Presque toute surface K3 contient une infinit\'e d'hypersurfaces Levi-plates lin\'eaires

TL;DR

This paper demonstrates that almost all K3 surfaces contain infinitely many real analytic Levi-flat hypersurfaces, constructed via images of real hyperplanes, building on recent methods and ergodic theory.

Contribution

It shows that nearly every K3 surface admits infinitely many Levi-flat hypersurfaces, extending previous constructions to a broad class of complex surfaces.

Findings

Almost all K3 surfaces contain infinitely many Levi-flat hypersurfaces.

Construction relies on images of real hyperplanes in complex tori.

Uses ergodic theory and recent geometric constructions.

Abstract

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 surfaces 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. -- On s'int\'eresse \`a la construction d'hypersurfaces Levi-plates analytiques r\'elles dans les surfaces K3. On peut en construire dans les tores complexes de dimension 2 en prenant des images d'hyperplans r\'eels. On montre que "presque toute" surface K3 contient une infinit\'e d'hypersurfaces Levi-plates de ce type. La preuve repose principalement sur une construction r\'ecente due \`a Koike-Uehara, ainsi que sur les id\'ees de Verbitsky sur les structures complexes ergodiques et une adaptation d'un argument d\^u \`a Ghys dans le cadre de l'\'etude de la topologie des feuilles g\'en\'eriques.