Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Monge-Ampère equations with prescribed singularities on compact Hermitian manifolds

Omar Alehyane, Chinh H. Lu, Mohammed Salouf

TL;DR

The paper advances pluripotential theory for Monge-Ampère equations on compact Hermitian manifolds by handling prescribed singularities under a semipositive, big reference form $\theta$ and RHS measures that may be non-pluripolar. It introduces the relative full mass class $\mathcal{E}(X,\theta,\phi)$ and proves a domination principle, enabling robust existence and uniqueness results via Perron envelopes and capacity convergence. The results cover RHS with $L^p$ densities and extend to general non-pluripolar measures, providing a complete framework for solving MAEs with prescribed singularities beyond the Kähler setting. These tools pave the way for canonical metrics and singular models on non-Kähler manifolds, with potential applications to complex geometry and mathematical physics.

Abstract

Given a compact complex manifold $X$, we study the existence and the uniqueness of weak solutions to degenerate Monge-Ampère equations on $X$ with prescribed singularities when the reference form is semipositive and big, while the right hand side is a non-pluripolar positive Radon measure. This generalizes our previous work to more general hermitian manifolds and also to the case of solutions with prescribed singularities.

Monge-Ampère equations with prescribed singularities on compact Hermitian manifolds

TL;DR

The paper advances pluripotential theory for Monge-Ampère equations on compact Hermitian manifolds by handling prescribed singularities under a semipositive, big reference form and RHS measures that may be non-pluripolar. It introduces the relative full mass class and proves a domination principle, enabling robust existence and uniqueness results via Perron envelopes and capacity convergence. The results cover RHS with densities and extend to general non-pluripolar measures, providing a complete framework for solving MAEs with prescribed singularities beyond the Kähler setting. These tools pave the way for canonical metrics and singular models on non-Kähler manifolds, with potential applications to complex geometry and mathematical physics.

Abstract

Given a compact complex manifold , we study the existence and the uniqueness of weak solutions to degenerate Monge-Ampère equations on with prescribed singularities when the reference form is semipositive and big, while the right hand side is a non-pluripolar positive Radon measure. This generalizes our previous work to more general hermitian manifolds and also to the case of solutions with prescribed singularities.

Paper Structure

This paper contains 14 sections, 40 theorems, 186 equations.

Table of Contents

  1. Introduction
  2. Non-pluripolar Monge-Ampère measures
  3. Quasi-plurisubharmonic functions
  4. Maximum principle and quasi-psh envelopes
  5. Convergence in capacity
  6. Bounds on Monge-Ampère volumes
  7. Model potentials
  8. The relative full mass class
  9. Definition and properties
  10. The domination principle
  11. Weak convergence of Monge-Ampère measures
  12. Monge-Ampère equations with prescribed singularities
  13. The case of measures with $L^p$-density
  14. The case of non-pluripolar measures

Key Result

Theorem 2.1

Let $u$, $v\in {\rm PSH}(X,\theta)$. We have If in addition $u \leq v$, then

Theorems & Definitions (80)

  • Theorem 2.1
  • Theorem 2.2: Theorem 2.3 in ALS24
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • proof
  • ...and 70 more