Formal principle for line bundles on neighborhoods of an analytic subset of a compact Kähler manifold

We investigate the formal principle for holomorphic line bundles on neighborhoods of an analytic subset of a complex manifold mainly in the case where it can be realized as an open subset of a compact Kähler manifold. Our approach identifies the obstruction as a global analytic class supported on a neighborhood of $Y$, and relates its vanishing to the solvability of a $\mathrm{\partial }\bar{\mathrm{\partial }}$-problem on neighborhoods of $Y$. As a consequence we obtain cohomological criteria ensuring the formal principle. We also construct a holomorphic family of compact Kähler surfaces containing a curve with topologically trivial normal bundle in which the formal principle holds for almost every fiber but fails for uncountably many fibers, exhibiting an instability phenomenon in families.