A boundedness theorem for cone singularities

TL;DR

This paper proves a boundedness theorem for cone singularities, showing that under certain conditions related to dimension, singularities, and isotropies, these varieties form a bounded family.

Contribution

It establishes a boundedness result for cone singularities based on their dimension, singularity type, and isotropy bounds, advancing classification efforts.

Findings

Boundedness of cone singularities with fixed dimension and isotropy bounds.

Class of epsilon-log canonical cone singularities is bounded.

Results contribute to the classification of algebraic varieties with 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.