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Hoang Hiep Pham

Publications and source records attributed to Hoang Hiep Pham.

6 recordsLinked to original sources
arXiv2026

Comparison principles for Monge-Ampère measures on pluripolar sets

In this paper, we introduce a notion of singularity comparison for plurisubharmonic functions based on the Bedford--Taylor capacity. We establish comparison principles for the complex Monge--Ampère operator on pluripolar sets in the Cegrell classes. As applications, we obtain a characterization of this relation via auxiliary functions in the energy class and prove a corresponding uniqueness result for the Monge--Ampère equation.

math.CV↗
arXiv2025

Singularities vs non-pluripolar Monge--Ampère masses

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.

math.CV↗
arXiv2018

Complex Monge-Ampère equation in strictly pseudoconvex domains

We study the complex Monge-Ampère equation $(dd^c u)^n=μ$ in a strictly pseudoconvex domain $Ω$ with the boundary condition $u=φ$, where $φ\in C(\partialΩ)$. We provide a non-trivial sufficient condition for continuity of the solution $u$ outside "small sets".

math.CV↗
arXiv2014

A sharp lower bound for the log canonical threshold

In this note, we prove a sharp lower bound for the log canonical threshold of a plurisubharmonic function $φ$ with an isolated singularity at $0$ in an open subset of ${\mathbb C}^n$. This threshold is defined as the supremum of constants $c>0$ such that $e^{-2cφ}$ is integrable on a neighborhood of $0$. We relate $c(φ)$ with the intermediate multiplicity numbers $e_j(φ)$, defined as the Lelong numbers of $(dd^cφ)^j$ at $0$ (so that in particular $e_0(φ)=1$). Our main result is that $c(φ)\ge\sum e_j(φ)/e_{j+1}(φ)$, $0\le j\le n-1$. This inequality is shown to be sharp; it simultaneously improves the classical result $c(φ)\ge 1/e_1(φ)$ due to Skoda, as well as the lower estimate $c(φ)\ge n/e_n(φ)^{1/n}$ which has received crucial applications to birational geometry in recent years. The proof consists in a reduction to the toric case, i.e. singularities arising from monomial ideals.

math.CV↗
arXiv2011

Hölder continuous solutions to Monge-Ampère equations

Let $(X,ω)$ be a compact Kähler manifold. We obtain uniform Hölder regularity for solutions to the complex Monge-Ampère equation on $X$ with $L^p$ right hand side, $p>1$. The same regularity is furthermore proved on the ample locus in any big cohomology class. We also study the range $\MAH(X,ω)$ of the complex Monge-Ampère operator acting on $ω$-plurisubharmonic Hölder continuous functions. We show that this set is convex, by sharpening Kołodziej's result that measures with $L^p$-density belong to $\MAH(X,ω)$ and proving that $\MAH(X,ω)$ has the "$L^p$-property", $p>1$. We also describe accurately the symmetric measures it contains.

math.CV↗