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The fibred density property and the automorphism group of the spectral ball

We generalize the notion of the density property for complex manifolds to holomorphic fibrations, and introduce the notion of the fibred density property. We prove that the natural fibration of the spectral ball over the symmetrized polydisc enjoys the fibred density property and describe the automorphism group of the spectral ball.


Introduction and Results
The density property for complex manifolds has been introduced by Varolin [Var01,Var00] in 2000 and has its orgin in the work of Andersén [And90] and Andersén-Lempert [AL92] in the early 1990ies. It characterizes complex manifolds with large automorphism groups. Since then the so-called Andersén-Lempert theory has developed rapidly. It has many applications for geometric questions in Several Complex Variables and has contributed a lot to a better understanding of large holomorphic automorphism groups. For a recent overview over the theory we refer to [KK11].
In this paper we introduce a parametrized version of the density property for holomorphic fibrations where the density property holds only in the direction of possibly singular fibres which are parametrized by the base space. For a comparison we restate the definition of the density property.
Definition 1.1. Let X be a complex manifold. We say that X has the density property if the Lie algebra generated by all holomorphic C-complete vector fields on X is dense (w.r.t. compact-open topology) in the Lie algebra of all holomorphic vector fields on X.
Definition 1.2. Let π : X → Y be a holomorphic fibration between complex manifolds. We say that the fibration has the fibred density property if the Lie algebra generated by all holomorphic C-complete vector fields on X tangent to the fibres of π is dense (w.r.t. compactopen topology) in the Lie algebra of all holomorphic vector fields on X tangent to π.
Example 1.3. The fibred density property is known for a trivial fibration: consider the projection π : U × V → V for Stein manifolds U and V where U has the density property. Then this fibration has the fibred density property, as a consequence of [Var01,Lemma 3.5].
In this paper we will first develop the general theory for the fibred density property (section 2) and then prove the fibred density property for a well-known quotient map from classical invariant theory. This enables us to describe the automorphism group of the spectral ball.
Concerning the general theory the following theorem is a fibred version of the classical Andersén-Lempert theorem [And90,AL92,Var01,Var00].
Theorem 1.4. Let X be a Stein manifold and let π : X → Y be a holomorphic fibration with the fibred density property. Let Ω ⊆ X be an open subset and ϕ t : Ω → X, t ∈ [0, 1], be a fibre-preserving C 1homotopy of injective holomorphic maps such that ϕ 0 is the natural embedding Ω ֒→ X and ϕ t (Ω) is Runge in X for all t ∈ [0, 1]. Then there exists a fibre-preserving homotopy Φ t : X → X of holomorphic automorphisms such that Φ 0 = id X and Φ t is arbitrarily close to ϕ t on Ω in the compact-open topology.
Moreover, Φ t can be chosen as a composition of flow maps corresponding to complete fibre-preserving vector fields which generate a dense Lie subalgebra of the Lie algebra of the fibre-preserving holomorphic vector fields.
By Aut π (X) we denote the group of fibre preserving holomorphic automorphisms of X, i.e. Aut π (X) := {f : X → X holomorphic automorphism, π • f = π} and by Aut π 0 (X) its path-connected component of identity. By choosing Ω = X and ϕ t a path in Aut π (X) starting at id X , we obtain: Corollary 1.5. Let X be a Stein manifold and let π : X → Y be a holomorphic fibration with the fibred density property. Then all fibrepreserving holomorphic automorphisms Aut π 0 (X) path-connected to the identity can be approximated by compositions of time-1 maps of flows of complete fibre-preserving vector fields which generate a dense Lie subalgebra of the Lie algebra of fibre-preserving holomorphic vector fields.
Next we present an application of our general result concerning the fibred density to a concrete example, the so-called spectral ball. It appears naturally in Control Theory [BFT89,BFT91], but is also of theoretical interest in Several Complex Variables.
Definition 1.6. The spectral ball of dimension n ∈ N is defined to be where ρ denotes the spectral radius, i.e. the modulus of the largest eigenvalue.
The study of the group of holomorphic automorphisms of the spectral ball has started with the work of Ransford and White [RW91] in 1991 and continued by various authors [BR00,Ros03,Tho08]. Recently, Kosiński [Kos13] described a dense subgroup of the 2 × 2 spectral ball Ω 2 . We generalize this result to Ω n for n ≥ 2 with another approach using the fibred density property.
