DOI: 10.1017/s1474748026101716 ISSN: 1474-7480
ON THE BOUNDEDNESS OF SINGULARITIES VIA NORMALIZED VOLUME
Yuchen Liu, Joaquin Moraga, Hendrik Suess Abstract
In this article we study conjectures regarding normalized volume and boundedness of singularities. We focus on singularities with a torus action of complexity
1
$1$
1
, threefold singularities, and hypersurface singularities. Given a real value
v
>
0
$v>0$
v greater than 0
, we prove that the class of K-semistable threefold singularities with normalized volume at least
v
forms a bounded family. Analogous statements are proved in the case of
n
-dimensional complexity-
1
$1$
1
and
n
-dimensional hypersurface singularities for arbitrary
n
. In the general case of klt singularities, i.e. without the assumption on K-semistability, we show that, up to special degenerations, the normalized volume bounds singularities with a complexity-
1
$1$
1
torus action. We exhibit a
3
$3$
3
-dimensional example which shows that this last statement is optimal.