Holomorphic actions on contractible domains without fixed points
It is known that every continuous action of \(\mathbb{Z}_{p^ n}\) (with \(p\) prime), or of a compact connected abelian Lie group, on a contractible manifold has a fixed point. In particular, this holds for holomorphic actions on contractible complex manifolds. On the other hand, there are examples of continuous actions of \(\mathbb{Z}_{pq}\) (with \(p,q\) relatively prime), and of any compact connected non-abelian Lie group, on an euclidean space of sufficiently high dimension without fixed points. The authors give examples of holomorphic (actually, complex linear) actions of these groups on bounded pseudoconvex taut contractible domains without fixed points. This is relevant, for instance, in iteration theory of holomorphic maps on taut manifolds, since it shows that there are holomorphic self-maps of contractible taut domains without fixed points and whose sequence of iterates is not compactly divergent. The proof starts modifying the topological constructions so to get a bounded contractible open set in some \(\mathbb{R}^ n\) where the group acts linearly without fixed points. Then one extends the action to the tube over it by complex linearity, and a suitably chosen invariant Stein neighbourhood in the tube of the open set yields the domain looked for.
scientific article; zbMATH DE number 126412
scientific article