The spectral ball Ω n can also be understood in the following way: Denote by σ 1 , . . . , σ n : C n → C the elementary symmetric polynomials in n complex variables. Let Eig : Mat (n × n; C) → C n assign to each matrix a vector of its eigenvalues. Then we denote by π 1 := σ 1 • Eig, . . . , π n := σ n • Eig the elementary symmetric polynomials in the eigenvalues. By symmetrizing we avoid any ambiguities of the order of eigenvalues in the definition of Eig and obtain a polynomial map π 1 , . . . , π n , symmetric in the entries of matrices in Mat (n × n; C), actually where χ A denotes the characteristic polynomial of A. Now we can consider the fibration π := (π 1 , . . . , π n ) : Ω n → G n of the spectral ball over the symmetrized polydisc G n := (σ 1 , . . . , σ n )(D n ). A generic fibre, i.e. a fibre above a base point with no multiple eigenvalues, consists exactly of one equivalence class of similar matrices. Therefore it is natural to study the action of SL n (C) on Ω n by conjugation.
A generic fibre is obviously a homogeneous space and hence smooth. A fibre above a base point with multiple eigenvalues decomposes into several strata of SL n (C) orbits where the largest orbit is the orbit of a matrix with the largest possible Jordan blocks. The structure of these fibres is well-known in classical invariant theory, see e.g. [Kra84].
Because a generic fibre is a homogeneous space of the complex Lie group SL n (C), it has the density property (according to [DDK10]). This does however not imply the fibred density property, since the dependence on the base point and the role of singular fibres is a priori not clear. However it motivates the investigation of fibre-preserving automorphisms by exploiting the homogeneity of the generic fibres. The most difficult part of our paper is to prove the fibred density property for π : Ω n → G n (see Theorem 4.6) that enables us to determine a dense subgroup of the holomorphic automorphism group Aut (Ω n ).
Theorem 1.7. The SL n (C)-shears and the SL n (C)-overshears together with matrix transposition and Möbius transformations generate a dense subgroup (in compact-open topology) of the holomorphic automorphism group Aut (Ω n ).
The precise definitions of shears and overshears will be given in section 3. They are obtained as time-1 maps of certain re-parametrizations of flow maps of complete vector fields; in case of SL n (C)-shears and SL n (C)-overshears they are certain re-parametrizations of 1-parameter subgroups of SL n (C) which act on Mat (n × n; C) by conjugation.
Similar to the situation for the holomorphic automorphism group of C n , n ≥ 2, it seems impossible to give an explicit set of algebraic generators for Aut (Ω n ), since we prove the following in the last section: Theorem 1.8. The dense subgroup of Aut (Ω n ) generated by the automorphisms in Theorem 1.7 is a meagre subset of Aut (Ω n ).

Andersén-Lempert theory for fibrations
We follow the original idea of the Andersén-Lempert Theorem, see [And90,AL92] and also the survey article [KK11]  . Let Θ be a vector field on a complex manifold X, and let (t, x) → A t (x) be a continuous map to X, defined on an open subset of R × X containing {0} ×X such that its t-derivative exists and is continuous. We say that A is algorithm for Θ if we have for all x ∈ X that Obviously, a flow map is always an algorithm whereas the converse does not need to be true. However, the following variant of Euler's method for solving an ODE works: Let Θ be a locally Lipschitz continuous vector field with flow ϕ t on a complex manifold X. Let Ω be the fundamental domain of Θ and Ω + := Ω∩(R + ×X). If A t is an algorithm for Θ, then for all (t, x) ∈ Ω + the n-th iterate A n t/n (x) of the map A t/n is defined for sufficiently large n = n(t, x) ∈ N and we have The convergence is uniform on compacts in Ω + . Conversely, if t 0 > 0 is such that A n t/n (x) is defined for all t ∈ [0, t 0 ] and all sufficiently large n ∈ N, and lim n→∞ A n t/n (x) exists, then (t 0 , x) ∈ Ω + .
For simplicity, we focus on the situation t ≥ 0, but the same results hold for negative times by replacing Θ with −Θ. The following lemma can be verified easily in local coordinates by Taylor series expansion, see e.g. [For11,Prop. 4.7.3].
