Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Uniform diameter estimates for Kaehler metrics in big cohomology classes

Duc-Bao Nguyen, Duc-Viet Vu

Abstract

We generalize previous diameter estimates and local non-vanishing of volumes for Kaehler metrics to the case of big cohomology classes. In our proof, among other things, we will prove a uniform diameter estimate for a family of smooth Kaehler metrics only involving an integrability condition. We also have to use fine stability properties of complex Monge-Ampere equations with prescribed singularities.

Uniform diameter estimates for Kaehler metrics in big cohomology classes

Abstract

We generalize previous diameter estimates and local non-vanishing of volumes for Kaehler metrics to the case of big cohomology classes. In our proof, among other things, we will prove a uniform diameter estimate for a family of smooth Kaehler metrics only involving an integrability condition. We also have to use fine stability properties of complex Monge-Ampere equations with prescribed singularities.

Paper Structure

This paper contains 8 sections, 30 theorems, 141 equations.

Table of Contents

  1. Introduction
  2. Uniform estimates for subharmonic functions
  3. Uniform estimates for Green's functions
  4. Sobolev inequalities for semi-positive classes
  5. Metrics on the space of singularity types
  6. --estimate for measures dominated by capacity
  7. Sobolev inequalities for big cohomology classes
  8. --condition

Key Result

Theorem 1.1

Let $(X,\omega_X)$ be a compact Kähler manifold of dimension $n$. Let $A , B$ be positive constants and $p > 1$ be a constant. Let $q\in (1,\frac{n}{n-1} )$. Then there exist positive constants $C_1= C_1(\omega_X,n,p,A,B)$ and $C_2 = C_2(\omega_X,n,p,A,B,q)$ such that for every $T\in \mathcal{W}_{\t for every $r\in (0,\mathop{\mathrm{diam}}\nolimits(\widehat{X},d_T)]$ and $x\in \widehat{X}$.

Theorems & Definitions (52)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Theorem 2.1
  • proof
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • ...and 42 more