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Almost Non-positive Kähler Manifolds

Yuguang Zhang

TL;DR

This work analyzes compact Kähler manifolds with small positive curvature by proving that a curvature–energy bound $K_{ ilde{g}} E_g( ilde{g}) \le \epsilon$ forces strong topological and algebro-geometric consequences: the universal cover is contractible and the holomorphic cotangent bundle is nef. It introduces the energy functional $E_g(\cdot)$ to replace diameter restrictions and proves a Schwarz-type inequality $K_{\tilde{g}}^h E_g(\tilde{g}) \le \mathcal{E}$ implying $\tilde{g} \le C E_g(\tilde{g}) g$, using the Chern–Lu inequality and a quantitative Schoen–Uhlenbeck framework. The main theorem is established by a contradiction argument leveraging Sacks–Uhlenbeck harmonic spheres and the Schwarz-type estimate, and the nefness conclusion follows from Griffiths positivity arguments applied to the induced curvature on tensor powers. The results yield corollaries about torus fibrations and almost flat Kähler manifolds, and provide a Kähler analogue to rigidity phenomena for manifolds with near non-positive curvature.

Abstract

This paper proves that the universal covering of a compact Kähler manifold with small positive sectional curvature in a certain sense is contractible.

Almost Non-positive Kähler Manifolds

TL;DR

This work analyzes compact Kähler manifolds with small positive curvature by proving that a curvature–energy bound forces strong topological and algebro-geometric consequences: the universal cover is contractible and the holomorphic cotangent bundle is nef. It introduces the energy functional to replace diameter restrictions and proves a Schwarz-type inequality implying , using the Chern–Lu inequality and a quantitative Schoen–Uhlenbeck framework. The main theorem is established by a contradiction argument leveraging Sacks–Uhlenbeck harmonic spheres and the Schwarz-type estimate, and the nefness conclusion follows from Griffiths positivity arguments applied to the induced curvature on tensor powers. The results yield corollaries about torus fibrations and almost flat Kähler manifolds, and provide a Kähler analogue to rigidity phenomena for manifolds with near non-positive curvature.

Abstract

This paper proves that the universal covering of a compact Kähler manifold with small positive sectional curvature in a certain sense is contractible.
Paper Structure (3 sections, 8 theorems, 77 equations)

This paper contains 3 sections, 8 theorems, 77 equations.

Table of Contents

  1. Introduction
  2. A Schwarz type inequality
  3. Proofs

Key Result

Theorem 1.1

Let $(X, \omega_g, g)$ be a compact Kähler manifold of complex dimension $n$. There exists a constant $\epsilon =\epsilon(X, [\omega_g])>0$ depending only on the complex manifold $X$ and the Kähler class $[\omega_g]\in H^{1,1}(X, \mathbb{R})$ satisfying that if there is a Kähler metric $\tilde{g}$ where $K_{\tilde{g}}$ is the Riemannian sectional curvature of $\tilde{g}$, then

Theorems & Definitions (14)

  • Theorem 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Proposition 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3: Lemma 4.3.2 in MS
  • proof : Proof of Proposition \ref{['prop1']}
  • Corollary 2.4
  • proof
  • ...and 4 more