Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Degenerate complex Monge-Ampère equations on some compact Hermitian manifolds

Mohammed Salouf

Abstract

Let $X$ be a compact complex manifold which admits a hermitian metric satisfying a curvature condition introduced by Guan-Li. Given a semipositive form $θ$ with positive volume, we define the Monge-Ampère operator for unbounded $θ$-psh functions and prove that it is continuous with respect to convergence in capacity. We then develop pluripotential tools to study degenerate complex Monge-Ampère equations in this context, extending recent results of Tosatti-Weinkove, Kolodziej-Nguyen, Guedj-Lu and many others who treat bounded solutions.

Degenerate complex Monge-Ampère equations on some compact Hermitian manifolds

Abstract

Let be a compact complex manifold which admits a hermitian metric satisfying a curvature condition introduced by Guan-Li. Given a semipositive form with positive volume, we define the Monge-Ampère operator for unbounded -psh functions and prove that it is continuous with respect to convergence in capacity. We then develop pluripotential tools to study degenerate complex Monge-Ampère equations in this context, extending recent results of Tosatti-Weinkove, Kolodziej-Nguyen, Guedj-Lu and many others who treat bounded solutions.
Paper Structure (8 sections, 26 theorems, 109 equations)

This paper contains 8 sections, 26 theorems, 109 equations.

Table of Contents

  1. Introduction
  2. Non-pluripolar Monge-Ampère measures
  3. Definitions
  4. Quasi-plurisubharmonic envelopes
  5. The full mass class
  6. The domination principle
  7. Continuity of non-pluripolar Monge-Ampère measures
  8. Weak solutions to Monge-Ampère equations

Key Result

Theorem 1.1

Assume $u_j,u \in \mathcal{E}(X,\theta)$ and $u_j\to u$ in $L^1(X,\omega^n)$.

Theorems & Definitions (49)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 2.1
  • proof
  • Corollary 2.2
  • proof
  • Theorem 2.3
  • proof
  • Corollary 2.4
  • ...and 39 more