Sur certains pseudogroupes de biholomorphismes locaux de \((\mathbb{C}^n,0)\). (On certain pseudogroups of germs of biholomorphisms of \((\mathbb{C}^n,0)\)). (French) Zbl 1006.58015

Belliart, Michel

Bull. Soc. Math. Fr. 129, No. 2, 259-284 (2001).

Author’s summary: Let \(\Gamma\) be a pseudogroup of local holomorphic transformations of \(\mathbb{C}^n\) fixing zero. We study the dynamics of \(\Gamma\). We show that if \(\Gamma\) contains two elements whose 2-jets are in “general position” and sufficiently near the identity, then:
1) \(\Gamma\) acts minimally on the bundle of infinite-order jests on some pointed neighbourhood \({\mathcal B}\) of 0 (that is to say: for any \(z_0\), \(z_1\in{\mathcal B}\) and any germ \(\varphi: z_0 \to z_1\) of biholomorphism, there exists a sequence \(\gamma_n\in\Gamma\) which converges to \(\varphi\) uniformly on some neighbourhood of \(z_0)\).
2) \(\Gamma\) preserves no geometric structure near 0 (this is a trivial consequence od 1).
3) For any holomorphic pseudogroup topologically conjugate to \(\Gamma\), the germ of conjugacy at 0 is either holomorphic or antiholomorphic.
The main feature of the proof is to attach to any pseudogroup \(\Gamma\) a sheaf \({\mathfrak g}_\Gamma\) of Lie algebras on \(\mathbb{C}^n\) such that \(\Gamma\) is “dense” in \({\mathfrak g}_\Gamma\) in a natural sense. Then we prove that under some natural assumption on \(\Gamma\), \({\mathfrak g}_\Gamma(U)\) must be the sheaf of all holomorphic vector fields for any \(U\) open in \({\mathcal B}\), where \({\mathcal B}\) is the (open) complement of 0 in its basin of attraction.

Reviewer: Jan Kubarski (Łódź)

MSC:

58H05	Pseudogroups and differentiable groupoids
58H15	Deformations of general structures on manifolds

Keywords:

conformal pseudo-groups; invariant geometric structure; holomorphic transformation; biholomorphism