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Convergence in $(ω,m)$-capacity in the class $\mathcal{E}(X, ω,m)$ on compact Kähler manifolds

Le Mau Hai, Nguyen Van Phu

Abstract

In this paper, we establish the weak*-convergence of a sequence of the complex Hessian measures $H_m(u_j)$ to the complex Hessian measure $H_m(u)$ in the class $\mathcal{E}(X,ω,m)$ under hypotheses that $u_j$ is convergent to $u$ in $(ω,m)$-capacity.

Convergence in $(ω,m)$-capacity in the class $\mathcal{E}(X, ω,m)$ on compact Kähler manifolds

Abstract

In this paper, we establish the weak*-convergence of a sequence of the complex Hessian measures to the complex Hessian measure in the class under hypotheses that is convergent to in -capacity.

Paper Structure

This paper contains 6 sections, 18 theorems, 93 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $m$-subharmonic functions
  4. $(\omega,m)$-subharmonic functions
  5. $(\omega,m)$-capacity
  6. Convergence in $(\omega,m)$-capacity of the class $\mathcal{E}(X, \omega,m)$

Key Result

Theorem 1.1

Let $u_j,u\in\mathcal{E}(X,\omega, m)$. Assume that $H_m({u_j})\leq \mu$ and $H_m(u)\leq \mu$ where $\mu$ is a positive measure satisfying $\mu(X)=\int_{X}\omega^n=1$ and $\mu\ll {\rm Cap}_{\omega,m}$. Let also $\sup\limits_Xu_j=\sup\limits_Xu=0$. Then the following three statements are equivalent:

Theorems & Definitions (33)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Theorem 2.6
  • Definition 2.7
  • Definition 2.8
  • Proposition 2.9
  • ...and 23 more