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A characterization of uniruled compact Kähler manifolds

Wenhao Ou

Abstract

We adapt Bost's algebraicity characterization to the situation of a germ in a compact Kähler manifold. As a consequence, we extend the algebraic integrability criteria of Campana-Păun and of Druel to foliations on compact Kähler manifolds. As an application, we prove that a compact Kähler manifold is uniruled if and only if its canonical line bundle is not pseudoeffective.

A characterization of uniruled compact Kähler manifolds

Abstract

We adapt Bost's algebraicity characterization to the situation of a germ in a compact Kähler manifold. As a consequence, we extend the algebraic integrability criteria of Campana-Păun and of Druel to foliations on compact Kähler manifolds. As an application, we prove that a compact Kähler manifold is uniruled if and only if its canonical line bundle is not pseudoeffective.

Paper Structure

This paper contains 15 sections, 38 theorems, 117 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Boucksom's divisorial Zariski decompositions
  4. Non pseudoeffective torsion-free sheaves
  5. Notion of formal extensions
  6. Quasi-psh functions with large restriction Lelong numbers
  7. Foliations induced by meromorphic maps
  8. Proof of Theorem \ref{['thm:alg-germ-nonpsef-normal']}
  9. Proof of Theorem \ref{['thm:foliation-psef-alg']} and Theorem \ref{['thm:BMQ']}
  10. Uniruled compact Kähler manifolds
  11. Relative Bergman kernel metrics
  12. Locally projective morphisms induced by holomorphic 2-forms
  13. Relative Albanese reductions
  14. Morphisms with relative Albanese dimension zero
  15. End of the proof

Key Result

Theorem 1.1

Let $X$ be a compact Kähler manifold. Then $X$ is uniruled if and only if the canonical line bundle $\omega_X$ is not pseudoeffective.

Theorems & Definitions (82)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Theorem 2.3
  • ...and 72 more