On the Levi-flats in complex tori of dimension two
Let \(M\) be a \({\mathcal C}^5\) Levi flat hypersurface in a homogeneous Kähler manifold \((X, \omega)\) of complex dimension \(2\). The first result of the paper consists in proving that there cannot exist any \({\mathcal C}^2\) function \(\varrho: X \setminus M \to ]-\infty, 0[\) that satisfies the following two conditions: a) there is a constant \(K > 0\) so that \(- i \partial \bar \partial \log(-\varrho)_z > K \omega_z\) for any \(z \in \mathcal U \setminus M\) of some neighborhood \(\mathcal U\) of \(M\); b) \(\rho_z = u_z \text{dist}(z, M)\) for some nowhere vanishing \({\mathcal C}^2\)-function \(u\) on a neighborhood of \(M\). This non-existence result is used by the author to prove that any \({\mathcal C}^5\) compact Levi hypersurface in a 2-dimensional complex torus has to contain a complex line segment. Then, using this result, he obtains a complete description of the real analytic, compact, Levi flat hypersurfaces in a 2-dimensional complex torus \(X\). In fact, he shows that any such hypersurface \(M\) is either \textit{holomorphically flat}, i.e. the pre-image \(\pi^{-1}(M)\) by the universal covering map \(\pi: {\mathbb C}^2 \to X\) is union of complex affine hypersurfaces of \({\mathbb C}^2\), or it is a \textit{Levi scroll}, i.e. it admits a surjective CR map \(\hat \pi: M \to C\) onto a compact Riemann surface \(C\) whose critical set is union of finitely many complex curves. This second case can occur only if the torus \(X\) is fibered holomorphically over an elliptic curve. [An addendum to this paper is given in ibid., 379--382 (2006; see the paper below, Zbl 1141.32017).]
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