Classify K-polystable smooth V22 threefolds

Determine exactly which smooth prime Fano threefolds of genus 12 (commonly denoted V22) are K-polystable, giving a complete classification of the K-polystable locus within the 6-dimensional moduli of smooth V22 threefolds.

Background

Prime Fano threefolds of genus 12 (V22) form a distinguished 6-dimensional family, and while many examples are known to be K-stable or K-polystable, a complete characterization is not available.

The authors emphasize that although certain special members (e.g., those with symmetries) are known to be K-polystable, a precise description of the entire K-polystable subset among smooth V22 is still missing and has been linked to a conjecture of Donaldson.

References

The precise description of which smooth $V_{22}$ are K-polystable is not known, and is the object of Donaldson's conjecture [Donaldson08].

— The boundary of K-moduli of prime Fano threefolds of genus twelve  (2603.29827 - Kaloghiros et al., 31 Mar 2026) in Introduction, K-moduli of V22 and Mukai's philosophy