Bounded deformations of $(ϵ,\delta )$-log canonical singularities

In this paper we study $(ϵ,\delta )$-lc singularites, i.e. $ϵ$-lc singularities admitting a $\delta$-plt blow-up. We prove that $n$-dimensional $(ϵ,\delta )$-lc singularities are bounded up to a deformation, and $2$-dimensional $(ϵ,\delta )$-lc singularities form a bounded family. Furthermore, we give an example which shows that $(ϵ,\delta )$-lc singularities are not bounded in higher dimensions, even in the analytic sense.