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 $3\leq n \leq 6$.

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 $3\leq n\leq 6$.

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 $\delta$-invariants of quintic Del Pezzo fourfolds and fivefolds which were shown to be K-unstable by K.

Fujita, and show that they admit Kähler-Ricci solitons.