Hermitian Volume Invariants for Plurisigned Metrics

Definitions and mathematical statement

The supplied record does not define a plurisigned Hermitian metric or the volume invariants $v_+(\omega)$ and $v_-(\omega)$. It also does not specify the underlying complex manifold, its dimension, compactness assumptions, or the hypotheses and normalizations used to define the invariants. These missing data prevent a self-contained mathematical formulation.

Let $\mathcal{P}$ denote, provisionally, the intended class of plurisigned Hermitian metrics in the supplied statement. This notation is a placeholder for an unspecified class, not a definition resolving the missing hypotheses. The question recorded is whether both assertions \[ \begin{aligned} &\text{For every }\omega\in\mathcal{P},\quad v_+(\omega)\text{ is finite},\\ &\text{For every }\omega\in\mathcal{P},\quad v_-(\omega)>0 \end{aligned} \] hold.

There is an additional scope qualification: the original excerpt asks whether these properties hold "always" after referring to unspecified restrictive conditions. It does not explicitly restrict the question to plurisigned metrics, whereas the supplied cleaned statement does. The preceding conditions and the intended scope must be recovered from the source before this schema can be made precise. No uniform bound over metrics, or uniform positive lower bound, is asserted here.

Short English statement

The supplied question asks whether every metric in the intended class has finite upper volume invariant and strictly positive lower volume invariant. The definitions, ambient hypotheses, and intended restriction to plurisigned metrics remain unresolved. The supplied open status is a dated curation claim, not an independently verified conclusion.

Sources

• Local UnsolvedMath registry: permanent ID $30005228$, source problem code catalogue entry, title Hermitian Volume Invariants for Plurisigned Metrics. The record labels the problem open and the statement corrected_verified; its verification note describes repairs to formula delimiters, notation, grammar, and typography, rather than independent mathematical verification.
• Supplied report citation: Complex Geometry and Dynamical Systems (2023), Oberwolfach Reports, DOI 10.4171/owr/2022/40, pp. 2335--2397. https://doi.org/10.4171/owr/2022/40. The record describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions, and records the proposed year as 2022.
• Attribution in the original excerpt: Vincent Guedj, Plurisigned hermitian metrics, joint work with Daniele Angella and Hoang Chinh Lu. The separate proposer field is unspecified.
• Supplied literature reference: Angella--Guedj--Lu, Plurisigned Hermitian metrics. https://www.math.univ-toulouse.fr/~guedj/fichierspdf/TAMS22.pdf.
• Supplied literature reference: Darvas--Di Nezza--Lu, Monge--Ampere volumes. https://arxiv.org/abs/2106.04272.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open, states that finiteness of the positive volume invariant and positivity of the negative one remain open outside restrictive hypotheses, and states that later envelope results do not give the requested general resolution. This is supplied dated curation derived from the OWR record, not independently verified here; its time-sensitive status and cited claims require independent checking.