K-stability of special Gushel-Mukai manifolds

TL;DR

This paper establishes K-stability for general special Gushel-Mukai manifolds of dimensions 3 to 6, analyzes the K-moduli walls, and computes delta-invariants, revealing their Kähler-Ricci soliton existence.

Contribution

It proves K-stability for general special Gushel-Mukai manifolds across dimensions 3 to 6 and describes the K-moduli walls for related pairs.

Findings

General special Gushel-Mukai n-folds are K-stable for 3 ≤ n ≤ 6.

Computed delta-invariants for quintic Del Pezzo fourfolds and fivefolds.

Identified Kähler-Ricci solitons on these Fano manifolds.

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 $\delta$-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.