Karpenko's Conjecture on the Special Schubert Class in the Gamma Filtration Fails for n = 8, 9 and 11: A Short Consequence of Printed Results
Year: 2026
Type: preprint
Abstract: Let φ be a non-degenerate quadratic form of dimension 2n+1 over a field F whose even Clifford algebra C_0(φ) is a division algebra, let X be its maximal orthogonal Grassmannian and S the Severi–Brauer variety of C_0(φ). N. Karpenko conjectured (Math. Z. 285 (2017), Conjecture 4.7, also Conjecture 3 of his abstract in Oberwolfach Report 5/2016) that for n ≥ 8 the class e_n ∈ K(X_{F(S)}) of the special Schubert variety of lowest dimension does not lie in the term number n+1−[(n+1)/2] of the gamma filtration. We observe that the conjecture is false: over every field of characteristic 0, for n = 8, 9 and 11 and for every such φ, the class e_n lies in this term (term 5 for n = 8, 9, term 6 for n = 11), and such forms exist. This is a short consequence of printed results, not found stated in print. The proof is Karpenko's own printed argument that Conjecture 4.7 implies his Conjecture 3.3 on indexes of orthogonal Grassmannians, written out in a general form with two implicit steps filled in, combined with Totaro's computation of the torsion indexes of Spin(20) and Spin(24) (for n = 9, 11) and with Karpenko's theorem on Spin(18)-torsors (for n = 8). A computer-assisted supplement, kept separate from the proof, finds by an exact computation in the Grothendieck ring of the split Grassmannian that e_8 lies in the seventh but not in the eighth gamma term, and e_9 in the eighth but not in the ninth. The cases n = 10 and n = 12 are not decided here, nor is any n ≥ 13. No printed answer to the question was found in the sources searched; a search that finds nothing is not a proof of novelty. This is an unrefereed note. Unrefereed preprint released for independent mathematical scrutiny. Publication on Zenodo does not constitute peer review. AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text. Corpus identifier: OWR-14213-008 (Karpenko, Grothendieck ring of maximal orthogonal Grassmannian, Oberwolfach Report 5/2016, Conjecture 3 = Math. Z. 285 (2017), Conjecture 4.7).
Author: Ferudun, Alper
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7223069976
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