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

Abstract

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 $\partial \overline{\partial }$-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.

Small modifications, no changes to the results