Holomorphic foliation associated with a semi-positive class of numerical dimension one

Let X be a compact Kähler manifold and α be a class in the Dolbeault cohomology class of bidegree (1, 1) on X . When α admits at least two smooth semipositive representatives, we show the existence of a family of real analytic Levi-flat hypersurfaces in X and a holomorphic foliation on a suitable domain of X along whose leaves any semi-positive representative of α is zero. As an application, we give the affirmative answer to [K2, Conjecture 2.1] on the relation between the semi-positivity of the line bundle [Y ] and the analytic structure of a neighborhood of Y for a smooth connected hypersurface Y of X .

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