Regularization of birational group operations in sense of Weil

The present paper deals with the classical results of A. Weil [11] on regularization of pre-groups and pre-transformation spaces (see Definitions 3.1 and 4.1). As pointed out in [4], those purely algebraic results appear to be very useful in the following complex analytic setting. Let D ⊂ C be a bounded domain and Aut(D) the group of all holomorphic automorphisms of D . By a theorem of H. Cartan ([1], see also [8]), Aut(D) is a real Lie group. In [10], Webster gave the conditions on D , such that all automorphisms extend to the birational transformations of the ambient C . Moreover, as shown in [13], the group Aut(D) has finitely many components in this case. Such properties are also valid for the automorphisms of bounded homogeneous domains, if they are realized as Siegel domains ([5]). The graph of every birational transformation defines an n-dimensional compact cycle in P2n . Thus we obtain an embedding of Aut(D) in the space Cn of n-dimensional cycles in P2n (the Chow scheme). The space Cn is a countable collection of projective varieties parameterized by the degrees of the cycles. In fact, it is proven in [4], that the degree of possible cycles is bounded, which means that Aut(D) lies in finitely many components of Cn (Theorem 3). The group operation of Aut(D) extends rationally to the Zariski closure Z of it in Cn and endows Z with a structure of a pre-group, which is in general not a group. The action Aut(D) × D → D extends also to a rational action Z × C → C . Again, this is a pre-transformation space which is not a transformation space in general. The pre-groups and pre-transformation spaces can be obtained by passing from algebraic groups and their regular actions on algebraic varieties to birationally equivalent algebraic varieties. The mentioned results of Weil imply that in this way we obtain all possible pre-groups and pre-transformation spaces ([11], p. 375). In the case of connected pre-groups and homogeneous pre-transformation spaces, a similar proof is given in the book of Merzlyakov [6]. The regularizations of pre-