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CscK metrics near the canonical class

Bin Guo, Wangjian Jian, Yalong Shi, Jian Song

Abstract

Let $X$ be a Kähler manifold with semi-ample canonical bundle $K_X$. It is proved by Jian-Shi-Song that for any Kähler class $γ$, there exists $δ>0$ such that for all $t\in (0, δ)$ there exists a unique cscK metric $g_t$ in $K_X+ t γ$. In this paper, we prove that $\{ (X, g_t) \}_{ t\in (0, δ)} $ have uniformly bounded Kähler potentials, volume forms and diameters. As a consequence, these metric spaces are pre-compact in the Gromov-Hausdorff sense.

CscK metrics near the canonical class

Abstract

Let be a Kähler manifold with semi-ample canonical bundle . It is proved by Jian-Shi-Song that for any Kähler class , there exists such that for all there exists a unique cscK metric in . In this paper, we prove that have uniformly bounded Kähler potentials, volume forms and diameters. As a consequence, these metric spaces are pre-compact in the Gromov-Hausdorff sense.
Paper Structure (4 sections, 10 theorems, 79 equations)

This paper contains 4 sections, 10 theorems, 79 equations.

Table of Contents

  1. Introduction
  2. Reduction to uniform a priori estimates for cscK system
  3. Uniform entropy bounds
  4. From entropy bound to $L^\infty$ estimates

Key Result

Theorem 1.1

Let $X$ be an $n$-dimensional compact Kähler manifold with semiample canonical bundle $K_X$. For any Kähler class $\gamma$, there exists $\delta=\delta(\gamma)>0$ such that there exists a unique cscK metric $\omega_t\in K_X+t\gamma$ for $t\in (0, \delta)$ as in JSS. Then there exists $\alpha=\alpha( for any $R\in (0,1)$, where $B_{\omega_t}(x,R)$ denotes the geodesic ball in $(X,\omega_t)$ with ce

Theorems & Definitions (17)

  • Conjecture 1.1
  • Theorem 1.1
  • Theorem 2.1
  • proof : Proof of Theorem \ref{['thm:main']} assuming Theorem \ref{['thm:tech']}:
  • Proposition 3.1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3: JSS
  • ...and 7 more