Open Problems in Pluripotential Theory — Question 28 (Dinew-Ko\l{}odziej)

Definitions and mathematical statement

Let $X$ be a compact complex manifold of complex dimension $n\geq 3$, and let $\omega$ be a smooth, strictly positive, real $(1,1)$-form on $X$, defining a Hermitian metric. No assumption that $d\omega=0$ is imposed. Fix the convention $dd^c=i\partial\bar\partial$; the supplied source does not specify a normalization.

A function $\varphi:X\to{\mathbb{R}}$ is bounded and $\omega$-plurisubharmonic if it is bounded, is locally the sum of a plurisubharmonic function and a smooth real-valued function, and satisfies \[ \omega+dd^c\varphi\geq 0 \] as a current. For such functions, interpret $(\omega+dd^c\varphi)^n$ as the complex Monge--Ampère measure defined using the Bedford--Taylor products for locally bounded potentials and the smooth form $\omega$.

For each compact complex manifold $X$ of complex dimension $n\geq 3$, characterize the Hermitian forms $\omega$ for which \[ \forall\,\varphi\in\operatorname{PSH}(X,\omega)\cap L^\infty(X),\qquad \int_X(\omega+dd^c\varphi)^n=\int_X\omega^n. \] Here $\operatorname{PSH}(X,\omega)\cap L^\infty(X)$ denotes the bounded $\omega$-plurisubharmonic functions just defined. The requested characterization is a necessary and sufficient condition on $\omega$. The normalization of $dd^c$ and the interpretation of the measure are explicit conventions for this draft; they are not specified in the supplied statement.

Short English statement

On a compact Hermitian manifold of complex dimension at least three, characterize the Hermitian forms whose total Monge--Ampère mass is unchanged by every bounded $\omega$-plurisubharmonic perturbation. The local registry labels this question open; that label is supplied source data and has not been independently verified.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $800897$, source problem code catalogue entry. The record lists Dinew, Guedj, and Zeriahi as proposers, with proposed year 2015, and labels the problem open.
• Dinew, Guedj, and Zeriahi, Open Problems in Pluripotential Theory (2015), Question 28. Supplied source URL: https://arxiv.org/abs/1511.00705. The supplied attribution/context is Dinew--Kołodziej.
• The source record reports access on 2026-10-16 and extraction by arxiv_latex. These are supplied provenance notes, not a claim of independent source access or verification. Supplied upstream revision: d371ee917ac9647fbb06cd6fc529deec6f3dda82.