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Zbl 1031.32030
Matsumoto, Kazuko; Ohsawa, Takeo
On the real analytic Levi flat hypersurfaces in complex tori of dimension two. (English)
[J] Ann. Inst. Fourier 52, No.5, 1525-1532 (2002). ISSN 0373-0956; ISSN 1777-5310/e

Recently it has been proved [{\it A. Lins Neto}, Ann. Inst. Fourier 49, 1369-1385 (1999; Zbl 0963.32022) and {\it T. Ohsawa}, Nagoya Math. J. 158, 95-98 (2000; Zbl 1023.32026)] that $\bbfP^n$ contains no compact real analytic Levi flat hypersurface if $n\ge 2$. 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.
[Shanyu Ji (Houston)]

MSC 2000:: *32V40 Real submanifolds in complex manifolds

Keywords: real analytic Levi flat hypersurface; complex tori

Citations: Zbl 0963.32022; Zbl 1023.32026

Cited in: Zbl 1062.32027

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