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Singularities vs non-pluripolar Monge--Ampère masses

Quang-Tuan Dang, Hoang-Son Do, Hoang Hiep Pham

TL;DR

The paper addresses how singularities of $\theta$-psh potentials influence the non-pluripolar Monge–Ampère mass $\int_X \theta_u^n$, establishing that masses decrease when singularity type becomes stronger in capacity. It develops envelope techniques and capacity arguments to prove a monotonicity result, including a short alternative proof of Witt–Nyström’s monotonicity and extends results to big cohomology classes. Key contributions include a characterization of membership in $\mathcal{E}(X,\theta,\phi)$ via singularity data and envelopes, a mass-invariance result for potentials of the same singularity type, and corollaries linking envelopes, Lelong numbers, and singularity types. The findings advance the understanding of how singularities govern non-pluripolar MA masses, with implications for variational approaches and complex geometric problems on Kähler manifolds.

Abstract

The aim of this paper is to compare singularities of closed positive currents whose non-pluripolar complex Monge--Ampère masses equal. We also provide a short alternative proof for the monotonicity of non-pluripolar complex Monge--Ampère masses, generalizing results of Witt-Nyström, Darvas--Di Nezza--Lu, Lu--Nguyên and Vu.

Singularities vs non-pluripolar Monge--Ampère masses

TL;DR

The paper addresses how singularities of -psh potentials influence the non-pluripolar Monge–Ampère mass , establishing that masses decrease when singularity type becomes stronger in capacity. It develops envelope techniques and capacity arguments to prove a monotonicity result, including a short alternative proof of Witt–Nyström’s monotonicity and extends results to big cohomology classes. Key contributions include a characterization of membership in via singularity data and envelopes, a mass-invariance result for potentials of the same singularity type, and corollaries linking envelopes, Lelong numbers, and singularity types. The findings advance the understanding of how singularities govern non-pluripolar MA masses, with implications for variational approaches and complex geometric problems on Kähler manifolds.

Abstract

The aim of this paper is to compare singularities of closed positive currents whose non-pluripolar complex Monge--Ampère masses equal. We also provide a short alternative proof for the monotonicity of non-pluripolar complex Monge--Ampère masses, generalizing results of Witt-Nyström, Darvas--Di Nezza--Lu, Lu--Nguyên and Vu.

Paper Structure

This paper contains 5 sections, 12 theorems, 38 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Non-pluripolar products
  4. Quasi-envelopes
  5. Proof of main theorems

Key Result

Theorem 1.1

Let $(X, \omega)$ be a compact Kähler manifold and $\theta$ a smooth closed real (1, 1)-form on $X$ whose cohomology class is big. Let $\phi\in{\rm PSH}(X,\theta)$. If $\varphi,\psi\in\mathcal{E}(X,\theta,\phi)$, then for each $c>0$, there exists a function $h\in\mathcal{E}(X,\omega)$ such that Conversely, if $\varphi,\psi\in{\rm PSH}(X,\theta)$ such that $|\varphi-\psi|\leq -ch$ for some $c>0$ a

Theorems & Definitions (26)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • Lemma 2.2
  • Definition 2.3
  • Theorem 2.4
  • proof
  • Proposition 2.5
  • proof
  • Corollary 2.6
  • ...and 16 more