We study curvature restrictions of Levi-flat real hypersurfaces in complex projective planes, whose existence is in question. We focus on its totally real Ricci curvature, the Ricci curvature of the real hypersurface in the direction of the Reeb vector field, and show that it cannot be greater than along a Levi-flat real hypersurface. We rely on a finiteness theorem for the space of square integrable holomorphic 2-forms on the complement of the Levi-flat real hypersurface, where the curvature plays the role of the size of the infinitesimal holonomy of its Levi foliation.
Nous étudions des restrictions sur une courbure des hypersurfaces réelles Levi-plates dans des plans projectifs complexes, dont l’existence est en question. Nous nous focalisons sur sa courbure de Ricci totalement réelle, c’est-â-dire la courbure de Ricci de l’hypersurface réelle dans la direction du champ de Reeb, et nous démontrons qu’elle ne peut pas être supérieure à le long de l’hypersurface réelle Levi-plate. Nous nous appuyons sur un théorème de finitude pour l’espace des 2-formes holomorphes de carrés intégrables sur le complément de l’hypersurface réelle Levi-plate, où la courbure joue le rôle de la taille de l’holonomie infinitésimale de son feuilletage de Levi.
Revised:
Accepted:
Published online:
Classification: 32V15, 32V40, 53B25, 53C12
Keywords: Levi-flat real hypersurface, totally real Ricci curvature, adjunction formula, integral formula
@article{AIF_2015__65_6_2547_0, author = {Adachi, Masanori and Brinkschulte, Judith}, title = {Curvature restrictions for {Levi-flat} real hypersurfaces in complex projective planes}, journal = {Annales de l'Institut Fourier}, pages = {2547--2569}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {65}, number = {6}, year = {2015}, doi = {10.5802/aif.2995}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2995/} }
TY - JOUR TI - Curvature restrictions for Levi-flat real hypersurfaces in complex projective planes JO - Annales de l'Institut Fourier PY - 2015 DA - 2015/// SP - 2547 EP - 2569 VL - 65 IS - 6 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.2995/ UR - https://doi.org/10.5802/aif.2995 DO - 10.5802/aif.2995 LA - en ID - AIF_2015__65_6_2547_0 ER -
%0 Journal Article %T Curvature restrictions for Levi-flat real hypersurfaces in complex projective planes %J Annales de l'Institut Fourier %D 2015 %P 2547-2569 %V 65 %N 6 %I Association des Annales de l’institut Fourier %U https://doi.org/10.5802/aif.2995 %R 10.5802/aif.2995 %G en %F AIF_2015__65_6_2547_0
Adachi, Masanori; Brinkschulte, Judith. Curvature restrictions for Levi-flat real hypersurfaces in complex projective planes. Annales de l'Institut Fourier, Volume 65 (2015) no. 6, pp. 2547-2569. doi : 10.5802/aif.2995. https://aif.centre-mersenne.org/articles/10.5802/aif.2995/
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