Higher Fano Manifolds as Higher Rational-Connectedness Conditions

Definitions and mathematical statement

The original question concerns an auxiliary condition $F_k$ on a variety $X$: \[ \text{Can the condition }F_k(X)\text{ be taken to mean that }X\text{ is }k\text{-Fano}? \] The supplied excerpt does not define $F_k$, the convention for $k$-Fano, the ground field, or the range of $k$. Consequently, this question cannot be converted into a fully specified mathematical assertion from the supplied evidence alone.

For clarity about the terminology in the supplied reconstruction, over a specified field $K$, a rational $k$-fold means an integral variety of dimension $k$ birational over $K$ to ${\mathbb{P}}^k_K$. Rationality after extension to an algebraic closure is a different condition; the record does not specify which is intended. If $f\colon Y\to B$ is a morphism with integral base $B$, a rational section is a rational map $s\colon B\dashrightarrow Y$, defined on a nonempty open subset $U\subseteq B$, such that $f\circ s|_U=\operatorname{id}_U$. These definitions explain terminology without imposing an unstated ground field or base hypothesis on the source question.

The registry reconstructs the intended consequences of the proposed choice of $F_k$ as the following two requests, for every relevant $k$:

1. Every $k$-Fano manifold $X$ is covered by rational subvarieties of dimension $k$.
2. A proper family $f\colon Y\to B$ over a base of dimension $k$, whose fibers are $k$-Fano, admits a rational section, subject to a Brauer obstruction.

These are qualified targets from the supplied reconstruction, not fully specified conjectures. The meaning of covered, the permitted dimensions of $X$, the hypotheses on the base and family, and whether the fiber condition applies to all fibers, geometric fibers, or a generic fiber are absent. In particular, no Brauer class, group containing it, or criterion expressing the qualification is supplied. Thus the phrase subject to a Brauer obstruction cannot yet be replaced by a precise hypothesis implying existence of a section. The defining properties required of $F_k$ must also be recovered before one can determine whether the two reconstructed consequences exhaust the original question.

Short English statement

The source asks whether being $k$-Fano can serve as an auxiliary higher rational-connectedness condition $F_k$. The registry interprets this as asking for coverage by rational $k$-folds and rational sections of proper families over $k$-dimensional bases, with a Brauer qualification. Missing definitions and hypotheses prevent a complete precise formulation; both the original question and the qualified reconstruction are retained for review.

Sources

• Local UnsolvedMath registry, supplied complete record for permanent ID $30005798$, title: Higher Fano Manifolds as Higher Rational-Connectedness Conditions. Exact source problem code: catalogue entry. The record belongs to the Oberwolfach Reports Open Problems collection and assigns proposed year 2024; no individual proposer is supplied.
• K-Stability, Birational Geometry and Mirror Symmetry (2024), Oberwolfach Reports, pp. 827--870, DOI 10.4171/owr/2024/14. Supplied source and literature link: https://doi.org/10.4171/owr/2024/14. The original excerpt identifies p. 849 and asks whether $F_k$ can be taken to mean that $X$ is $k$-Fano. Its reference label [1] is retained here; the corresponding bibliography entry is not supplied.
• Supplied literature assessment dated exactly 2026-08-22: the record labels the question open and reports only partial implications, including coverage by rational $k$-folds for low $k$. This is supplied dated curation derived from the Oberwolfach material, not an independently verified literature assessment. The supplied background expressly requests independent verification of time-sensitive status and cited claims.
• The registry labels its reconstructed statement corrected_verified and reports that it was reconstructed and checked against the official report. This is supplied provenance, not independent source verification performed for this draft.