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Continuity of the complex Monge-Ampère operator on compact Hermitian manifolds

Le Mau Hai, Nguyen Van Phu, Trinh Tung

Abstract

In this note, we establish several results concerning the continuity (or weak convergence) of the complex Monge-Ampère operator on compact Hermitian manifolds. At the end of this note, we find a weak solution of the complex Monge-Ampère equation on a compact Hermitian manifold under the assumption of the existence of a smooth subsolution.

Continuity of the complex Monge-Ampère operator on compact Hermitian manifolds

Abstract

In this note, we establish several results concerning the continuity (or weak convergence) of the complex Monge-Ampère operator on compact Hermitian manifolds. At the end of this note, we find a weak solution of the complex Monge-Ampère equation on a compact Hermitian manifold under the assumption of the existence of a smooth subsolution.

Paper Structure

This paper contains 6 sections, 12 theorems, 39 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Organization
  4. Preliminairies
  5. Continuity of the complex Monge-Ammpère operator on compact Hermitian manifolds
  6. Applications

Key Result

Proposition 2.5

Let $u\in PSH(\omega)$ be a $\omega$-psh function. Then for each $\varepsilon >0$ there exists an open subset $G\subset X$ with $Cap_{\omega}(G) <\varepsilon$ and $u|_{X\setminus G}$ is continuous.

Theorems & Definitions (25)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Proposition 2.5
  • Theorem 3.1
  • proof
  • Corollary 3.2
  • proof
  • Theorem 3.3
  • ...and 15 more