On the group of holomorphic automorphisms of \(\mathbb{C}{}^ n\)
The paper deals with the problem of the structure of the group of holomorphic automorphisms \(\text{Aut}(\mathbb{C}^ n)\) of \(\mathbb{C}^ n\) (with the topology of locally uniform convergence). The authors consider two classes of holomorphic automorphisms of \(\mathbb{C}^ n\): 1. shears --- of the form \((z_ 1,\dots,z_ n)\mapsto(z_ 1,\dots,z_{n-1}\), \(f(z_ 1,\dots,z_{n-1})+z_ n)\), 2. overshears --- of the form \((z_ 1,\dots,z_ n)\mapsto(z_ 1,\dots,z_{n-1}, f(z_ 1,\dots,z_{n-1})+h(z_ 1,\dots,z_{n- 1})z_ n)\), where \(f, h\) are holomorphic functions on \(\mathbb{C}^{n-1}\), \(h\) nowhere 0. Let \(\text{Aut}_ 1(\mathbb{C}^ n)\) be the group of holomorphic automorphisms of \(\mathbb{C}^ n\) with Jacobi determinant 1. Denote by \(G(\mathbb{C}^ n)\) (resp. \(G_ 1(\mathbb{C}^ n))\) the subgroup of \(\text{Aut}(\mathbb{C}^ n)\) (resp. \(\text{Aut}_ 1(\mathbb{C}^ n))\) generated by overshears (resp. shears) i.e. all finite compositions of overshears (resp. shears). The first author in Complex Variables, Theory Appl. 14, No. 1-4, 223-235 (1990; Zbl 0705.58008) proved that for \(n\geq 2\) \(G_ 1(\mathbb{C}^ n)\) is dense in \(\text{Aut}_ 1(\mathbb{C}^ n)\) and that \(G_ 1(\mathbb{C}^ 2)\neq\text{Aut}_ 1(\mathbb{C}^ 2)\). In the paper the authors push further these results. Namely, they prove: 1. \(G_ 1(\mathbb{C}^ n)\neq\text{Aut}_ 1(\mathbb{C}^ n)\), 2. \(G(\mathbb{C}^ n)\neq\text{Aut}(\mathbb{C}^ n)\), 3. \(G(\mathbb{C}^ n)\) is dense in \(\text{Aut}(\mathbb{C}^ n)\).
scientific article; zbMATH DE number 167399
scientific article