A boundedness theorem for cone singularities

Abstract

A cone singularity is a normal affine variety $X$ with an effective one-dimensional torus action with a unique fixed point $x\in X$ which lies in the closure of any orbit of the $k^*$-action. In this article, we prove a boundedness theorem for cone singularities in terms of their dimension, singularities, and isotropies. Given $d$ and $N$ two positive integers and $\epsilon$ a positive real number, we prove that the class of $d$-dimensional $\epsilon$-log canonical cone singularities with isotropies bounded by $N$ forms a bounded family.