Open Problems in Pluripotential Theory — Question 36

Definitions and mathematical statement

The supplied question refers to an ambient space $\mathcal X$, a distinguished subspace $X_0$, a line bundle $\mathcal L$ on $\mathcal X$, and prescribed metric weight data $\varphi_0$ on $\mathcal L|_{X_0}$. The record does not supply the hypotheses on these objects or the regularity and positivity assumptions on $\varphi_0$. Consequently, the intended class of data over which the question ranges is unresolved.

Extension question (attributed to Păun). For the intended data $(\mathcal X,X_0,\mathcal L,\varphi_0)$, does there exist a Hermitian metric $h=e^{-\varphi}$ on $\mathcal L$ such that \[ dd^c\varphi\geq 0, \qquad \varphi|_{X_0}=\varphi_0? \] Here $\varphi$ denotes metric weights in local trivializations, and the restriction equality is understood in compatible trivializations on $X_0$. The displayed inequality is the source's condition for a semi-positive extension. The source excerpt does not specify whether singular Hermitian metrics are allowed, how restrictions of such weights are defined, or whether the inequality is interpreted for smooth forms or currents. These qualifications must be recovered before this can be treated as a fully specified mathematical statement.

Short English statement

The question asks whether prescribed metric data on $X_0$ extend to a Hermitian metric on $\mathcal L$ over $\mathcal X$ with semi-positive curvature and exactly the prescribed restriction. The supplied excerpt omits the geometric hypotheses and the permitted metric regularity.

Sources

• Local UnsolvedMath registry, permanent ID $800905$, source problem code catalogue entry, titled Open Problems in Pluripotential Theory --- Question 36. The registry labels the problem open; this is supplied source metadata, not an independently verified status assessment.
• Dinew, Guedj, Zeriahi, Open Problems in Pluripotential Theory (2015), Question 36. Supplied source URL: https://arxiv.org/abs/1511.00705. The question itself is attributed to Păun in the supplied statement.
• Supplied provenance records access on 2026-10-16 and extraction by arxiv_latex. These are supplied provenance claims; the source was not independently accessed or verified for this draft.