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 $$x\in X$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>X</mml:mi> </mml:mrow> </mml:math> which lies in the closure of any orbit of the $$k^*$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>k</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> </mml:math> -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 $$\epsilon $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ϵ</mml:mi> </mml:math> a positive real number, we prove that the class of d -dimensional $$\epsilon $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ϵ</mml:mi> </mml:math> -log canonical cone singularities with isotropies bounded by N forms a bounded family.