On the real analytic Levi flat hypersurfaces in complex tori of dimension two
type: Item
instance of: scholarly article
title: On the real analytic Levi flat hypersurfaces in complex tori of dimension two
zbMATH Open document ID: 1031.32030
author: Kazuko Matsumoto
author: Takeo Ohsawa
publication date: 2003-03-12T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/116016
review text: Recently it has been proved [\textit{A. Lins Neto}, Ann. Inst. Fourier 49, 1369-1385 (1999; Zbl 0963.32022) and \textit{T. Ohsawa}, Nagoya Math. J. 158, 95-98 (2000; Zbl 1023.32026)] that \(\mathbb{P}^n\) contains no compact real analytic Levi flat hypersurface if \(n\geq 2\). On complex tori \(T\), the authors raised a conjecture: Let \(M\) be a compact Levi flat hypersurface of \(T\). Then \(\pi^{-1}(M)\) is a union of complex affine hyperplanes, where \(\pi\) is the canonical projection. If moreover \(T\) contains no proper complex tori of positive dimension, \(M\) is flat. A partial result is proved in the paper. Let \(M, \pi\) and \(T\) be as above. If \(M\) is real analytic and \(\dim T=2\), then \(\pi^{-1}(M)\) is a union of complex affine line. Moreover, if \(M\) does not contain any elliptic curve, \(M\) is flat.
reviewed by: Shanyu Ji
Mathematics Subject Classification ID: 32V40
zbMATH DE Number: 1880703
zbMATH Keywords: real analytic Levi flat hypersurface
zbMATH Keywords: complex tori
MaRDI profile type: Publication
OpenAlex ID: W2468935008
cites work: Levisches Problem und Rungescher Satz für Teilgebiete Steinscher Mannigfaltigkeiten
cites work: A note on projective Levi flats and minimal sets of algebraic foliations
cites work: Q4126649
cites work: Nonexistence of real analytic Levi flat hypersurfaces in ℙ2
cites work: Nonexistence of smooth Levi-flat hypersurfaces in complex projective spaces of dimension \(\geq 3\)
DOI: 10.5802/AIF.1923
published in: Annales de l’institut Fourier
description: scientific article; zbMATH DE number 1880703