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A reduction theorem for stable sets of holomorphic foliations on complex tori. (English) Zbl 1187.32009

The following theorem is proved: let \(T\) be a complex torus equipped with a flat Hermitian metric and let \(A\subset T\) be a closed subset. Assume that there exists a neighborhood \(U \supset A\) and a one-codimensional holomorphic foliation \(\mathcal{F} \subset \Omega_T^1 \) (= the sheaf of holomorphic 1-forms on \(T\)) with possibly singular leaves on \(U\) such that \(A\) is a stable set of \(\mathcal{F}\) (i.e. \(A\) is the union of leaves of \(\mathcal{F}\) intersecting with \(A\)) and that \(\mathcal{F}\) is topologically trivial on a neighborhood of \(A.\) Then, either \(\mathcal{F}\) is generated by a holomorphic 1-form on \(T\) and \(A\) is totally geodesic, or there exists a complex 2-torus \(T',\) a holomorphic map \(\pi : T \longrightarrow T'\) and a closed subset \(A' \subset T'\) such that \(A=\pi^{-1} (A').\) In the latter case, \(\mathcal{F}\) is the pull back of some foliation \(\mathcal{F}'\) on a neighborhood of \(A'\) whenever \(A\) is not complex analytic.
A crucial step in the proof is an application of the Hodge theory on pseudoconvex manifolds.

MSC:

32E40 The Levi problem
53C40 Global submanifolds
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References:

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[9] T. Ohsawa, \(\overline\partial\)-cohomology and geometry of the boundary of pseudoconvex domains , Ann. Polon. Math., 19 (2007), 249–262. Added in proof. Examples in the appendix has already been known by E. Ghys, essentially (cf. Ann. Fac. Sci. Toulouse Math. (6), 5 (1996), 493–519). As for a somewhat more advanced result for (singular) foliations on tori, the reader is recommended to read M. Brunella: Codimension one foliations on complex tori, preprint. (See also Brunella’s another paper in Indiana Univ. Math. J., 57 (2008), 3101–3114.) · Zbl 1140.32023 · doi:10.4064/ap91-2-12
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