On affine group actions on Stein manifolds (Q2882754)

The present paper is concerned with actions of the affine group \(\text{Aff}(\mathbb{C})\cong \mathbb{C}^*\times \mathbb{C}\) on an \(n\)-dimensional complex manifold \(M\). Any such action is given by two complete holomorphic vector fields \(X,Y\in {\mathcal X}(M)\), such that \([X,Y]=Y\), the flow of \(X\) is periodic and the action is foliated where \(X\) and \(Y\) are linearly independent (an action \(\phi\) is \textsl{foliated} if all orbits have the same dimension, in which case they are the leaves of a unique holomorphic foliation \({\mathcal F}_\phi\)). The main result of the paper is the following linearization theorem for an action admitting a \textsl{regular dicritic singularity} \(p\in M\) (i.e. the vectors \(X(p)\) and \(Y(p)\) are linearly dependent with \(X(p)=0\) and \(Y(p)\not =0\), and for some neighborhood \(U\) of \(p\) there are infinitely many orbits on \(U\) accumulating only at~\(p\) ).NEWLINENEWLINETheorem. Let \(\phi\) be an affine group action on an \(n\)-dimensional Stein manifold \(M\) and suppose there is a regular dicritic singularity \(p\in {\mathcal F}_\phi\). Then \(\phi\) is globally conjugate to an affine group action on \({\mathbb C}^n\) given in affine coordinates by \((z_1,z_2,\dots,z_n)\mapsto (t+s^{-1}z_1,s^{\lambda_2}z_2,\dots,s^{\lambda_n}z_n)\), with \(\lambda_i\in{\mathbb Z}\). In particular, \(M\) is biholomorphic to \({\mathbb C}^n\).NEWLINENEWLINEThe following \textsl{stability theorem} is also obtained.NEWLINENEWLINETheorem. Let \({\mathcal F}\) be a foliation given by an affine group action on a 3-dimensional Stein manifold \(M^3\), and assume that \({\mathcal F}\) admits a closed leaf \(L\) biholomorphic to \({\mathbb C}^*\times {\mathbb C}^*\) with finite holonomy group. Then there is an invariant open subset \(U\supset L\) such that every leaf in \(U\) is biholomorphic to \(L\).