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Smooth deformation limit of Moishezon manifolds is Moishezon

Mu-Lin Li, Sheng Rao, Kai Wang, Meng-jiao Wang

TL;DR

The paper proves that the deformation limit of Moishezon manifolds under a smooth deformation over a unit disk is Moishezon, resolving a central conjecture. The authors develop a framework combining Bott–Chern cohomology, Kodaira–Spencer theory, Gauduchon metrics, and Barlet’s theory of relative cycle spaces to construct a global object controlling $(n-1)$-cycles across the family and to extend a good filling from the punctured disk to the whole base. This leads to a semi-continuity result for the algebraic dimension in smooth families and yields applications including deformation-invariance of plurigenera and the pseudo-projective structure of families with many Moishezon fibers. The work extends prior results of Popovici, Barlet, and Rao–Tsai, and provides a robust mechanism to analyze Moishezon-ness in higher dimensions via cycle-space methods and cohomological control.

Abstract

We prove the conjecture that the deformation limit of Moishezon manifolds under a smooth deformation over a unit disk in $\mathbb{C}$ is Moishezon.

Smooth deformation limit of Moishezon manifolds is Moishezon

TL;DR

The paper proves that the deformation limit of Moishezon manifolds under a smooth deformation over a unit disk is Moishezon, resolving a central conjecture. The authors develop a framework combining Bott–Chern cohomology, Kodaira–Spencer theory, Gauduchon metrics, and Barlet’s theory of relative cycle spaces to construct a global object controlling -cycles across the family and to extend a good filling from the punctured disk to the whole base. This leads to a semi-continuity result for the algebraic dimension in smooth families and yields applications including deformation-invariance of plurigenera and the pseudo-projective structure of families with many Moishezon fibers. The work extends prior results of Popovici, Barlet, and Rao–Tsai, and provides a robust mechanism to analyze Moishezon-ness in higher dimensions via cycle-space methods and cohomological control.

Abstract

We prove the conjecture that the deformation limit of Moishezon manifolds under a smooth deformation over a unit disk in is Moishezon.
Paper Structure (14 sections, 17 theorems, 65 equations)

This paper contains 14 sections, 17 theorems, 65 equations.

Table of Contents

  1. Introduction
  2. Preliminaries: constant sheaves and Barlet's cycle spaces
  3. Constant sheaves and trivial bundles
  4. Cohomologies of complex manifolds
  5. Barlet's theory of cycle spaces
  6. Absolute case
  7. Relative case
  8. Deformation limit of Moishezon manifolds
  9. Construction via Bott--Chern and Kodaira--Spencer theories
  10. Proofs of Theorems \ref{['projective']} and \ref{['Moishezon']}
  11. Applications to Moishezon families
  12. Deformation invariance of plurigenera
  13. Pseudo-projective families and projective loci
  14. Barlet's theory of relative codimension 1 cycle-space

Key Result

Theorem 1.1

Assume that there exist uncountably many Moishezon fibers in a smooth family over $\Delta$. Then all fibers of this family are Moishezon.

Theorems & Definitions (31)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 2.1
  • Lemma 2.2: ES10
  • Theorem 2.3: Lt90
  • Theorem 2.4: Lt90
  • Lemma 2.5: e.g. Sch
  • Proposition 2.6
  • proof
  • ...and 21 more