K-stability of special Gushel-Mukai manifolds
Yuchen Liu, Linsheng Wang
TL;DR
The paper advances the understanding of K-stability for Gushel-Mukai manifolds by focusing on general special GM n-folds (n=4,5,6) and establishing their K-stability via delta-invariants, wall-crossing analyses, and cone-degeneration techniques. It computes precise δ-invariants for quintic Del Pezzo fourfolds and fivefolds (δ(M_4)=25/27 minimized by E_S; δ(M_5)=15/16 minimized by E_W), and determines the K-semistable domains for pairs (M,cQ), including explicit last-wall and first-wall phenomena and soliton candidates. By combining Abban–Zhuang-type estimates with cone degeneration and weighted K-stability, it proves that general special GM manifolds are K-stable, and uses openness to extend to all GM n-folds in the range 3≤n≤6. Additionally, the work demonstrates that M_4 and M_5 admit Kähler–Ricci solitons, contributing to the broader program linking canonical metrics with algebro-geometric stability. The results have implications for K-moduli and soliton existence in higher-dimensional Fano geometries.
Abstract
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 $δ$-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.