Complete algebraic vector fields on Danielewski surfaces (Q332182)

A holomorphic vector field on a complex manifold is complete if admits holomorphic global flow map (i.e., for all initial condition the solution exists for all complex times). As a seminal result, \textit{M. Brunella} [Topology 43, No. 2, 433--445 (2004; Zbl 1047.32015)] gave an explicit classification of the complete algebraic vector fields on \(\mathbb{C}^2\), up to polynomial authomorphisms.NEWLINENEWLINEIn the present work, the author provides the classification of complete algebraic vector fields on Danielewski surfaces (smooth complex surfaces given by \(\{ xy = p(z) \}\)).NEWLINENEWLINEIn a way similar to Brunella's work, a basic fact is that each complete algebraic vector field preserves the fibers of a regular function with \(\mathbb{C}^*\) or \(\mathbb{C}\) generic fibers. It turns out to be true in almost all normal affine surfaces, see also [\textit{S. Kaliman, F. Kutzschebauch} and \textit{M. Leuenberger}, ``Complete algebraic vector fields on affine surfaces'', Preprint, \url{arXiv:1411.5484}].NEWLINENEWLINEIn order to get the explicit list of complete vector fields, a classification of regular functions with general fiber \(\mathbb{C}\) or \(\mathbb{C}^*\) is required. In this paper, results about such fibrations on Gizatullin surfaces are obtained. Hence a precise description of these fibrations for Danielewski surfaces follows.