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On compact Kähler manifolds with pseudo-effective tangent bundle

Shin-ichi Matsumura, Chenghao Qing

TL;DR

The paper addresses the structure of compact Kähler manifolds with pseudo-effective tangent bundles by proving the existence of a smooth fibration $φ: X o Y$ onto a finite étale torus quotient $Y$, with a very general fiber $F$ that is rationally connected and has pseudo-effective $T_F$. The approach centers on an Albanese-map-based fibration and an induction on dimension, establishing that fibers become projective and rationally connected after finite étale covers, and that the base $Y$ is a finite étale quotient of a torus; if $T_X$ admits a positively curved singular Hermitian metric, the fibration is locally constant. This result extends the projective structure theorem to compact Kähler manifolds, implies virtual abelianity of $ obreak π_1(X)$, and fits into Campana’s framework of special varieties, providing a Kähler analogue of uniformization results for RC fibrations. Overall, it advances the classification of Kähler manifolds with pseudo-effective tangent bundles and clarifies how curvature hypotheses constrain the fibration structure.

Abstract

In this paper, we prove that a compact Kähler manifold $X$ with pseudo-effective (resp. singular positively curved) tangent bundle admits a smooth (resp. locally constant) rationally connected fibration $φ\colon X \to Y$ onto a finite étale quotient $Y$ of a compact complex torus. This result extends the structure theorem previously established for smooth projective varieties to compact Kähler manifolds.

On compact Kähler manifolds with pseudo-effective tangent bundle

TL;DR

The paper addresses the structure of compact Kähler manifolds with pseudo-effective tangent bundles by proving the existence of a smooth fibration onto a finite étale torus quotient , with a very general fiber that is rationally connected and has pseudo-effective . The approach centers on an Albanese-map-based fibration and an induction on dimension, establishing that fibers become projective and rationally connected after finite étale covers, and that the base is a finite étale quotient of a torus; if admits a positively curved singular Hermitian metric, the fibration is locally constant. This result extends the projective structure theorem to compact Kähler manifolds, implies virtual abelianity of , and fits into Campana’s framework of special varieties, providing a Kähler analogue of uniformization results for RC fibrations. Overall, it advances the classification of Kähler manifolds with pseudo-effective tangent bundles and clarifies how curvature hypotheses constrain the fibration structure.

Abstract

In this paper, we prove that a compact Kähler manifold with pseudo-effective (resp. singular positively curved) tangent bundle admits a smooth (resp. locally constant) rationally connected fibration onto a finite étale quotient of a compact complex torus. This result extends the structure theorem previously established for smooth projective varieties to compact Kähler manifolds.

Paper Structure

This paper contains 3 sections, 5 theorems, 11 equations.

Table of Contents

  1. Introduction
  2. Notation and Conventions
  3. Proof of the main result

Key Result

Theorem 1.1

Let $X$ be a compact Kähler manifold with pseudo-effective tangent bundle $($see Subsection subsec-notation for our definition of pseudo-effective vector bundles$)$. Then $X$ admits a fibration $\phi: X \to Y$$($i.e., a surjective holomorphic map with connected fibers$)$ with the following propertie Moreover, if we further assume that $T_X$ admits a $($possibly singular$)$ positively curved singul

Theorems & Definitions (10)

  • Theorem 1.1
  • Remark 1.2
  • Corollary 1.3
  • Lemma 2.1
  • proof
  • Theorem 2.2
  • proof
  • Theorem 2.3
  • proof
  • proof : Proof of Theorem \ref{['thm-main']}