Intrinsic $L^2$ Extension Beyond Nilpotent Sections

Definitions and mathematical statement

Let $X$ denote the complex analytic ambient space implicit in the supplied record, and let $\mathcal O_X$ denote its structure sheaf. The record gives a multiplier-ideal filtration $\mathcal I(m\psi)$, with jumping parameters \[ 0=m_0<m_1<m_2<\cdots. \] Here jumping parameters refer to parameter values at which the multiplier ideal changes; the record does not specify the hypotheses on $X$ or $\psi$, or the precise parameter convention for the filtration.

For a chosen index $p\geq 1$, set \[ A=\mathcal I(m_{p-1}\psi),\qquad B=\mathcal I(m_p\psi),\qquad Q=A/B. \] The decreasing filtration gives $B\subseteq A$. Define the ideal quotient \[ J=B:A, \qquad J_x=\{g\in\mathcal O_{X,x}:gA_x\subseteq B_x\} \quad (x\in X), \] and write $Y=V(J)$, as in the supplied statement. Thus $J$ annihilates $Q$. The record does not specify whether $Y$ is to carry its reduced structure or the possibly nonreduced structure defined by $J$.

The requested problem is to determine whether, for sections $f$ on $Y$, there is a reasonably intrinsic $L^2$ norm and a corresponding intrinsic $L^2$ extension estimate, without imposing the source's requirement that $f$ be nilpotent and associated with \[ \mathcal I(m_{p-1}\psi)/\mathcal I(m_p\psi). \] Both the intrinsic norm and the extension estimate are requested objects. The supplied record does not define the coefficient sheaf of $f$, the meaning of nilpotence for such sections, the precise meaning of being associated with this quotient, or the ambient extension problem. In particular, it gives no extension target, metrics, measures, curvature assumptions, admissible class of sections, or bound with specified constants. Consequently this is a qualified formulation of the supplied question; a fully quantified analytic inequality cannot be recovered from this record alone.

Short English statement

For the space defined by the ideal quotient at two consecutive multiplier-ideal jumping numbers, seek an intrinsic $L^2$ norm and an extension estimate for sections after removing the stated nilpotence and previous-jump quotient requirements. The ambient hypotheses, section space, and meaning of the estimate require clarification.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30003574$, exact source problem code catalogue entry, title: Intrinsic $L^2$ Extension Beyond Nilpotent Sections. Collection: Oberwolfach Reports Open Problems. No individual proposer is supplied; the proposed year is recorded as 2017.
• Supplied report citation: Komplexe Analysis (2018), Oberwolfach Reports, DOI 10.4171/owr/2017/39, pp. 2427--2473. https://doi.org/10.4171/owr/2017/39. The supplied background describes these reports as records of mathematical workshops and their open-problem discussions.
• Supplied literature source labelled Demailly: https://www-fourier.ujf-grenoble.fr/~demailly/source_files/holom_extension/holom_extension_ltx1.pdf. No bibliographic title or pinpoint reference is supplied.
• Supplied literature source labelled OWR report: https://doi.org/10.4171/owr/2017/39.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and states that the intrinsic estimate without the nilpotent-section hypothesis is explicitly open, while available extension results cover important special cases but not the general formulation. This is supplied dated curation derived from the OWR material, not an independently verified literature assessment; its status and cited claims require independent verification.
• The supplied registry labels the statement corrected_verified and reports that the official report was checked to restore the jumping-number setup, ideal quotient, space $Y$, and the two requested objects. This records the supplied verification claim only; no source was independently visited for this draft.