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Criteria for a fiberwise Fujiki/Kähler family to be locally Moishezon/projective

Jian Chen

TL;DR

The paper develops cohomological and semi-positivity criteria to measure how far a fiberwise Fujiki or Kähler family is from being locally Moishezon or locally projective. It uses Campana's ideas and Boucksom's modified Kähler classes to translate fiberwise positivity into global conclusions, providing criteria that guarantee local Moishezonness for Fujiki total spaces and local projectivity under suitable cohomological and semi-positivity conditions, including Takegoshi's torsion-freeness and relative $(1,1)$-class results. Key contributions include explicit local Moishezon criteria for fiberwise Fujiki families, corollaries giving Moishezon/projective status under various extendability and positivity hypotheses, and a framework connecting cohomology class extension with extensions of representatives via $L^2$-estimates. The findings unify bimeromorphic Kodaira-type criteria with relative Lefschetz-type theorems, yielding concrete criteria for local projectivity and providing tools for understanding when degenerations of projective fibers remain projective.

Abstract

Inspired by certain topics in local deformation theory, we primarily utilize F. Campana's methods to investigate how far a fiberwise Fujiki family is from being locally Moishezon and how far a fiberwise Kähler family is from being locally projective. We investigate these questions from two main perspectives: cohomological data and global semi-positivity data on the total space.

Criteria for a fiberwise Fujiki/Kähler family to be locally Moishezon/projective

TL;DR

The paper develops cohomological and semi-positivity criteria to measure how far a fiberwise Fujiki or Kähler family is from being locally Moishezon or locally projective. It uses Campana's ideas and Boucksom's modified Kähler classes to translate fiberwise positivity into global conclusions, providing criteria that guarantee local Moishezonness for Fujiki total spaces and local projectivity under suitable cohomological and semi-positivity conditions, including Takegoshi's torsion-freeness and relative -class results. Key contributions include explicit local Moishezon criteria for fiberwise Fujiki families, corollaries giving Moishezon/projective status under various extendability and positivity hypotheses, and a framework connecting cohomology class extension with extensions of representatives via -estimates. The findings unify bimeromorphic Kodaira-type criteria with relative Lefschetz-type theorems, yielding concrete criteria for local projectivity and providing tools for understanding when degenerations of projective fibers remain projective.

Abstract

Inspired by certain topics in local deformation theory, we primarily utilize F. Campana's methods to investigate how far a fiberwise Fujiki family is from being locally Moishezon and how far a fiberwise Kähler family is from being locally projective. We investigate these questions from two main perspectives: cohomological data and global semi-positivity data on the total space.

Paper Structure

This paper contains 7 sections, 25 theorems, 13 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Criteria for a fiberwise Fujiki family to be locally Moishezon
  4. Fiberwise Fujiki families
  5. Corollaries of Lemma \ref{['pair version']}
  6. On the extension of cohomology classes
  7. Criteria for a fiberwise Kähler morphism to be locally projective

Key Result

Theorem 1.1

Let $f: X\to S$ be a smooth family, where $X$ is a Fujiki manifold, $S$ is a compact smooth curve, and the fiber over $t\in S$ is denoted by $X_t:=f^{-1}(t)$. Assume that the natural composed map has its image in $H^{1,1}(X_t, \mathbb{R})$ for any $t\in S$ (e.g., when any holomorphic $(2,0)$-form on $X$ vanishes along the direction of each fiber). Then $f$ is locally Moishezon.

Theorems & Definitions (52)

  • Theorem 1.1: =Theorem \ref{['thm- fiber divisor']}
  • Theorem 1.2: =Theorem \ref{['thm- fiber divisor-refine-appli']}
  • Theorem 1.3: =Theorem \ref{['rela 1-1 class']}
  • Theorem 1.4: =Theorem \ref{['projective criterian']}
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3: e.g., DP04
  • Lemma 2.4: e.g., C22
  • Definition 2.5: P94
  • Lemma 2.6: JS93
  • ...and 42 more