On the real analytic Levi flat hypersurfaces in complex tori of dimension two
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.
scientific article; zbMATH DE number 1880703