Open Problems in Pluripotential Theory — Question 7

Definitions and mathematical statement

The supplied record attributes to Demailly the question \[ (dd^c\varphi_j)^n(0)\longrightarrow(dd^c\varphi)^n(0) \qquad (j\to\infty). \] Here the approximation sequence $(\varphi_j)$, the function $\varphi$, their domain, the hypotheses on their singularities, the normalization of $d^c$, and the meaning and existence of the displayed Monge--Ampère quantities are not defined in the supplied excerpt.

A conditional interpretation of the question can be made precise as follows. Let $n\geq1$, let $U\subseteq{\mathbb{C}}^n$ be an open neighborhood of the origin, and suppose that the intended functions and Monge--Ampère construction yield positive Borel measures \[ \mu_j=(dd^c\varphi_j)^n,\qquad \mu=(dd^c\varphi)^n \] on $U$, each finite on compact subsets of $U$. Define the atomic mass of such a measure $\nu$ at the origin by $\nu(\{0\})$. If the source's notation $(dd^c\psi)^n(0)$ denotes this atomic mass, the question is whether the intended approximation satisfies \[ \lim_{j\to\infty}\mu_j(\{0\})=\mu(\{0\}). \] This is only a conditional reading: the supplied excerpt does not establish these hypotheses, specify the approximation, identify the Monge--Ampère construction, or confirm the atomic-mass interpretation. Consequently, a fully quantified statement for the intended class of functions cannot be recovered from this record alone.

Short English statement

Demailly asks whether the Monge--Ampère masses at the origin of the intended approximants $\varphi_j$ converge to that of $\varphi$. The excerpt omits the approximation construction and analytic hypotheses, and does not define the point-mass notation. The displayed precise version is therefore conditional and requires human mathematical review.

Sources

• Local UnsolvedMath registry, permanent ID $800876$, source code catalogue entry, title Open Problems in Pluripotential Theory --- Question 7; supplied set: AMR Open Problem Lists. The registry labels this problem open; this label is supplied source data, not an independently verified assessment.
• Dinew, Guedj, Zeriahi, Open Problems in Pluripotential Theory (2015), Question 7. Supplied source URL: https://arxiv.org/abs/1511.00705. The supplied provenance records access on 2026-10-16 and extraction method arxiv_latex; these are supplied provenance statements, not a claim of independent consultation.
• The supplied question is explicitly attributed to Demailly. Its introductory sentence attributes to Demailly in reference [Dem93] the observation that the Ohsawa--Takegoshi theorem gives a very accurate approximation of $\varphi$ by the $\varphi_j$. The record supplies neither a definition of this approximation nor bibliographic details for [Dem93]; the observation is retained as source attribution without independent verification.