Formal principle for line bundles on neighborhoods of an analytic subset of a compact K\"ahler manifold

TL;DR

This paper studies the formal principle for holomorphic line bundles near analytic subsets in compact K"ahler manifolds, linking obstructions to a global class and solving a $ar{ ext{d}}$-$ ext{d}$ problem, with applications to cohomology and family stability.

Contribution

It introduces a cohomological criterion for the formal principle and demonstrates an instability phenomenon in families of K"ahler surfaces.

Findings

Obstruction characterized by a global analytic class.

Cohomological criteria for the formal principle.

Existence of families where the principle holds for almost all fibers but not all.

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 K\"ahler 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 K\"ahler 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.