Convex Exhaustions of Curve Moduli Spaces

Definitions and mathematical statement

Under the usual complex-analytic interpretation, $\mathcal M_g$ denotes the moduli orbifold of smooth, connected, projective complex curves of genus $g$. This interpretation is conventional; the supplied record does not explicitly specify the ground field or the allowed values of $g$.

A function $f:\mathcal M_g\to{\mathbb{R}}$ is orbifold-smooth if its lift to each orbifold chart is a smooth function invariant under the chart's local group. It is proper if $f^{-1}(K)$ is compact for every compact subset $K\subseteq{\mathbb{R}}$, and bounded below if \[ \exists B\in{\mathbb{R}}\ \forall x\in\mathcal M_g,\qquad f(x)\ge B. \] Together, properness and boundedness below imply that every sublevel set $\{x\in\mathcal M_g:f(x)\le c\}$, for $c\in{\mathbb{R}}$, is compact.

The supplied question is whether there exists such a function that is also $(g-2)$-convex: \[ \exists f:\mathcal M_g\longrightarrow{\mathbb{R}}\quad \text{proper, bounded below, orbifold-smooth, and $(g-2)$-convex}. \] The meaning of $(g-2)$-convex is unresolved in the supplied record. For example, one standard complex-analytic convention calls a smooth function on a complex orbifold of dimension $n$ $q$-convex when its Levi form has at least $n-q+1$ strictly positive eigenvalues at every point, computed in local orbifold charts. The record does not establish that this is the intended convention, whether the condition is required everywhere or only outside a compact subset, or how boundary values of $g-2$ are to be treated. Consequently, the displayed question preserves the source target but requires a specified convexity convention, genus range, and quantification over $g$ before it becomes a fully determined mathematical statement.

Short English statement

The question asks whether the curve moduli orbifold $\mathcal M_g$ has a smooth exhaustion function with the specified $(g-2)$-convexity property. The supplied record leaves the genus range and precise convexity condition unspecified. Its open status and comparison with cohomological dimension bounds are supplied dated assessments, not independently verified conclusions.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30003302$, title Convex Exhaustions of Curve Moduli Spaces, Oberwolfach Reports Open Problems collection; exact source problem code catalogue entry. The record lists proposed year 2016 and supplies no named proposer.
• Supplied report citation: Surface Bundles (2017), Oberwolfach Reports, pp. 3149--3195, DOI 10.4171/owr/2016/56. https://doi.org/10.4171/owr/2016/56. The supplied background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature source label: OWR 56 (2017). https://doi.org/10.4171/owr/2016/56.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and describes the requested proper, bounded-below $(g-2)$-convex exhaustion as an unverified strengthening of known cohomological dimension bounds. This is supplied dated literature triage derived from the OWR curation, not an independently verified assessment; time-sensitive status and cited claims require independent verification.
• The registry supplies statement status corrected_verified and reports a manual semantic, typographic, and LaTeX review. These are source metadata and do not resolve the formulation issues identified here or constitute mathematical review of this draft.