Delta-invariants for Fano varieties with large automorphism groups

Abstract

For a variety [Formula: see text], a big [Formula: see text]-divisor [Formula: see text] and a closed connected subgroup [Formula: see text] we define a [Formula: see text]-invariant version of the [Formula: see text]-threshold. We prove that for a Fano variety [Formula: see text] and a connected subgroup [Formula: see text] this invariant characterizes [Formula: see text]-equivariant uniform [Formula: see text]-stability. We also use this invariant to investigate [Formula: see text]-equivariant [Formula: see text]-stability of some Fano varieties with large groups of symmetries, including spherical Fano varieties. We also consider the case of [Formula: see text] being a finite group.