Open Problems in Pluripotential Theory — Question 19 (Berman)

Definitions and mathematical statement

The source asks about a smooth totally real submanifold $S\subset X$ of maximal real dimension $n$. Here total reality means that $X$ has complex dimension $n$ and \[ T_pS\cap J(T_pS)=\{0\}\qquad(p\in S), \] where $J$ is the complex structure of $X$.

The supplied excerpt does not specify the ambient hypotheses on $(X,\omega)$, the metric defining Hausdorff measure, or finiteness of that measure. A conditional formulation in the usual compact Kähler setting is as follows: let $(X,\omega)$ be a compact Kähler manifold of complex dimension $n$, with $\int_X\omega^n=1$, and fix the metric used to define $\mathcal H^n$. Suppose $S\subset X$ is a smooth totally real submanifold of real dimension $n$, with $0<\mathcal H^n(S)<\infty$. Define its normalized Hausdorff measure by \[ \mu_S(A)=\frac{\mathcal H^n(A\cap S)}{\mathcal H^n(S)} \] for Borel sets $A\subset X$.

An $\omega$-plurisubharmonic function is an upper semicontinuous, locally integrable function $u$, locally equal to a plurisubharmonic function plus a smooth function, such that $\omega+dd^cu\geq0$ as a current. Write $\mathcal E(X,\omega)$ for the full Monge--Ampère mass class, consisting of such functions with \[ \int_X\left\langle(\omega+dd^cu)^n\right\rangle=\int_X\omega^n. \] The brackets denote the non-pluripolar Monge--Ampère product; the normalization of $dd^c$ must be fixed consistently.

The supplied source asserts that there is a unique $\varphi_S\in\mathcal E(X,\omega)$ satisfying \[ \left\langle(\omega+dd^c\varphi_S)^n\right\rangle=\mu_S, \qquad \sup_X\varphi_S=0. \] Question (Berman, as attributed in the source). Determine the optimal regularity of $\varphi_S$.

This remains an incompletely specified regularity question: the excerpt does not identify a regularity scale, distinguish global regularity from regularity on or away from $S$, or specify compactness, closedness, or boundary conditions for $S$. The compact Kähler and measure hypotheses above are explicit conditional reconstruction assumptions requiring confirmation from the source context. The source attributes non-pluripolarity of $S$ to Sadullaev, reference [Sad], and presents existence and uniqueness as a consequence. Non-pluripolarity of a set means that it is not contained in the $-\infty$ locus of a plurisubharmonic function locally, with the usual local interpretation on $X$; this assertion alone does not state that its Hausdorff measure gives zero mass to every pluripolar subset. The needed measure-level justification is not supplied in the excerpt.

Short English statement

Determine the best regularity of the normalized $\omega$-plurisubharmonic potential whose Monge--Ampère measure is normalized Hausdorff measure on a smooth, maximally totally real submanifold. The source leaves the precise ambient assumptions, boundary conditions, and intended regularity scale unresolved.

Sources

• Local UnsolvedMath registry, supplied complete source record: permanent ID $800888$, exact source code catalogue entry, AMR Open Problem Lists. The registry labels the problem open; this is a supplied status, not an independently verified assessment.
• Dinew, Guedj, and Zeriahi, Open Problems in Pluripotential Theory (2015), Question 19. Supplied source URL: https://arxiv.org/abs/1511.00705. The record lists Dinew, Guedj, and Zeriahi as proposers and attributes the question/context to Berman.
• Supplied provenance: extraction method arxiv_latex; access date 2026-10-16. These are supplied provenance claims; no source visit or independent verification is asserted here.
• Sadullaev, reference [Sad] in the supplied excerpt, cited there for non-pluripolarity of $S$. No title, publication details, or URL for [Sad] is supplied.