Relative Kähler–Ricci Flow on Projective-Space Fibrations

Definitions and mathematical statement

The supplied record concerns a projective-space fibration $f:X={\mathbb{P}}(E)\to S$ associated with a rank-$r$ bundle $E$, where $r\geq 1$ and the fibres are ${\mathbb{P}}^{r-1}$. To interpret the notation, use the convention that ${\mathbb{P}}(E)$ parametrizes one-dimensional quotients and $\mathcal O_E(1)$ is its tautological quotient line bundle; put $\mathcal O_E(r)=\mathcal O_E(1)^{\otimes r}$ and $\det E=\bigwedge^r E$. This convention is compatible with the supplied identity.

In a smooth holomorphic interpretation, the relative canonical line bundle is $K_{X/S}=\bigwedge^{r-1}\Omega^1_{X/S}$, and the stated relation is \[ K_{X/S}^{-1}\cong\mathcal O_E(r)\otimes f^*(\det E)^{-1}. \] The source does not explicitly specify the geometric hypotheses on $S$ and $E$ needed for its intended flow; the holomorphic interpretation above explains the notation without resolving those missing hypotheses.

Research problem. Develop a relative Kähler--Ricci flow directly on $\mathcal O_E(r)$, using the displayed relation, in place of a formulation on $K_{X/S}$.

This is a proposed research direction rather than a fully specified existence or convergence assertion. The record supplies no evolution equation, normalization, initial data, time interval, boundary conditions, or precise meaning of a flow "on" the line bundle. It also does not specify how the pullback determinant factor enters the proposed evolution. The comparison with $K_{X/S}$ is retained exactly as supplied, although the displayed identity concerns its inverse.

Short English statement

The question asks for a formulation of relative Kähler--Ricci flow on the degree-$r$ tautological line bundle of a projective bundle, using its relation to the relative anticanonical bundle. The supplied entry describes a research program; its hypotheses and intended flow equation remain unspecified.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30003571$, source problem code catalogue entry, titled "Relative Kähler--Ricci Flow on Projective-Space Fibrations." Collection: Oberwolfach Reports Open Problems. The recorded proposed year is 2017; no individual proposer is supplied.
• Supplied report citation: Komplexe Analysis (2018), Oberwolfach Reports, pp. 2427--2473, DOI 10.4171/owr/2017/39. https://doi.org/10.4171/owr/2017/39. The background attributes the entry to workshop open-problem discussions documented in this report; no exact problem page or named speaker is supplied.
• Supplied literature reference labelled "Berman": https://arxiv.org/abs/1002.3717. No title, theorem number, or precise supporting passage is supplied.
• Supplied literature reference 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 describes it as a proposed relative Kähler--Ricci-flow research direction, stating that special relative flows are understood while the general program remains unresolved. This is supplied dated curation, not an independently verified literature assessment. The record itself requests independent verification of time-sensitive status and cited claims.
• The supplied statement status is corrected_verified; its accompanying explanation concerns preservation of mathematical content while repairing notation, typography, or omitted definitions for publication. This supplied editorial label does not establish mathematical correctness or resolution.