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Pham Hoang Hiep

Publications and source records attributed to Pham Hoang Hiep.

12 recordsLinked to original sources
arXiv2026

Local-to-global maximality for plurisubharmonic functions with locally analytic singularities

We prove that a locally maximal plurisubharmonic function on a domain in $\bC^n$, $n\geq2$, is maximal if it has locally analytic singularities with a two-sided locally bounded remainder. Using McAdam's theorem, we show that local maximality forces the analytic spread of each defining ideal to be less than $n$. We use normalized blow-ups and Siu's analyticity theorem to compare singularities through generic divisorial Lelong numbers. We also obtain a criterion for maximality in terms of maximality outside the pole set and the analytic spread of the local defining ideals.

math.CV↗
arXiv2026

Comparison principles for local plurisubharmonic potentials on complex manifolds and applications

We prove a weighted comparison principle for local plurisubharmonic potentials in Cegrell's class on an arbitrary complex manifold under suitable boundary and gluing conditions. We obtain domination and uniqueness when the manifold admits a global psh function in $\mathcal E_{\mathrm{loc}}$ whose Monge-Ampère measure has an absolutely continuous part with positive density almost everywhere. We apply these results to plurisubharmonic functions on domains in $\mathbb C^2$ whose singularities satisfy local inequalities involving $α\log\|f\|$ and a Cegrell potential. For such functions, maximality is equivalent to the vanishing of the Monge-Ampère measure of the current remaining after subtraction of the Siu divisorial part. Consequently, maximality is a local property in this class.

math.CV↗
arXiv2025

Singularity invariants of plurisubharmonic functions and complex spaces

In this paper, we combine tools from pluripotential theory and commutative algebra to study singularity invariants of plurisubharmonic functions. We establish several relationships between the singularity invariants of plurisubharmonic functions and those of holomorphic functions. These results yield a sharp lower bound for the log canonical threshold of a plurisubharmonic function. Our bound simultaneously improves upon the main result of Demailly and Pham (Acta Math. 212: 1--9, 2014), the classical result of Skoda (Bull. Soc. Math. France 100: 353--408, 1972), and the lower estimate of T. de Fernex, L. Ein and M. Mustaţǎ (Math. Res. Lett. 10: 219--236, 2003), which has played a crucial role in recent developments in birational geometry. Finally, we explore how singularity invariants associated with plurisubharmonic functions can be extended to complex spaces.

math.CV↗
arXiv2025

On Lelong numbers of plurisubharmonic functions on complex spaces

In this paper, we introduce the notion of strong locally irreducible complex spaces $\widetilde{X}$. Based on this notion we prove the equality $\barν_φ(x)=$ mult$(\widetilde{X},x). ν_φ(x)$ for all $x\in \widetilde{X}$, where $\barν_φ(x)$ is the projective mass of a plurisubharmonic function $φ$ at $x$ and mult$(\widetilde{X},x)$ is the multiplicity of $\widetilde{X}$ at $x$ and $ν_φ(x)$ is Lelong number of $φ$ at $x$. Moreover, we show that the closure of the upper-level sets $\{z\in \widetilde{X}:ν_φ(z)\geq c\}$ of a plurisubharmonic function $φ$ on a strong locally irreducible complex space $\widetilde{X}$ is a subvariety of $\widetilde{X}$ for all $c\geq 0$.

math.CV↗
arXiv2016

Log canonical thresholds and Monge-Ampere masses

In this paper, we prove an inequality for log canonical thresholds and Monge-Ampere masses. The idea of proof is a combination of the Ohsawa-Takegoshi $L^2$-extension theorem and inequalities in [ACKHZ09], [DH14]. It also give an analytic proof for the main result in [DH14] in the case of dimension 2.

math.CV↗
arXiv2014

Monge-Ampere measures on subvarieties

In this article we address the question whether the complex Monge-Ampère equation is solvable for measures with large singular part. We prove that under some conditions there are no solution when the right-hand side is carried by a smooth subvariety in $\CEP{n}$ of dimension $k<n$.

math.CV↗
arXiv2014

The weighted log canonical threshold

In this note, we show how to apply the original $L^2$-extension theorem of Ohsawa and Takegoshi to the standard basis of a multiplier ideal sheaf associated with a plurisubharmonic function. In this way, we are able to reprove the strong openness conjecture and to obtain an effective version of the semicontinuity theorem for weighted log canonical thresholds.

math.CV↗
arXiv2009

On some weighted energy classes of plurisubharmonic functions

In this paper we study the relation between the weighted energy class $\mathcal{E}_χ$ introduced by S. Benelkouchi, V. Guedj and A. Zeriahi recently with the classes $\mathcal{E}$ and $\mathcal{N}$ studied by Cegrell. Moreover, we establish a generalized comparison principle for the operator $\text{M}_χ$ and, as an application, we prove a slight version of existence of solutions of Monge-Ampère type equation in the class $\mathcal{E}_χ(H,Ø)$.

math.CV↗
arXiv2009

Convergence in capacity on compact Kähler manifolds

The aim of this note is to study the convergence in capacity for functions in the class $\mathcal E(X,ω)$. We obtain several stability theorems. Some of these are (optimal) generalizations of results of Xing, while others are new.

math.CV↗
arXiv2008

Monge-Ampere measures on pluripolar sets

In this article we solve the complex Monge-Ampere equation for measures with large singular part. This result generalizes classical results by Demailly, Lelong and Lempert a.o., who considered singular parts carried on discrete sets. By using our result we obtain a generalization of Kolodziej's subsolution theorem. More precisely, we prove that if a non-negative Borel measure is dominated by a complex Monge-Ampere measure, then it is a complex Monge-Ampere measure.

math.CV↗