Open Problems in Pluripotential Theory — Question 10

Definitions and mathematical statement

The supplied excerpt does not explicitly specify the ambient geometry or define its function classes. Under the standard compact Kähler interpretation, let $X$ be a compact Kähler manifold of complex dimension $n\geq 1$, let $\omega$ be a Kähler form, and put $V=\int_X\omega^n$. These ambient hypotheses and the definitions below require confirmation against the complete source.

Fix the convention $dd^c=\frac{i}{\pi}\partial\bar\partial$. Write $\operatorname{PSH}(X,\omega)$ for the upper semicontinuous, locally integrable functions $\varphi:X\to[-\infty,\infty)$ satisfying $\omega+dd^c\varphi\geq 0$ in the sense of currents. For bounded such functions, the measure $(\omega+dd^c\varphi)^n$ is understood in the Bedford--Taylor sense.

The global domain of definition $DMA(X,\omega)$ consists of those $\varphi\in\operatorname{PSH}(X,\omega)$ for which there exists a positive Borel measure $MA(\varphi)$ such that, for every sequence $\varphi_j\in\operatorname{PSH}(X,\omega)\cap L^\infty(X)$ decreasing pointwise to $\varphi$, \[ (\omega+dd^c\varphi_j)^n\rightharpoonup MA(\varphi). \] In particular, the limiting measure must be independent of the approximating sequence.

For $\varphi\in\operatorname{PSH}(X,\omega)$, set $\varphi^{(j)}=\max\{\varphi,-j\}$. The full non-pluripolar Monge--Ampère mass class is \[ \mathcal E(X,\omega)=\left\{\varphi\in\operatorname{PSH}(X,\omega):\lim_{j\to\infty}\int_{\{\varphi>-j\}}(\omega+dd^c\varphi^{(j)})^n=V\right\}. \]

The question attributed in the supplied source to Coman--Guedj--Zeriahi asks whether, for every $\varphi\in DMA(X,\omega)$, \[ MA(\varphi)\left(\{x\in X:\varphi(x)=-\infty\}\right)=0 \quad\Longrightarrow\quad \varphi\in\mathcal E(X,\omega). \] Here the hypothesis concerns only the pole set of $\varphi$; it does not assert that $MA(\varphi)$ vanishes on every pluripolar set. The supplied registry labels the question open; this draft does not independently verify that status.

Short English statement

In the standard compact Kähler setting, suppose the Monge--Ampère measure of a function in the global domain of definition gives no mass to the set where that function equals $-\infty$. Must the function have full non-pluripolar Monge--Ampère mass? The excerpt leaves the ambient hypotheses and class definitions implicit, so the stated interpretation requires source confirmation.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $800879$, exact source code catalogue entry, title "Open Problems in Pluripotential Theory --- Question 10." The registry attributes the problem to Dinew, Guedj, and Zeriahi, gives proposed year 2015, and labels it open. These are supplied metadata, not independently verified conclusions.
• Dinew, Guedj, and Zeriahi, Open Problems in Pluripotential Theory (2015), Question 10: https://arxiv.org/abs/1511.00705. The supplied provenance records access on 2026-10-16 and extraction by arxiv_latex; neither access nor extraction was independently performed for this draft.
• The supplied question carries the attribution "Coman--Guedj--Zeriahi" and cites the reference label [CGZ08]. The excerpt describes the investigation in [CGZ08] as having limited success; this is the supplied survey's assessment. No full bibliographic entry or separate URL for [CGZ08] is supplied.