Kawamata log terminal (klt) singularities form an important class of singularities due to its fundamental roles in MMP, Kähler-Einstein geometry, and K-stability. Recently, Chi Li invented a new invariant called the local volume of a klt singularity which encodes lots of interesting geometric and topological information. In this talk, we will explore the relation between local volumes and certain boundedness condition of singularities related to the existence of ε-plt blow-ups. As a main result, we show that the set of local volumes of klt singularities is discrete away from zero (resp. satisfies ACC) if the coefficient set is finite (resp. satisfies DCC) and the ambient spaces are analytically bounded.
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