Regularization of birational group operations in sense of Weil.

*(English)*Zbl 0873.14012The present paper deals with the classical results of A. Weil [Am. J. Math. 77, 355-391 (1955; Zbl 0065.14201)] on regularization of pre-groups and pre-transformation spaces.

Let \(D\subset \mathbb{C}^n\) be a bounded domain and \(\operatorname{Aut} (D)\) the group of all holomorphic automorphisms of \(D\). \(\operatorname{Aut} (D)\) is a real Lie group. S. M. Webster [Invent. Math. 43, 53-68 (1977; Zbl 0348.32005)] gave the conditions on \(D\), such that all automorphisms extend to the birational transformations of the ambient \(\mathbb{C}^n\). The graph of every birational transformation defines an \(n\)-dimensional compact cycle in \(\mathbb{P}_{2n}\). Thus we obtain an embedding of \(\operatorname{Aut} (D)\) in the space \(C_n\) of \(n\)-dimensional cycles in \(\mathbb{P}_{2n}\) (the Chow scheme). \(\operatorname{Aut} (D)\) lies in finitely many components of \(C_n\). The group operation of \(\operatorname{Aut} (D)\) extends rationally to the Zariski closure \(Z\) of it in \(C_n\) and endows \(Z\) with a structure of a pre-group, which is in general not a group. The action \(\operatorname{Aut} (D)\times D\to D\) extends also to a rational action \(Z\times \mathbb{C}^n\to \mathbb{C}^n\). 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. We propose here a geometrical way to study pre-groups and pre-transformation spaces. This allows to reprove the above classical results without use of generic points and to generalize them to the case of several components:

Theorem 3.7. For every algebraic pre-group \(V\) there exists a regularization \(\widetilde V\) which is unique up to isomorphisms.

Theorem 4.9. For every algebraic pre-group \(V\) and algebraic pre-transformation \(V\)-space \(X\), there exists a regularization \(\psi: X\to \widetilde X\).

Let \(D\subset \mathbb{C}^n\) be a bounded domain and \(\operatorname{Aut} (D)\) the group of all holomorphic automorphisms of \(D\). \(\operatorname{Aut} (D)\) is a real Lie group. S. M. Webster [Invent. Math. 43, 53-68 (1977; Zbl 0348.32005)] gave the conditions on \(D\), such that all automorphisms extend to the birational transformations of the ambient \(\mathbb{C}^n\). The graph of every birational transformation defines an \(n\)-dimensional compact cycle in \(\mathbb{P}_{2n}\). Thus we obtain an embedding of \(\operatorname{Aut} (D)\) in the space \(C_n\) of \(n\)-dimensional cycles in \(\mathbb{P}_{2n}\) (the Chow scheme). \(\operatorname{Aut} (D)\) lies in finitely many components of \(C_n\). The group operation of \(\operatorname{Aut} (D)\) extends rationally to the Zariski closure \(Z\) of it in \(C_n\) and endows \(Z\) with a structure of a pre-group, which is in general not a group. The action \(\operatorname{Aut} (D)\times D\to D\) extends also to a rational action \(Z\times \mathbb{C}^n\to \mathbb{C}^n\). 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. We propose here a geometrical way to study pre-groups and pre-transformation spaces. This allows to reprove the above classical results without use of generic points and to generalize them to the case of several components:

Theorem 3.7. For every algebraic pre-group \(V\) there exists a regularization \(\widetilde V\) which is unique up to isomorphisms.

Theorem 4.9. For every algebraic pre-group \(V\) and algebraic pre-transformation \(V\)-space \(X\), there exists a regularization \(\psi: X\to \widetilde X\).