Analytic Torsion under Holomorphic Degeneration

Definitions and mathematical statement

Let $\pi:X\to S$ be the proper holomorphic degeneration in the supplied record, with distinguished parameter $0\in S$. For $s\in S$, write $X_s=\pi^{-1}(s)$; the central fiber $X_0$ is allowed to be singular. The source asks for the behavior of \[ T(s):=\tau\left(X_s,\omega_{X_s}(\xi_s)\right). \] Here $\tau$ is called analytic torsion in the source. The objects $\xi_s$, the meaning of $\omega_{X_s}(\xi_s)$, and the geometric data defining this torsion are not specified. Under a usual interpretation, on a smooth complex fiber of dimension $n$, the canonical bundle is $\omega_{X_s}=\bigwedge^n T^{*(1,0)}X_s$, and twisting by a holomorphic vector bundle $\xi_s$ would mean $\omega_{X_s}\otimes\xi_s$. This interpretation is conditional and is not an additional hypothesis supplied by the record.

Source-level asymptotic question. For such a proper holomorphic degeneration, determine the asymptotic behavior \[ \tau\left(X_s,\omega_{X_s}(\xi_s)\right) \qquad\text{as }s\to0. \] The source does not specify the parameter space near $0$, the parameters for which the torsion is defined, smoothness of the nearby fibers, metric hypotheses, a normalization of torsion, or the desired form of the asymptotic statement. Consequently, this is an asymptotic research question whose fully quantified analytic formulation remains unresolved from the supplied evidence. The supplied literature assessment says that the question is settled under additional assumptions, but those assumptions and the corresponding results are not stated.

Short English statement

Determine how analytic torsion of the indicated twisted canonical object on the fibers of a proper holomorphic degeneration behaves as the parameter approaches a possibly singular central fiber. The bundle, metrics, regularity hypotheses, torsion convention, and intended asymptotic conclusion require clarification. The supplied dated curation labels the problem partially solved under unspecified additional assumptions.

Sources

• Local UnsolvedMath registry, permanent ID $30001711$, source problem code catalogue entry, title Analytic Torsion under Holomorphic Degeneration; supplied record in the Oberwolfach Reports Open Problems collection. The project identifies data/unsolvedmath/problems.json and catalog/problems.json as local registries; their contents were not independently inspected for this draft.
• Supplied report citation: Geometric Quantization in the Non-compact Setting (2011), Oberwolfach Reports, pp. 425--521, DOI 10.4171/owr/2011/09. Source URL and supplied literature-source link labeled "original Oberwolfach report": https://doi.org/10.4171/owr/2011/09. The record attributes the question to workshop open-problem discussions and supplies no individual proposer or precise location within the report.
• Literature assessment dated exactly 2026-08-21: supplied status "partially solved." The supplied assessment says that the report settles the analytic-torsion degeneration question under additional assumptions and does not give a general resolution for the extracted setup. This is supplied dated curation, not an independently verified literature assessment; its time-sensitive status and cited claims require independent verification.