K-stability of special Gushel-Mukai manifolds

Gushel-Mukai manifolds are specific families of n-dimensional Fano manifolds of Picard rank $1$ and index $n-2$ where 3≤ n ≤ 6. A Gushel-Mukai n-fold is either ordinary, i.e. a hyperquadric section of a quintic Del Pezzo $(n+1)$-fold, or special, i.e. it admits a double cover over the quintic Del Pezzo n-fold branched along an ordinary Gushel-Mukai $(n-1)$-fold. In this paper, we prove that a general special Gushel-Mukai n-fold is K-stable for every 3≤ n≤ 6. Furthermore, we give a description of the first and last walls of the K-moduli of the pair $(M,cQ)$, where M is the quintic Del Pezzo fourfold (or fivefold) and Q is an ordinary Gushel-Mukai threefold (or fourfold). Besides, we compute δ-invariants of quintic Del Pezzo fourfolds and fivefolds which were shown to be K-unstable by K. Fujita, and show that they admit K\"ahler-Ricci solitons.