On the linearization of the automorphism groups of algebraic domains

On the linearization of the automorphism groups of algebraic domains. 1 Linearization Theorem and applications 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. Since G acts effectively, the representation in C N +1 must be faithful. In our previous paper [11], however, we considered an example of a bounded domain D ⊂ C 2 with an effective action of a finite covering G of the group SL 2 (R). In this case the group G doesn't admit a faithful representation. The example shows that a linearization in the above sense doesn't exist in general. In the present paper we give a criterion for the existence of the projective linearization for birational automorphisms. The domains we discuss here are open connected sets defined by finitely many real polynomial inequalities or connected finite unions of such sets. These domains are called algebraic. For instance, in the above example the domain D is algebraic. Definition 1.1 1. A Nash map is a real analytic map f = (f 1 ,. .. , f m): U → R m (where U ⊂ R n is open) such that for each of the components f k there is a non-trivial polynomial P k with 2. A Nash manifold M is a real analytic manifold with finitely many coordinate charts φ i : U i → V i such that V i ⊂ R n is Nash for all i and the transition functions are Nash (a Nash atlas).