Open Problems in Pluripotential Theory — Question 34

Definitions and mathematical statement

For notation, on a complex manifold $X$ with a smooth real $(1,1)$-form $\omega$, an $\omega$-plurisubharmonic function is an upper semicontinuous, locally integrable function $\varphi:X\to[-\infty,\infty)$ satisfying \[ \omega+dd^c\varphi\geq 0 \] in the sense of currents, where $dd^c=\frac{i}{\pi}\partial\bar\partial$. For such a function, its Lelong number at $x$, normalized so that $\log\lVert z\rVert$ has Lelong number one at the origin, is \[ \nu(\varphi,x)=\liminf_{\substack{z\to x\\z\ne x}}\frac{\varphi(z)}{\log\lVert z-x\rVert} \] in local coordinates centered at $x$. Equivalently, one may first add a smooth local function making $\varphi$ plurisubharmonic; this does not change the Lelong number.

The supplied excerpt does not state the ambient hypotheses on $X$ and $\omega$, define $\mathcal E(X,\omega)$, or specify the meaning of finite energy. The notation above explains the local terminology; it does not supply those missing hypotheses. Retain $\mathcal E(X,\omega)$ as the source-designated class and finite energy as the source-designated condition.

Question 34, attributed in the source to Berman--Boucksom--Eyssidieux--Guedj--Zeriahi. Under the ambient hypotheses and energy convention intended by the source, is the following assertion true? \[ \forall\varphi\in\mathcal E(X,\omega),\qquad \left[\varphi\text{ is }\omega\text{-plurisubharmonic and has finite energy}\right] \Longrightarrow \left[\forall x\in X,\ \nu(\varphi,x)=0\right]. \] A fully specified mathematical formulation requires recovery of the omitted ambient hypotheses and energy definitions. In particular, the conclusion concerns every point of $X$, with no exceptional locus stated in the supplied excerpt.

Short English statement

Question 34 asks whether every finite-energy $\omega$-plurisubharmonic function in the source's class $\mathcal E(X,\omega)$ has zero Lelong number at every point of $X$. The ambient assumptions and the energy convention are missing from the supplied excerpt, so this is a conditional statement-only draft requiring mathematical review.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $800903$, exact source problem code catalogue entry, title Open Problems in Pluripotential Theory --- Question 34. The registry lists Dinew, Guedj, and Zeriahi as proposers, with proposed year 2015.
• Dinew, Guedj, and Zeriahi, Open Problems in Pluripotential Theory (2015), Question 34. Supplied source URL: https://arxiv.org/abs/1511.00705. The supplied background records access on 2026-10-16 and extraction by arxiv_latex; these are supplied provenance details, not a claim of independent source access or verification.
• The supplied question explicitly attributes the problem to Berman--Boucksom--Eyssidieux--Guedj--Zeriahi. No separate bibliographic reference for that attribution is supplied.
• The local registry labels the problem open. This is a supplied status label, not an independently verified literature assessment.