Answered Reconstruction of Contact Structures from Contact Lines

Definitions and mathematical statement

The supplied record asks a reconstruction question, but marks its statement as unrecoverable and also reports that the motivating question was answered. The formulation below is therefore a conditional interpretation of the retained question, requiring mathematical and source review.

Under the usual complex projective interpretation, let $X$ be a smooth complex projective Fano manifold of dimension $2n+1$, with $n\geq 1$, and suppose \[ b_2(X):=\dim_{{\mathbb{Q}}}H^2(X,{\mathbb{Q}})=1. \] Here Fano means that the anticanonical line bundle $-K_X$ is ample. A holomorphic contact structure is a rank-$2n$ subbundle $F\subset T_X$ whose quotient $L_F=T_X/F$ is a line bundle and whose bracket pairing \[ \bigwedge^2F\longrightarrow L_F,\qquad u\wedge v\longmapsto [u,v]\bmod F, \] is nondegenerate at every point.

For purposes of interpreting the question, call an irreducible rational curve $C\subset X$ an $F$-contact line if its normalization $\nu:{\mathbb{P}}^1\to C\subset X$ satisfies \[ \deg(\nu^*L_F)=1, \qquad {\,\mathrm{d}}\nu(T_{{\mathbb{P}}^1})\subseteq\nu^*F. \] Write $\mathcal K(F)$ for the collection of these curves. At a general point $x\in X$, let \[ \mathcal C_x(F)=\overline{\{[T_xC]\in{\mathbb{P}}(T_xX):C\in\mathcal K(F),\ x\in C_{\mathrm{sm}}\}} \] be the Zariski closure of their tangent directions. Identifying this set with the variety of minimal rational tangents requires a specified minimal rational curve family, which the supplied record does not provide.

Retained reconstruction question. For every $X$ as above and every pair of contact structures $F,G\subset T_X$, does \[ \mathcal K(F)=\mathcal K(G)\quad\Longrightarrow\quad F=G \] hold? The source also describes reconstruction from the variety of minimal rational tangents as equivalent. A possible precise reading of that clause is whether \[ \mathcal C_x(F)=\mathcal C_x(G)\ \text{for general }x\in X \quad\Longrightarrow\quad F=G. \] The equivalence of these readings, the intended curve family, and any necessary exclusions are unresolved in the supplied formulation. The complex ground field, positive contact dimension, contact-line convention, and equality on a fixed manifold are explicit interpretive conventions here, rather than independently established source hypotheses. In particular, the record states no hypothesis guaranteeing that these contact lines exist or cover $X$.

The supplied verification note says that the official report proves that contact lines generate the contact distribution and explicitly answers the motivating question. This is recorded as a supplied source claim, not asserted here as an independently verified theorem or resolution.

Short English statement

The retained question asks whether the rational contact lines of a Fano contact manifold with $b_2=1$, or their tangent directions at general points, uniquely determine its contact distribution. The record simultaneously labels the item open and unrecoverable and says that the report answered the question. The precise hypotheses, curve family, and intended uniqueness notion require human review.

Sources

• Local registry: supplied complete source record, permanent UnsolvedMath ID $30002443$, exact source problem code catalogue entry. No individual proposer is supplied; the proposed year is recorded as 2013.
• Supplied report citation: Mini-Workshop: Quaternion Kähler Structures in Riemannian and Algebraic Geometry (2014), Oberwolfach Reports, pp. 3115--3145, DOI 10.4171/owr/2013/53. Supplied source URL and literature-source link labelled "original OWR report": https://doi.org/10.4171/owr/2013/53. The citation year 2014 and the record's proposed year and report identifier referring to 2013 are retained without reconciliation.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the record open because the extracted fragment allegedly did not identify the mathematical claim. This is supplied dated curation, not independently verified. The background expressly requests independent verification of time-sensitive status and cited claims.
• The supplied statement-verification field says that the report proves that contact lines generate the contact distribution and answers the motivating question. It also says that no separate clean statement is promoted and that the original record is retained publicly for provenance. No theorem number, exact passage, or narrower page reference is supplied. The report has not been independently consulted for this draft.