# Delta-invariants for Fano varieties with large automorphism groups

## TL;DR

This paper introduces a $G$-invariant delta-threshold for Fano varieties, linking it to $G$-equivariant K-stability and exploring its implications for varieties with large automorphism groups, including spherical Fano varieties.

## Contribution

It defines a new $G$-invariant delta-threshold and proves its equivalence to $G$-equivariant K-stability for Fano varieties, extending stability analysis to varieties with large symmetry groups.

## Key findings

- The $G$-invariant delta-threshold characterizes $G$-equivariant K-stability.
- Application to spherical Fano varieties with large automorphism groups.
- Analysis of $G$-equivariant stability for varieties with finite automorphism groups.

## Abstract

For a variety $X$, a big $\mathbb{Q}$-divisor $L$ and a closed connected subgroup $G \subset \mathrm{Aut}(X, L)$ we define a $G$-invariant version of the $\delta$-threshold. We prove that for a Fano variety $(X, -K_X)$ and a connected subgroup $G \subset \mathrm{Aut}(X)$ this invariant characterizes $G$-equivariant uniform $K$-stability. We also use this invariant to investigate $G$-equivariant $K$-stability of some Fano varieties with large groups of symmetries, including spherical Fano varieties. We also consider the case of $G$ being a finite group.