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

Abstract

Let $X$ be a compact Kähler manifold and $\alpha$ be a Dolbeault cohomology class of bi-degree $(1, 1)$ on $X$. When the numerical dimension of $\alpha$ is one and $\alpha$ 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 $W$ of $X$ along whose leaves any semi-positive representative of $\alpha$ is zero. We also investigate the maximal choice of such a domain $W$ by considering the analytic continuation of the holomorphic foliation. As an application, we give a criterion for the $U(1)$-flatness of line bundles which particularly gives an 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 $X$.