Asymptotics of Intermediate Jet-Bundle Cohomology

Definitions and mathematical statement

The supplied question concerns the expression \[ h^0\!\left(X_k,\mathcal O_{X_k}(m)\otimes\pi_k^*\mathcal O(-A)\right). \] Its notation presupposes spaces $X_k$ and $X$, a morphism $\pi_k:X_k\to X$, and sheaves denoted by $\mathcal O_{X_k}(m)$ and $\mathcal O(-A)$, the latter on the base of $\pi_k$. The supplied record does not define these objects or their hypotheses. In its stated jet-differential context, it does not specify the jet-space construction, the meaning of the index $k$, the twisting convention, or the nature and positivity assumptions on $A$.

Whenever these data are specified and the relevant cohomology groups are finite-dimensional over the intended ground field, write \[ \mathcal L_{k,m}:=\mathcal O_{X_k}(m)\otimes\pi_k^*\mathcal O(-A), \qquad h^q(X_k,\mathcal L_{k,m}):=\dim H^q(X_k,\mathcal L_{k,m}). \] Here $H^q$ denotes sheaf cohomology. In the usual terminology, if $X_k$ has finite dimension $d$, intermediate cohomology refers to degrees $0<q<d$; the supplied record does not specify the degrees intended or the nonvanishing conditions under consideration.

Source question, with unresolved qualifications. Determine the asymptotic behavior of $h^0(X_k,\mathcal L_{k,m})$ in the presence of intermediate cohomology, and determine whether the resulting control can establish the generalized Green--Griffiths--Lang conjecture intended by the source. The notation suggests an asymptotic problem as $m\to+\infty$, but the record does not explicitly specify this limit, whether $k$ and $A$ remain fixed, or any dependence between these parameters. It supplies neither a proposed asymptotic formula or bound nor a precise formulation of the generalized conjecture and the hypotheses under which it is to follow. Consequently, this is a qualified research question, not a fully quantified conjecture; specifying those missing data is necessary before a precise mathematical assertion can be extracted.

Short English statement

The question asks how the dimension of a space of negatively twisted jet-differential sections grows when intermediate cohomology is present, and whether suitable estimates could prove the intended generalized Green--Griffiths--Lang conjecture. The supplied record leaves the geometric setup, asymptotic regime, and target conjecture unspecified.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30001916$, title Asymptotics of Intermediate Jet-Bundle Cohomology, exact source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The record gives proposed year $2011$ and names no proposer.
• Supplied report citation: Complex Algebraic Geometry (2012), Oberwolfach Reports, DOI 10.4171/owr/2011/47, pp. 2669--2733. Supplied source URL: https://doi.org/10.4171/owr/2011/47. The background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions. The supplied citation's year $2012$ differs from the record's proposed year $2011$ and the year appearing in the DOI; the relationship is not explained in the supplied material.
• Supplied literature source, labelled "original OWR report": https://doi.org/10.4171/owr/2011/47.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and describes the required asymptotic control of intermediate cohomology of jet-differential bundles as a central unresolved technical problem in the generalized Green--Griffiths--Lang program. This is supplied dated curation, not an independently verified literature assessment. The background expressly requests independent verification of time-sensitive status and cited claims.
• The registry supplies statement status corrected_verified and reports that its mathematical content was checked against the official Oberwolfach report, with notation, context, and question form normalized without changing the source-level claim. That verification claim is part of the supplied record; no independent source consultation or mathematical verification is asserted here.