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Abstract A cone singularity is a normal affine variety X with an effective one-dimensional torus action with a unique fixed point x X x ∈ X which lies in the closure of any orbit of the k^* 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 ϵ a positive real number, we prove that the class of d -dimensional ϵ -log canonical cone singularities with isotropies bounded by N forms a bounded family.
Joaquín Moraga (Sat,) studied this question.