CscK metrics on rank one spherical Fano fourfolds

TL;DR

This paper investigates explicit K-stability of low-dimensional, low-rank spherical varieties, applying a combinatorial criterion to a specific example, suggesting it admits cscK metrics across all K"ahler classes.

Contribution

It introduces a simple combinatorial criterion for K-stability of rank one spherical varieties and applies it to a specific example, providing evidence for the existence of cscK metrics.

Findings

The example likely admits cscK metrics in all K"ahler classes.

The combinatorial criterion effectively assesses K-stability in this context.

Strong indication of universal cscK metrics for the studied variety.

Abstract

This four-pages note is an invitation to explore explicit K-stability for arbitrary K\"ahler classes of low dimension and low rank spherical varieties. We apply our simple combinatorial criterion of K-stability of rank one spherical varieties to the example of the blowup of the product of two copies of the projective planes along the diagonal, and obtain strong indication that it admits cscK metrics in every K\"ahler classes.