On the group of holomorphic automorphisms of ? n

Algebraic density property of homogeneous spaces

“…Now one has a 4 > a 3 = 5 > b 2 = 4 which settles case (1) by Proposition 5.1. Then b 3 , d 4 , a 6 > 5 which settles case (2). Similarly, a 2l > b l for l ≥ 3, a 2l−1 > c l for l ≥ 3, d l > b l−1 for l ≥ 4, and 21 > 16 which settles cases (3)- (6) and concludes the proof.…”

“…Since all components of G/R are isomorphic as varieties we can suppose that G is connected. Furthermore, by Corollary 4.2 and Remark 3.18 (2) we are done with a semi-simple G.…”

“…The locally nilpotent vector fields associated with these actions of H 1 and H 2 are, obviously, semi-compatible since they are compatible on SL 2 (see Example 2.11). Consider the SL 2 -equivariant morphism G × V → G × Gx V where V , G = SL 2 , and G x are as in diagram (2). By definition G × Gx V is the quotient of G × V with respect to the G x -action whose restriction to the first factor is the multiplication from the right.…”

“…Denote the corresponding compatible locally nilpotent derivations byδ ψ 2 1 andδ ψ 2 2 on G 1 ⊕ G 2 (and also by abusing notation on G). Note that the vectorsδ ϕ 2 2 (e 1 ⊕ e 2 ) andδ ψ 2 2 (e 1 ⊕ e 2 ) can be assumed different with an appropriate choice of ψ 2 . Hence these two vectors form a generating subset of T e 1 ⊕e 2 G 1 ⊕ G 2 .…”

“…In this paper we develop further methods introduced by F. Kutzschebauch and the third author in [16] which they used to obtain new results in the Andersén-Lempert theory ( [1], [2]). D. Varolin was the first one to realize the importance of the following notion in that theory ( [27]).…”