Open Problems in Pluripotential Theory — Question 8

Definitions and mathematical statement

Let $X={\mathbb{C}}\mathbb{P}^{n}$, with $n\geq 1$, and let $\omega$ be its Fubini--Study Kähler form. An $\omega$-plurisubharmonic function is an upper semicontinuous, locally integrable function $u:X\to[-\infty,\infty)$ for which $\omega+dd^{c}u$ is a positive current. Here $dd^{c}$ and the scale of $\omega$ are to follow the source conventions, which the supplied excerpt does not specify.

The global domain of definition $DMA(X,\omega)$ consists of those $\omega$-plurisubharmonic functions $u$ for which every decreasing sequence of bounded $\omega$-plurisubharmonic functions $u_j\downarrow u$ has Bedford--Taylor measures $(\omega+dd^{c}u_j)^n$ converging weakly to a positive measure independent of the approximating sequence. Denote that measure by $(\omega+dd^{c}u)^n$. Write $\delta_a$ for the unit Dirac measure at $a$, and $\nu(u,a)$ for the Lelong number at $a$ of a local plurisubharmonic potential for $\omega+dd^{c}u$; adding a smooth local potential does not affect this number.

Question (attributed in the source to Coman--Guedj). Can one find a point $a\in X$ and a function $G_a\in DMA(X,\omega)$ such that \[ (\omega+dd^{c}G_a)^n =\left(\int_X\omega^n\right)\delta_a, \qquad \nu(G_a,a)\in{\mathbb{R}}\setminus{\mathbb{Q}}? \] Thus the entire Monge--Ampère mass must be concentrated at the single point $a$. The volume factor retains the unspecified normalization of the Fubini--Study form. The excerpt does not state whether examples are sought in some dimension, in every dimension, or for a particular fixed dimension; $n$ is therefore left as a dimension parameter. The normalization of Lelong numbers must also be checked against the original source before interpreting the numerical irrationality condition.

Short English statement

The question attributed to Coman--Guedj asks for an $\omega$-plurisubharmonic function on complex projective space, in the global domain of definition of the Monge--Ampère operator, whose entire Monge--Ampère measure is supported at one point and whose Lelong number there is irrational. The dimension quantifier and numerical normalizations require source confirmation. The registry labels the problem open; that label is supplied metadata, not an independently verified assessment.

Sources

• Local UnsolvedMath registry: permanent ID $800877$, exact source problem code catalogue entry, title Open Problems in Pluripotential Theory --- Question 8. The supplied record lists Dinew, Guedj, Zeriahi as proposers and gives the year 2015.
• Dinew, Guedj, Zeriahi, Open Problems in Pluripotential Theory (2015), Question 8: https://arxiv.org/abs/1511.00705. The question itself carries the attribution [Coman--Guedj] in the supplied excerpt.
• Supplied provenance: accessed 2026-10-16; extraction method arxiv_latex; upstream revision d371ee917ac9647fbb06cd6fc529deec6f3dda82. These are supplied provenance details; no source access or literature verification was performed for this draft.