Linearization of transition functions of a semi-positive line bundle along a certain submanifold

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 Kähler 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.

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