups from $G$ to
$G_{a}$ and lemma 11.2 implies that the $G_{a}$-action on $B$ lifts to an
action on $E$ via an embedding of $G_{a}$ into $G$. ∎

######  Theorem 11.4.

Let $E\to B$ be an algebraic vector bundle. Assume that $B$ is a flexible
smooth affine variety.

Then $E$ is flexible, too.

###### Proof.

For $p\in E$ let $U_{p}$ denote the set of all tangent vectors $v\in T_{p}E$
for which there exists a corresponding $G_{a}$-action. Recall that $p$ is a
flexible point if $U_{p}=T_{p}E$ and that $E$ is flexible if every $p\in E$ is
a flexible point.

Since $B$ is flexible, Proposition 11.3 implies that $\pi_{*}:U_{p}\to
T_{\pi(p)}B$ is surjective for every $p\in E$.

Next we employ the work of [GS19]. We consider the natural
${\mathbb{C}}^{*}$-action on the vector bundle $E$. In the language used in
[GS19] $E$ is a parabolic non-hyperbolic $G_{m}$-space. Using Proposition 3 in
§3 of [GS19] we may conclude that every point $p$ in the zero-section of
$\pi:E\to B$ is a flexible point in $E$.

Now we observe that the set of flexible points is open and invariant under all
algebraic automorphisms of $E$. The automorphisms of $E$ include fiberwise
multiplication with scalars $\lambda\in{\mathbb{C}}^{*}$. Evidently the only
open neighborhood of the zero-section which is invariant under this
${\mathbb{C}}^{*}$-action is $E$ itself.

Thus every point of $E$ is flexible, i.e., $E$ is a flexible variety. ∎

The following is an additional result about existence of vector fields.

######  Proposition 11.5.

Let $X$ be a smooth affine variety with a non-trivial action of
$U=({\mathbb{C}},+)$. Let $\Theta$ denote the corresponding LND-Vector field.

Then there exists a $U$-invariant Zariski open subset $\Omega\subset X$ such
that for every $p\in\Omega$ and $v\in T_{p}X$ there is a vector field $W$ on
$X$ with $[W,\Theta]=0$ and $W_{p}=v$.

###### Proof.

By a result of Rosenlicht (?) there is a $U$-invariant Zariski open subset
$\Omega_{0}\subset X$ with $\Omega_{0}\simeq{\mathbb{C}}\times M_{0}$ such
that the $U$-action on $\Omega_{0}$ is simply translation in the first factor.
Let $I\subset{\mathbb{C}}[X]$ be the ideal of functions vanishing on
$X\setminus\Omega$. Now $U$ is unipotent and acts linearly on
${\mathbb{C}}[X]$, stabilizing $I$. Therefore there is a non-zero
$U$-invariant element $f$ in $I$. Define $\Omega=\\{x\in\Omega_{0}:f(x)\neq
0\\}$. By construction, $\Omega$ is an open invariant subset of $\Omega_{0}$.
Therefore $\Omega\simeq{\mathbb{C}}\times M$ for some Zariski open subset
$M\subset M_{0}$. Choose $p\in\Omega$, $v\in T_{p}X$. Then $v=c(\Theta_{p})+w$
with $w\in T_{p}M$. We choose a vector field $W_{0}$ on $M$ with
$(W_{0})_{p}=w$. Now $W_{1}=c\Theta+W_{0}$ is a vector field on $\Omega$ with
$(W_{1})_{p}=v$ and $[W_{1},\Theta]=0$. It extends to a rational vector field
on $X$. Recall that $f$ is a $U$-invariant function which vanishes outside
$\Omega$. Therefore $W=f^{N}W_{1}$ for $N$ sufficiently large is a regular
vector field with $[W,\Theta]=0$ and $W_{p}=v$. ∎

##  12\. Tame sets

The notion of tame set was first introduced by Rosay and Rudin [RR88] as they
were studying the properties of the group of holomorphic automorphisms of
complex Euclidean spaces.

######  Definition 12.1 ([RR88],Def. 3.3).

Let $e_{1}$ be the first standard basis vector of $\mathbb{C}^{n}$. A set
$A\subset\mathbb{C}^{n}$ is called _tame_ (in the sense of Rosay and Rudin) if
there exists an $F\in\mathop{\mathrm{Aut}}(\mathbb{C}^{n})$ such that
$F(A)=\mathbb{N}\cdot e_{1}$. It is very tame if we can choose
$F\in\mathop{\mathrm{Aut}}_{1}(\mathbb{C}^{n})$ to be volume preserving.

In recent years, this notion was generalized to arbitrary complex manifolds.
The definition of strongly tame set is due to Andrist and the first named
author [AU19]; weakly tame sets were introduced by the second named author
[Win19].

