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Compact Kähler manifolds with partially semi-positive curvature

Shiyu Zhang, Xi Zhang

TL;DR

The paper develops a differential-geometric program to understand compact Kähler manifolds under partially semi-positive curvature via MRC fibrations. It introduces BC-$p$ positivity for the tangent bundle and proves that BC-$p$ positivity for all $p$ forces rational connectedness, yielding a practical criterion that subsumes several RC-positivity conditions and applications to orthogonal Ricci curvature and conformally Kähler metrics. It also establishes structure theorems for two immediate curvature regimes, showing that either the rational dimension satisfies $\mathrm{rd}(X)\ge n-k+1$ or $X$ fibers locally with a Ricci-flat base and a rationally connected fiber, with a holomorphic-isometric splitting of the universal cover. The analysis relies on Bochner-type integral inequalities for Hermitian ratios of pseudo-effective subsheaves and currents, connecting curvature conditions to holomorphic foliation structures and fibration theory to derive splitting and quasi-product structures. These results extend and unify previous structure theorems (e.g., Campana–Demailly–Peternell and Matsumura) to the setting of BC-$p$ positivity and mixed curvature, providing new tools for probing the geometry of partially positive Kähler manifolds.

Abstract

In this paper, we study MRC fibrations of compact Kähler manifolds with partially semi-positive curvature. We first prove that a compact Kähler manifold is rationally connected if its tangent bundle is BC-$p$ positive for all $1\leq p\leq \dim X$. As applications, we confirm a conjecture of Lei Ni that any compact Kähler manifold with positive orthogonal Ricci curvature must be rationally connected, and generalize a result of Heier-Wong and Yang to the conformally Kähler case. The second result concern structure theorems for two immediate curvature conditions. We prove that, a compact Kähler manifold with $k$-semi-positive Ricci curvature or semi-positive $k$-scalar curvature, either the rational dimension $\geq n-k+1$ or it admits a locally constant fibration $f: X\rightarrow Y$ such that the fibre is rationally connected and the image $Y$ is Ricci-flat.

Compact Kähler manifolds with partially semi-positive curvature

TL;DR

The paper develops a differential-geometric program to understand compact Kähler manifolds under partially semi-positive curvature via MRC fibrations. It introduces BC- positivity for the tangent bundle and proves that BC- positivity for all forces rational connectedness, yielding a practical criterion that subsumes several RC-positivity conditions and applications to orthogonal Ricci curvature and conformally Kähler metrics. It also establishes structure theorems for two immediate curvature regimes, showing that either the rational dimension satisfies or fibers locally with a Ricci-flat base and a rationally connected fiber, with a holomorphic-isometric splitting of the universal cover. The analysis relies on Bochner-type integral inequalities for Hermitian ratios of pseudo-effective subsheaves and currents, connecting curvature conditions to holomorphic foliation structures and fibration theory to derive splitting and quasi-product structures. These results extend and unify previous structure theorems (e.g., Campana–Demailly–Peternell and Matsumura) to the setting of BC- positivity and mixed curvature, providing new tools for probing the geometry of partially positive Kähler manifolds.

Abstract

In this paper, we study MRC fibrations of compact Kähler manifolds with partially semi-positive curvature. We first prove that a compact Kähler manifold is rationally connected if its tangent bundle is BC- positive for all . As applications, we confirm a conjecture of Lei Ni that any compact Kähler manifold with positive orthogonal Ricci curvature must be rationally connected, and generalize a result of Heier-Wong and Yang to the conformally Kähler case. The second result concern structure theorems for two immediate curvature conditions. We prove that, a compact Kähler manifold with -semi-positive Ricci curvature or semi-positive -scalar curvature, either the rational dimension or it admits a locally constant fibration such that the fibre is rationally connected and the image is Ricci-flat.

Paper Structure

This paper contains 16 sections, 34 theorems, 67 equations.

Table of Contents

  1. Introduction and statements
  2. BC positivity and rational connectedness
  3. Structure theorems for immediate curvature conditions
  4. Basic materials
  5. Currents associated to a plurisubharmonic function
  6. MRC fibrations
  7. Positivity of vector bundles
  8. BC-positivity and Rational connectedness
  9. Bochner-type formula
  10. Proof of Theorem \ref{['main-thm-BC-p-1']} and applications
  11. Structure theorems for immediate curvature conditions
  12. Integral inequality
  13. Splitting theorem of the tangent bundle
  14. Proof of Theorem \ref{['main-thm-partially']}
  15. Structure theorems
  16. ...and 1 more sections

Key Result

Theorem 1.2

Let $X$ be a compact complex manifold with BC-$p$ positive tangent bundle for some $p\geq1$. Suppose that $f:X\dashrightarrow Y$ is a dominant meromorphic map between compact complex manifolds such that $\dim Y= p\leq \dim X$, then $K_Y$ is not pseudo-effective.

Theorems & Definitions (68)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Conjecture 1.7
  • Conjecture 1.8: yang2018
  • Definition 1.9
  • Theorem 1.10
  • ...and 58 more