Lemma 2.3. Let X be a complex manifold and let Θ and Ξ be holomorphic vector fields on X with flow maps or algorithms ϕ t and ψ t . Then (1) ϕ t • ψ t is an algorithm for Θ + Ξ . Proof of Theorem 1.4. By A we denote a dense Lie subalgebra of complete fibre-preserving vector fields which is dense in the Lie algebra of fibre-preserving holomorphic vector fields. This Lie subalgebra exists by assertion (fibred density property).
We define a time-dependent vector field Θ t z :=φ t (ϕ −1 t (z)) which is still tangent to the fibres of π : X → Y . For any n ∈ N we can partition the interval [0, 1] in n intervals [k/n, (k + 1)/n] of length 1/n and consider the piecewise constant vector field Θ t z := Θ k/n z for t ∈ [k/n, (k + 1)/n) Let ϕ k t denote the flow map of Θ k/n z . Because Ω ⊆ X is Runge and X is Stein, we can approximate any fibre preserving vector field on Ω by a fibre preserving vector field on X, uniformly on compacts of Ω. We remark that the sheaf of sections of fibre-preserving vector fields is (as the kernel of the map induced by π between the coherent sheafs of sections of the tangent bundles) a coherent sheaf of O(X) modules. A standard application of Cartan's Theorems A and B implies that the sections of any coherent sheaf over a Runge subset in a Stein space can be approximated by global sections.
By assertion we know that every such vector field can be approximated by vector fields from the Lie algebra A. Proposition 2.2 and Lemma 2.3 then show that we are able to approximate the flows of all the vector fields in the closure of A in the compact-open topology by the flows of the complete vector fields which generate A.
The composition of the flows ϕ k ... • · · · • ϕ 1 ... is the flow of Θ t z . It remains only to show that in the limit n → ∞ the flow of Θ t converges uniformly on compacts to the flow of Θ t which follows from the fact that these flows are tangent to each other at the times k/n.

Shears and Overshears for SL n (C)
As a preparation for proving a fibred density property for the spectral ball, we need to study a special type of automorphisms, the so-called shears and overshears. They will serve us a building blocks for general automorphisms.
Definition 3.1. Let X be a complex manifold and let Θ be a Ccomplete vector field on X, i.e. such that its flow-map exists for all complex times.
Lemma 3.3. Let X be a complex manifold and let Θ be a C-complete vector field on X, then all Θ-shear and Θ-overshear vector fields are C-complete as well. In fact, if φ t denotes the flow map of Θ, the flows of the shears and overshears are given by Notice that flow map for the shear vector field can be obtained as a limit of the flow map for the overshear vector field with Θ z f → 0.

Proof.
Let φ t denote the flow map of Θ, i.e.
The calculation is analogous to the previous case: We can compare the higher order derivatives: For t = 0 all the derivatives of these two terms agree, hence they the terms describe the same holomorphic function.
We will call the time-1 maps of such C-complete vector fields shears resp. overshears.
In our context here we will focus on shears and overshears arising from the SL n (C)-action on Ω n by conjugation. A Θ-shear of vector field Θ arising from the SL n (C)-action will be called a SL n (C)-shear and a Θ-overshear of such a vector field will be called a SL n (C)-overshear.
We denote the following commutators as H a := [E a,a+1 , E a+1,a ] = (δ ak δ aℓ − δ a+1,k δ a+1,ℓ ) n k,ℓ=1 for a = 1, . . . , n − 1. It is well-known that the E ab with a = b together with the H a span the matrix Lie algebra sl n (C) as vector space over C. We need to write down explicitly the adjoint representation of SL n (C) with vector fields and determine the action of these vector fields on polynomials.
For a matrix V ∈ sl n (C) and X ∈ Mat (n × n; C) it well known that We consider now X as the matrix of linear monomials x kℓ and V as a vector field which acts on X. For V = E ab , a = b, we obtain: Hence, we define Definition 3.4. (1) (2) We will frequently refer to the vector fields Ξ a as hyperbolic vector fields.
