Complete algebraic vector fields on Danielewski surfaces

Leuenberger, Matthias

Annales de l'Institut Fourier, Tome 66 (2016) no. 2, pp. 433-454.

Nous donnons une classification de tous les champs de vecteurs algébriques complets sur les surfaces de Danielewski (surface lisse donnée par $xy=p\left(z\right)$). Nous utilisons le fait que pour chaque tel champ vectoriel, il existe une fibration préservée par son flot. Une classification des fonctions régulières avec fibre générique $ℂ$ or ${ℂ}^{*}$ est requise pour obtenir la liste explicite des champs vectoriels. Nous présentons des résultats sur de telles fibrations définies sur des surfaces de Gizatullin et donnons une description précise de ces fibrés pour les surfaces de Danielewski.

We give the classification of all complete algebraic vector fields on Danielewski surfaces (smooth surfaces given by $xy=p\left(z\right)$). We use the fact that for each such vector field there exists a certain fibration that is preserved under its flow. In order to get the explicit list of vector fields a classification of regular function with general fiber $ℂ$ or ${ℂ}^{*}$ is required. In this text we present results about such fibrations on Gizatullin surfaces and we give a precise description of these fibrations for Danielewski surfaces.

Classification : 32M25, 37F75, 14R25
Keywords: affine surfaces, complete vector fields, algebraic fibrations
Mot clés : Surfaces affines, champs de vecteurs complets, fibrés algébriques

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