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Remarks on Relative Canonical Bundles and Algebraicity Criteria for Foliations in Kähler context

Junyan Cao, Mihai Păun

TL;DR

The work develops an analytic approach to positivity questions for relative canonical bundles in the Kähler setting, proving a pseudo-effectiveness result for $K_{X/Y}+L- D(p)$ under fiberwise pseudo-effectiveness assumptions and a rational-type curvature data. It provides an analytic proof of Ou's algebraicity criterion by establishing the pseudo-effectiveness of conormal bundles via Lelong-number techniques and mass-concentration arguments, and extends Ou's uniruledness criterion within the Kähler context. A key feature is the use of Bergman-type fiberwise metrics, Ohsawa–Takegoshi extension, and Gauduchon-stability tools to translate positivity properties into algebraicity and rational connectedness of leaves. Collectively, the results supply a robust analytic framework for the positivity of relative canonical bundles and algebraicity of foliations in non-projective (Kähler) settings, with implications for meromorphic fibrations and uniruledness phenomena.

Abstract

In this note, motivated by the recent preprint of W. Ou, we pursue three main objectives. The first is to make progress towards the positivity of the relative canonical bundle in the Kähler setting. In the second part, we provide a proof of Ou's algebraicity criterion. Finally, based on the two previous parts, we slightly extend his uniruledness criterion.

Remarks on Relative Canonical Bundles and Algebraicity Criteria for Foliations in Kähler context

TL;DR

The work develops an analytic approach to positivity questions for relative canonical bundles in the Kähler setting, proving a pseudo-effectiveness result for under fiberwise pseudo-effectiveness assumptions and a rational-type curvature data. It provides an analytic proof of Ou's algebraicity criterion by establishing the pseudo-effectiveness of conormal bundles via Lelong-number techniques and mass-concentration arguments, and extends Ou's uniruledness criterion within the Kähler context. A key feature is the use of Bergman-type fiberwise metrics, Ohsawa–Takegoshi extension, and Gauduchon-stability tools to translate positivity properties into algebraicity and rational connectedness of leaves. Collectively, the results supply a robust analytic framework for the positivity of relative canonical bundles and algebraicity of foliations in non-projective (Kähler) settings, with implications for meromorphic fibrations and uniruledness phenomena.

Abstract

In this note, motivated by the recent preprint of W. Ou, we pursue three main objectives. The first is to make progress towards the positivity of the relative canonical bundle in the Kähler setting. In the second part, we provide a proof of Ou's algebraicity criterion. Finally, based on the two previous parts, we slightly extend his uniruledness criterion.

Paper Structure

This paper contains 6 sections, 27 theorems, 99 equations.

Table of Contents

  1. Introduction
  2. Some Preliminary Results
  3. Positivity of relative canonical bundles in Kähler context
  4. Proof of Theorem \ref{['mainthm']}
  5. Algebraicity of positive foliations.
  6. Appendix: stability with respect to the degree induced by a Gauduchon metric

Key Result

Theorem 1.2

In the setting of Conjecture conjc-h, further assume that there exists a Kähler metric $\omega$ and a holomorphic 2-form $\sigma$ on $X$ such that $\omega+ \sigma + \overline{\sigma}$ is a rational class and that the restriction of $\sigma$ to the generic fiber of $p$ vanishes. Then $K_{X/Y}+ L- D(p

Theorems & Definitions (53)

  • Conjecture 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Proposition 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • ...and 43 more