Presque toute surface K3 contient une infinité d’hypersurfaces Levi-plates linéaires

Lequen, Félix

Journal de l’École polytechnique — Mathématiques, Tome 10 (2023), pp. 815-836

On s’intéresse à la construction d’hypersurfaces Levi-plates analytiques réelles dans les surfaces K3. On peut en construire dans les tores complexes de dimension $2$ en prenant des images d’hyperplans réels. On montre que « presque toute » surface K3 contient une infinité d’hypersurfaces Levi-plates de ce type. La preuve repose principalement sur une construction récente due à Koike-Uehara, ainsi que sur les idées de Verbitsky sur les structures complexes ergodiques et une adaptation d’un argument dû à Ghys dans le cadre de l’étude de la topologie des feuilles génériques.

We investigate the construction of real analytic Levi-flat hypersurfaces in K3 surfaces. By taking images of real hyperplanes, one can construct such hypersurfaces in two-dimensional complex tori. We show that “almost every” K3 surface contains infinitely many Levi-flat hypersurfaces of this type. The proof relies mainly on a recent construction of Koike and Uehara, ideas of Verbitsky on ergodic complex structures, as well as an argument due to Ghys in the context of the study of the topology of generic leaves.

Classification : 14J28, 37A25
Mots-clés : Surfaces K3, hypersurfaces Levi-plates, périodes, théorie ergodique
Keywords: K3 surfaces, Levi-flat hypersurfaces, periods, ergodic theory

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