Open Problems in Pluripotential Theory — Question 30 (Dinew-Ko\l{}odziej)

Definitions and mathematical statement

Let $X$ be a compact complex manifold of complex dimension $n$, and let $\omega$ be its Hermitian $(1,1)$-form; no assumption that $d\omega=0$ is imposed. Write $\operatorname{PSH}(X,\omega)$ for the upper semicontinuous quasi-plurisubharmonic functions $\varphi$ satisfying \[ \omega+dd^c\varphi\geq 0 \] in the sense of currents. Here quasi-plurisubharmonic means locally the sum of a plurisubharmonic function and a smooth function. The normalization of $dd^c$ must be fixed consistently; the supplied record does not specify it. For bounded $\omega$-plurisubharmonic functions, interpret $(\omega+dd^c\varphi)^n$ as the complex Monge--Ampère measure defined using Bedford--Taylor products and the smooth form $\omega$.

Define \[ \operatorname{cap}(X,\omega) :=\inf\left\{ \int_X(\omega+dd^c\varphi)^n \;:\; \varphi\in\operatorname{PSH}(X,\omega),\quad 0\leq\varphi\leq1\ \text{on }X \right\}. \] The pointwise bounds are part of the admissibility condition.

Question (source Question 30). Relate $\operatorname{cap}(X,\omega)$ to the geometry of $(X,\omega)$. In particular, can this quantity be computed in terms of geometric data, and how does it behave under deformations of $(X,\omega)$?

The supplied question is an open-ended research problem: it specifies neither the geometric data sought nor the class or regularity of deformations. In particular, it does not formulate a specific continuity, semicontinuity, or invariance assertion.

Short English statement

For a compact Hermitian manifold, take the infimum of the total complex Monge--Ampère mass over all $\omega$-plurisubharmonic functions with values in $[0,1]$. Describe this quantity geometrically, determine whether geometric data can compute it, and study its behavior under deformations. The source leaves the geometric data, deformation framework, and normalization of $dd^c$ unspecified.

Sources

• Local UnsolvedMath registry, permanent ID $800899$, source problem code catalogue entry; title: Open Problems in Pluripotential Theory --- Question 30 (Dinew--Kołodziej). The supplied registry labels the problem open; this label is supplied curation, not an independently verified assessment.
• Dinew, Guedj, Zeriahi, Open Problems in Pluripotential Theory (2015), Question 30. Supplied source link: https://arxiv.org/abs/1511.00705. The record reports 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 background attributes the question or its context to Dinew--Kołodziej. It gives no separate reference clarifying that attribution.