The anti-canonical polar cylinder criterion for K-polystability of Fano varieties

The conjecture asserts that a Fano variety with at most Kawamata log terminal singularities and no anti-canonical polar cylinder is $K$-polystable. No proposer or original date is identified in the retrieved sources.

Known results

• For log-canonical Fano varieties, $\alpha(X)\geq 1$ implies absence of anti-canonical polar cylinders.
• $\delta(X)\geq 1$ implies $K$-semistability, while $\delta(X)>1$ implies $K$-stability; neither implication proves the cylinder criterion.

March 13, 2026 claimed counterexamples

Adrien Dubouloz, In-Kyun Kim, Takashi Kishimoto, and Joonyeong Won claim that del Pezzo hypersurfaces in families $1$ through $27$, for $n\geq 4$, are $K$-unstable despite having no anti-canonical polar cylinders. Family $4$ is given explicitly; if correct, this refutes even the version with $K$-semistability. The claim is unverified.

Current status (as of September 2026): The conjecture has a claimed counterexample, but its correctness is unverified, so the general problem remains unresolved.