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

Takayuki Koike

TL;DR

This work develops a formal principle for holomorphic line bundles on neighborhoods of a complex analytic subset $Y$ inside a compact Kähler manifold $X$ by translating the triviality problem into a global obstruction class supported near $Y$, and linking solvability of a $\partial\overline{\partial}$-problem to the vanishing of this obstruction. It introduces the obstruction class $v(\mathcal{I},L)$ and related cohomological data, establishes conditions under which unitary flatness and local trivializations propagate from the infinitesimal neighborhood data to a genuine neighborhood, and proves the main theorem along with corollaries that connect to divisor geometry and positivity. The paper also demonstrates instability phenomena in families and places the results in the context of Ueda theory and Ohsawa-type convexity results, providing both general criteria and concrete cases (e.g., curves in surfaces) where the formal principle holds or fails. Collectively, the approach yields practical cohomological criteria for the FPLB and clarifies the interplay between analytic obstructions, curvature, and the topology of the complement $M=X\setminus Y$.

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.

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

TL;DR

This work develops a formal principle for holomorphic line bundles on neighborhoods of a complex analytic subset inside a compact Kähler manifold by translating the triviality problem into a global obstruction class supported near , and linking solvability of a -problem to the vanishing of this obstruction. It introduces the obstruction class and related cohomological data, establishes conditions under which unitary flatness and local trivializations propagate from the infinitesimal neighborhood data to a genuine neighborhood, and proves the main theorem along with corollaries that connect to divisor geometry and positivity. The paper also demonstrates instability phenomena in families and places the results in the context of Ueda theory and Ohsawa-type convexity results, providing both general criteria and concrete cases (e.g., curves in surfaces) where the formal principle holds or fails. Collectively, the approach yields practical cohomological criteria for the FPLB and clarifies the interplay between analytic obstructions, curvature, and the topology of the complement .

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 , and relates its vanishing to the solvability of a -problem on neighborhoods of . 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.
Paper Structure (18 sections, 42 equations)

This paper contains 18 sections, 42 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Some fundamental facts on cohomologies
  5. Unitary flatness of holomorphic line bundles and the $\partial\overline{\partial}$-Lemma
  6. Obstruction classes and some sufficient conditions for the FPLB
  7. The FPLB Problem in terms of $\mathcal{I}$
  8. Obstruction classes and some fundamental sufficient conditions for the FPLB
  9. Proof of a weak variant of Theorem \ref{['thm:main']} $(i)$ and definition of $v(\mathcal{I}, L)$
  10. Proof of a weak variant of Theorem \ref{['thm:main']} $(i)$
  11. Definition of $v(\mathcal{I}, L)$
  12. Proof of Theorem \ref{['thm:main']} and its corollaries
  13. Hermitian metrics on a line bundle whose restriction to the $\mu$-th infinitesimal neighborhood is trivial
  14. The $\partial\overline{\partial}$-problem on a neighborhood $Y$ and the obstruction class $v(\mathcal{I}, L)$
  15. Proof of Theorem \ref{['thm:main']} and its corollaries
  16. ...and 3 more sections

Theorems & Definitions (24)

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  • ...and 14 more