The vector fields Θ ab and their commutators Ξ a = [Θ a,a+1 , Θ a+1,a ] span the adoint represention of the Lie algebra sl n (C) written as vector fields and obey of course the same commutation relations as the E ab and H a . In particular, Example 3.5. Let n ≥ 2. Since Θ 12 (x 21 ) = 0, the vector field x 21 Θ 12 is a shear vector field. And its flow map is given by The semi-group property is satisfied because x 21 is conjugation invariant under the action of the one-parameter subgroup generated by Θ 12 . Now we consider the overshear vector field x 11 Θ 12 with Θ 12 (x 11 ) = x 21 and Θ 2 12 (x 11 ) = 0. The flow map is given by The semi-group property is less obvious, but can be verified by direct calculation or the more abstract argument in Lemma 3.3. Note that this map has no (non-removable) singularities; for x 21 → 0 it degenerates to a shear map.

Fibred Density Property for the spectral ball
In this section we prove the fibred density property for the spectral ball and determine its automorphism group. The crucial technical part is Proposition 4.5, and the following lemmas will be needed for the induction in the proof of this proposition.
We illustrate these results by summarizing them for sl 3 (C) in Tables 1 and 2. Proof.
Thus, we obtain all such differences except x 22 −x 11 , but instead the scalar multiple x 11 −x 22 . For X ∈ (sl n (C)) * we further have Table 1. Vector fields for the adjoint representation of sl 3 (C) x 13 x 23 0 0 Table 2. Action of vector fields on linear monomials for sl 3 (C).
the trace condition x 11 + x 22 + · · · + x nn = 0. Therefore we can realize all diagonal terms up to equivalence.
We are now prepared to proof our main proposition. The proof is by induction over the degree. To understand the proof and its structure it might be helpful to look first at the induction step which starts after degree two. L n := {f · Θ : f polynomial, Θ ∈ Θ kℓ , Ξ m } By A n we denote the Lie algebra generated by all vector fields which are SL n (C)-shears or SL n (C)-overshears with monomial coefficients of degree at most 2.
Proposition 4.5. Let n ≥ 2, then L n = A n .
Proof. The proof is by induction on the degree d of the polynomial coefficients. The induction hypothesis for d = 0 is true by assertion.
We treat the case d = 1 separately: The only missing vector fields are x kℓ Θ kℓ and the linear monomials in front of the hyperbolic vector fields. For (k, ℓ) = (1, 2) a short calculation shows: Because x 22 Ξ 1 is a shear vector field, we can conclude that x 12 Θ 12 is in A n . By symmetry (index permutation in Θ kℓ ), this is true for all x kℓ Θ kℓ . For the hyperbolic vector fields, see the general case, step 3.
We also need to treat the case d = 2 separately: (1) Let a and f be monomials of degree one. We first remark that if Θ(a) = 0 and Θ 2 (f ) = 0, then Θ 2 (af ) = 0, i.e. under these assertions af · Θ is a Θ-overshear vector field and hence in A n . Consider the following difference of Lie brackets: For the quadratic terms in front of Θ = Θ kℓ we can restrict us without loss of generality to the case Θ = Θ 12 . By Lemma 4.3 we obtain all terms of the form f Λ(a) · Θ = x ab x cd Θ 12 with x ab = x 12 = x cd .
(2) We focus on the most difficult term, i.e. x 2 kℓ Θ kℓ . It is sufficient to consider Θ 12 : We make a detour to a hyperbolic vector field and aim to obtain x 2 12 Ξ 1 . Note that x 2 12 · Θ 21 is a shear vector field.
(4) For other hyperbolic vector fields, see again the general case, step 3.
(1) Let f and g be monomials of degree d − 1 (or less) and let a be a monomial of degree one.
To obtain the coefficients in front of Θ kℓ it is by symmetry sufficient to consider Θ = Θ 12 . Choose a ∈ ker Θ 12 . For Λ we can choose any other vector field in sl n . From Lemma 4.3 we know that all linear monomials except x 12 are obtained as Λ(a) in case of dimension n ≥ 3. In dimension n = 2 we only have Θ 12 (x 21 ) = 0, but -using also the hyperbolic vector fieldstill obtain Θ 21 (x 21 ) = x 11 − x 22 and Ξ 1 (x 21 ) = 2x 21 ; note that x 11 + x 22 = 0. We therefore obtain all monomial coefficients f gΛ(a) of degree d + 1 (actually, up to 2d − 1) in front of Θ 12 except x d+1 12 Θ 12 .
