Mathematics – Algebraic Geometry
Scientific paper
1994-10-13
Mathematics
Algebraic Geometry
10 pages, LaTeX
Scientific paper
Let $D$ be a domain in $C^n$ and $G$ a topological group which acts effectively on $D$ by holomorphic automorphisms. In this paper we are interested in projective linearizations of the action of $G$, i.e. a linear representation of $G$ in some $C^{N+1}$ and an equivariant imbedding of $D$ into $\P^N$ with respect to this representation. The domains we discuss here are open connected sets defined by finitely many real polynomial inequalities or connected finite unions of such sets. Assume that the group $G$ acts by birational automorphisms. Our main result is the equivalence of the following conditions: 1) there exists a projective linearization, i.e. a linear representation of $G$ in some $\C^{N+1}$ and a biregular imbedding $i\colon \P^n \hookrightarrow \P^N$ such that the restriction $i|_D$ is $G$-equivariant. 2) $G$ is a subgroup of a Lie group $\hat G$ of birational automorphisms of $D$ which extends the action of $G$ and has finitely many connected components; 3) $G$ is a subgroup of a Nash group $\hat G$ of birational automorphisms of $D$ which extends the action of $G$ to a Nash action $\hat G\times D\to D$; 4) $G$ is a subgroup of a Nash group $\hat G$ such that the action $G\times D\to D$ extends to a Nash action $\hat G\times D\to D$; 5) the degree of the automorphism $\phi_g\colon D\to D$
No associations
LandOfFree
On the linearization of the automorphism groups of algebraic domains does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with On the linearization of the automorphism groups of algebraic domains, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and On the linearization of the automorphism groups of algebraic domains will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-324259