  Let $X$ be a complex manifold and $L$ be a holomorphic line bundle on $X$.
Assume that $L$ is semi-positive, namely $L$ admits a smooth Hermitian metric
with semi-positive Chern curvature. Let $Y$ be a compact K\"ahler submanifold
of $X$ such that the restriction of $L$ to $Y$ is topologically trivial. We
investigate the obstruction for $L$ to be unitary flat on a neighborhood of $Y$
in $X$. As an application, for example, we show the existence of nef, big, and
non semi-positive line bundle on a non-singular projective surface.