(2) To obtain x d+1 12 Θ 12 we calculate: 12 Θ 12 and we have found all monomial coefficients of degree d + 1 in front of the Θ kℓ .
(3) Now we turn to the hyperbolic vector fields. Again by symmetry it is sufficient to consider Ξ 1 = [Θ 12 , Θ 21 ]. [ Hence, we obtain all polynomials f of degree d + 1 in front of Ξ 1 which are such that both f · Θ 12 and Θ 21 (f ) · Θ 12 are already known to be in A n . Since Θ 21 (f ) does not increase the degree of f , we are done.
Corollary 4.7. The natural fibration π : Ω n → G n has the fibred density property.
Proof. The condition π −1 (π(Ω)) = Ω is clear, because Ω = π −1 (G n ). The domain Ω n ⊂ Mat (n × n; C) is balanced since obviously λz ∈ Ω n for all z ∈ Ω n and all λ ∈ D. Therefore it is Runge and then by the preceding remark it enjoys the fibred density property.
We will need the following terminology from representation theory and a result of Dixmier [Dix79].  (1) Θ annulates the locally invariant functions on U.
(4) There exists a holomorphic map g : U → g such that Θ(x) = [x, g(x)] for all x ∈ U.
We are now able to prove the fibred density property for the spectral ball and to determine its automorphism group.
Proof of Theorem 4.6. Let Θ be a fibre-preserving holomorphic vector field on Ω ⊆ Mat (n × n; C). This is equivalent to annihilate SL n (C)-invariant functions on Ω. Note that a fibre in Ω is a fibre in Mat (n × n; C) due to the assertion Ω = π −1 (π(Ω)). According to Theorem 4.9 there exists a holomorphic map g : sl n (C) → sl n (C) such that Θ(x) = [x, g(x)]. Hence we can write Θ as a holomorphic linear combination of the vector fields Θ ab and Ξ c . In Mat (n × n; C) ∼ = C n 2 we can approximate these holomorphic linear combinations by polynomial linear combinations. This works as well for any Runge domain Ω ⊆ Mat (n × n; C). Note that all holomorphic linear combinations of such vector fields are automatically tangent to the fibres of σ. By Proposition 4.5 all polynomials in the Lie algebra generated by the vector fields in A n which correspond to SL n (C)-shears and SL n (C)overshears with at most quadratic order.
Proof of Theorem 1.7. Let f : Ω n → Ω n be a holomorphic automorphism. By [RW91, Thm. 4] we find a Möbius transformation h : such that the composition g := f • h −1 : Ω n → Ω n is fibre-preserving, i.e. π • g = π, and in addition g(0) = 0. By [RW91, Thm. 2] we know that under these circumstances g ′ (0) is a linear automorphism of Ω n . According to [RW91,Thm. 4,Cor.] it is of the form It is clear that we can connect g to its linear part by a C 1 -homotopy [0, 1] ∋ s → g(sA) s → g ′ (0)(A) and that any conjugation by a G ∈ SL n (C) can be connected to the identity as well.
Therefore we may assume that f , after composition with a Möbius transformation and possibly a transposition is in the identity component of Aut π (Ω n ). By Theorem 4.6 the spectral ball has the fibred density property and we can apply Theorem 1.4 with A = A n generated by at most quadratic SL n (C)-shears and at most quadratic SL n (C)overshears as the dense Lie subalgebra generated by complete vector fields.
We would like to compare our result to a question asked by Ransford and White [RW91] when they started the study of holomorphic automorphisms of the spectral ball. They asked whether the automorphisms of the spectral ball are compositions of the Möbius transformations, the transposition and conjugations of the following form X → exp(f (X)) · X · exp(f (X)) with f : Ω n → sl n (C) such that it is SL n (C)-invariant, i.e. f (GXG −1 ) = f (X) for all X ∈ Ω n and all G ∈ SL n (C) (invariance). A counterexample was already given in [Kos12].
