Dicritical holomorphic flows on Stein manifolds (Q2474114)

For a complete holomorphic vector field \(X\) with isolated singularities and a dicritical singularity \(P\) (i.e. for some neighborhood \(V\) of \(P\) there are infinitely many orbits of \(X|_{V}\) accumulating only at \(P\)) on a Stein manifold \(M\) of dimension \(n\geq 2\), the authors give sufficient conditions which ensure that \(X\) is globally analytically linearizable and therefore \(M\) is biholomorphic to \(\mathbb{C}^{n}\). The case \(n=2\) was already proven by the second author. Then the authors show that if \(X\) is a complete holomorphic field with isolated singularities on the affine space \(\mathbb{C}^{n}\) (\(n \geq 2\)), if for some point \(p \in \mathbb{C}^{n}\) the corresponding orbit is closed, isomorphic to \(\mathbb{C}^ \ast\) and has finite holonomy group, then the generic orbit is isomorphic to \(\mathbb{C}^ \ast\). Several conjectures and open questions on the subject are also presented.