$L^2$ Eta Forms Without Determinant-Class Fibers

Definitions and mathematical statement

Write the family of normal coverings schematically as $\widetilde M_b\to M_b$, indexed by $b\in B$, with deck transformation group $\Gamma$. Let $A_t$, for $t>0$, denote the rescaled Bismut superconnection associated with the family of signature operators. Using the source's conventions for the operator $\tau$ and the $\Gamma$-trace, set \[ \omega_t:=\operatorname{tr}_{\Gamma}\!\left(\tau\,\frac{dA_t}{dt}e^{-A_t^2}\right),\qquad t\geq1. \] This is the differential-form-valued expression whose large-time integrability is in question.

Proposed extension, subject to unresolved source qualifications. For every family satisfying all the other hypotheses of the report's integrability theorem for the $L^2$ eta form, does $\omega_t$ remain integrable on $[1,\infty)$ after removing the assumption that the fibers are of determinant class, so that the $L^2$ eta form is well-defined and continuous? The associated large-time contribution is represented formally by \[ \int_1^{\infty}\omega_t\,{\,\mathrm{d}} t. \] No determinant-class hypothesis is to be imposed in this proposed extension.

The supplied record does not reproduce the theorem's remaining geometric and analytic hypotheses, its definition of determinant class, the conventions for $A_t$, $\tau$, and the trace, or the normalization of the eta form. It also does not specify the sense of integrability of differential forms, whether absolute or locally uniform convergence is required, or the topology and parameter dependence intended by continuity. Consequently, the displayed expression identifies the requested extension but does not determine a complete, independently usable quantified theorem.

The original registry wording instead asks to remove determinant class from a large-time limit theorem. The supplied verification note states that the report already gives that limit without this hypothesis and that a subsequent conjectural reference contains a numbering slip referring to the determinant-class-dependent integrability theorem. The formulation above follows that supplied correction; the correction has not been independently verified here.

Short English statement

The question asks whether the large-time integrability needed for a continuous $L^2$ eta form persists for families of signature operators on normal coverings when the determinant-class assumption on the fibers is removed. The supplied correction distinguishes this question from a large-time limit theorem. The remaining hypotheses and the precise meanings of integrability and continuity require source review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30001500$, exact source problem code catalogue entry, title "$L^2$ Eta Forms Without Determinant-Class Fibers," in the Oberwolfach Reports Open Problems collection. The record gives proposed year 2010 and identifies no individual proposer.
• Supplied report citation: Analysis and Geometric Singularities (2010), Oberwolfach Reports, DOI 10.4171/owr/2010/28, pp. 1625--1690. Source URL and supplied literature source labeled "original OWR Report 28/2010": https://doi.org/10.4171/owr/2010/28. No specific contribution, theorem text, or pinpoint page is supplied.
• Literature assessment dated 2026-08-21: the supplied curation labels the problem open and reports that a general proof of the proposed extension of Theorem 2 was not identified. This is supplied dated curation, not an independently verified literature assessment; the record explicitly requests independent verification of time-sensitive status and cited claims.
• The record labels its statement corrected_verified. Its supplied verification note says that the official report already states the large-time limit without determinant class and interprets the subsequent conjectural reference as a numbering slip pointing to the integrability theorem for the $L^2$ eta form. This source interpretation is supplied provenance and has not been independently checked here.