It is however easy to see that the invariance condition is too restrictive: Choose a generic fibre X λ . It is a SL n (C)-homogeneous complex manifold and hence isomorphic to SL n (C)/H for a reductive subgroup H of SL n (C). Hence there exists a (up to a scalar constant) unique algebraic volume form ω ′ on X λ which is invariant under left-multiplication of SL n (C), see [KK14,Appendix]. This volume form ω ′ corresponds to a SL n (C)-conjugation invariant volume form ω. The invariance condition for the automorphisms of Ransford and White is equivalent to saying that G(X) depends only on σ(X). Therefore it necessarily preserves the volume form ω. However, the overshears never preserve this volume form, as can be seen for the product formula for the divergence: div(f Θ) = f div Θ + Θ(f ) For a vector field with algebraic flow map (e.g. all the Θ ab ), the divergence necessarily vanishes and by the overshear condition Θ(f ) = 0.
Since all the automorphisms of Ransford and White however have vanishing ω-divergence, no finite or infinite composition of them can yield such an overshear. In the next section we will prove a much stronger result.

Dense, but meagre
In Theorem 1.7 we have determined a dense subgroup of Aut (Ω n ). In this section we will prove that this dense subgroup is not the whole Aut (Ω n ), in fact a meagre subset. We follow the original strategy of Andersén and Lempert [AL92, Sec. 7] Definition 5.1. By P m we denote the vector space of polynomials on sl n (C) of total degree at most m ∈ N 0 .
Proof. ByP m we denote the vector space of homogeneous polynomials on sl n (C) of degree m ∈ N 0 . We prove the following claims by induction: We first prove the claim for the kernel of Θ ab . For m = 1 this is just Corollary 4.2. Consider the induction step m → m + 1: The ab |P m+1 is clear by the Leibniz rule. Assume there exists a polynomial p ∈ P m+1 such that Θ 2 ab (p) = 0 but p is not of the form claimed above. Since the kernel is a vector space, we may assume that we have already subtracted the terms claimed to be in the kernel, hence p = x d ab · q with q ∈ P m and d ≥ 1 maximal. We calculate ab q If we assume q ∈ ker Θ 2 ab , then necessarily q = 0, a contradiction. But if q ∈ ker Θ 2 ab , then x ab must divide q which contradicts the maximality of d.
Taking into account the trace condition of sl n (C), there are n 2 − 2 monomials of the form {x c k d k : (c k , d k ) = (a, b)} for given a = b. Therefore we obtain dim ker Θ 2 ab |P m = n 2 −2+m−1 m and dim ker Θ 2 ab |P m = m k=0 Now we prove the claim for the kernel of Ξ 2 a . For m = 1 this is again Corollary 4.2 and its preceding Lemma 4.1 which reveals that ker Ξ a |P 1 = ker Ξ 2 a |P 1 . We proceed similarly as before: Assume there exists a polynomial p ∈ P m+1 such that Ξ 2 a (p) = 0, but p is not of the form claimed above. We can assume that p = k p k with linearly independent monomials p k ∈P m+1 . In view of Lemma 4.1, each monomial p k will be multiplied by a sum of square numbers when Ξ 2 a acts on it. Hence, Ξ 2 a (p k ) = 0 for all k, and it is sufficient to consider a monomial p ∈P m+1 . We can write p = r · q with monomials r ∈P 1 and q ∈P m with Ξ 2 a (r) = 0. We calculate We divide by r and obtain Ξ 2 a (q) = −ε 2 q. In view of Lemma 4.1, q is multiplied by a sum of square numbers when Ξ 2 a acts on it. But the r.h.s. is negative, hence q = 0, a contradiction.
Taking into account the trace condition of sl n (C), there are (n−2) 2 + 2 − 1 monomials of the form {x c k d k : c k = d k or c k , d k ∈ {a, a + 1}}.
(1) If you compare this Lemma to [AL92, Sec. 7] and their notation, note that ker (Θ 2 ab |P m ) resp. ker (Ξ 2 a |P m ) correspond to the vector space of polynomials Θ m 0 × Θ m−1 in their notation.
(2) Note one subtlety about the degree: For a p ∈ ker (Θ ab |P m ) ⊂ ker (Θ 2 ab |P m ) the polynomial vector field p · Θ ab is of the degree m + 1, but the corresponding matrix exp(p · E ab ) will contain polynomials of degree m, since E 2 ab for a = b. However, the polynomial shear automorphism arising by conjugation with this matrix will be of degree 2m + 1.
It is convenient to introduce the notion of density property also for Lie algebras.