######  Definition 12.2 (Andrist, Ugolini).

Let $X$ be a complex manifold and let $G\subseteq\mathop{\mathrm{Aut}}(X)$ be
a subgroup of its group of holomorphic automorphisms. We call a closed
discrete infinite set $A\subset X$ a _strongly $G$-tame set_ if for every
injective mapping $f\colon A\to A$ there exists a holomorphic automorphism
$F\in G$ such that $F|_{A}=f$. If $G=\mathop{\mathrm{Aut}}(X)$, we only say
strongly tame set.

######  Definition 12.3 (Winkelmann).

Let $X$ be a complex manifold. An infinite discrete subset $D$ is called
weakly tame if for every exhaustion function $\rho\colon X\to\mathbb{R}$ and
every function $\zeta\colon D\to\mathbb{R}$ there exists an automorphism
$\Phi$ of $X$ such that $\rho(\Phi(x))\geq\zeta(x)$ for all $x\in D$.

Any strongly tame set is automatically weakly tame, hence their names. When
$X={\mathbb{C}}^{n},\ n>1$, both notions coincide with the classical
definition by Rosay and Rudin.

It is clear that the existence of a tame set (strong or weak) requires the
group of holomorphic automorphisms to be somewhat large. In this section, we
provide existence results for total spaces of holomorphic vector bundles.

######  Proposition 12.4.

Let $E\to B$ be a holomorphic vector bundle over a Stein manifold $B$. Then
$E$ admits a weakly tame set.

See [Win19], Proposition 8.4 for a closely related result.

###### Proof.

Choose a closed discrete sequence $A=\\{a_{n}\\}_{n\in{\mathbb{N}}}\subset E$
such that $\pi(a_{n})\neq\pi(a_{m})$ for $n\neq m$, $a_{n}\neq 0\in
E_{\pi(a_{n})}$, and such that $\pi(A)$ is also closed and discrete in $B$.

Fix an exhaustion function $\rho$. Given a function $\zeta\colon
A\to\mathbb{R}$, we can find a holomorphic function $h$ on $B$ such that
$\rho(e^{h(\pi(a_{n}))}\cdot a_{n})>\zeta(a_{n})$, because $A$ is a discrete
subset in the Stein manifold $B$.

Let $\Theta$ be the Euler vector field, then the time-1 flow $\Phi$ of the
vector field $h\Theta$ is such that $\rho(\Phi(x))\geq\zeta(x)$ for all $x\in
A$ i.e. $A$ is weakly tame. ∎

As we saw, the existence of a weakly tame set in the total space of a vector
bundle over a Stein manifold does not require any additional hypothesis. The
picture is quite different for strongly tame sets.

Given a complex manifold $X$, we denote by $\mathop{\mathrm{Aut}}(X)^{0}$ the
connected component of the identity in $\mathop{\mathrm{Aut}}(X)$.

######  Proposition 12.5.

Let $E\to B$ be a holomorphic vector bundle over a Stein manifold $B$ and
assume that $B$ admits a strongly $\mathop{\mathrm{Aut}}(B)^{0}$-tame set.
Then, $E$ admits a strongly tame set.

###### Proof.

Let $A^{\prime}\subset B$ be a strongly $\mathop{\mathrm{Aut}}(B)^{0}$-tame
set. Denote by $A=\\{a_{n}\\}_{n\in{\mathbb{N}}}\subset E$ the image of
$A^{\prime}$ under the zero section of $E\to B$.

Let $f\colon A\to A$ be an injective self-map. Since $A^{\prime}$ is
$\mathop{\mathrm{Aut}}(B)^{0}$-tame, there exists an automorphism
$\Phi\in\mathop{\mathrm{Aut}}(B)^{0}$ such that
$\Phi(\pi(a_{n}))=\pi(f(a_{n}))$. The map $\Phi$ induces a commutative diagram

${\Phi^{\ast}E}$${E}$${B}$${B}$$\scriptstyle{\tilde{\Phi}}$$\scriptstyle{\Phi}$

Since the map $\Phi$ is connected to the identity, $E$ and $\Phi^{\ast}E$ are
topologically isomorphic as vector bundles over $B$. As the topological and
holomorphic classification of vector bundles over Stein manifolds is
equivalent, there exists a holomorphic vector bundle isomorphism $F\colon
E\to\Phi^{\ast}E$ (covering the identity).

The map $H=\tilde{\Phi}\circ F$ satisfies $H(a_{n})=f(a_{n})$, hence $A$ is
strongly tame.

∎

It is not clear whether reducing ourselves to the connected component of the
identity is particularly restrictive. When the base space $B$ is an affine
homogeneous space of a linear complex Lie group, the existence of strongly
$\mathop{\mathrm{Aut}}(B)^{0}$-tame sets is proved in [AU22].

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