Exact Indices of Orthogonal Grassmannians

Definitions and mathematical statement

Let $m\geq 18$ be an even integer and put $r=\lfloor m/4\rfloor$. For a quadratic form $\varphi$ on an $m$-dimensional vector space $V$ over a field $F$, write $X_r(\varphi)$ for the orthogonal Grassmannian parameterizing $r$-dimensional totally isotropic subspaces: for a field extension $E/F$, its $E$-points correspond to subspaces $U\subseteq V\otimes_F E$ with $\dim_E U=r$ and $\varphi_E|_U=0$. The usual smooth projective interpretation requires appropriate nonsingularity assumptions, which the supplied record does not explicitly specify.

For a nonempty proper variety $X/F$, define its index by \[ \operatorname{ind}(X)=\gcd\{[\kappa(x):F]:x\text{ is a closed point of }X\}, \] where $\kappa(x)$ is the residue field of $x$.

The source uses $I_q^3$ for a condition in the fundamental-ideal filtration of quadratic Witt theory. In characteristic different from $2$, the usual interpretation is that the Witt class $[\varphi]$ belongs to $I(F)^3$, where $W(F)$ is the Witt ring and $I(F)$ is its fundamental ideal of even-dimensional classes. The supplied record gives neither a characteristic restriction on $F$ nor a characteristic-dependent definition of $I_q^3$. It also gives no definition of a generic form or of the family relative to which genericity is taken. These qualifications remain unresolved and must be fixed from the source before the assertion has a fully specified scope.

Conjecture as supplied. For every even integer $m\geq18$, every field $F$, and every generic $m$-dimensional quadratic form $\varphi/F$ in $I_q^3$, with the admissible fields, genericity, and quadratic-form conventions understood as intended by the source, \[ \operatorname{ind}\left(X_{\lfloor m/4\rfloor}(\varphi)\right) =2^{\lfloor m/4\rfloor}. \] The quantification over fields is subject to the unresolved source conventions just described; this display does not assert that every characteristic is admissible.

Short English statement

The supplied conjecture says that, for every even $m\geq18$, the variety of $\lfloor m/4\rfloor$-dimensional totally isotropic subspaces of a generic $m$-dimensional quadratic form in $I_q^3$ has index exactly $2^{\lfloor m/4\rfloor}$. The field restrictions, nonsingularity assumptions, and meaning of genericity require clarification from the source.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30003043$, title Exact Indices of Orthogonal Grassmannians, exact source problem code catalogue entry. The record gives proposed year $2016$ but names no individual proposer.
• Supplied primary-source citation: Algebraic Cobordism and Projective Homogeneous Varieties (2016), Oberwolfach Reports, pp. 215--265, DOI 10.4171/owr/2016/5. Source URL and literature-source link labeled "original OWR report": https://doi.org/10.4171/owr/2016/5. No precise page or named contributor for this conjecture is supplied.
• Supplied literature-source link labeled "Karpenko on 16-dimensional forms": https://sites.ualberta.ca/~karpenko/publ/a16i3.pdf. The record supplies no further bibliographic details for this link.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open, describes the exact index formula as explicitly conjectural in the primary source, and states that related work on $I$- and quadratic-form invariants does not verify it. This is supplied dated curation, not an independently verified literature assessment. The background expressly requests independent verification of time-sensitive status and cited claims.