Definition 5.4 ([Var00, Definition 0.1]). Let g be a Lie algebra of holomorphic vector fields on a complex manifold X. We say that g has the density property if the Lie subalgebra of g generated by the complete holomorphic vector fields is dense in g.
Definition 5.5. Following [Var00, Section 0] we recall the definition of jet spaces: (1) Let J k g (X) be the space of k-jets of local biholomorphisms of the form ϕ Θm tm • · · · • ϕ Θ 1 t 1 where ϕ Θ t is the local flow map at time t ∈ C of the vector field Θ for vector fields Θ 1 , . . . , Θ m ∈ g and times t 1 , . . . , t m small enough. For a jet γ ∈ J k g (X) we denote by σ(γ) its source and denote J k,0 g = {γ ∈ J k g : σ(γ) = 0}. (2) By Aut g (X) we denote the subgroup of Aut (X) generated by the time-1-maps of flows of complete holomorphic vector fields.
(3) For a holomorphic map f : X → X, its k-jet at x ∈ X is denoted by j k x (f ). Theorem 5.6 ([Var00, Theorem 0.1]). Let g be the Lie algebra of holomorphic vector fields on a complex manifold X with the density property. Then for each jet γ ∈ J k g (X) there exists f ∈ Aut g (X) such that j k σ(γ) (f ) = γ Lemma 5.7. Let g be the Lie algebra of all fibre preserving vector fields of π : Ω n → G n . The jet space J 0,2m g (Ω n ) is a finite-dimensional complex manifold and dim J 0,2m+1 g (Ω n ) ≥ (n 2 − 1) m + n 2 n 2 (4) Proof. The jet space is a complex-affine variety and actually smooth since it is SL n (C)-homogeneous. All conjugations with matrices A(z) ∈ SL n (C) depending polynomially on the points z ∈ Ω n are polynomial endomorphisms. The degree of the jet will be 2m + 1 if the polynomial entries in the matrix A(z) are of degree m. Such an endomorphism preserves the fibres and the origin and it is locally invertible. Actually, φ(z) := A(z) · z · A −1 (z) can be connected to its linear part by a path 1/s · φ(s · z) and letting s → 0, and can then be connected further to the identity, as shown already in the proof of Theorem 1.7 on page 15. By the density property, the vector field can then be approximated by linear combinations and Lie brackets of complete vector fields, hence the jet of φ indeed belongs to J 0,2m+1 g (Ω n ). Now, dim J 0,2m+1 g (Ω n ) ≥ d(m) · (n 2 − 1) where d(m) is the dimension of the vector space of polynomials of degree at most m, i.e. d(m) = m+n 2 n 2 .
Corollary 5.8. For fixed k ∈ N and m ∈ N large enough: dim dim J 0,2m+1 g (Ω n ) ≥ k · max dim ker Θ 2 ab |P m , dim ker Ξ 2 a |P m Proof. The l.h.s. is a polynomial in m of degree n 2 and the r.h.s. is of degree at most n 2 − 2 since n 2 +m−2 m = n 2 +m−2 n 2 −2 .
Definition 5.9. For the vector fields, we define a truncation map tr m : O(Ω n ) → P m which sends a holomorphic function to its Taylor polynomial about 0 of degree m.
Proof. We first need to show that the maps really map into the given targets. This is clear for Ψ k . Since the Λ 2 1 , . . . , Λ 2 k preserve the total degree of polynomials, the truncation tr m actually maps the kernel into itself. The diagram commutes by definition of ψ k .
Proof of Theorem 1.8. We can restrict us to the case of the subgroup Aut π (Ω n ) of fibre-preserving automorphisms. We follow the idea of the proof of [AL92, Theorem 7.1].
We denote by C k , k ∈ N, the set of automorphisms obtained by the composition of k overshears of Θ ab resp. Ξ a . Using the notation of Lemma 5.10 above, we see that C k is the image of the map Ψ k . We claim that the set C k is meagre in Aut π (Ω n ) for all k ∈ N. By Baire's theorem it would then follow that ∪ k∈N C k is also meagre, hence ∪ k∈N C k = Aut π (Ω n ).
In order to establish the claim, we first need to show that j 2m+1 g is an open mapping. By Theorem 5.6 the map j 2m+1 g is surjective. Forming the jet j 2m+1 g (f ) of a map f is a linear mapping and therefore surjectivity implies openness.