Closedness of algebraic solutions to the Grassmannian C-space equation

Determine whether integrability conditions, presumably under the \wedge^2-nondegeneracy hypothesis, force an algebraic solution of the Grassmannian C-space equation to be closed, and thereby clarify its possible relation to a generalized EinsteinWeyl structure and the associated BGG sequences.

Background

For a torsion-free Grassmannian structure, weak non-degeneracy permits solving algebraically for a unique one-form U from the trace of the Grassmannian C-space equation. Substitution into the untraced equation yields necessary and sufficient conditions for the existence of an algebraic solution, but the paper does not establish whether that solution is closed.

In the anti-self-dual four-dimensional case, an analogous integrability argument shows that the relevant algebraic one-form is closed, which is essential for converting the Einstein-Weyl-type condition into an actual Einstein scale. The unresolved higher-dimensional question is whether an analogous result follows from integrability under \wedge2-nondegeneracy, potentially through a generalized Einstein-Weyl structure and the BGG machinery underlying the anti-self-dual result.

References

The next natural question is whether there are integrability conditions (presumably using $\wedge2$-non-degeneracy) on an algebraic solution of (\ref{para-C-space}) that imply it is closed. We leave this for future work, and one may speculate the answer is related to an appropriate generalisation of Einstein-Weyl structure to this setting, and the BGG sequences that underlie the corresponding fact in the ASD case, as explained in .

— Conformally Einstein anti-self-dual spaces and their generalisations  (2608.17847 - Moy, 18 Aug 2026) in Section 8, Einstein scales in Grassmannian geometry, immediately following equation (para-C-space-trace)