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 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.
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Mot clés : Surfaces K3, hypersurfaces Levi-plates, périodes, théorie ergodique
Keywords: K3 surfaces, Levi-flat hypersurfaces, periods, ergodic theory
Félix Lequen 1
CC-BY 4.0
@article{JEP_2023__10__815_0,
author = {F\'elix Lequen},
title = {Presque toute surface {K3} contient une infinit\'e d{\textquoteright}hypersurfaces {Levi-plates} lin\'eaires},
journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
pages = {815--836},
publisher = {\'Ecole polytechnique},
volume = {10},
year = {2023},
doi = {10.5802/jep.233},
language = {fr},
url = {https://jep.centre-mersenne.org/articles/10.5802/jep.233/}
}
TY - JOUR AU - Félix Lequen TI - Presque toute surface K3 contient une infinité d’hypersurfaces Levi-plates linéaires JO - Journal de l’École polytechnique — Mathématiques PY - 2023 SP - 815 EP - 836 VL - 10 PB - École polytechnique UR - https://jep.centre-mersenne.org/articles/10.5802/jep.233/ DO - 10.5802/jep.233 LA - fr ID - JEP_2023__10__815_0 ER -
%0 Journal Article %A Félix Lequen %T Presque toute surface K3 contient une infinité d’hypersurfaces Levi-plates linéaires %J Journal de l’École polytechnique — Mathématiques %D 2023 %P 815-836 %V 10 %I École polytechnique %U https://jep.centre-mersenne.org/articles/10.5802/jep.233/ %R 10.5802/jep.233 %G fr %F JEP_2023__10__815_0
Félix Lequen. Presque toute surface K3 contient une infinité d’hypersurfaces Levi-plates linéaires. Journal de l’École polytechnique — Mathématiques, Tome 10 (2023), pp. 815-836. doi : 10.5802/jep.233. https://jep.centre-mersenne.org/articles/10.5802/jep.233/
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